Thomism and Mathematical Physics

No, d'ordre:

DISSERTATION

PRESENTED

TO THE FACULTY OP PHILOSOPHY
OP LAVAL UNIVERSITY

TO OBTAIN .
THE DEG REE OP DOCTOR OF PHILOSOPHY
BY
BERNARD I, MULLAHY, CS.C,

LICENTIATUS IN PHILOSOPHY

GREGOPJAN UNIVERSITY, ROME

Thomism and Mathematical Physics o
JULY 1946

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THOMISM AMD MATH EMATICAL PHYSICS

TABLE OF CONTENTS.
Section OttQ i

1» Introduction : The problem of Mathematical Phy8ios, ....... ..Chap, I

v 1 A Symbol of Progress....... .. . •■ 1

v 2 Historical Perspective i,. «. ...... . .n 4

v 3 . \ Relevance of Thomism .... .*.'.«*«'"• •• 3?

\/4 Some Implications of the Problem...., 61

2 . Body; "

/ lo The ^Specification of the Sciences, ...... .cOnap.J.J.

"bM". ^ >«>^V v^i The Problem....... 67

^ ' \/2 Speculative and Practical Knowledge* . . 7±-
\/?> . The Hierarchy of Speculative Knowledge 77 ■

\/4 Ultimate Specification, „ . . . : <• '^ )i -

v/5 Natural Doctrine and Practical Knovfledge 120

V" Specification and Method, 127

U-A.\» cvs.

2, The Subalternation of the Sciences... . ., Chap, III

1 The Species of Subalternation. . » 132

1

«*> *■ >p^;«-> 2 vThe Essence of Subalternation,, 139

\s 3 Subalternation and Soientia Media , , . , 146

\X 4- Scientia Media and Mathematical Physics , , . . . 151

B a/ | Development of the Principles:)
1. Antithesis:

a. The Study of Nature:

1) Cosmos and Logos,, Chap , IV

\sl Movement towards Concretion, ....... . 165

V^2 Thomism and Experience,, ,,,,.,.,...»...,. .,, 175

\/?> Experience and Certitude «.,; ...,*.....,...,,. 188

V/4 Philosophy and Experimental Science,.*.,,,.. 196

\/o The Interrogation of Nature,,.. ..,,..,. ,...i 206

\/& Operationalism, . , , ................... 213

\Z*7 Laws and Theories... .. o .............. , 219

\/8 Objective and Subjective Logos,, ,, 227

2) Experimental Science and Dialectics. . Chap .V

' s/' 1 The Problem.,,.. '.. ................. 237

\/2 The Nature of Dialectics,,,...,,..'. 240

\/ 3 Dialectics and Experimental Science, ,, 251

V 1 Mathematical Abstraction, „,..,.. » 260

V 2 Mathematics and Existence ...................0 270

V 3 Mathematics and the Intuitive Imagination «... ' 281

V 4 Mathematics and the Human Hindoo ,. .....o. ..«<> 286

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CHAPTER ONE

THE PROBLEM OF MATIIEirATICAL PHYSICS.

1, A.SZ?S!i2i.i :! iL&2Sres_So_

"On the second floor of the Hall of Science at the Cent-
ury of Progress Exposition, held at Chicago in the summers of 1933
and 1934, reaching up into the great tower of the building was a
smaller iovrer designed to symbolize the interrelations and interde-
pendence of the physical sciences , The huge base on which the remain-
ing sciences were supported and uplifted was assigned to mathematics ,
Astronomy, physics^ chemistry, the medical sciences, geology, geo-
graphy, engineering, architecture, the industrial arts — all had their
roots in the science by whose methods and attainments they have learned
and continue to learn to express themselves," (l)

The milling throngs that crowded the pavillions of Chicago's
Exposition found a great many things 'to make their visit rewarding. For
there, under a great variety of forms, were the concrete and tangible
results of a century 'of amazing scientific and technological progress
which had gone to almost incredible lengths in penetrating into the
inner secrets of Nature and in controlling its hidden forces. But for
those who were interested not merely in things, but in their meanings,
the tower of the sciences resting upon the base of mathematics was the
most significant object in the whole Exposition,, For it was a symbol of
a human triumph that v/as the source from which 'had come all the other
remarkable achievements on display — a source so fruitful that it
reached beyond the limitations of these particular achievements, and^
would ever continue to reach beyond the even more remarkable accomplish-
ments that would come from it in the future. More than that, it was the
symbol of something that was far too great to be put on display: the
amazing theoretical attainments of Einstein, Planck, Bohr, Heisenberg,
Schrcdinger. Dirac, and De Broglie — to mention only a few of the na-
mes which have made modern physics great.

But there were even more f ar-reaching implications in
this symbolism. For it was a revelation of what has happened to the
human intellect in modern. times. And hore we have in mind, not merely

a question of scientific methodology, but something far deeper „ In
this symbolism could be found an indication of the precise direction
in which the, mind of men has progressed in the modem era<, For in so
far as the (speculative) intellect is concerned, modern progress has not
been a progress "in" wisdom, but in sciences; and not in science in the
full and perfect sense of the term in which it was understood by the
Greeks and the Medievalists — the sense in which it signifies an
intellectual triumph over the obscurity of matter to the extent of
laying hold of the objective JLogo_s of nature with clarity and certitude
— ^ u ^_^- n that dialectical rt^pS^of knowledge into which science neces-
sarily_ i s sue §T as^TtT pur sue s lis ~deveT.opm.ent 'in The - dxrectTon"' of "increas-
:mgj^qncreti6n in matTer^ And'in'^s^^^^

concerned, modern progress has not been a pr6gr&SS~"±iT"pl*udence , but in
art; and., once again, not in the higher form of art, the art of imitation
or fine art, in which the darkness of matter is transfused by the light
of the mind, but in technological art, in which the intellect is bent
upon the exploitation of matter, and at best achieves only a kind of
compromise with it. And as this development has gone on, not only has
dialectical science tended to dispute the hegemony of wisdom in the
speculative order^ and_ technological art that of prudence in_ the
practical jDrdor^but science" and^aiT"have^6en fdrarai closer and closer,
and_united in a now_ahd strange intimacy^,,)

Obviously, the matrix of this distinctive intellectual
/ growth, so characteristic of our times, is something highly complex,
and it v/ould be a naive oversimplification to attribute it to any one
factor,, Nevertheless, we feel that the source, which has contributed
most to xi, and given it its strongest impetus, and dictated its precise
direction has been the erection of the tower of the sciences upon the
base of mathematics: the interpretation of the physical Yrorld in the
V light of the world of mathematics, '

For the moment we shall not attempt to establish this
point. It has been suggested here merely to orientate properly the_
problem we are' undertaking to discuss, and further development of it
now would take us too far afield and make it necessary to. anticipate
much of what is to follow. But perhaps it would not be irrelevant to
qu:3te a passage from one of the greatest contemporary mathematical
physicists, in which Y^hat wo have been saying finds at least a general
confirmation. In the introduction to his E lectr ons. P rotons, Neutro ns^
and Cosmic Rays^ Professor Millikan points out that it is only through
the~applicItion of mathematics to the physical world that the secrets
of nature can bo effectively laid bare, and the road thrown open to
man's control over nature through technological arts

"For it usually happens that when nature's inner workings have

once been laid bare, man sooner or later finds a way to put his
brains inside the machine and to drive it whither he wills. Every
increase in man's knowledge of the way in which nature works must,
in the long run, increase by just so inuch man's ability to control
nature and to turn her hidden forces to his own account, ■ , ,
In this presentation I shall not shun the discussion of exact
quantitative experiments, for it is only upon such a basis, as
Pythagoras asserted more than two thousand years ago, that any
real scientific treatment of physical phenomena is possible
Indeed, from the point of view of that ancien/philosopher, the
problem of all nature philosophy is to driv e out _ qualitative
9£BS e Hii°S s _ - a J3^_3i2_£9PiL a ° e . jkh en l ty Q l ^'' c %5&QyGJ?§3-&tions „ And
this point of view has been enrpliasTied by the farseeing throughout
all the history of physics clear dovm to the present,, One of the
greatest of modem physicists, Lord Kelvin, writes: "When you
can measure what you are speaking about and express it in numbers,
you know something about it, when you cannot express it in numbers,
your knowledge is of a meagre and unsatisfactory kind,, It may be
the beginning of knowledge , but you have scarcely in your thought
advanced to the stage of a science „" (2)

Perhaps enough has been said to suggest that. there is
hardly a more important or more pressing task confronting contemporary
philosophy, nor one which promises greater intellectual fruitfulness,
than the analysis of the significance of the symbolism of the scientific
tower resting upon the base of mathematics, the attempt to unfold one
by one its manifold implications in their proper focus. Such is the
purpose of this study, Vfc shall not attempt to unravel completely the
whole complicated maze of epistemological problems that have arisen out
of mathematical physics, and particularly out of its more recent devel-
opment. The state of this development is still too fluid perhaps to
make any attempt of that kind feasible „ We shall content ourselves
with an analysis of the basic significance of the interpretation of
nature in terms of mathematics

It would be interesting to know ho?/ many of the hundreds
of thousands of visitors at the Chicago Exposition found the tower
within the tower worthy of special interest, and how many grasped the
profound meaning of its symbolism. Pr ima faci e, it would undoubtedly
seem preposterous to suggest that no one among those who had reaped
the fruits of modern progress, or even among those whose genius had
been immediately responsible for its great achievements, could understand
this symbolism quite so well as some who lived centuries before the
Century of Progress'begari, Te ; B" it does not seem necessary, or even
possible to "rule "out" such a supposition in ajoripri fashions And if_
this supposition could be proved to be true, it would provide striking
evidence" that not everything that has happened in the century of progress

has been progress,, In any case, it is important to understand that
modem progress has not been ambiogenetic, The mathematical interpret-
ation of nature is indeed characteristic of the modem mind, but not
in the sense that it was first discovered or. created in recent times „
Like most modern . things it has its roots deep in the past. This has al-
ready been suggested in the passage just quoted from Millilcan, and it
will be one of the main purposes of this essay to show how important
these roots are. But for the present it is necessary to examine its
historical background only in a summary way, so that our problem will
be thrown into proper focus.

Historical Perspective ,

Not a few historians have considered the Renaissance as
the origin of the physico-mathematical method in science and have
generally accorded to Galileo or to Descartes the honor of being its
creator. But history is there to contradict the historians, and Pierre
Duhem, among others, has shown with that remarkable clarity of outline
the so-called modem scientific method had already been conceived in
ancient times,, We shall nave occasion, later to show that this is true
of all the major elements in this scientific method, but for the moment
we are interested only in the application of mathematics to physics.
It is true, of course, that only in modem times have the far-reaching
possibilities and remarkable fruitfulness of this application been
fully realized — realized both conceptually and practicality. That is
why Duhem himself could write: "Creee au XVII siecle, la physique
matheraatique a prouve qu'elle etait la saine methode physique par les
progres prodigieux et incessants qu'elle a faits dans 1' etude de la
nature, "^ (3) It is also true that the, modern developments of mathe-
matical physics have brought to light, or thrown into sharper outline,
certain new epistemological aspects of the general physico-mathematical
method, And ""it is" probably "these" "hew aspects that have led Sir James
Jeans to declare: "The fact that the mathematical picture fits nature
must, I think, be conceded to be a new discovery of science, embodying
a new knowledge of nature such as could not have been predicted by any
sort of general argument„" (4) But these new aspects do not change
the essence of the method. And it is this essence which has its roots
in the -oast. It is, moreover, this essence which lias the deepest and
most interesting philosophical implications. That is why we must, if
wo would see things in their proper perspective, try to situate our
problem in its historical context.

Already among the ancient Greeks the physico-mathematical
method was clearly conceived, and actually put to considerable use.

In this connection the name of Archimedes comes readily to mind, for
it was through him that this method achieved its fullest fruitfulnoss •
in ancient times, and actually led to the definite and clear cut form-
ulation of the sciences of mechanics and hydrostatics, But_Archimedes
^s_not _the__inventor of the method. Long before his time","" the" Greek
astronomers, such as Eudoxus of Chidos, had united mathematics and
physics by attempting to "save the phenomena" through deduction drawn
from geometrical hypotheses, (5) In the same way mathematics had
been applied successfully in other sciences, such as optics. But, Since
the purpose of this historical sketch is to orientate a philosophical
problem, we ore interested less in those who actually applied rasMie"
ma tic's to nature, than in those who in some reflective way attempted .
to bring to light the philosophical significance of this applica-tioiia;
And in this connection it has become customary to designate, two Greek
philosophers as the ones who in ancient times grasped more clearly'' '
than any others the meaning of the mathematical interpretation of
nature and the reach of its possibilities. They are Pythagoras and
Plato

' The basic doctrine of the Pythagoreans is well known,*
The ultimate reality of things was for them essentially mAthematic'al; —
the structure of the universe was based on numbers and their relations,
Aristotle characterizes their position in the following terms:

"Contemporaneously with these philosophers and before
them, the so-called Pythagoreans, who were the first to take up
mathematics, not only advanced this N study, but also having been
brought up in.it they thought its principles were the principle's,
of all things a' Since of these principal e 3 numbers are by nature
the first, and in number's" they seemed' to see many resemblances
to the things that exist and come into being — more than in fire
and earth and water ( such and such a .'modification of numbers being
justice, another being soul and reason^ e^nother being opportunity
— and similarly almost all other things being numerically expres-
sible; since, again, they saw that the modifications .and the ratios
of the musical scales were expressible); in' numbers; ?'*» since,
then all other things seemed in their whole nature to be modelled
on numbers, and numbers seemed'' to be the first .things in the whole
of nature, they supposed the elements of numbers- to be, the elements
of all things, and the whole heaven to be a musical scale and a
number. And all the properties of numbers, and scales which they
could show to agree with; -the attributes and, parts and the whole
arrangement of the heavens, they collected and fitted into their _
scheme; and if there was a gap anywhere, they readily made additions
so as to make their whole, theory coherent, (6)

For the Pythagoreans the devine One was a mathematical god;

he was the supremo number, and the source ana. cause of all the numbers
that constituted the universe, (V) All this seems to be a distant
anticipation of the conclusions that one of the greatest contemporary
mathematical physicists lias arrived at as the result of his many years
of work in the field and of his philosophical reflections upon its
meaning,, "Our contention", writes Sir James Jeans, "is that the universe
now appears to be mathematical in a sense different from any which
Kant contemplated or possibly could have contemplated — in brief,
the mathematics enter the universe from above rather than from below,," (8))
"Prom the intrinsic evidence of his creation, the Great Architect of .
the universe now begins to appear as a pure mathematician," (9) More
and more modern scientists' are looking back to Pythagoras as to the
one who first conceived the vision that they are laboring to realize
Villi tohead, for example, has this to say:

So today when Einstein, and his followers proclaim that
physical facts, such as gravitation, are to be construed as exhibit-
ionsof local peculiarities of spatio-temporal properties, they are
following the pure Pythagorean tradition,, Truly, Pythagoras in
founding European philosophy and European mathematics, endowed
them with the luckiest of lucky guesses — or, was it a flash of
divine genius, penetrating to the inmost nature of things „ , •

I Finally, our last reflection must be, that we have in the end
come back to a version of the doctrine of old Pythagoras, from
whom mathematics and mathematical physics, took their rise,, (10)

Ernst Cassirer also sees in Pythagoras the progenitor
of modern science:

In the times of Pythagoras and the first Pythagoreans
Greek philosophy had discovered a new language, the language of
numbers „ This discovery marked thj3_natal Li hour of our modern -con-
ception of science , , .

The Pythagorean thinkers were the first to conceive number
as an all-embracing, a really universal element,, Its use is no
longer confined within the limits of a special field of investigation.
It extends over the whole realm of being When Pythagoras made his
first great discovery, when he found the dependence of the pitch
of sound on the length of the vibrating chords, it was not the
fact itself but the interpretation of the fact which became deci-
sive for the future orientation of philosophical and mathematical
thought, Pythagoras could not think of this discovery as an isolated
phenomenon. One of the most profound mysteries, the mystery of
beauty, seemed to be disclosed here. To the Greek mind beauty
always had an entirely objective meaning. Beauty is truth; it is
a fundamental character of reality. If the beauty which we feel
in the harmony, of sounds is reducible to a simple numerical ratio
it is number that reveals to us the fundamental structure of the

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cosmic order, "Number", says one of the Pythagorean texts, "is
the guide and master of human thought. Without its power every-
thing would remain obscure and confused,," We would not live in a
world of truth, but in a world of deception and illusion. In num-
ber, and in number alone, we find an intelligible universe, , „
In this general methodological ideal we find no antago-
/ nism between classical and modem physics,, Quantum mechanics is
j in a sense the true renaissance, the renovation and confirmation
\ of the classical Pythagorean ideal, (ll)

But Pythagoras is not the only one among the ancient Greeks
to whom modern scientists and philosophers of sciences are looking back
for inspiration. In the question of the mathematical interpretation of
nature he is made to share his honors with Plato 5

An intense belief that a knowledge of mathematical rela-
tions would prove the key to unlock the mysteries of the related-
ness within nature was ever at the back of Plato's cosmological
speculations, . .

His own speculations as to the course of nature are all
founded upon the conjectural application of some mathematical
construction, . .

Plato's mathematical speculations have been treated as
sheer mysticism by scholars who follow the literary traditions
of the Italian Renaissance » In truth, they are the products of
genius brooding on the future of intellect exploring a world of
mystery, (12)

The Platonic doctrine on the question of mathematical
physics is considerably more difficult to define than the Pythagorean,
For in the time that had elapsed between Pythagoras and Plato the
development of the philosophical mind had gone a long way: it had
gone far enough to reach a high degree of complexity, but not far
enough to reduce this complexity to the clarityof an accurately defined
and well articulated system. Historians have presented the position of
Plato in a way which makes it appear extremely paradoxical. On the one
hand, it is often identified with that of Pythagoras „ It is in this
way that it is presented by Emile Meyerson: "Pour Platon, le fin
fond de la nature, ce que nous appelons actuellement, d'un terme
kantien, la chose en soi, est mathematique et n'est que mathematique.
Tout le reel se compose uniquement de figures de geometrie." (15)
Since mathematics is in a sense the most perfect form of rationality
for the human mind, it would seem to follow that for Plato nature
was in itself something perfectly rational,, And Meyerson seems to
accept in substance this inescapable consequence, for he writes:
"Platon. . . croyait fermement a l'explicabilite de 1'univers, . .
Pour lui, en eff'et, la regularity de la nature, sa legalite, n'etait

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precisement qu'un corollaire de cette rationalite," (14)

On the other hand, nature would seom to have been in a
scnse^completely irrational for Plato, for he held that no true science
(episteme) of it was possible „ About the material universe man could
have oa^y opinion (doxa) „ (15) And it has been customary to draw a
sharp contrast between the irrationality of the universe of Plato and
the rationality of the tiniverse of Aristotle, who made a science of
nature possible by incarnating, so to speak, the Platonic ideas in
the world of sense o The paradox could scarcely bo more incisive; on
the one hand the transparent intelligibility of mathematics, the most
rational of all the sciences; on the other an unintelligibility so
complete as to preclude the possibility of any true science

We are evidently faced here with the traditional problem
of the conflict between the rationality and the irrationality of the
cosmos which had been so acute for the philosophers who had preceded ■
Plato, especially Heraclitus and Parmenides, In a sense it is this
conflict that is at the bottom of the problem we are undertaking to
solve. But we feel that in so far as Plato himself is concerned the
paradox has been rendered more acute than it actually is by the more
or less arbitrary oversimplifications of historians,.

In the first place, though it is true that Plato borrowed
Heavily from the Pythagoreans, his position cannot be identified with
theirs o The impact upon the Platonic physics of other systems, especially
that of Heraclitus, was too strong to allow such an identification, (16)
For Plato the mathematical world was not realized as such in the yroi-ld
of sense; the ideal mathematical forms were not given in nature, but
merely suggested by it, in so far as nature in some more or less obscure
way participated in them. The world of mathematics was not simply im-
manent in the physical world, but to some extent trahscendant from it.
Yet it was not so far removed from it as the world of pure ideas. It
occupied, in fact, a kind of intermediary position between the ideas
and the world of changing things , That is why the mathematical forms
were realized in nature more easily and more perfectly than the other
ideas, But at the same time this realization came from without.

The following passage of Aristotle brings out the dif-
ference between the position of Plato and that of the Pytagoreans:

But he agreed with the Pythagoreans in saying that the
One is substance and not a predicate of something else; and in
saying that the Numbers are the causes of the reality of other
things he agreed with them; but positing a dyad and constructing
the infinite out of grea.t and small, instead of treating the infinite
as one, is peculiar to him; and so is his view that the Numbers

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exist apart from sensible things, while they say that the things
themselves are Numbers, and do not place the objects of mathematics
between forms and sensible things. His divergence from the Pythagor-
eans in making the One and the Numbers separate from things, and
his ^ introduction of the Forms, were due to his inquiries in the
region of definitions (for the earlier thinkers had no tincture
of dialectic), , „ (17)

It is clear from this text that the reason why Plato separated the
mathematical forms from the pliysical world was that the absolute,
universal, and necessary definitions characteristic of mathematics
could not be realized as such in the essentially mutable world of
sense. Nevertheless, physical reality in some way participated in
these niathematical forms, and it seems that for Plato our knowledge
of nature could approximate to the true scientific knowledge that
is characteristic of the intelligible world in so far as it could
take on the form of precise measurement and mathematical formulation.
In the Philebus (18) for example, he distinguishes between the arts
"which have a greater participation in true scientific knowledge and
those which have less." And to illustrate his point he says, "If we
took away the numbering and measuring and weighing from all the arts,
what would be left in each case would be called a poor thing,.,"

Ernst Cassirer has . characterized the position of Plato
in the following terms:

/ It is rooted in Plato's interpretation of mathematics,
j which is for him the 'mediator' between the ideas and the things
\ of sense. The transformation of empirical connections into ideal
ones cannot _ take pla_ce_ without this middle- term. The first and
necessary step througholit""is~f6"' trMisf orAi "the" sensuous indefinite,
which as such cannot be grasped and enclosed in fixed limits,
into something that is quantitatively definite, that can be mastered
by measure and number'. It is especially the later Platonic dialogues,
as for example the Philebus , which most clearly developed this
postulate. The chaos of sense perception. must be confined in strict
limits, by applying the pure concepts of quantity, before it can
become an object of knowle dgg. We cannot rest with the indefinite
'more' or 'less', with the'stronger' or 'weaker' , which we think
we discern in sensation, but we must strive throughout for exact
measurement of being and process. In this measurement, being is
grasped and explained (cf . Philebus, 16, 24f) Thus we stand before

(a new ideal of knowledge, one which Plato himself recognized as
in immediate harmony with his teleological thought , and combining
.with it a unified view. Being is a cosmo s, a purposively ordered
whole, only in so far as its structure is characterized by strict
mathematical laws. The mathematical order is at once the condition

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Idf^Sf^ ° f f° existance °f reality, it is the numerical
g^mffitengss of the universe that secures its inner self-pre-

_ Plato's doctrine here, as in so many questions, is far
from being easily definable. But perhaps enough has been said to show
that his position can be identified with that of Pythagoras only by
considerable oversimplification. On the other hand, it is perhaps an
evBi^greaoer oversimplification to draw the contrast between him and
Arisuotle so incisively that the peripatetic world appears as something
completely rational and the Platonic world as something completely
irra clonal. >fe shall point out later what a large part the jparalogon <
played m T;he system of Aristotle. It was. precisely because off the "
^ r Q^o^i^yJi^awJ.n_^ J cosmos that he conce'ived'"Sf TnathSmatical
P_"ys?-P-S.as a scienfria media, an InTermedlSy'iHen5e"In'wHc¥TrTijas
, necessary to reach out beyond the realm of physics to that of .mathematics
1". order. to ^ nationalise mture. Paradoxical as it may appear, the
Aristotelian cosmos is at once both less rational and more rational
than the Platonic, and the solution of this antinomy lies in the distinct-
ion between two types of rationality. Yfe consider that distinction to be
of capital importance; it will, in fact, be one of the keys for the
solution of our own problem.

The first type of rationality is that proper to the phy-
sical world itself. It is a rationality that arises out of the existence
of foci of intelligibility in the obscure mass of materiality, of
rallying points of intellectual stability in the flux of contingency.
Because the mand can discover and disengage these intelligible forms,
in a confused way at least, a science of nature in the strict sense
of the. word, in the sense of episteme , is possible. It would seem
that Plato never arrived' at the realization of this possibility, and
it remained' for Aristotle to find the philosophy of nature. Prom this
point of view, the Platonic cosmos was irrational; it was the Heraclitean
cosmos of change and obscurity. Of it the mind could not have true
egisteme, but only doxa, ' •

The second type of rationality is the mathematical ratio-
nality of which we have already spoken. Prom this point of view the
Platonic world was extremely rational. For even though in the scheme
of Plato nature was not composed intrinsically of mathematical forms.,
and the process of mathematization came in some way from without,
nevertheless nature was profoundly mathematical in the sense of being
highly amenable, perhap3 indefinitely amenable', to this process of
mathematization. Professor A.E, Taylor sums up Plato's doctrine on
this point in the following terms s

The identification of the forms ( 6i6n ) with numbers

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racaiis that the "manifold* of nature is only accessible to scientific
knowledge in so far as we can correlate its CvSriet^. with definite
numerical <S Sctions ) of "arguments" r~The 'arguments" have then them-
selves to bo correlated with numerical functions of "arguments"
y pf higher degree"^ If this process could be carried through without
remainder, the sensible world would be finally resolved into combin-
ations of numbers, and so into the transparently—int elligible „
This would be the coiiiplete " rationalization" of nature ,, The process
cannot in fact be completed, because nature is always a "becoming",
always unfinished; in other words, because. there is real contingency, '
/ But our business in science is always to carry the process one
V step further, We can never completely arithmetize nature, but
it is our duty to continue steadily arithmetising her. "And still
beyond the sea there is more sea" j but the mariner is never to
arrest his vessels The " surd" never quite Mjomes out " . but we can
carry the 'evaluation a "place" ^further , and we must. If we Y/ill
not, we become "ageometretes" , (20)

Plato seems to have considered this mathematization as
the revelation of "a logos that was proper^ tojiature , That is why in
his system mathematical rationality could supplant, physical rationality,
and his mathematical interpretation of nature become a philosophy of
nature From this poinF~oT"v : iew7"Sristotle' , s attribution of mathematicism
to the Platonists would seem to apply to Plato himself: "Mathematics
has been turned by our present day thinkers into the whole of philo-
sophy". (21)

Aristotle' sfliscovery of the physical rationality of nature
did not make him lose sight of two important facts. The first fact
was that this rationality is only partial, indeed extremely meager.
He too recognized a doxa of nature along with the episteme he had
discovered. As we have already suggested, and as we shall explain
more fully later, it is only _as_ long as the mind _ remains , ingenerali-
tios that it.. is,„able^icrTay hold of an oFjecTive ■'logos of ^nature with
certitudejC.and^s^ it follows 'its

concretion, this certitude very quickly fades into a dialectical know-
ledge that is similar to ~~ 'the Platonic doxa ft The 'second fact was^
that Aristotle also recognized the. part played by mathematical ration- ■
ality in the study of nature. Indeed, one of the main objectives of
this study is to show with what clarity and precision he recognized
it. But we shall not take time out now in this brief historical sketch
to sot forth his position on this point. For besides the general fact
that all that is to follow will b'e an explanation and development of <
it, we intend later in -this chapter to give special attention to the
question of the relevance of Poripatcticism in the problem of mathe-
matical physios. Let it suffice for the moment to have pointed out
why the Aristotelian cosmos was at once both more x-ational and loss

-12-

rational than the Platonic, The universe of Plato seems to have been
completely rational from the mathematical- point of view, at least in
the sense of being indefinitely amenable to rnathematization. It was
at the same time completelyirrational from the purely physical point
of. view „ The .universe of Aristotle was at once partially rational and
partially irrational from both points of view.

Another interesting paradox emerges from a comparison of
the positions of Plato and Aristotle, In the doctrine of Plato the
mathematical world is closer to the physical world and at the same
time farther away from it than in the doctrine of Aristotle. It is
closer to it, for the raasons just indicated: for Plato the physical
world is indefinitely amenable to rnathematization, and this mathema-
tization is a revelation of a logos that is proper to nature; for
[Aristotle only one (aspect) of nature is susceptible of the application
of mathematics, and even with regard to this one aspect, the appli-
cation always remains e_s^entially extrinsic in the sense of providing
only a substitute retionality^)

The mathematical vrorld is at the same time farther away
from the physical world in the position of Plato than in that of
Aristotle, In separating the mathematical , world from the physical
world with which it was identified in the doctrine of the pythagoreans,
Plato gave to it an ontological existence that was independent of
the material cosmos, Aristotle also separated the mathematical world
from the physical vrorld, but in, doing sg _ho ;,.gaye ^^ it gnly..a,_conceptual
existence, For him mathematical forms are abstracted by the mind from
'the quantitative determinations of the material cosmos. As such they
can only exist in the mind. In so far as ontological existence can be
attributed to them at all, this existence must be found in the material
\ cosmos. (22) But they can have this existence only at the expense
of being robbed of the specific state of abstraction that is proper :
to them, and_that is why, in themselves, t^they always /remain essentially
extrinsic "to ratureT Since,"" then, the mathematical forms of Aristotle
have no ontologi-cal existence apart from sensible things and always
have an essential physical reference they are closer to the physical
world than'those of Plato, But" since the abstraction that is proper
to them makes it impossible for their properties to be attributed to
the things of nature, they are at the same time farther away from the
physical world, »

It is clear, then, why Aristotle was justified in claiming
that the Platonists had turned mathematics into the whole of philosophy,
P021 because of the closeness of the mathematical world to the physical
world in the doctrine of Plato, his physics was a kind of mathematical
physics. On the other hand, because of the ontological existence at-
tributed to the mathematical world, hi 3 mathematics took on a metaphy-

-13-

sical character, and to that extent his metaphysics was a kind of
mathematical metaphysics. That is why so much of his speculation about
reality, whether physical or metaphysical, is involved in mathematics.
And that is why on the face of things his system might appear as the
best philosophical explanation of the mathematical interpretation of
nature. But we feel that a deeper analysis will reveal that this is
not true , For his mathematical physics is far from being the mathema-
tical physics of modern science , Strange as it may seem, the very
proximity of his mathematical world to the physical world prevents
his' doctrine from being the true explanation of modern mathematical
physics. On the other hand, the very fact that he invested the mathe-
matical world with an ontological existence of its own drew, mathematics
out of its proper sphere and away from its proper function, and got
it involved, in intellectual situations alien to its true character
and to the role it plays in modern science.

The following lines of Professor Strong are extremely
pertinent heres '

To substitute mathematical objects for the "fiction" of
Forms makes ideal and mathematical number the same and destroys
the distinction by which mathematical number is valid no matter
what metaphysical theory of the universe is advanced; "for they
state hypoteses peculiar to themselves and not to those of mathe-
matics" . (Aristotle: Met, XIII, 1086 a 9)

The hypotheses in respect to the metaphysical status of
number are peculiar to metaphysics and not to mathematics. To
make ideal and mathematical number' the same is a verbalism, a
figurative way of speech disguising the fact that the ideal number
is not the mathematician's science nor the use of mathematics
in dealing withphysical phenomena. Optics, music, and astronomy
are open to mathematical treatment or involve a mathematical ele-
j ment. Their subject-matter is mathematically formulable, because -
I objects can be designated by number' and'ean 'present quantitative
laspects) Further to posit mathematical objects and relations a»
having substantial existence not only does not advance mathematical
sciencg but also results in a confusion of mathematical procedures
and properties with the first principles of being, , .

Plato, if we may judge from Aristotle's account proposes
a scientific myth. Aristotle would object to identifying mathematics 3
the demonstrative science, with the con- ■ •

jectural theories of existential number; at least he objects to
supposing that "ideal" mathematical number is, in fact, what mathe-
matics is before going to the length of paying it metaphysical
compliments.

If we suppose that God is a geometer who geometrizes
continually, wo have carried mathematical certainty to the throne

"14-

of metaphysical or theological certainty. It will thence be de=
livered back to us in the creation of things, by figure and number.
It will enter into knowledge, since the soul itself will be a
number „ What actually returns in the philosopher's account is
the discretion and the classification of Intelligences, Ideas,
the soul, and the existences which make up the world after the
patterns, paradigms exemplars, divine or seminal numbers in the
mind of God, The procedures without which there is no demonstrative
science do not come back from this journey. Numbers arid figures
are valued in respect to their reality and this depends upon their
1 status in respect to God and not to mathematical use. In the face

of such a transformation, arithmetic and geometry are propaedeutic
: to theological arithmetic, ancillary sciences for a kind of super-
'. science in which they become metaphores and analogues, (23)

As has already been noted, it is being frequently urged
by contemporary philosophers of science that the doctrine of Plato
and the Platonic tradition are the metaphysical forebear of modern
mathematical physics, "In modern times," writes Cassirer, "mathematical,
physics first seeks to prove its claims by going back from the philo-
sophy of Aristotle to that of Plato," (24) This claim might mean
several things. In the first place, it might mean that historically
it was Platonic tradition that actually gave birth to modern mathema-
tical physics, that it provided the metaphysical basis and the intel-
lectual impetus which brought about its origin and development. It
is in this way that the claim is understood by, many modern critics,
and Professor Burtt, among others, has gone, to some lengths in his
Metaphysical Foundations of Mo dern Physical Science to give it substan-
ce, (25) We do not think that the claim, understood in this sense,
has as much importance as might first appear. For history is not logic;
nor generally speaking, is its development shaped by per-se determined
causes. There is consequently no reason why a philosophical sj stem
which is wholly inadequate to explain the true meaning of mathematical
physics might not have been the actual historical impetus which brought
about the origin of modern physical science <>

Yet it is interesting to note that an accurate and detailed
study of -Uhia question recently undertaken by Professor Strong has
made the claim that the Platonic tradition sired modern science appear
extremely dubious. Strong undertook this study with the intention
of consolidating the opinion of Burtt, but all the evidence that emerged
from a close examination of the work of the scientists of the early-
modern period forced him to arrive ■ at the opposite conclusion. In his
Procedures an d Meta physics he writes:

A Pythagorean-Platoni?. (or Neo-Platonic) conception of
mathematics is regarded by some present-day critics as the roalistio

-15-

and rationalistic doctrine of a mathematical structure of nature.
This may mean that we are today (in the light of contemporary
Platonic scholarship) in a position to establish critically analogies
between Plato's writings and prominent characteristics of ' modern
science and philosophy. If, however, it is asserted that the early-
| modem mathematical investigators based their science upon metaphy-
sical foundations, Platonic or otherwise, the weight of evidence
gleaned from a survey of some of the Italian scientists is opposed
v to such an assertion. The historical problem should here be dis-
entangled from modern critical exposition. By such exposition,
it can be maintained that a Pythagorean-Platonic metaphysics is
compatible with the mathematical treatment of nature. In the light
of historical evidence, however, we may question whether the Plato-
nism of the fifteenth and sixteenth centuries had at that time
the role and significance which philosophers now critically assign
to it in connection with modern science. The assertion that the
Platonic metaphysics laid the foundations for the mathematical
science of Galileo is at odds with the positive evidence already
presented. Furthermore, it appears highly questionable when' the
tradition of Platonism is examined. The Neo-Pla tonic doctrines
of Picino, Giovanni Pico, and Reuchlin, and of the mathematical
writers — Zamberti, Domenico and Dee — express' metomathematical
doctrines carried over from Proclus and his predecessors with
additional cabalistic embroideries. If this archaic tradition is
characteristic, we are in a position to recall the objections and
difficulties raised against Nicomachus,. Theon, and Proclus. The
main intention of this chapter is to expose the definitely archaic ■
character of the Platonizing tradition of mathematics preserved ■
in several mathematical .writers — archaic that is, in the sense
of its ineptness and nonconnection with the scientific work of
the period in which it is reinvoked. . .

The Noo-Pythagoreans and Meo-Platonists were impressed
vri.th the mathematical disciplines, particularly arithmetic. Mathe-
matics is -taken over and given a cosmological significance, but
the doctrines presented, the metamathematics of Platonizing thinkers,
are foreign to the -method arid use of mathematics. The role attributed
to number satisfied the assertions of metaphysics, but these assert-
ions could not be applied or substantiated by either the logic
or the practice of the mathematician. The metamathematicians assume
a being and function for. mathematical objects superior to the
subject-matter and procedure of the science proper and assume .
that this metaphysical status is more real and important. Mathematics
and mathematical science could not and were not expected to subs-
stantiate the ' assertion that one could by mathematics mount to
a knowledge of a superior realm of beingj yet a propaedeutic value
was supposed to lie in this initiative capacity of mathematical
study. The converse of this aaaertion is oq.ual.ly unsubstantiated,

-16-

namely, that he who knows the mysteries of ontological and cosrao-
logical number forms is able to penetrate into the inner signifi-
cance of natural things. This is not a hypothesis for mathematical
procedure. The basic supposition is the notion that natural things
are the created copies of a creating form, inferior effects in
an^individual of a superior, unitary cause. Thus, although the
nctamathematicians employed a number-symbolism, the symbolism
stood for forms and efficacies not mathematically conceived. . ,

It is a sobering reflection to consider how long the
Pythagorean arithmotology and its constitution in the Neo-Platonic
system persisted in claims unsubstantiated in fact. Demands of
logical and doctrinal consistency were satisfied so far as the
purpose and end of the metaphysician wore concerned. To suit a
metaphysical purpose, mathematics was thrown into a status and
assigned a role divorced from mathematical conception and meaning-
less for procedure. The metaphysical end of cosmological status
and divine residence was assumed to be the goal for which mathe-
matics was preparatory as an intellectual purification} and since
the One is casual of the many and the archetypal number-form is
the unity of the individual, created thing, the use of mathematics
is supposed to depend upon the constitution of natural things by
the raetamathematical patterns. Modern mathematical-physical science
established its method and achieved its results in spite of, rathex
than because of, this kind of metamathematical tradition. Had
the early modern, mathematical investigators in general, rather
j than by exception, taken the philosophical tradition seriously,
.history might have seen more mixtures of metaphysics and science
i similar to Kepler's, without, perhaps, the saving conditions that
j brought Kepler's metaphysical predispositions to a scientific
; issue, (26)

But the modern critics' insistence upon the relevance of
the doctrine of Plato for modern science might also be taken to mean
that among all philosophical systems, or at least among those which
have come down to us from antiquity, this doctrine provides the most
adequate explanation of the true meaning of mathematical physics.
Understood in this sense, the claim is of extreme importance , _And
it is the_ purpose of this study J^o_d^pu^e_i^jmlldi;ky, But in doing
SO we have no "intention" J co minimize the genius of Plato or his contri-
butions to the philosophy of science. In his doctrine the philosophical
mind made ' a great advance towards providing the true explanation of
the mathematical interpretation of nature. The concept of the world
of mathematics as occupying a kind of intermediary position (between
the physical world and the world of pure' "ideas) was' "a "significant con-
tribution. Even more significant' was ! the' corollary that naturally
flowed from it; the mathematization of the cosmos is_in_some sense
imposed upon nature from without. Moreover, there are a number of

-17-

sti-ilcing analogies between prominent features of modem science and
points of Platonic doctrine . The view now generally accepted by the
best scientists and philosophers that experimental science can never
giy. e . more ..than pjrobable knowledge would seem to be a confirmation of
the Platonic doxa. The' increasingly evident fact that modern science
is essentially constructed of idealizations, that is to say of ideal
form and limit ceases which are not, given in nature but merely sug-
gested by it, that scientific laws are not discovered in the objective
universe but imposed by the mind in its attempt to rationalize experience
would seem to be reminiscent of the Platonic doctrine of the relation
between ideas and physical reality. Out of this mathematization and
ratiqnalj^ation of 'experience (through_thc_jprocess of idealizations has.
come the ever increasing use of ''h^bthesis7"wMoh _ playeTrsu6h'an'''es-
sential role in the method of Plato. (27) And there would seem to
be something kindred to Platonism in the a priori character of the
modern scientific world, which is made up so largely of constructs of
the mind. All of these points are significant, but we do not feel
that they suffice to constitute the doctrine of Plato as an adequate
philosophy of science.

Continuing nov/ our historical sketch, we- find that in
the middle ages the problem of the mathematical interpretation of
nature received comparatively little attention, though, as we shall
see, its true nature was far from being ignored by the Thomistic school.
Grosseteste at Oxford seems to have had considerable interest in the
possibilities of mathematical physics. We are told tliat_he...tried_tg
reduce all the sciences of nature to the one universa2_science of
optics ) that he considered mathematical principles as the key to all
knowledge of the physical universe, and consequently tried to explain
natural phenomena in terms, of geometrical lines, figures and angles.
This same interest is found in Roger Bacon, who in this, as in so
many ways, anticipated the so-called modern mind. Bacon held that the
book of nature is, written in the language of, geometry, and that mathe-
matics is "the alphabet of all philosophy," How accurately , he had con-
ceived the mathamatico-observational method of modern physics may be
gathered from the following linos:'

It is true that mathematics possesses useful experience
with regard to its own problems of figure and number, which apply
to all the sciences and experience itself, for no science can be
known without mathematics. But if we wish to have complete and
thoroughly verified knowledge, we must proceed by the methods of
experimental science. (28)

With the dawn of the early modern period a new, spontaneous
enthusiasm for .mathematics began to make itoelf manifest. And this
gravitation of the mind towards mathematical science soon became all

-13-

of a piece with tho general pattern of renaissance philosophy, which
was so profoundly humanistic. For, as wo shall explain later on, raathe-
'^^J-JL..'* most "te'i-ai"' af all the "scicncos,<:in "the" sense" that it
has the groa^st'OThMturaXity'with the human intellect J It is also
uho science in which tho'm^nd'crLQ' in' some way nMtate the a J>rjiori
and creative character of divine know ledge, and as a consequence it
offers to the i.iind a great measure of' autonomy. That is why it was
almost inevitable that there should be a natural gravitation towards
niathciiatios in the period of humanism in which the intellect of man
tended to become tho measure of all things and to' that extent necos-
SC i r i?" y divine » and ^ which there was such a universal vindication
of the complete autono;ay of the mind. "Through Copernicus' , Kepler's
and Galileo's great discoveries," writes Dilthey, "and through the
accompanying the ory. of constructing nature by means of ' mathematical
^r?!H?5?_ S ' SiY? 1 ? AJ^i°;?.i was thus founded the sovereign conciousness
of the autononiy of the human intellect and of its 'power over nature j
a doctrine which became the prevailing conviction of tho most advanced
minds." (29)

This gravitation towards mathematics is already found
in the doctrine of Cardinal Nicholas of Cusa, in whom were burgeoning
practically all the trends, which were subsequently to give direction
to the development of the modem mind He hold that "knowledge is (30)
always measurement", that "number is the first model of things in
the mind of the Creator", (31) and that "There is nothing certain
in our knowledge except mathematics o " (32) From these principles
he derived the idea of a universal mathematical structure and dotermi-
nation of reality, or a reality whoso spiritual ooro and origin is
revealed in its being the subject of universal laws, laws of number
and magnitude". (33)

In the early modem period the one ; who grasped most clearly
the sigiiif icance of mathematics for the study of nature was undoubtedly
Leonardo da Vinci. For Leonardo science was genuine only in the measure
in which it was mathehiatical. "Ho human investigation can call itself
true science unless it proceeds through mathematical demonstrations,"
"There is no certainty in sciences where one of ■ the mathematical sciences
cannot be o.pplicd, or which aro not in relations with these mathematics,"
(34) "Oh, students, study mathematics, .and do not build without a
f oundation. " This enthusiasm for mathematics did not, however, lead
him to believe that nature itself was mathematical; he attributed to
the mathematical worjd only conceptual existence: e tuta montalo.
And he v.'as insistent upon combining observation with mathematical
speculation. "Those sciences aro vain and full of errors which aro
not bom.from experiment, tho mother of all certainty, and which do
not end with one" clear experiment," (35) That all this was not pure
theory in the i.iind of Leonardo is well known. His important contributions

-19-

to the dovelopment of mechanics, hydraulics, and optics were" an impres-
sive confirmation of his belief in tho fruitfulness of tho raathemati-
co-observational method.

This method was taken up by Kepler and applied with great
success to problems of astronomy, "Astronomy is subordinate to the
genus of Mathematical discipline and uses Geometry and Arithmetic as
two wings: through them, it considers quantities and figures of mundane
bodies and movements, and enumerates times, and in this way prepares
its own demonstrations: and it brings all speculations into use or
practice," (36) We have already remarked that there is no conclusive
evidence. to show that Platonic Philosophy, provided a foundation for
the scientific work of any of the early-modern scientists . It might
seem, however, that a case could be built up for Kepler, For his writings
are saturated with a deep conviction that ' the cosmos is made up of
hidden mathematical harmonies, a conviction that seems impregnated
with the quasi mystical attitude of the Pythagoreans and Neo-Platonists,
which attached a recondite religious significance to the mathematical
character of reality, "Geometry" he writes, "was the form of creation
and entered into man with the image of God", (37) There can be no
doubt that a great deal of philosophical reflection distinctively
Neo-Platonio in tone accompanied the scientific work of Kepler, but
it remains extremely questionable to what extent, if any, the former
provided a foundation for the latter, or exercised any true casual
influence upon it, (38)

In the work of Galileo the mathematico-observational
method became a well-defined scientific procedure. In his famous ex-
periment of rolling a ball dewn an incline plane at the tower of Pisa
and of describing the phenomenon in terms of a mathematical equation,
modern scientific method was clearly' crystallized. And he pointed out,
the fundamental principle of this method when he wrote: "To be placed
on the title-page of my collected works: Here it Trill be percexved
from innumerable examples what is tho use of mathematics for judgements
in the natural sciences and how impossible it is to philosophise o<j^W
without the guidance of Geometry, as the wise maxim of Plato iias it. (6V)
"Philosophy is written in that great book which ever lies before our
cyos ._ x i m ean the universe ~ but vrcrcannot understand it if we do
not first learn the language and grasp the symbols, in which it is
written. This book is written in the mathematical language, and the _
symbols are triangles, circles, and other geometrical figures, withouG
whose help it is impossible to comprehend a single word of i",; without
which one wanders in vain through a dark labyrinth." (40)

jMl scientific method involves selection, and it was
inevitable that the growing concibusness of the fruitfulness of mathe-
matics in the explanation of natural phenomnna should result in •.

-20-

an increasing concentration of attention upon the quantitative aspects
of nature. But scientific methods all too easily tend to become tyrannical,
and what begins as a mere selection for tho purpose of explaining
phenomena often issues into an explaining away of the elements left
out of the selection, Galileo was probably the first in modern times
to call into question the existence of the non- quantitative aspects
of reality. Kepler seems to have supposed that the non-mathematical
properties of nature were in some way less real, but he did not deny
their objective existence,, TM.3 'denial is found explicitly in Galileo,
for whom the qualitative properties of nature had existence as such
only in the faculties of man, ^

I feci myself impelled by necessity, as soon as I conceive
! a piece of matter or corporal substance, of conceiving that in
\ its own nature it is bounded and figured by such and such a figure,
, that in relation to others it is large or small, that it is in
| this or that place, in this or that time, that it is in motion
j or remains at rest, that it touches or does not touch another
i body, that it is single, few or many; in short by no imagination
\ can a body be separated from such conditions. But that it must
bo white or r^d, bitter' or sweet, sounding or mute, of a pleasant
or unpleasant odour, I ' do not perceive my mind forced to acknowledge
it accompanied by such conditions; so if the sense were not the
escorts perhaps tho reason or the imagination by itself would
never have arrived at them. Hence I think that those tastes, odours,
colours, etc, on the sido of the object in which they seem to
exist, are nothing, else but mere names, but hold their residence
solely/In the sensitive body; so that if tho animal were removed,
every such quality would be abolished and annihilated, (4-±)
7*
This qualification of nature found its full realization in the philosophy
of Rene Descartes,

It has been oustomary to consider Descartes as the Phi-
losopher of modern mathematical physics, Meyerson writes: "C'est Des-
cartes, incontestablcment, qui a ete le veritable legislateur de la
science moderne," (42) This opinion is shared by Marl tain:

„..il (Descartes) a eu la clairc vue intellectuelle du constitutif
propre et des droits do la science physico-mathematique du monde,
avec toutes ses exigences ,'et, si jo puis dire, sa ferocite de
discipline originalo, d'habitus irreductiblo, II merite vraiement,
a ce point de vue, d'etre regarde commo le fondatour de la science
inodeme, non qu'i'l l'ait oreee de toutes pieces, mais parce que
c'est lui qui 1*5. tireo a la lumi&re du pie in jour ot etablie a
son compte dans la repubiique de la penseo, (43)

-21-

We believe that this passage is filled with errors and ambiguities.
It. will eventually become clear, "we" hope, that Descartes ' intellectual
[view of the "constitutif propre" of mathematical physics was extremely
•confused and profpuiidly erroneous, l^As a consequence he could "have no
just notion of its rights''aM"cxigenbloso'rAs"a"mtter"6?"'fac : t,"''the
Vgxtent to which he exaggerated "them 'was nothing, less than monstruous ,
Since mathematical physics is, as \{Q shall see, an jmtenjiediary science,
and since it is, in fact, not_a science_tojbhe j3trict and J'ormal sense
of the word ,(but dialec tics, ,)"nothiiiK"" could J;e_mo"re" false "than to eg ply
i°J-_V. ^k® JierasJ'discipliiie'originale" and "habi^slrce^uctible 11 ,
Much could be said, moreover,' "In criticism of the expression "republi-
que de la penseo" for taken as it stands it could easily lead to a
false notion of the independence of the sciences, but this is not the
place to develop such a criticism.

We do not believe that Descartes deserves to be called
the founder of modem science. Nevertheless, his doctrine had an ex-
tremely important historical influence upon the development of mathe-
matical physics and for that reason it merits considerable attention,

For Descartes the mathematization of nature was not a
mere scientific method; it 'was a world vision,, The story of how that
vision came to him on that winter's night aiyflewburg on the Danube >
is one of the best known events in the history of philosophy. It had!
been preceded by another great discovery which was to play an all
important part in the fruitful development of mathematical physics —
the discovery of Analytical Geometry.^ Having succeeded in reducing
geometry to arithmetic an&_algebra, (jln spite of the fact that the
ArTstbTe"ITa^s'"had" always insisted on "their'T'orml" distihctiori,;) the
next step" was to jreducc^ physics^ comple"tly_to math'ematicJ7'~It" was a
^rel5end^us"~i"tep~, but Descartes dTd 'nbT~hesitate~t"6"take it. In actual
fact he Trent much farther than this and reduced the whole of philosophy
jto mathematics in the sense that his universal method was .the geometrical
I method of beginning with a clear and distinct intuition and proceeding
[by means of deduction^ All this lay behind the "Cogito," That is why
his whole "philosophy may'be considered a kind of mathematicism. But
we are not interested in this aspect of Cartesianism here.

The vision of which we have spoken is summed up in the
epitaph written by his closest friend, Chanut: "In his winter furlough
comparing the mysteries of nature with the laws of mathematics he
dared hope that the secrets of both could be unlocked with the same
key," And he has himself described this vision for us in the following
terms :

As I considered the matter carefully it gradually came
to light that all those matters only are referred to mathematics

-22-

in which order and measurement are investigated, and' that it
makes no difference whether it be in numbers, figures^ stars,
sounds, or any other object that the question of measurement ariseso -
1 1 saw consequently that there must be some general science to
I explain that element as a whole which gives rise to problems about
order and .measureitont, r'e8trioted~as — thedo~aro to no special subject
^ matter, This^ I perceived* was called universal mathematics

Such a soicnoi/sliould contain the primary rudiments of
human reason, anG. its province ought to extend to/the eliciting
of true results in every subjeotb To speak freely, I am convinced
that it is a more powerful instrument of knowledge than any other
that has been bequeathed to us by human agency, as being the sourco
of all others.

3qv;ca
(44)

Having once laid down this principle, Descartes did not
hesitate to follow its consequences to the very endi "My whole._physics'' ,
he wrotejbo. his i friend Merseme^iS _JH5lE:??s~-H5_-i?-? m . e ^^ ^ • " ^ 45 ^ "^
accept no principles in physics which are not at the same time accepted
in mathematics. " And he goes on to explain:

Nam plane profiteor, me nullam aliam rerum oorporearum
materiam agnoscere, quam illam orariimode divisibilem, figurabilom
et mobilem quam Geometrae quantitatem vocant et pro objecto suarum
demons trationum assumunt; ac nihil plane in ipsa considerare,
praeter istas divisiones, figuras et motus; nihilque do ipsis ut
verum admittere, quod non ex communibus illis notionibus de quarum
veritate non possumus dubitaro, tarn' evidentur, deducatur, ut pro
mathematica demons tratione sit habendum, Et_ quia^.sic. oninm.,riaturae.

1 phaenomena possunt explicari, ut in sequentibus apparebit, nulla
alia Phy'sicae principia" puto esse admittenda s nee alia .e tiara

\optanda,"(46)

The immediate^consequenc? of the transformation of physios
into mathematics was the identification of the nature of bodies with
extension, (6f matter" with "quantity.') What Is matter,' "asks" Descartes
in the Principlao And Ms answer is that "Its nature consists neither
in hardness^ rior in weight, nor in heat, nor in any other qualities,
but only in extension in length, breadth, and depth, which the geome-
tricians call quantity," "Those who distinguish between substance and
extension or quantity, either have no real idea corresponding to the
name of substance,; or else^have 3 confused idea of material substance, ;

Motion had traditionally been the main stumbling block_
for those who had tried to mathematicize nature, Aristotle's criticism
of the Pythagoreans and the Platonists had been that iTathema.ti2a.ti0n
means the exclusion of movement, and he' who is ignorant of movement^
cannot understand nature," And" Saint Thomas had said: "Ex mathematics ■

-23-

non potest aliquid efficaciter de motu concludi." (48) This problem
proved no obstacle to Descartes. He was convinced that even movement
could be mathematiciaed, not in the sense in which it would /be rnathe-
raaticized later by the : calculus "of Newton and Leibniz, but in a sense
far "more radical , Descartes thought that motion was in its very _ essence
mathematical, Qbhat in the last analysis it could be reduced to the
displacement of a point on a plane, j'Ahd this seemed so evident to
him, and the nature" of motion seemed so immediately clear that he
scorned the definition of Aristotle whose profundity appeared to him
to be nothing but the obscuration of something essentially simple
and transparent^

Some modern philosophers find in this difference in the
concept of motion the best expression of the difference between the
ancient and the modern mind. Thus, Mo Brunschvicg believes that in
the modern concept of motion ""une forme de 1' intelligence apparait ,
qui remplace une autre forme de 1' intelligen ce, aveo qui elle est
s ans aucun _ rapport „" (50) Whatever, may be thought of this view,
it is certain that in this difference between the obscurity of the
Aristotelian definition of motion and the clarity of Cartesian motion
we have" a striking symbol of the vast change wrought by Descartes
in the history of philosophy. Reality which for the Greeks and the
Mediavelists had always been something profoundly complex , suddenl y
became transparently clear ,, This is a very significant point ,

But in a particular way, we find in this question of
motion the sharpest contrast between Aristotelian and Cartesian physics.
In fact, a more incisive antinomy could hardly be imagined, -'For Aristotle
movement was a becoming ; for Descax^tes it was a state ; for Aristotle
it was a_process; for Descartes it, was a relation ,, For Aristotle it
was self-evident that because of the principle of inertia the cessation
of a body in motion demanded a cause. We shall return to this antinomy
in the course of our analysis.

With these two clear intuitions of matter and motion
as -ooints of departure > Descartes set out to deduce the whole co smos
even to its smallest detail. He felt confident that with matter and
motion alone hei could construct the world. In commenting upon this
attempt of Descartes, Duhem writes: . -

Ainsi, dans tout l'un^vers, est repandue une roati&re
unique, homogenc, incompressible et indilatable dont nous ne con-*
naissons rien sinon qu'elle est etendue; cette matierc est divi-
sible en parties do divers figures, et ces parties peuvent se
mouvoir le s unes par rapport aux autx-os ; tellos sont les seules ^
proprieties veritables do ce qui forme les corps; a cos proprietes
doivent se ramener toutes les apparantes qualites qui affectent

-24-

nos sens, L'objet de la physique Cartesienne est d'expliquer comment
se faitoette reduction ,

Qu'est-ce que la gravite? L'effet produit stir les corps
par des tourbillons de matiere subtile, Qu'est-ce qu'un corps
chaud? Un corps 'compose de petites parties qui se rerauent sepa-
reraent l'une de 1' autre d'un mouvement tres prompt et trhs violent,'
, Qu'est-oo que la lumiere? Une pression exercee sur 1' ether par
le mouvement des corps en flammes et transmise instantanement
aux plus grandes distances, Toutes les qualites des corps, sans
aucune omission, se trouvent expliquees par une theorie ou l'on
no considere que 1'etendue georaetrique,, les figures qu'on y peut
tracer et les divers mouvements dont ces figures sont susceptibles,
I 'L'univers est une machine en laquelle il n'y a rien du tout a
> considerer que les figures et les mouvements de ses parties, '
Ainsi la science entiere de la nature materielle est reduite a
une sorte d'Arithmetique universelle d'ou la categorie de la qua-
lite est radicalement bannie," (51)

When he had finished his task, Descartes stopped to con-
template it with pride and satisfaction, and he declared that nothing
was lacking, that his' work was perfect. One of the last paragraphs
in the Prinoipia has as mts title; 'That thei-e is no, phenomenon that
is not "included in what has been explained in this treatise," (52)
It was no slight claim on the part of Descartes to pretend to have
a direct intuition of the inner essence of physical reality and to
be able to embrace all its phenomena in a type of knowledge that was
clear and exhaustive, \ . ■ ■

The proclamation of Descartes as the founder or legislator
of modern mathematical physios is susceptible of a variety of interpret-
ations. It may, in the first place, bo taken to mean that his philoso-
phical system affords the truest explanation of the meaning of physi-
co-mathematical knowledge. We believe that any claim of this kind as
far from being justified, but it would be premature to embark upon
a discussion of this point here. It may also be taken to mean that
he formulated with accuracy and clarity the method that has been res-
ponsible for the development of modern physics. Wo do not think that
even this ctLaim is admissible, Cartesian physics as a system was ex-
tremely short-lived. This in itself is not necessarily a condemnation
of Cartesian method, for it is possible for a thinker, Jo work out
a true scientific method, and yet in spite of it be fasnd-into numerous
errors in the order of application, and this faulty application may
be due to circumstances beyond control. But in the case of Descartes
the errors were for the most part because of his method rather than
in spite of it. His physics is a tissue of arbitrary as sumptxons pre-
cisely because he refused to recognize the inductive character of •
physical science, Modern sciSnco is constituted, essentially of both

-25-

a priori and a posteriori elements and Descavtes was -as blind to the
latter as Francis Bacon was to the former,,

Nevertheless there is something to be said for Descartes.
His discovery of analytical geometry provideii an extremely useful
instrument for the mathematization of nature,, even though he failed
to recognize the true nature of his oral creation. But more than that,

I his ambition of a complete^ mathematioizted physics bequeathed to -phy-
sicists a_ dialectical_ goal towards vrtiich the-r would never cease to
strive ; to bring all the phenomena of nature .under the control of
number. That is why it may be said that in the philosophy of Descartes
the mathematical interpretation of nature sesmed to have received
its official character, Prom then on there isas never any question of
the role that physics would follow in its development.

Added to the general inspiration given to mathematical
physics by cartesian philosophy, was the tremendous impetus coming
from the new discoveries in mathematics :

No picture however generalized * of the achievements of
scientific thought in this century can nmrait the advance in mathe-
matics. Here as elsewhere the genius of, the epoch made itself
evident. Three great Frenchmen, Des cart'is, Desargues, Pasca l,
initiated the modern period in geometry , Another Frenchman, Fermat,
laid the foundations of modern analysis , and all but perfected
the method of the differential calculus, Newton and Leibniz, between
them, actually did create the different ial calculus as a physical
method of mathematical reasoning. When the century ended, mathematics
as an instrument for application to physical problems was well
established in something of its modern proficiency, (53)

As a result of the philosophic jal influence that stemmed
from Descartes and of the discovery of moro powerful mathematical
instruments, the role of mathematics in physics continued to grow
with ever increasing fruitfulraess. There wsr© a few reactionary attempts
made, particularly in Germany by Goethe , S chelling and Hege l, but
they had no lasting success, and left behind them no positive ti-ace
\ in science.

In the physics of Newton the mathematical interpretation
of nature seemed to have reached its crow ling achievement, "The out- _
standing fact that colors every other belief in this age of the Newtonian
world." Writes Randall, "was the success 3f the mathematical interpret-
ation of nature," (54) The part that ma thematics played in the work
of Newton himself is aptly expressed by -the title he chose for his
classical work, The Mathematical Prinoi p] es of NatoajJPtojggophg,
and by the brief interpretation he gave c f its significance m -one

-26-

preface:

We offer this work as mathematical principles of philosophy.
. » , . By the propositions mathematically demonstrated in the
first book, we then derive from the celestial phenomena the forces
of gravity with which bodies tend to the sun and the several planets.
Then, from these forces, by other propositions which are also
mathematical, we deduce the motions of the planets, the comets,
s the moon, and the sea... (55)

Although throughout his 'work Newton acted as though in nature there
were a possibility of infinite determinatio n, it may be doubted perhaps
just what significance he attached to this methodological principle „
"To Newton, at any rate," says J.W.N. Sullivan, "the attempt to describe
nature mathematically was an adventure that might or might not be
successful," (56) And Dingle writes:

In the matter of fitting observations into a mathematical
framework, Newton was both more or less thoroughgoing than Galileo,
He himself enlarged the framework considerably, so that while to
Galileo mathematics was mainly geometry, to Newton geometry oc-
cupied only a subordinate place. Thus he was a ble to' conduct a
mathematical treatment of the phenomena of colour which Galile o
Sid relegated to the rank of a subjective qualit y. On the other
hand, he did not regard the whole of external Nature as necessarily
mathematical in character, although he hoped it might prove to
be soi (57) •

It would be too long and tedious to trace the subsequent
development of mathematical physics in full detail. Much could evidently
be said about Leibniz whose doctrine, in so far as it related to the
physical universe, was, in the last analysis, a kind of mathematicism,
Mich could be said in particular about Kant, whose Transcendental
aesthetics deals with the question of pure mathematics, and whose
Transcendental Analytic is an explanation of the mathematical science .
of nature. One of the greatest contemporary philosophers of physical
science, Sir Arthur Eddington, has this to say about the doctrine
of Kant:

If it were necessary to choose a leader from among the
older philosophers, there can be no doubt that our choice would
be Kant. We do not accept the Kantian label; butj as a matter of
acknowledgement, it is right to say that Kant anticipated to a
remarkable extent the ideas to which we are now being impelled
by the modem developments of physics, (58)

We shall not stop to evaluate" this statement now, nor

-27-

oo discuss in detail the relation of mathematical physics to the philo-
sophy of Kant. This we hope to do in chapter XII, By that time we
saruj. be in a positi on to see how many large concessions must be made
to Kantianism if we are to understand the true nature of •ph ysioo^riathe-
matical knowledge. For the present let it suffice to point out that
Rant considoi-od Newtonian physics as the only. genuine type of science,
and chat there is a sense in which it is true to soy that he made it
the foundation of his whole elaborate philosophical system. Prom the
following lines it is evident that for hiraihe physical world can be
known scientifically only through mathematics:

Les suppositions de la geometrie ne sont pas des deter-
minations d'une simple creation de notre fanteasie poetique, ne
pouvant ainsi etre rapportees avec certitude' a des objets reels,
mais elles sont necessairement valablespour l'espace,, et par suite
pour tout ce qui peut se rencontrer dans l'espace, parce que l'es-
I pace, n'est pas autre chose que la forme de tous les phenomenes
/ exterieurs sous laquelle des objets des sens peuvent nous etre
\ donnes. La sensibilite sur la forme 'de laquelle se. fonde la geo-
metrie, est ce dont depend la possiMli+.p. des phonoci'spa oxteri.fiuivjj
ceux-ci ne peuvent done jamais renrernier autx-c chose que ce que
la geometrie leur prescritj (59)

For Kant space and time which are the a priori forms that determine
all our scientific knowledge of the material world are reducible to
the abstract concepts of continuous and discrete quantit y. In his ,
?ir^^IIsisehtM±c23- Principles of the Science of Nature he writes:
''In every particular theory of nature the only "thing that is scientific
in the strict sense of the world is the quantity of mathematics it
contains o" (60)

The progress of physics in recent years, particularly
since the advent of the theory of relativity, the quantum theory and
wave-mechanics, has resulted in a' mathematization of nature never
dreamed of by the most enthusiastic of the classical physicists. (61)
In one sense at least, the mathematical element seems to be supplanting
more and more the purely physical. An obvious example of this is the
way in which the problem of gravitation, which in classical physics
was a question of dynaniios ( involving the notion of force7 ) has in Eins -
teinian physics been reduced to a problem of pure geometr y. Moreover,
in the comparison with classical physics, the conceptual mathematical
implements now being used are of a much more abstract nature, and
are taken from what is sometimes known as "pure mathematics. " Sir
James Jeans sees in this application of "pure mathematics" to the
physical universe a n=sw epistemological phenomenon which constitute
a major difference between contemporary and classical mathematical
physics. (62)

-28-

On the other hand, paradoxical as it may seem, Relativity
and Quantum physics are at the same time less mathematical and more

, physical than classical physics. Cartesian and Newtonian physics were
in many ways extremely simplicist. They attempted to impose upon the
physical universe absolute quantitative determinations such as they
may be conceived of by a mathematician who does not have' to worry
about concrete physical processes of observation and concrete physical

, procedures of measurement, Einstein brought to light the vast difference

I between a pure mathematician and a matheinatical physicist by showing
how much is involved in the concrete procedures of observation and

, measurement. As a result, science has been brought closer to the object-
ive physical universe. Moreover, contemporary physics has become less
mathematical and more physical in the sense that it has come to realize
more clearly that nature overflows any geometrical frame that we may
attempt to impose upon it, that there is a greater irrational element
in nature than was suspected before. However, underneath this revolu-
tionary character of contemporary physics there is,. of course, a
fundamental continuity with the past, as we shall try to make clear
later on, (63)

One of the characteristic features of recent physics
which is of particular interest to us is its self-conciousness. Clas-
sical physics was self-concious but it was, so to speak, the naive
self-conciousness of adolescence. In recent years physical science
has begun to achieve the self-conciousness of maturity, which consists
chiefly in a detached self-criticism. All of the greatest contemporary
mathematical physicists, those who have contributed most to the ad-
vancement of science, such as Einstein, Planck'JjiDe Broglie, Weil,
Dirac, Heisenberg, Schrodinger, Eddington and. Jeans, have felt the
need of doing some serious reflective thinking about the nature of '
their science. This thinking is of unequal philosophical Value, to .
be sure, but out of it has come a wealth of helpful insights into
the nature of physical science. At this point we can do no more than
select from these contributions a few typical observations on the
general nature of mathematical physics. These will be sufficient to
situate our problem accurately in its contemporary context, and that
is all that interests us for the moment.

But before indicating the characteristic positions taken
by some of the more recent mathematical physicists as to the general
nature of their science, perhaps it would be worth while to consider
here a highly significant passage of one of the most outstanding of
nineteenth century biologists, Claude Bernard. Bernard was one of
those who made the greatest contributions to the growth of the crit i-
cal view of science , and his observations on the general character
of natural science - are of the greatest value:

-29-

The absolute principle of the experimental sciences is
a necessary and conscious determinism An the conditions of the
Phenomena. It is of such a sort that a natural phenomenon, whatever
it is, being given, the experimenter can never admit that there
is a variation in the expression of this phenomenon, unless at
-ohc same time there bo the intervention of the new conditions
m xts manifestation; moreover, he has an a priori certitude that
these variations are determined by rigorous and mathematical con-
nections. Experience simply shows us the form of the phenomena;
but the connection of the phenomena to a determined cause is neces-
sary and independent of experience, and it is necessarily mathe-

|matically absolute,, 'We thus see that tho principle of the criterion
of the experimental sciences is in reality identical with that
of the mathematical sciences, since in each of them this principle

^is expressed by a' necessary and absolute relation of things. However,
in the experimental science/these connections are surrounded by
numerous, complex, and infinitely varied phenomena, which hide
the connections from our view,. By the aid of experience we analyze,
we dissociate the phenomena, in order to reduce them to relations
and conditions that are more simple. We wish in this way to seise
the form of scientific truth, that is to say, to find the law
which should give us the key to all the variations of the pheno mena „
This experimental analysis is the only means , that we have for '
searching out the truths in the experimental sciences; and the
absolute determinism of the phenomena, of which we have an a priori
consciousness , is the sole criterion or the sole principle which
directs and supports us. In spite of our efforts, we are still
very far from this absolute truth; and it is probable, especially
in the biological sciences that we shall never see it in its
nudity. (64-)

When the scientists speak of the general question of
determinism in nature, it is sometimes difficult to know whether they
are talking of determinism as a methodological principle or as a physical
principle, . In fact the two are often enough confused in the mind of
the scientists themselves. Determinism is , of course , legitimate and
necessary as a methodological principle . Without it there could be
no science . But it is evident from the passage just quoted that for
Bernard determinism is not merely a method existing in the mind of
the scientist and in the process tlirough 4 he studies nature,, but a
reality existing in nature itself , y in the physical universe is object -
ively realized tho infinite r i gor of the mathematical world .j This
view of Bernard seems to have been the generally accepted opinion
of the classical physicists, though among them there was this difference.,
that while for some the infinite determination of nature could be
arrived at by science, at least theoretically, for others it was an
objective limit towards which science must ever move. The ever increasing

-50-

success of the application of mathematics to nature tends almost inevi-
tably to lead scientists to some position of this kind, for as Professor
Bridgman has pointed out:

...it is a result of every day experience that as we refine the
accuracy of our physical measurements the quantitative statements
of geometry are verified within an ever decreasing margin of error.
From this arises that view of the nature of mathematics Y/hich
apparently is more commonly held; namely that if we could eliminate
the imperfections of our measurements, the relations of mathematics
would he exactly verified. Abstract mathematical principles are
supposed to be active in nature , controlling natural phenomena,
as Pythagoras long ago tried to express wi th his harmony of the
spheres and the mystic relation of numbers. (65)

And although Heiseriberg's principle of uncertainty, which expresses
the high degree of indeterminisra recently discovered by scientists
on the level of microscopic phenomena , has thrown wide' open the whole
problem of the determination of nature, there are _ still many scientists
w ho hold that this indeterminisn is purely subjective and that it
gives no reason for doubting the objective existence. of a mathematical
determination in the universe.

In the annals of modern. science there is no greater name
than that of Albert Einstein, and consequently his opinion on the
nature of mathematical physics is of the utmost interest. Of the many
important statements he has made on the subject the following is perhaps
the most significant for us and the most relevant to our present
purpose.

On the contrary, the scientists of those times were for
the most part convinced that the basic concepts and laws of physics
were not in a logical sense free inventions of the human mind,
but rather that they were derivable by abstraction, i.e. by a
logical process, from experiments. It was the general theory of
Relativity that showed in a convincing manner the . incorrectness
of this view. For this theory revealed that it was pessible for
us, using basic principles very far removed from those of Newton,
to do justice to the entire range of the data of experience in
a nanner even more complete and satisfactory than was possible
with Newton's principles. But quite apart from the question of
compara Live merits, the fictitious character of the principles
is made quite obvious by the fact that it is possible to exibit
two essentially different bases, each of which in its consequences
Vleads to a large measure of agreement with experience. This indicate s
that any attempt logically to derive the basic concepts and laws
of mechanics from the ultimate data of experience is doomed to ~

-51-

f allure . If then, it is the case that the axioiit.tic basis of thcore-
cical physics cannot he an inference from experience , hut must
he free invention , have wo any right to hope that wo shall find
tho correct way? Still more — does this correct approach exist
at all, save in our imagination? Havo wo any right to hope that
experience will guide us aright, when there arc theories (like
classical mechanics) which agree with experience to a very great
extent, even without comprehending tho subjects in its depths?
To this I answer with complete assurance, that in ray opinion there
is the correct path, and, moreover, that it is in our power to
find it. Our experience up to date .justifies us in feeling sur e
that in Nature is actualized the idea of mathematical simplicity ,,
lit is ray conviction that pure mathematical construction enables
/us to discover the concepts and laws connecting them which give
Vus the key to the un derst anding of the phenomena of Nature . Expe-
rience can of course guide us in our choice of . serviceable mathe-
matical concepts,' it cannot possibly be the source from which
they are derived ; experience of course remains the sole criterion
or the serviceability of a mathematical construction for physics
but the truly creative principle resides in mathematics . [ In a
certain sense, therefore , I hold it to be true that pure thought
is competent to comprehend the real, as the ancients dreamed,. ) (66)

This passage is so lucid and precise that it scarcely needs a commentary.
The important point to be drawn from it is that although the. mathe-
matical concepts and principles used in physics are not derived directl y
from nature oCpj it come from the p roductive activity of the mincQ never-
theless there ■exist in the cosmos a. basic matfrematxeai structure and
through the progress of science the mathematical construction of the
^j nind can ultimately be brought into exact, conformity with itT)

Allusion has already been made to the views of Sir James
Jeans .on the significance of tho application of mathematics . to nature.
For Jeans recent developments in physics have produced a new and highly
significant epistemological phenomenon: the successful application
of "pure mathematics" to the physical universe In classical physics
the use of mathematics had been large and fruitful, but the mathematic s
used was something that had been previously drawn from nature ; it
was not "pure mathematics" deriving solely from the creative activity

^of the intellect. By 'pure mathematics 1 is meant those departments
of mathematics which are creations of pure thought, or reason operating
solely within her own. sphere, as contrasted with 'applied mathematics '
which reasons about the external world, after first taking some sup-
posed property of the external as its raw material," (67) It is this
"pure mathematics" v/hich 1! now used in Relativity and Quantum physics,

I And the great mystery is that nature seems to conform to these free

\ creations of pure thoughts

-32-

\7o could not of course draw any conclusions from this
if the concepts of pure mathematics which we find to be inherent
in the structure of the universe were merely part of, or had been
introduced through, the concepts of applied mathematics which
we used to discover the workings of the universe. It would prove
nothing if nature had merely been found to act in accordance with
the concepts of applied mathematics; these concepts wera specially
and deliberately designed by man to fit the workings of nature,
| Thus it may still be objected that even our pure mathematics does
not in actual fact represent a creation of our own minds so much
I as an effort, based on forgotten or subconscious memories, to
\ understand the workings of nature. If so, it is not surprising
.that nature should be found to work according to the laws of pure
mathematics. It cannot of course be denied that some of the concepts
with which the pure mathematician works are token direct from
his experience of nature. An obvious instance is the concept of
quantity, but this is so fundamental that it is hard to imagine
any scheme of nature from which it was entirely excluded. Other
concepts borrow at least something from experience; for instance
multidimensional geometry, which clearly originated out of the
experience of the three dimensions of space. If, however, the more
intricate concepts of pure mathematics have been transplanted
from the workings of nature, they must have been buried very deep
indeed in our subconscious minds. This very controversial possibility
is one whi ch_ cannot be entirely dismissed , but it is exceedingly
hard to believe that such intricate concepts as a finite curved
space and an expanding space can have entered into pure mathematics
through any worth of unconscious or subconscious experience of
Vthe workings of the actual universe. In any event, it can hardl y
be disputed that nature and our conscious mathematical minds work
acc ording to the same laws . She does not model her behaviour,
so to speak, on that forced on us by our whims and passions, or
oil that of our muscles and joints, but on that of our thinking
minds. This remains true whether our minds impress their laws
on nature, or she impresses her laws on us, and provides a suf-
ficient justification for thinking of the universe as being o f
ma thematical design . Lapsing back again into the crudely anthro- '
pomorphic language - we have already used, we may say that we have
already considered with disfavour the possibility of the universe'
having been planned by a biologist or an engineer; from the intrinsic
evidence of his creation, the Great Architect of the Universe
^now begins to appear as a pure mathematician, (68)

It is to be noted that for Jeans the mathematical interpretation of
nature gives exhaustive knowledge of it, for ho says: "The final truth
about a phenomenon resides in the mathematical description of it; so
long as there is no imperfection in this our knowledge of the phenomenon

-33-

is complete, " (69)

If wc were to stop at this point and look back over the
historical sketch we have been giving, we would find this one central
thought running through the various opinions discussed: the fundamental
reason why np.theraa tj. 03 can he applied to nature C ls that nature ±3
uJjjjSajgly-Jaathe'''Tatioal,) that in the phvsicalT~univer s e there is realized
a basic mathematical structur e; ^math ematicalphysics simply means that
in the last analysis mathematics and physics are in some n«nse identified^
Most of the authors we have mentioned would suscribe to the opinion
of Juvets "Sans preoio^r davantage notre pensee, nous dirons que le
monde physique n" est qu' nn reflet on una section du monde mathema-
tique." (70) ~ '

But at the present time a large number of authors are
advancing an opinion which on the surface at, least seems to be directly
opposed to the position just stated „ For many modern, philosophers of
science, mathematics is nothing but formal log ic, ( and the part that
it play s_in_ ph ysics has no other significance than the part that logic plays
in all the sciences^ Vassily Pavlov has summed up this position in
the following terms :

It were well, then, to introduce briefly the claim that
mathematics at bottom is only logic. To many this claim has been
demonstrated for all time in the work of Frege, Peano, Bertrand
Russ ell, A.N. Whitehead, and othe rs, Cwho_ developed the subjec t'
oF^ymbolic" or "mathematical" iogic^ ) Matheira.tics and formal
logic have been declared to be identical . Both have bben pictured
as vast systems of so-called "tautologies.-." substitutions, identities ,
^possessing novelty only in a psychological sense The entire system
of mathematics (or logic) i s said to be contained in its postulate
sets, which are nothing but the 'rules of the game", a game con-
ventional to the core, possibly derived from reality but nastily
indifferent to it. In short, there has occured an apotheosis of
the rules, th^_rul^_wxjthout. the_game»

Many of us are very uncomfortable over the sharp separation
which has occored between the rules of the game and the game it-
self. Every application of mathematics-logic to nature, then,
seems to us a promise of a happy reunion. We return to nature
only that which belonged to it in the first place. The mystery,
if any, lies in the original separation, rather than in the ap-
plication, (71)

Taken as it is presented here, this opinion means that
mathematics is used in physics merely as an instrume nt that remain s
extrinsic to the essence of the science m whxch it is em ployed,
3u3t as logic is a mere instrument that remains essentially extrlnsxc

~34~

to the inner constitution of the sciences which employ it. But it
must be noted that not all the authors who teach that mathematics
is only a tool in physics necessarily hold that it is a purely extrinsic
j instrument, For, as we shall explain presently, it is possible to
hold that the mathematics employed in physics constitutes an essential
part of the object of physical science and still consider it as purely
instrumental \in the sense that\the 7/hole purpose of physical scienc e
is to know the physical universe and not the mathematical worl dj and
consequently the whole raison d'etre of the use of mathematics is to
enable the mind to come into closer contact with the objective cosmos.
Perhaps it is in this light that we must interpret the opinion of
Dirac:

Prom the mathematical side the approach to the new theories
presents no difficulties, as the mathematics required (at any rate
that which is required for the development of physics up to the
present) is not essentially different from what has been cuirent
for a considerable time,, Matheinatics is a tool specially suitable
for dealing with abstract concepts of any kind and there is no
limit to its power in this field. For this reason a book on the
new physics, if not purely descriptive of experimental work, must
be essentially mathematical,! All _the same t he .ma thematics is onl y
a tool (and one should learn to h o ld the physical ideas in one's
i mind without reference to the mathematical foray ) (72)

It seems quite probable that it is also in this light
that the position of Sir Arthur Eddington must be understood. Contrar y
to the opinion of Jeans , he holds that the physical universe is no t
mathemat ical, ( and that if mathematics enters into physical science
it is only because the mind has introduced it from without^ ) Nor can
the role of mathematics be reduced to a question of mere symbolism.
Mathematics is able to get a grip on the cosmos because physical realit y
can by processes of measurement be t ransformed into series of measure -
numbers, (and the relation between these measure-numbers can be built
up~into a mathematical system principally through the instrumentalTt y
of the theory of Groups? In The Philosoph y of Physical Science he
has this to say:

(Theoretical physics today- is highly mathematical. Where
does the mathematics come from? I cannot accept Joans' view that
mathematical conceptions appear in physics because it deals with
a un iverse created by a Pure Mathematician ; ( my opinion of pure
mathematicians 8 though respectful, is not so exalted as that," )
An unbiased consideration of human experience as a whole does ,
'not suggest that either the experience itself or the truth re«
vealed in it is of such a nature as to resolve itself spontaneously
into mathematical conceptions. The mathematics, is not there till

-55-

we put it thei~e. The question to be discussed in this chapter
is» A^bjsrtiat_p_oiivt does the mathematician contrive to get a grip
on material which_ iiitrinsicall y does not of itself render a sub-
ject mathcraatical.~if in a public lecture I uso the common abbre-
viation No. for a number, nobody protests; but if I abreviate it
as N. 5 it mil be reported that 'at this point the lecturer deviated
into higher mathematics" „ Disregarding such prejudice's, we must
recognize that the allocation of symbols A, B, C, ,... to various
entities or qualities is merely an abbreviated nomenclature which
involves no mathematical conceptionsT) (73)

And he goes on to explain how the Theory of Groups is employed in
> tivoisf orming physical science, into a mathematical system. (74)

There is still another opinion which in the mind of many
of the authors who advance it may not. represent anything substantially
different from the position of those who hold . that mathematics is
nothing more than a logical tool, but which if taken litterally amounts
to something quite different. It is the view that the role played by
mathematics in physics, is that of a universal and extremely conveni ght
language , ( in so far as it is used in physj:cs 3 mathematics is just a
code, a"~ki nd, of symbolic lan guage, a sort of esperanto of science .J ( 75)

{"Mathematics," says Herzfelt, "is only a tool, fa shorthand 'way of
expression ^ but_ cannot add anything to the physical concept , although
it might occasionally suggest a physical law because its mathematical
expr ession might b e_ particularly simple iT j)

For some who hold this opinion, the role of , raathetnatios
in physics is reduced to that of a stenographic method ; and just as
short-hand i s a mere substitute for long-hand , (j md everything it expres -
ses can be ex pressed v/ith equal fulness and accuracy^ ) though not with
equal convenience, by the ordinary mode of writing, so everything
contained in a v/orld geometry could, strictly speaking, be expressed
in purely "phjrsical language." For others the symbolism of number
has- advantages over the symbolism of ordinary language which reach
far beyond mere convenience, and which are the source of the fruifa __ — ^^
fu lness of the application of mathematics to physics . For the ( symbolisrn )
of ordinairy language can represent reality only in a dispersed and
isolated vfav . ^whereas the s ymbolism of ' number is essentially a relational
syrabolism fond that is why it is able to represent the structure of
the univer se~and thus open up its secrets . .Perhaps the clearest expres-
sion of this opinion is found in Ernest Cassiror:

The symbols of language themselves have no definite sys-
tematic order. Every single linguistic term has a special "area
of meaning". It is, as Gardiner says, "a beam of light, illuminating
first this portion and then that portion of the field within which
the thing, or rather the complex concatenation of things signified

-36-

by a Sentence lies." But all these different beams of light do
not have a common focus. The y are dispersed and isolated . In the
"synthesis of the manifold" every new word makes a new start.

This state of affairs is completely changed as soon as
ye enter into the realm of number,, Yfc cannot speak of single and
isolated numbers , ^The essence of number is always relative, not
absolute J A single number is only a single place in a g eneral
systematic o^^r ^ It has no being o f its own, no self-contF.i""-*
reality- J lta meaning j s <-"!,. fiwpd "iv y thn .nm^j-m f t — wupies in"
\th e whole numerical system. ,~7j Wo uuiuiexve it as a new and powerful
symbolism which, for all scientific purposes , is definitely superior
to the sy mboli sm of speech, For what ■" we find here are no longer
detached Affords ) but terms that proceed according to one and the
same fundamental plan an d that, therefore, show us a clear and
d efinite structural lawT ) (76) fa*.. M , (/w-vv.—Y>p. i.u,%\z

This view, which at first glance, at least, seems to
reduce the role of mathematics in physics to" a question of language,
differs from the opinion of those who. identify mathematics with formal
logic to the extent that language differs from logic , though perhaps
the distance between logic and mathematical language would not be so
great as that between logic and ordinary language, it might be argue d
that in the measure in which mathematics would be considered a universa l
ahguage it would be lifted out of, the materiality of individuality

and brought closer to the universal laws of thought „) At first sight,
this position would seem to 1 be at the other extreme from the opinion
which sees the mathematical world realized in the physical world,
but perhaps if, we looked deeper we 1 might find ourselves in the presence
of a case where extremes meet, if mathematical language is but a subs--
ui'oute for "physical language" might not the reason bo that the ma the-
maticol wort-Id and the physical world aro really one?

I

-37-

3, Rel evance of Thomisra

In under talcing to establish the significance of Thomisra
(77) - for the problem of mathematical physics we are not insensible
to the fact that such' an undertaking calls for an apologia. For histo-
rians_aljnost ^without _ exception haye_ represented^ the [..rise anflTdevilop-
i .™H5_££j2^^r n _Bh5I?,iS?„ .. as _ sor ']?^ :5 -?'S cong?letely, antithetical "to"the "'

Russell says: "His few factsY sufficed' tocbstroy the whole vast system
of supposed knowledge handed' down from Aristotle , as even the palest
morning sun suffices to, extinguish the stars." ' (78) And Professor
Burtt writes: ..'.■■-..

But now, of course, the question which Copernicus has
thus easily answered carries with it a tremendous metaphysical
assumption. Nor were' people slew to see it and bring to the fore-
front of discussion. Is it legitimate to take any other point of ■
reference in astronomy than the earth? Mathematicians who were
themselves subject to all the influences working in Copernicus'
mind, would, so,' he hoped^ be apt to say yes. But of course the whole
Aristotelian and empirical philosophy of the age rose up and said
no. For the question went pretty deep, it meant .not only, is the
astronomical realm fundamentally geometrical, which almost any
one would grant, but is the universe as a whole, including our
earth, fundamentally geometrical in its structure? Just because
this shift of the point of reference, gives a simpler geometrical
expression for facts, is it legitimate to make it? To admit this
point is to overthrovf the whole Aristotelian physics and cosmo-
logy. (79), , ! ■ ; :

We are dealing! here not merely with those who hold it.
as an indjputable methodological, principle that enlightenment first
dawned upon the world at the time of the Renaissance, Such as these
we could afford to ignore „ But there, are many others who while they
have a sincere admiration for all that Greek and Medieval culture
has to offer us in the way of- art, of metaphysics, and of morals,
nevertheless„believe.-. .that...if _.ther.e....i.s„one _ f ^^ield...in^vhich, both. Aristotle
and the Medievalists are completely barren, it is .the field of science.
Most 'of' 'these mxghT be willing : enbugh' "to "concede to Professor whitehead
that scholastic logic and theology prepared the soil in which modern
t science took its roots, (80) but this, could scarcely serve as a
sufficient basis to constitute Thomism a_^ J^sj-gjiif icant ^^ philosophy
of science

-38-

Among contemporary philosophers of science few have won
for themselves wider recognition and a greater name than Eraile Meyerson,
particularly in questions of the relation between modern science and
its historical background, Xet if there is one theme which runs through
all of Meyerson' s voluminous works it is that peripateticism has ab-
SjOJLutely^jiothing to offer to scienceT~Tn 'Id'en'tite e t Realite he writes:
"Le retour au peri'patetlcisme, preconise avec tan t de force et de
savoir par Duhem nous parait impossible, II ne nous scnble pas,, en
effet, que la pure doctrine d'Aristote ait ete une doctrine verita-
blement scientifique," (81) And again in Du Cheminemcnt de la P ens is,
he says: "La science peripatetique, assurement a peri et, quoi qu'en
pensent certains partisans extremes du . retour au moyen age, peri to-
talement et irreuiajiThleiaeiatJl est aussi impossible do la maintenir
en face du triorapho de la physique moderne qu'il I 1 est de la concilier,
fut-ce meme partiellement avec celle-ci," (82)

In recent years, a few historians have, indeed, come to
recognize the eminence of the scientific spirit and method of Aristotle,
and the worthwhile significance of the accomplishments which were
the fruit of that spirit, and method; but J^e jtrj^utes p_f_ these ,fe W
are , entirel y restricted to the field g£.Mgfp^i9J^_. s i ) J:g5° e ,° That these
tributes are merited is evident to. anyone who has taken the pains
to read the physical treatises of Aristotle, but they leave unsolved
the question in which we are directly interested^ In fact some have
seen in the intense devotion of the Stagirite to research in the field
of biology an argument against the contention we have set out to subs-
tantiate, Dopp, for example, writes:

II est arrive qu'Aristote s'est senti peu de gout pour
les mathematiques, ne s'est point consacre a ces sciences qui
les utilisaient, mais s'est donne surtout a des recherches d'his-
i toire naturelle et do biologie, lesquelles consistaient essentiel-
i lenient, en descriptions ou en analyses de qualites oud' activates
i plus ou inbins discontinues, done " qualita'tiyes ,.' ._ '_ .

Cette 'doctrine' ava'mt en sdimvie pqurpbrt'ee do liberer

lc physician a l'egard de la pensee mathematique „ Elle pesera

sur toute la tradition philosophique du Moyen Age et, par certaines

de ses consequences, sur la philosophic moderne jusqu'a nos jours, (8

The view is how being advanced by more than one philosopher
of science that there is a direct connection between Aristotle's pre-
dominant interest in biological sciences and the type of logic he
evolved, and that Aristotelian logic is not only of little use for
the development of mathematical physics, but in some sense an obstacle
to 3ti For biology is essentially qualitative and class if icatory,
that is to say, it attempts to classify living beings in a schema of
genera and species that is based upon qualitative characteristics.

-59-

And that explains, we are told, why Aristotelian logic is essentially
classificatory, and not relational like /modern mathematical logic.
Professor Whitehead has laid considerable emphasis on : this point:

In a sense, Plato and Py/thagoras stand nearer to modern
physical science than does Aristotle, The tvp former were mathe-
maticians, whereas Aristotle was the son of a doctor, though of
course he was not thereby ignorant of mathematics. The practical
counsel to be derived from Pythagoras is to measure, and thus to
express quality in terms of numerically' determined quantity. But
the biological sciences then and till our own time, have been
overwhelmingly classificatory. Accordingly, Aristotle by his logic
throws the emphasis oil classification. The popularity of Aristotelia n
Logic retarded the advance of -physical science throughout the
Middle Ages . If only the schoolmen had measured instead of clas-

sifying, how much they might have learnt, (84)

Professor Etionne Gilsoiij who is considered by many to
be one of the most eminent modern champions of Thomism, has gone far
boyond either Ddpp or Whitehead by claiming that peripateticism has
been utterly sterile in the realm of_p_hysi.cs because Aristotle attempted
to biologize the whole of ph ysical realit y, (t hat he actuall y made
physical bodies into so many animals ^ In his essay, "Concerning Christian
Philosophy" (85) we find the following devastating criticism:

... We are bound to condemn the scientific sterilit y of the middle
Ages for those very reasons which today make us condemn the phi-
losophic sterilit y of 'scienticism". Aristotle also had exaggerated
the scope of one science and the value of its method, to the de-
triment of the others; and in a sense he Yra.s less excusable than
Descartes, for in this he came into open contradiction with the
requirements of his own method, whereas Descartes was only carrying
his through. And yet, philosophically, Aristotle's was the less
dangerous error, for it wa s an error of fact , and left the question

Iof principle untouched: to biologize the inorganic as he and the
medieval philosophers did , was to condemn oneself to ignorance
about those sciences of the inorganic world whose present popularity
comes chiefly from the inexhaustible fertility which- they, display
in things practical; but to mathematize knowledge entirely, and
on principle,' was to set fetronge limits to physics and chemistry,
and" to mage impossible biology, metaphysics, and consequently
moral theory... Aristotle's error lay in not being true to his
principle of a science of ; the real for every order of the real,
and the error of medieval philosophy lay in f ollowing him in this „
Committing the opposite mistake to that of Descartes, Aristotle
set up t he biological method as a physical me thod. It is generally
adiiiitted that the only positive' kinds of knowledge in which

-40-

Amstotelianism achieved any i:>rogress are those which treat of
the morphology and the functions of the living "beings. The fact
is that Aristotle was before everything a naturalist just as Des-
cartes was before everything a Mathematician; so much so indeed
that instead of reducing .the organic 'to the inorganic like Descartes,
Aristotle claimed to include the ino rga nic in the organic . Struck
by the dominance of form in the living being, he made it not only
a principle of the explanation of the phenomena of life, but even
extended it from living beings to mobile beings in general. Hence
the famous theory of substantial forms, the elimination of which
was to be the first care of Descartes. For a scholastic philosopher,
as a matter of fact, physical bodies are endowed with forms from
which they derive their movement and their properties; and just
as the soul is a certain species of form — that of a living being —
so is fordp. certain genus of soul — the genus which includes
both the forms of inorganic beings and the forms or souls of or-
ganized beings.

This explains the relative sterility of the scholastic
philosophy in the order of physics and even chemistry, as well
as the inadequacy of Cartesianisra in the order of the natural
sciences. If there is in the living being anything other than
pure mechanism, Descartes is foredoomed to miss it; but if there
is not in physical reality that,, which defines the living being
as such, then the scholastic philosophy will not only fail to
find it there, but will never discover even what is there. Never-
theless it wasted its time in looking for what was not there;
; and as it was convinced that all the operations of inorganic bodies
I are explained by forms, it strove with all its might against those

who claimed to see there something else, and clung to that impossible
V position until, in losing it, it lost itself . Three centuries ;
spent__in classing wh at must be measured , as today some persist
in measuring what must be classed, produced only a kind of pseudo-
physics, as dangerous to the -future of science as to that of the
philosophy v/hich imagined itself bound to it; scholasticism was
unable to extract from its own principles the physics which could
and should have flowed from it , ■„ Pormae natural.e s sunt actuosae
et quasi viva e, said the scholastics: between the Cartesian art-
ificialism which makes animals into so many machines, and the
Aristotelian vitalism which makes physical bodies into so many
animals, there, must be room for a mechanism in physics and a vi-
talism in biology, (86)

To this criticism Gilson appends the following interesting
footnote:

It is clear that Aristotle's error, less serious than
that of Descartes from the point of view of philosophy, was more

-41-

serious -from the point of view of science. To extend, like Descartes,
a more general science to the less general sciences, leaves it
possible to reach in these last what they have', in common with the
first; hence a mechanization, always possible though always partial,
of biology; but to turn the method, of a more particular science
back upon a more general science amounts to. leaving the more ge-
neral without an object. Nowj in missing the real objects of physics
and chemistry, Aristotle missed at the same time all that bio-che-
mistry teaches us concerning biological facts — which, although
it is neither the whole nor the most important part, is possibly
the part which ftost useful. And this, as well as being a serious
gap in his theory, is the thing that human utilitarianism will
never forgive him. (87)

It is to be noted that these lines are written by an historian, who
does not oij e so much as one .text to substantiate his criticism . More-
over, tins only thing that presents' the semblance of a reason for the
assertions made is that Aristotle extended his doctrine of substantial
form to inorganic as ■■well as organic bodies, "and just as the soul
is a certain species of form — that of a living being — so is form
a certain genus of soul — the genus which includes both the forms
of inorganic beings and' the. forms or souls of organized beings." The]
sophistry of this argument is so obvious that it does not have to J
be pointed out,

Gilson holds that peripatetic sterility in the realm
of physics derives from the fact that Aristotle failed to recognize
or at least to follow the .■; principles that were inherent in his doctrine,
but he admits that these principles could provide a fruitful philosophy
of science o This, however, has been denied by M., Augustih Mansion,
who in a long, article entitled "La Physique Aristotelicienne et la.
philosophic," (88) has tried to show not only why nothing of any
consequence for mathematical physics is found in the doctrine of Aris-
totle, but even why it was 'theoretically impossible for it to be found
therein, According to Mansion, mathematical, physics could find no
proper place in the doctrine" of Arist6tle because by an unfortunate
and hi ghl y arbitrary division of the, science s ( he created an ab yss
between physic s and mathematics) by placing them in foimally different
degrees of abstract ion. Having once made . this fatal blunder , he could

(notTuFlSe* embarrassed by the actual : existence of certain physical
sciences already to some extent mathenmt:k;ize&,,such as astronomy, optics,
etc, and recognizing the utter impossibility of finding a special
place for them in the schema* he had conceived a priori , he was forced
to clas s them__among the jrathematical sciences, ( while at the same time
attempting to save" the situat ion in some fashion by p ointing out tha t
they were 'More physical " jjhan p ure mathem atics,) In this way he removed
these" s"c i ence s~f r om the realm of physics proper. This, added to the

-4-2-

fact that Aristotle had a personal aversion for mathematical speculation,
explains why peripateticism is completely barren from the point of viev/
of mathematical physics „

Voila done ecartees de l'oeuvre d'Aristote, avant tout
physicien et naturaliste, — quand il n'est pas logicien et meta-
physicien, — - los sciences mathematiques proprement dites, Mais
il est alle plus loin, et, cette fois, il a, de fagon expresse,
fait appel a ses principes, pour alleger son programme de certai-
nes sciences auxquelles on ne peut guere denier le caract&re
de sciences physiques, Ge sont celles precisenent qui, de son
temps, se trouvaient Stre les plus avancees et qui avaient pris
deja la forme qui leur fait reconnaitre la qualite de sciences
au sens modeme du mot: astronomie, optique, harmonique ou acous-
tique, mecanique, La superiacite caracteristique de ces discipli-
nes, comparees a d'autres encore moins developpees, provenait du
fait que le cSte quantitatif des phenomenes envisages etait non
settlement reconnu et decrit en termes generaux, mais etait etudie
en detail, par 1 ! application poussec aussi loin que possible,
Des lors, il fallait une competence suffisante en mathematiques
pour aborder ces branches, de savoir, qui par le fait m&me etaient
devenues 1' apanage des mathematiciens, Aussi Aristote l gg__classe-
t-il san s he sitat ion parmi Ig j* lin.^ujrca. ■*- les sciences ma-
thematiques, — tout en leur~aTtribuant un caractere "plus phy-
, sique" qu'aux mathematiques pures (Physic, B<,2, 194 a 7 - 12)

On touche du doigt ici les consequences de la doctrine
des deux premiers' degres d'abstraction, en meme temps que de l'e-
loignement qu'eprouvait Aristote pour la speculation mathematique,
Les sciences ou branches de la physique deja mathematisees auraient
du constituer pour lui le type le plus acheve des sciences phy-
siques particulieres, a condition, bien entendu, d'assigner a
chacune d'elles 1' etude comp_lete des phenora&nes d'un domaine
bien delimite, celui de 1'astronomie ou de la mecanique par exem-
ple, „ ,

On voit done comment, en ecartant de la physique, pour
les assigner au domaine mathematique , les sciences mentionnees
a 1' instant, Aristote a manque' 1' occasion de traiter a fond sur
des cas coricrets parfaitement adaptes, le problems de la diffe-
rence entre une etude philosophique et une etude purement scien-
tifique de telle ou telle portion du monde materiel,, Ses vues
sur le degre d'abstraction de l'objet mathematique en sont res-
ponsables pour une part; mais, d'un autre cote, une fois admises,
elles eussent aussi bien postule une astronomie ou une mecanique
complete, a la fois mathematique . et physique, en effet, de l'aveu
meme du Stagirite, lea entites matheiV iati ques sont £g &6cu_peoecog

; ce sont des abstrafts ou des extraits d'un en*
semble plus complexe, qui constitue precis ement l'objet physique.

-43-

Done elles en font partie, et pour etudier oe dernier objoi) de
facon mtegrale, le physicien lui-meme n'en peut negliger 1' as-
pect quantitatif jusque dans ses dernieres determinations,,

Nous savon3 done pourquoi, — ' touchant la question de*
fait, nous ne trouvons pas et nous ne pouvons pas trouver,
dans l'oeuvre d'Aristote,, des exposes ou des traites ressortis-
sant au doraaine physique et repondant a des sciences particulie-
res assez avancees pour avoir revetu une forme mathematique quel-
que peu developpee, (89)

Some authors have sought for a source of this barrenness
in the Aristotelian doctrine on sensible knowledge which establishe s
an absolute identity between the sensible and the physical , thus pre-
cluding the passibility of a physical science that .would be based
not on the sensible qualities of nature, but upon its quantitative
^relations, Speaking of the physico-matheraatical sciences in relation
to the system of Aristotle, Salmon writes:

Elles ne deri-^ent pas en effet normalement de la theo-
rie des degres d" abstraction, mais sont des donnees de fait, as-
sez genantes d'ailleurs, que le theoricien integxe comme il le
peut dans une synthase qui ne les prevoyait pas» Pour les auteurs
scolastiqueS il n'y avait done qu'une physique unique, homogene
et uniforme, qui expliquait tout, depuis le Premier Moteur jusqu'a.
la aalure des mers, et le regime des vents,, Et ces conceptions

/ epistemologiques etaient f ondees sur une doctrine deliberee de
la connaissance sensible, qui identif iait resolument le physiq ue

V _et le sensible , (90) '

Salman makes much of this scholastic identification between the physical
and the sensible. He finds in it a reason to reject not only that
part of scholastic natural doctrine which corresponds to modern physics,
but even the whole philosophy of nature:

Les scolastiques croyaient deboucher de plain-pied dans
le reel, en percevoir d'emblee et par les sens, 1* organisation
intiiiK3 Gratifies d'une donnee immediate et parfaitement simple,
ils pouvaient edifier une scientia : naturalis unique et homogene
qui epuisait la connaissance de l'univers sensible, Les modernes
sont moins bien partages. Ils savent qu'il leur faut traverser
la zone du sensible; qui est physiquement impure avant de retrou-
ver un monde materiel vraiment objectif; ce n'est qu'ensuite,
lorsqu'une penible reconstruction leur aura rendu des donnees
authentiquement physiques, qu'ils pourjront songer a en faire la

(philosophie. La "Philosophic de la Nature" si eventuellement elle
se reconstitue, sera l 1 analogue de la philosophia naturalis me-
dievale; tandis que la science physique moderne, malgre ses res-

-44-

ser.iblances suporficielles avec l'ancienne, est d'un type episte-
mologxquo radxcalcnent nouveau, dpnt.i l serai t naif de cherch er

me coup la portee veritable de la pliysique scolastique, et ses
possxbxlxtes d'adaptation. II est manifestenent futile, en effet,
de multxplier les "objets fomels", dont les nuances plus subtit-
les devraxent renplacer les vues insuffisamment differenciees
des ancxenso Car, pour user de ce langage scolastique, e'est l"'ob-
. < jet raaterxel" lui-nene qui se derobc Ces qualites sensibles,
■■ surJ^quelj ; es_repos^Jgute^^ n edievalo, n'ont point

la porteo ontologique - q~u'on leur accbrdait,, EHes~~nT5xistent pas
I dans les copjs de la nature, mais seuler.ient dans la perception
Vde qui les connait. La Physique ancienne n'est done pas seuloment
erronee dans telle ou telle de ses conclusions, elle est atteinte,
des son point de depart, d'un sub -jectivisnie radical dont se res-
sent profondenont lo systerae dans son ensemble. Plusieurs de ses
theses essentielles conservent sans doute une valeur permanente,
et^seront peut-Stre sauvees. Mais elles ne pourront revivre qu'a-
presde nouvelles demonstrations fondees sur de nouvelles donnees,
oxprimees surtout dans un langage et avec une technique concep-
tuelle inspires du reel physique et- non par 'la vaine imagerie
du sensible. Le soul parti raisonnable des lors est de renonoer
definitivement aux rapprochements superficiels et de reprendre
1' elaboration d'une philosophie naturelle sur les bases toutes
nouvelles que nous iraposent une connaissance plus nuancee du mon-
de physique et de son difficile acces„ (90 a)

. Other arguments of this kind could be easily adduced „
One of the most telling consists iu this that for Aristotle, physics
is the study of mobile being (ens mobile), and every things it considers
must be studied in the light of nobility; yet the Aristotelians have
always taught that raathema tics necessarily "excludes notion . As we
have already pointed out, Aristotle hinself used this argument against
the nathenatization of nature taught by the Pythagoreans and the Pla-
tonists and St-sThoi.ias stated explicitly: . "ex mathenaticis non potest
aliquid efficaciter de notu concludi" . It . would seem impossible, then ,
for a science to exist which would bo at onc e ph ysical and math eraa-
i tioal„

Montaigne once said of Aristotle that he had an "oar
in every Water and meddled with all things," However, the arguments
we have just considered seem cogent enough to force the conclusion
upon us that there was ono expanse of water in which the Aristotelian
oar never dipped: that of mathematical physics

These are serious charges. They question the competence .

of Thomism in the whole realm of thought where philosophy comes to I ,>

-45-

grips with science and with the multitudinous epistemological problems
which have arisen out of its modern development , They go far deeper
tha n even those who proffer them may suspect . In a sense they touch
lhomsn at its heart, For if there is ono thing upon which Thomlsm
prides itself, it is its preeminence in that .part of philosophy that
is truly wisdom. Now it pertains to wisdom not only to have a critical
knowledge of its own nature, but also to have that same critical know-
ledge of all the other sciences and of all their manifold interrelations.
If Thor.as m_cannot find within itself the principles which will be
ajjlg_to_ogerP up the inner meaning of mathematical physics and to situate
it_acgurately in_thc_ wholc epistemological schem e, it must renounce
its , claim to the possession of integral wisdom ,

Y/e do not propose to answer here all the charges indicated
above. The whole study. Yre are undertaking will be an answer to them.
Yet it seems necessary at this point to purify, the atmosphere of ir-
relevant considerations so that the real issue mil be thrown into
sharper focus,

. In the first plsce, it must be pointed out that in seeking
to establish the significance of Thomism as a_phJlQssph y of science
we hold no brief for the decadent scholasticism which first felt the
impact of the rise of modern science and which has persisted in so
many ways down to our own day. It is a sign of a singular lack of
discernment on the part of historians- to confuse Thomism with this
grotesque caricature, Galileo, who has traditionally been held up
as the direct antithesis of all that Peripateticism stands for, realized
the necessity of distinguishing between them. In his .Lettere Intorno
A lle Macchie Solari he says: "Nee sum ignarus, quara haec opinio sit
ininica philosophiae Aristotelic'ae : sectae raagis quam principi est
diversa. Da mihi redivivura AristotelemT " (9l)

This does not mean that the advancement • of physical science
has not resulted in the liquidation of a good many of the theories
proposed by Aristotle in his treatises which deal with nature and
its concretion. But only those who are utterly ignorant of the meaning
of experimental science con find in this a reason to condemn hin.
In dealing with nature in its concretion error is normal. As we pointed
out in considering the philosophy of Descartes, it is important, when
one Y/ishes to evaluate the work of a thinker of the palst, to dis-
tinguish between the errors for which his system and method are in-
trinsically responsible, and those over which he had no control. The
historians who are so eloquent in' ridiculing the physics of Aristotle
fail to realize that the only goal that experimental science can attain
is, in the Last analysis, to "save the phenomena", and that the physics
of Aristotle saved the phenomena that were known in his time just as
accurately and as perfectly as the theory of Relativity saves the

"46« , |

phenomena that are known todays And we nay well wonder how much of
Einstein's work will be still standing after as many thousands of
years have passed over it as have elapsed since the tiue of Aristotle,

We think that the following passage of Charles Singer
is extremely discerning:

Against Aristotle it has beon urged that he obstructed
tho progress of astronomy by not . identifying terrestrial and ce -
lestial mech anics, and by laying down the principle that celestial
notions were regulated by p eculiar laws. He placed the heavens
beyond the possibility of experimental research, and at the sane
time impeded the progress of mechanics -by his assumption of a
- distinction between "natural" and "unnatural" motion. On the other
hand, we 'should remenbor that Aristotle gave an interest to the
study of Nature by his provision of a positive and tangible scheme.
It seems unfair to bring his own greatness as a charge against
him„ All our conceptions of the material world --"scientific theo-
ries" as we call then — are but temporary devices to be abandoned
when occasion demands. That the scheme propounded by Aristotle
lasted more than' 'two thousand years is evidence of its symmetry
and beauty and of the greatness of the mind that wrought it. That
it received no effective criticism is no fault of Aristotle's,
but is evidence of what dwarfs tho men who followed him were by
comparison, with him (92)

| It is evident that the first one to call into question Aristotles

theory of the heavens seems to have been Thomas Aquinas, v/ho considered
^Aristotle's doctrine as a mere opinion, (93)

It is clear, then, that in attempting to establish the
relevance of Thonism for mathematical physics, we are not seeking
to revive outmoded physical theories. Nor are we presuming to maintain
that Aristotle or any of the Medievalists wero great mathematical
physicists, Tho point is that Aristotle was something greater than
a mathematical physicist: he was a great philosopher . Unquestionably,
a full and exact knowledge of mathematical physics is indispensable
for any philosopher who attempts to come to grips with the highly
specific and concrete cpistemological problems that arise out of the
advanced development of physical science. But this knowledge is not
necessary in order to dis cover tho key which wiflTbpen up a clear
and precise view of t he tru e nature of mathematical physics and it s
*2 3?ti°5!L- to all the other sciences,, ^We bolieyo t hat Aristotle dis-
covered that key ,J Wo believe that ihhat key is necessary today if wo
ore - tcTfind our way out of the epistcnological maze into which the
progress of science has led us

-47"

It may readily be admitted that from a purely mat erial point of view
Aristotle had very little to say about mathematical physics. The few
passages in which he touches upon the subject are almost swallowed
up m the great bulk of his writings. But that point of view is entirely
irrelevant. Moreover, there are other reasons to explain this phenomenon
,QJh£r_fchjUjJ|hcy?ur cly extrinsic reasons pyhioh delight so many historians^.
It lias oftcnBeeiTmaintained that AristotTe~knew very lTftle mathema= '
Vticsj and that he had a particular aversion for mathematical speculations,

Gdlsonj for example, tells us that if Aristotle did not get Very far
wfEKysCientific enquiry in terns of quantity and measurement, "it may
be simply because of his ignorance of mathematics, of which he seems
to_ have known only simple pro portion. It is possible that this fact
had a ^ considerable influence on the general trend of his labours," (94)
This is also the opinion of Mansion, as we have seen, Gilson gives
us neither reasons nor references to support his assertion . And all
that Mansion has to offer is an allusion to a text in the twelfth
.book of the Meta physics where Aristotle, speaking of the movements
of the heavenly bodies, writes:

That the movements are more numerous than the bodies
that are moved is evident to those who have given even moderate
attention to the matter; for each of the planets has more than
one movement. But as to the actual number of these movements,
we now — to give some notion of the subject — quote what some
of" the r.iathematicians say, that our thought may have some definite
number to grasp; but, for the rest, we must partly investigate
for ourselves, partly learn from other investigators, and if those
who study this subject from an opinion contrary to what we have
now stated, we must esteem both parties indeed, but follow the
k more accurate a (95)

Of this text Mansion says: "temoin la confession a. peine voilee qu'il
en a fait au Xlle livre de la Metaphysique a propos des astronomes,
traites corar.e des specialistes, devant la competence desquels il s'in-
cline sans vouloir discuter ni leur titre ni leurs hypotheses," (96)
Even a casua l reading of the text of Aristotle reveals the utter gra-
tuity_^_Mans ion's inference. No one who is at all acquainted with
the writings of Aristotle is unaware of the fact that it is customary
for him .to introduce a question by considering what authorities in
the field /have had to say about it, and that he always has respect
for the opinion of these authorities unless his own reasoning lias
produced evidence to induce him to differ from them. In this case,,
it is evident from the text and context in question that he is inter-
ested merely in arriving at some probable opinion about the number
of the movements of the heavenly bodies so th at the mi nd will be ab le
to fix_ itself upon a def in ite number . (97) And since the opinions
of Eudoxus and Callipus seem probable to him he accepts them.

-48-

.. A f a matter of faot, scholars are now coning to recognize

that Aristotle's knowledge of mathematics was far advanced for his
day. ^ It was knowledge, rather than ignorance of the mathematics of
his time, ' writes F.S.C. Northrop, "which supported Aristotle in the
^formulation of his logic."' (98)

Aristotle's polemic against the raathenaticism of the
Platomsts was not a polemic -against the existence of mathematical
science, as some seen to think, but against the ontolog iGal existence
of ^ mathematical entities i_. By dissipating the confusiorTb'f mathematics •.
|i*--^2ik^E5yiics^^me^pjiy^i^s Q bhat was characteristic_of_the-doc= ~~~~ '
^ine _of the platonists ^ Aristotle est ablished (h ijrtgue ep istemologica l
status. He thus freed it of all the associations \McTrtena~ed to dra w'
it awa y from its proper_f unction . and made of it a more apt instrument
for the use of scientists. Professor Strong has brought out this point
with remarkable clarity, and we cannot refrain from quoting the fol-
lowing passage in spite of its length:

Critics can criticize Aristotle for his refusal to accept
the doctrine of Form as metaph ysical number , but certainly not
upon the ground that lie failed to consider the meaning of mathe-
matics. Rather, one may say, it was because Aristotle refused
to confuse mathematical science with metaphysical principles,
and because he insisted upon the o peration al character and ph ysical
reference of mathematics (that he' refused to identif y mathematical
num ber with ideal number existi ng in a separate realm of reality .)
This means that Aristotle did not advocate the formulation of
a metaphysics in mathematical terms and relations and saw such
a metaphysics as a confusion of the notion of mathematics with
ontological realities. Hence Aristotle had no doctrine of the

(universe framed in i.iathematical universals of relation, for he j

regarded the ratios and proportions of mathematics as constituting
no class of existences-in- themselves. They are relational only
of entities of a mathematical character- in arithmetic, geometry,
or some more physical science such as mechanics.

The Physica, De Caelo, and Problenata reveal passages
in which he used mathematics in connection with physical problems .
This is of course not equivalent to saying that the basic prin-
ciples of Aristotle's physical science were i.iathematical. Aristotle
recogniaed mathematics as a self-contained science and as an ins-
trument in the physical sciences. So far as he ma inly dire oted
his own treatment of nature to the probl em of g rowth(where mathe-
matical fo rmulation was not relevant , >o far we may say tEaTTTiis
interest and approach were directed to other than the quantitative
aspects and concepts of nature.

It is characteristic of Aristotle's approach to his
predecessors that he regards them as men striving for the theore-

-49-

tical view, His analyses of his predecessors are thus a source
of knowledge with respect to their "metaphysics". His ovm inquiry
ends m apposition opposed to the view of Democritus ( and ) Plato .
The opposition, in accordance v/ith the view presented in the fore-
going analysis, is not to mathematics or to the use of mathematics
.in natural science, but to the role which number and mathematical
/objects §rc.s^gsca b |q have as ontological existences. To insist

upon thc^-^lnMP^ c \il% e |fib!e 1 §t-r. 1 atter and the substantial and
V ideal lumber attributed to Plato, does not involve a re j ection
of mathematics proper . It docs involve a rejection of theories
about the '-'real" existence of number-forms. Those who assume that a
mathematical metaphysics is fundamentally important in a regulative
and interpretative role to the development of mechanics and mathe-
matical physics charge Aristotle, upon basis of his different
conclusion in metaphysics, v/ith having obstructed the progress
that would supposedly have followed from his acceptance of the
Platonic theories of e xistential number . So far as Plato and the
Academy were actually^erigaged in mathematical work, the argument
appears to carry weight, Nevertheless, the assumption that meta-
physics is important in respect to subject-matter and procedure j
must first be established before Aristotle can be held responsible]
for obstructing the development of mathematical science, (99) /

It. is cleaij then, that there must be other reasons besides
a lack of knowledge of mathematics to explain why Aristotle, having
once discovered the true principles of mathematical physics, did not
devote hinself to their development. In the first place, in order
for any substantial progress to be made in the application of mathe-
matics to nature two kinds of instruments are essential; conceptual
mathematical instruments and physical instruments of exact experime nt
and measurement „ Without these only extremely meager progress can
be made, and Aristotle lacked both. It was only after the Renaissance
that the necessary physical instruments were invented, and the conceptual
instruments which were to prove so fruitful, such as analytical geometry
and the calculus, were discovered. The development of mathematical
physics depends completely upon these instruments, and, as Moyerson
has pointed out, "si le3 mathematiques accomplissaient a l'heure ac-
tuelle un progres comparable, ne fut-ce que dans une certaine mesure,
a celui qui a ete effoctue par la creation du cajcul infinitesimal,
la physique a. son tour ferait, presquo immediatement, un bond en avant
immense," (100)

Another possible explanation of why Aristotle failed
(to give more attention to the exploitation of the fruitful principles

he had discovered may be that ho was far from realizing the vast extent
^of the applicability of his own principles. But before considering

this possibility it is necessary to examine the major texts in which

-50-

thesc principles are laid down.

There are two capital texts in which Aristotle deals
explicitly with the nature of mathematical physics. These will 'cons -
ti tute the seed out of which our whole study will grow ; The first
of these two texts is found in the Posterior Analyt ics. This whole
work is devoted to a discussion of the principles that are common
to all the sciences,, In chapter thirteen of tho first book Aristotle
explains how knowledge of the fact ( scientia qui a^ differs fron know-
ledge of the reasoned fact ( soientia prop ter quid") . After showing
how. they differ within the sane science, 'he goes on to explain how
■ they differ when they are found in different sciences: and in raking
(this explanation he brings in the question of the subalternation o f
| the scien ces (^u,ch_V7e_cgnsider t he key to the whole problem of mathe -
^mtical physics.) ~~

But there is another way too in which the fact and the
reasoned fact differ, and that is when they, are investigated respect-
ively by different sciences. This occurs in the case of problems
related to one another as subalternate and superior, as when optical
problems are subalternated to geometry, mechanical problems to
stereometry, harmonic problems to arithmetic*, the data of observation
to astronomy. Some of these sciences are almost sy nonymous, e.g,,
mathematical and nautical astronomy, mathematical and acoustical
harmonics. Here it is the business of the empirical observers
to know tho fact , of the mathematician t o kno w the reason fo r
the fact.l^For the latter are in possession of the demonstrations
giving" the caus es,;) and~are often ignorant of the simple fac t;
just as those who know universals are often ignorant of some of
its particular instances through lack of observation. Such are
all the sciences which, though differing by their essence, use
forms. For the mathematical sciences have to do with forms; they
are not concerned with a subject , since, even though geometrical
properties are predicable of a subjec t, [ it is not as predicable
of a subject that t hey consider theriu) As optics is related to
ge one try, "so another science is related to optics, namely the
theory of the rainbow. Here it pertains to the physician to know
the fact, but to the optician to know the reason for the fact,

I either _gua optician or _qua mathematician. Many sciences, though
not subalternated, are mutually related in a similar" way, e.g,
medicine and geometry: it is the business of tho student of medicine
to know that circular wounds heal more slowly, but it pertains
to tho geometer to know the reason why, (101)

The second important text is found in cliaptor two of
the second book of the Physics. Since some historians have failed
to see why this passage should be in this, particular place a nd have
preferdto seek for some exteinsic^ rgasj^_tj3_explain its presence her e,

-51-

lt is worthwhile to point out its connection with the context. After
haying discussed in book one the problem of the principles of nature,
Aristotle takes up in book two the principles of the science of nature.
The general principles common to all science had already been considered
in the Posterior y Analytica, But each science has its -own proper method,
and consequently it was necessary for Aristotle after having determined
upon the principles of nature to discuss the method to bo used in the
investigation of nature. It was necessary to consider the causes ac-
Vcordin£_^wh^i(aemonstration my be had )in natural science . Now
it happens That tHe~natura'I~scient'ist in seeking for the cause of
natural phenomena often turns to mathematics for light. Aristotle
^i5^-J°^ x 3? lain ^ le significance of this recou rse t o mathomat: ics , In
other words, after having "discussed in~"the Poster i or Analytics the
general principles governing the subaltomation of one science to
another, ho now applies these principles to the subaltemation of
physics to mathematics .

Having determined the different ways in which the term
"nature" is used, we must now consider how the mathematician dif-
fers from the physicisto For physical bodies contain surfaces
and volumes, lines and points, and these are the object of the
mathematician. Moreover, astronomy is either different from physics
or a part of it, For it seems strange that it should pertain to
the physicist to know the nature of the sun or the moon, but not
to know any of their accidents, especially since writers on physics
obviously do discuss their shape also and whether the earth and
the world are spherical or not, Nov; the mathematician, though
he treats of these things, nevertheless does not treat of them
as the limits of a physical body ; nor does he consider the accidents
• precisely as accidents of such bodies . That is why ho abstracts
them; for in thoug ht they are abstractable from motion , and it
makes no difference, nor is any falsity involved if he so abstracts
them. The holders of the theory of Forms are unaware of this, .
For they abstract physical things , even though these are less
abstractable than mathematical things. This becomes plain if one
tries to state in each of the Wo cases the definitions of the
things and of their attributes. 'Odd' and 'even', 'straight' and
'curved 1 , and likewise 'number', 'line', and 'figure', do not inv olve
motion; not so 'flesh' and 'bono' and 'nan' — these are define d
like ' snub nose ' , not like 'curved^ , Similar evidence is supplied
by~the~sciences which are more p"hysical than mathematical , such
as optics, harmonics, and astronomy. These are in a way the converse
of geometry, for while geometry investigates physical lines but
not qua physical,, optics investigates ma thema tical lines , but
_gua physical, not _gua mathematical, (102)

The central idea that emerges f rom these two texts is

-52-

that mathematical physics is a hybrid science in which p hysics is
subaltern ated to mathematics ,, It is, to use tho~te^555aT Thomistic
expression, a s cientia media, an intermediary science between physics
arid mathematics; it involves a kind of noetic hylemor phism (in which
the ma tonal _g lement is drawn" from phys ics" and the formal element
from ■/.Tathematics,) Thepurposeof this study is to analyze the uniqu e
type of lmow3^ a|o^ [t~ii"l3orn of" this union . As wo have already in-
dicatcd, it is" not our intention to attempt 'to come to grips with
all the complicated epistemological problems which have evolved out
of the development of mathematical physics. Rather we have in mind
to take this one idea of this scientia media ( and explore all of its
^im plications""^ But we hope to draw out those implications far enoug h
to m ake it clear th at in this one idea is found the central ke y which
will open up the meaning of all the other problems encountered in
ph ysics o — ~~~ — ~

Before undertaking the detailed analysis of these texts
several general considerations are in order. In the first place, for-
the purpose of. indicating the direction that this analysis will follow,
it is helpful to try ±o orientate the position of Aristotle in relation
to the other positions outlined earlier in this chapter. As we have
already suggested, most of these opinions can be reduced to two cate-
gories: the role of mathematics in physics is either considered to
be that of a. pure instrument (whe_ther_ logical or merely linguistic,)
that is employed by the scientist in order to work more effectively
upon his sole direct object which is nature; or it is considered to
be that__of th e direct object of thejjoience itself(in_ th e sense th at
the m athematical wor23~is"~l ! d~entif"ied with or realized in the physica l
worldoJNow the position of Aristotle is located squarely between these
two extreme positions.

In the first place, the role of mathematics in physics
I is essentially instrumental in the sense that the whole raison d'etre
of its introduction into physics is to enable the mind to get to know
\the phys ical u niverse better ,, The goal at which the whole of mathe-
matical physics aims is not 'to know the m ath ematical world (for that
is already known) but the physical vrorld. Mathematics is employed
as a means to that endo

On the other hand, mathematics is much more thaii a mere
tool in physics, that is to sayj CLt does not remain extrinsic to the
science |) on the contrary^ it enters i ntrinsically into _jLts very consti-
tivELon* AnoTTir^nteri _ into it intrinsically not morelyTn the sense
of providing the principles from which physics may draw conclusions
concerning its own proper object which in itself remains untouched
by j'.iather.atics, fbut_in_the sense_o£_entering_into the very object
of "the" soionce JPor7'"as we~shall see in chapter three, the type of

-55-

subal.ernation found in mathematical physics is not merely subaltern-
ation accordxng to principles, such as is found in the dependence of
Hl^LQSpipon^bhg^s cienco of tho ~BT^s^ed7^ ul~su5arge r nat i bn accor d i ng
to cho object.) This means that thTfSSal object of nathenia"ti5aT : 5hv5ic R
is constituted by a combination of both a. mathematical and a physical
clement ,

But the nature of this combination must be rightly under-
stood. It does not mean that mathematical physics studies as such
the quantitative detcn-.iinations found in nature from the point of
view that is proper to them. Such a study is possible, but it will
be either pure physics ( if the quantit ative ..de terminations are oon -
gJJgggcLHL r glaM°A . to mobi lity) "or~metaphysics (if the nature ~~oT
quantity and its properties are considered) . Mathematical physics
studies the quantitative determinations found in nature, not just
in the light of their ontological status, but in the light of the
status that is proper to mathematical abstraction. For example, when
the physicist says that light is propagated in a straight line, the
line he is talking about i s neither a mere phy sical, _sg nsible lin e,
(such_ as is found in nature~ 7) nor is it merely a mathematical line ;
it__is a combination of the two: | the sensible line is considerecTin
^^-■iifeL-Q f a nathera atical lineT)

In this way mathematics enters into the very essence
of the object of physics, but it does so in such a fashion that th e
mathematical Yrorld,._i.B_not iden tified with th e physical world . I It retains
t he extrinsic_ _char.acter that is pro per to it, M.nd this is extremely
important. For only by reraianing extrinsic can it fulfill its essen-
tially functional and instrumental role, b y retaining all the plianc y
and inexhaustibl e virtuosit y that is pro per to mathematical abstraction <,

This brings us to a delicate point that must be touched
upon before proceeding further in our analysis It would seem that
for Aristotle and the medieval Thomists the combination between the
mathematical and the physical element in the object of mathematical
physics was in a sense more intimate than it is possible to admit
^today. Because of a lack of refinement in their means of observation,
they seem to have held that there are quantitative determinations in
nature which come sufficiently close to the absolute state of perfection
that they enjoy in the mathematical world to allow for a true scienti-
fic (105) handling of them in terms of mathematics. The heavenly
bodies, for example, were for them perfect spheres, and consequently
there was sufficient conformity between them and mathematical spheres
to allow the mathematical properties of sphericity to be applied to
them directly and adequately. This does not moan, of course, that
mathematical entities were realized as_^uch in the physical universe,
for that would involve a confusion of mathematics and physics, and

-54-

+w : wJ St, Thomas go to groat lengths in inveighing against
those who proposed sueh a confusion. .(±04) But it docs mean that
some physical entities' possessed a determination which was in close
enough conformity- with the perfect determination of mathematical en- '
titles for mathematics to give on adequate explanation of them. That
^is why Aristotle and St. Thomas could look upon the combination of
mathematics and physics as giving rise to a science in the strict
sense of the term,,

Jt would seen that this particular aspect of their doc-
trine is open to modification. Because of our more highly refined
instruments of research., we are no longer inclined to believe that
such a conformity exists between physical and mathematical entities.
As a consequence, the mathematical interpretation of nature is never
more than an extrinsic approach to nature , [An d that is why ffrom this
p oint of v iow >athematical physics cannot be~con s idered a scienceTn
^e^ triciTEusto jelian_s ense of the term j fbut a~species of dialectics^ )

There is another closely related point that must be under-
scored here in order to establish accurately the connection between
Thomistic doctrine and modem mathematical physics. When Aristotle
and the medieval Thonists speak of mathematics they understand it
in the sense in which it was generally understood until recent years —
that is to say, as a science which deals with quantitative relations
that are capable of realization in the sensible world though not in
the state of a bstraction that is pro per to t hem -- "oportet salvari
principia mathematica in omnibus naturalibus, ut dicitur III Caeli
et Mundiu" As is well known, modern mathematics is no longer restricted
to these limits* It now embraces a great ra ngp of conceptual cons truction
which reach far beyond these quantitative relations. Now it is bootless
to dispute about names s but it is extremely important to keep in mind
what they are meant to signify. And in so far as our problem is con-
cerned, it is necessary. to recognize the fact that from the point
of view of Thomistic terminology, the part of 'modern mathematics which
does not deal with quantitative relations abstracted from the 'sensible

world is not mathematics , Cbut a tissu e of dialectical constructions"^

Now these dialectical constructions have been employed with great
success in the recent developments of physics. The obvious example •
which immediately suggests itself is the use of non-Euclidian geometry
' in the theory of relativity,, Does this mean that the Thomistic doctrine
I of scientia m edia has no relevance for recent mathematical physics.
We, do not believe that such a conclusion is legitimate, For the ap-
plication of the dialectical constructions of modern mathematics to
nature follows the same general pattern as the application of mathe-
m atics in~the re strictedsen se in wh ich it was understood by Arist otle
and the MedXaveliata , Cand is governed by many of the same general
principles;;) Nevertheless it is necessary to keep in mind that in so

-55-

far as these conceptual constructions are employed, mathematical physics
^_a^alcctical in a sense never envisaged by Aristotle and St„ Thomas,
tha-o is to say, although their notion of dialectics is applicable,
Uhey never envisaged this application.

In connection with this question of the meaning of the
tem "mathematical" it will be helpful to determine here what breadth
of meaning the phrase "mathematical physics" yd.ll have throughout
this study „ This is a double problem, involving the range of applica-
bility of both the term "mathematical" and the term "physics" , In so
far as the first aspect of the question is concerned, it is to be
noted that some authors restrict the phrase "mathematical physics"
to those parts of physics which have attained the highest degree of
nathemtization. Professor Lenzcn, for example, divides physics into
experimental physics, theoretical physics, ideal theoretical physics,
Vand mathematical physics, , (105) The Thoraistic acceptance of the
phrase is much broader. It includes any part of physics in which a
mathematical element is introduce d to determine the objeo t^ in such
a way)Fhat_new si gnificant truths result (which wo uld_ n ot "arise without
this determinationT]

The second question which must be determined is the meaning
of the term "physics" „ A ■• reading of the texts of Aristotle cited above
raises a problem about the range of applicability which the principles
laid down in then had for Aristotle and the medieval Thomists, The
examples given in these passages are restricted to a very few especially
privileged cases in which the presupposition of all raatheraatization,
namely ; order and regularity, is found in a particularly high degree - .
whether it be the geometrical order that is found in astronomy, for
lexanple, or the arithmetical order that is found in music. It would,
/seem that the exanples given are more 1 than examples, that they are
yan exhaustive indication of the fields in >7hich physics had to some
extent been subalte mated to' mathematics,. Did Aristotle or the medieval
Thomists looked beyond these fields? Did they concieve the possibility
of a universal interpretation of nature in terras of mathematics? It/
seem s quite possible that they did not . It is probable that, the honor
of this discovery must be accorded to the scientists and philosophers
,of the Renaissance. But this admission in no way compromises the ob-
jective o.pplicability of these principles, nor their real fecundity.

Mathematics is almost synonymous 'with determination , '
and as a consequence nature is refractory to raathematization to the
i extent in which it participates in some form of indetemination, That
is why it is necessary to understand the ways in which nature is subject
to indetcrmination(if we are to see the ext ent to which mathematics
r^"bo~'ap^lie^nio^naturep"MoYf there are two" typos"of inde termination:
pas"siTe"'IndeTclTOS"tT6irwhich an imperfection arising out of the potent-

-56-

laliuy of mtter, and active inde termination which is a perfection
deriving iron the actuality of form. Passive inde termination is found
in a^.j bcxngs which have any share in potentiality; active indoter -
uination xs found in its fullness only in the liberty proper to spiritual
beings, but it is also found anticipated to a greater 01- lesser degree
m the spontaneity of all living tilings.

Now in Aristotle's and the medieval Thomists' concept
of the cosmos, the heavenly bodies occupied a very privileged position,
Though_^iobile) they were incorruptible , | and they consequently occupie d
a positi on between the metaphysical realm of Cjmmobile) beings and the
terrestrial world of C orruptib le? beings J (106) Though inanimate.
xhoy were in a sense more perfect than the living beings of the earth,
even than nan, i n that they we re ... sub .iect to no intrinsic corruption ,
[but_orig _ to the extrinsic mobility of local motion, ) (107) They were
/thus free of both the passive indetermination that is proper to cor-
ruptible things, and. the active indetemination that is found in living
I hoingsT ) That is why , for the, ancients ( Ehey constituted the part of
=?( ncturo that was most highly amenable to i-.iathoraatizatioi p.lt would

(be difficult to say just what possibilities of mather.iatizatioh Aristotle
and St, Thomas saw in the terrestrial world of corruptible thin gs
fc nwhich ■both p assive and -active indotcrm ination pl ay such a largg
P^ES oj-But a-f = Teasir~this raucn can" be said: they would readily grant
the possibility of a mathematical interpretation of the corruptible
world to the extent in which definite regularity and or der could be
discovered in its phenomena. (103 ) ] ' ~ ~ ] ~~~~

But whatever Aristotle or Saint Thomas may have thought
about the extent to which nature may be raatheraitized, there is no
douh-i that their principles aire applicable to the whole range of mathe -
raatizo.tion Which modern physics has achieved ,, And that is all that
is of any real importance. This universal applicability of Thomistic
principles is so true that in this study we shall, when speaking of
mathematical physics, take the term "physics" in its primitive Aris-
totelian meaning in which it is coterminous With the whole of nature .
[in this sense it includes not only chemistry but even biology and
rpsyoholcjgrjjAB we shall see, according to Thomistic principles of the
unity and distinction of the sciences, all the sciences which deal _
with nature , (w hether it be, inanimate , animat e, or even psychic nature )
constitute one" indivisible science , ^In recent years there has been
an attempt made by many Thomists to depart from this doctrine, but ^_^^ .
we shall point out in Chapter Two. -the error involved in" this attemp t.) lv,A " **'
Thcih mathematics lias been successfully and fruitfully applied to ail
of these different fields of study is well known. And all of these
applications (and whatever new applications the future may discover)
constitute -She scientia media of which Aristotle and Saint Thoma s
.speak. (109)

-57-

+o nM-h^nM^ all m5^ el ^ S ^ the Btud V of nature are equally amenable
to ,.nthe,natization. This is evident a posteriori from the history .

of m&inH^ m ° re evMent ^^3£r±. For the ^objective basis,

found X »,+ f T 1 !i as we sha11 a^r^hgag ^^oiii-Mc^oW ^

bSffS ' fe^i^^^_tl^3in^hioh the- object of a c ertain
branch of natural doctrine has. to do with homogeneous exteriority
and in che measure in which it excludes heterogeneous int erioritv.
to that extent mathematization is possible . The field in which this
°gnaitioji^sjFound_ in its highest degree is, of cours e physics, in
^gJ^HLgHSg^fj^Jerm, And that explains not only why m athg-
maoizationis possible to such a large extent in physics, but also
why-iu is (gecessary> For, to the extent in which heterogeneous inte-
riority is excluded, physical rationality loses ground . That is why,
if scientific investigation in the realm of physics is to advance
at all, it must proceed in the light of mathematical rationalit y.

For experimental scientists, physics realizes the ideal
type of science. And it is perfectly legitimate and natural for them
to make every effort to bring the other branches of natural doctrine
into as close conformity with physics as possible. As we shall see
later, homogeneity is from one point of view more knowable than he-
terogeneit y, and as Aristotle and 'St. Thomas point out, i t is natural
for the intellect to r educe_ the less knowable to the more knowable .
But there is no doubt that this conformity, will never be complete „ (110)
Mathematics is not competent to treat adequately of all natural bein g»
For the s ubject of mathematics is. quantity, | which is the order of
the parts of the substance ) in which it inheres . But the parts in q uest-
ion are always material parts , and hence must not be confused with
the form of the substance ( This confusion would lead to a denial o f
what is best in naturar~thiHg go )

/ In other words, in the measure in which beings are onto-

lo gicall y more perfect, they lend themselves le ss to mathematica l
interpretation . For a being is perfect in proportion to the extent
that its form emerges above the potentiality of matter, that is to
say, triumphs over the potentiality of matter. Now, in the structure
of material being, while quantity follows upon matter , equality follow s
upon forrig ) That is why as ws ascend the scale of material being quali-
tative determinations assume an ever increasing importance. This is
particularly true of living beings. For the formal principle of life
is form , (and if a thing is living it is because its form has emerged
to a sufflicTent~ext"e nT^bove the potentiality of matter ^ That is why-
qualities and qualification play such an important role in biolog y.
Moreover, in living beings we find not only the passive inde termination
common to all material things, but also the active indetermination
of their vital spo ntane ityaljrh is double indetermination will alway s
provide" great resistance^to mathematization^

-58-

i * -u • , j amounts to saying that as we ascend the scale
oi being heteroge neous interio rity constantly increases., Within the
cosmos it finds ltFfullest realisation limn, the most perfect cosmic

Vbein^JAnd vre are referring here not merely to ' thTpSydSxo-sIdo'of ~~

man, but_gl3g_to tho somatic part of_ his make-up . Of all the bodies
in the universe, tne oody of man'nas'the greatest heterogeneous inte-
^ 1 °Ii'' y '' 3j_ is the farthest removed from the Cartesian bod y.rwhich
H_fche_idg al of an aut onomo us and self-suf fic ient ph ysios.) ItfisPbhis
heterogeneity of living beings that makes it possible for us to have
a valid science of biology without m at hematig ation - a s cience o f >.'.'•>
clas s if ication . ' '

It. is interesting to note here in passing that whereas
for physical science, (in the modern sense of the term) heterogeneity.
*?^.J:^^iPffi-L-^ it is homogeneity that is ' .. •

[in some sense) irrational^ Here' we" "are"t6uching'"up*6n' an 'important point
(to which Yre shall return in' chapter nine: The difference in the raeasure-
ment tha t is proper to each science For every" 'soienceT^even metaphysics, ^
is in a way based upon measurement, but in each science there is a
vast difference in the measure which provides the norm in relation
•to which everything that falls wi thin its object is determined.

The important point to be born in mind for the present
is that in spite of the great heterogeneity found in nature, a ll natural
thing s are spatio-temporal being s ( and conseq uently) subject to a common
measuri. In discussing the problem of Indeterminism, Professor
DeKoninck has emphasized this point:

. Qu'on ne croit pas echapper a cette consequence en disant
que 1* animal et la plante sont heterogenes et rebelles a une mosur e
homogene . Ne peut-on pasmcsurer leur duree par une jiieme horloge?
Cependant, puisque l 1 existence est proportionelle a 1' essence - -
quantum unicuique inest de f orma, ,tantu m ihest e i de virt ute essend i
- - la duree des Stres cosmiqups esT"aussi de plus~en plus simple,
de moins en moins temporelle; il existe aussi toute line hierarchie
de du rees cosmiques ii Mais cette heterogeneite ontologique n'empeche
pas le temps physique, que l'on definit par la description de
son procede de mesure, d'enlacer tous les etres spatio-tempoiels
par_ce qu'ils ont d 1 homogene entre eux au point de vue duree .
Cette commune~mesure est fondee |sur le genre commun de corporeitej
dang lequel conviennent tous les cStres natufels. Le temps physique
- n'atteint que leur bas-fond, et encore n'y. touche-t-il que du^
dehors, L'homoge neite est .fondement de toute mesure quantitative ) V
ce genre" physique commun explique suffisamment l'unite specif ique
du temps experimental et pourquoi 1' heterogeneite des ^durees^ echap-
po aux prises d'une metrique calquee sur l'exteriorite homogene.
La science .oxperimentalo debouche la ou tous les etres se touchent
et se confondent: 1'echelle graduee sur la balance n'indiq uo au-
oune difference entre 150 livres d'homme et 150 livres de briques.

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Si maintenance temps physique touchait les etres dans leur fond
ontologiguc e^aecifique, si ce temps epuisait le reel, ne fut-oe
qu au point de vuecEla duree, les different degres d' etres
ne seraient que des epiphenomenes de complexite materielle crois-
sante, Meme si les choses sont plus que du dehora, cela n'empe-
che pas quo la mesuro de leur exteriorite homogene soit commune '
/et vraie. Ces deux_pj gspectivos ne sont point oont raires, elles
/ se completetrrrninelTautre. Sans_co niTaltro la complexite exp e-
I rimentale d' une chose on ne peuT laiisir la richesse de son uni-
\te ontologique, (Hi)

The same author has elsewhere summed up the 'question
at issue:

La biologie experimentale est une science exacte. Les
sciences^ experimen tales pouvent etre appelees exactes dans la
mes ure-ou elles nous permottent de faire des predictio ns » G'est
en ce' sens que la physique peut etre dite la plus RYnrTte fip.e. scien-
ces experimentales. En astronomie on peut predire des eclipses
qui n'auront lieu que dans plusieurs Siecles, a une fracti on
de seconde pres. La science experimentale estf^sen tigllement')
me trique , Elle ne peut|Jif^iir_J : es_j2ropjcietesJque par la descrip-
tion de "leur lprocede de mesurej Aucune loi experimentale - - re-
lation algebrique entre des nombre-mesures - - n'est absolument
rigoureuse, Cependant, dans l'ensemble, les loi s striotement p hy-
siq ues sont plus rigour e use s que les lois biologiques . Nulle rai-
son de s'en etonner. Nous venons de dire qu'il y a dans les Stres
vivants une spontaneite toujours croissante qui dans l'homne a-
boutit a. une veritable liberte, II est absolument impossible a
un physicien de predire d'avance quel mouvement de bras je ferai
dans les cinq minutes a venir, _si j'y pr§te attention,, II peut.
mesurer le mouvement que je fais quand jo le fais. Mais de cette
mesure il ne peut pas deduire le mouvement suivant, Chaque mou-
vement que j'effectue librement est quelque chose d 1 absolument
nouveau dans le monde, Des lors on peut dire que plus un etre
est parfait, plus il echa upe a. la rigu eur metrique. Plus il est
concentre au-dessus deTTespace. temps, "plus il eohappe aux prises
de la science experimentale, Ainsi, de toutes les sciences expe-
rimentales , [j ^jsjrcholgj^ie^jyeriiiientale ost la^lus ^ jnpajrfaite j
la pl^~ina^puate, Cbion~^ r oTJe~e r E uEe~Ia~ J pTus h aut e foi-me~5 T or -
.gan isation nature llej

(En philosop hic j e'est le contraire qui est vrai.-. Plus
nous nous Hoignons de l'homme pour descendre 1'echelle des vi-
vants, plus leur vie devient obscure, Ainsi, la vie des plantes
est plus obscure pour nous que la vie animale. Nous reviendrons
la-dcssus, II suffit de remarquer pour le moment qu'il existera
une certaine complementarite comparaatrice entre cos deux ordros

-60-

de connaissance si profondemeht diatinotsb- Et par cette comple-
mentante compensatrice, jj n'entends pas qu'a un certain point
ces deux ordres fie connaissance se fusionnent l'un dans l'autre,
Non, lis no sont jamais plus eloignes l'un de l'autre qu'au point
ouils se touchcnt: comme des points sur'une droite non euclidienne
qui sonc infiniment proches, mais aussi infiniment eloignes." (112)

In chemistry we already find an element which is refrac-
tgJ3LJg_SailgMjB^lgEgJi^^g n « g ° r ^h e P^t that qualitative di ~
versity plays in chemistry is essential, (113) And even though history
has made short shrift of C omte|s_(reje ctior^ of the possibility o fthe
ma thematiiation of" chemistry . (114) as it has nf imry other flnmt.-LW
theory, it is safe to conclude that in this science there will always
remain ajinrgin impenatrable to complete mathematization .

In biology this margin will always be immeasurably larger
than in chemistry, for the reasons indicated above. Nevertheless, the
attempts already made towards mathematization in this field have been
surprisin gly fruitful , ( and there is no .w fl.Y_Qf laying down any well
defined limits beyond which this" mathematization may not"~"g o_ n ) As Whyte
has pointed out, "if the laws of lifo were independent of physical
laws, life could neithe r_exist w ithin the physical universe nor dis -
cover its laws ." (115) And just as it is the duty of every scientist
to proceed (in practice ) as thoug h there were no limit to the determinati on
coming from per se causality , that is to say, as though there were
no chance in natur e, (so it is the duty of the biologist to act as
though there were no limit t oT tethe m atization in biology^ ) even though
he may realize that the immanence Tjfiat is characteristic of lifg wil l
always remain superior to pure corporality, and thus (to some ext ent)
^e:Scape measurabilityy

It does not fall within the scope of this study to discuss
in detail the various (wa^ in which mathematics have been applied
to biology, (116) But the work already carried on in biomathematic s
by such men as D'Arcy Thompson, W.R. Thompson, Janisch, ~A. J, Lotka,
Vito Volterra, and E.A. Fisher, for example, has been sufficient to
demonstrate how _ promising this line of research in biology is . To
cite only a few typical examples, mathematics have been applied suc-
cessfully and fruitfully to problems of organic structure, (117)
laws of growth,- laws of reproduction, etc. Of particular interest
are the attempts being made to relate biological phenomena with the
discoveries of modern physics. In this connection the experiments
carried on by Timofeef-Ressovsky, Zimrner and Delbruck on the relation
between genes and molecules , and those carried on by Stanley on the
relation between viru s individuals and molecu les seem especially sug-
gestive. Moreover, recent experimentation on the biological effects
of radiation seem to indicate some promise on the general usefulness

-61-

of an atom c^Elffigical and_guantum-ph, 7aical interpretation of funda-
men tal life processes . And it is interesting to note that BotaHSi
lent ^e great weight of his name to the belief that the new physics
will ultimately have profound repercussions upon biological science „
ihore can be no doubt that by abandoning the mechanism of the nine-
teenth century m favor of the analysis of phenomena in terms of cons-
t ituent functional r ej : atignship_s , physics has immeasurably increased'
its significance for biology, and opened up in the latter science
^great possibilities of mathematization.

As we have already suggested, experimental psychology
is °f all the fields of jiatur al doctrine the least congenial to ma-
thematical interpretation. Yet even here the application of mathema-
tics has been large and fruitful, (118) The use of mathematical
formulations in the intelligence tests of Binet and his followers
is well known, (119)

The Weber-Fechner law for the intensity of sensation,

(blieJLoga rithmic la ws g overning rote memory and forge ttin g,") The Spearman

anal ysis of mental abilities are only a few 5T~the results

of the application of mathematics to experimental psychology. And
what we have said of biology applies here as well: there is no way
of laying down definite limits beyond which this mathematization may
not go

4, Some Implications of the Problem.

In the beginning of this essay we alluded to the importance
of the philosophical study of the' nature of mathematical physics.
Perhaps it would be well, before bringing this chapter to a close,
to try to round out our introductory considerations by indicating
briefly some of the major issues involved in the study we are under-
taking.

In the first place, this study is of vital importance
for physical science itself. There was a time when philosophy was
hermetically sealed off from science. Even when scientists did not
feel it necessary to be inimical to philosophy', they thought that
they could remain completely aloof from it. That time has passed,

"It is a well-founded historical generalization, "says
V/hitehead in a somewhat different context, "that the last thing to
bo discovered in any science is what the science is really about.
Men go on groping for centuries, guided merely by a dim instinot and
a puzzled curiosity, till at last 'some great truth is loosened'" (120)

-62-

us realSo thSin TrS "T^ ±n m ° dern ^ sics and tho ^ ^e nvxde
necessarvto f^fln, f f ^° ^^ ° n the P^ess of science it is
pointed out how nil /I'f S ° ience is rea11 ^ about ' We ^ already
forced bv hol™ n V h Vr test contemporary physicists have b^en
soX ^|r^^f d ^M^- ir scie " c - 3 to invade the realm of philo-
is winSL l n g ^ S1 g«ificant phenomenon. It means that science

HeisenbSg ^iW° 8nlZe " nCCd f ° r Vdsd0m " In tMs COnnec ti0n

Many of the abstractions that are characteristic of
modern theoretical physics are ,to be found discussed in the phi-
losophy of past centuries. At that time these abstractions could
be disregarded as mere mental exercises by. those scientists whose
onty concern was with reality, but ioday we are compelled by the
refin ements of experi mgntalart to consider them serious^y7(12l)

0f tne many great physicists who have felt the need of
turning to philosophy, no one has contributed more to scientific
epistemology than Sir Arthur Eddington, In his Philosophy of Phy -
sical Science Eddington. discusses the significance of the need that
science has of philosophy:

It is however, important to recognize that about twenty
five years ago the invasion of philosophy by physics assumed a
different character. Up till then traffic with philosophy had
been a luxury for those scientists whose dispositions happened
/to turn that way, I can find no indication that ihhe scientific
researches of Pearson and Poincare were in any way inspired or
Vguided by tloi 1 .? particular philosophical outlook. They had no
opportunity to put their philosophy into practice. Converse^,
their philosophical conclusions were the outcome of general
scientific training, and were not- to any extent dependant on fa-
miliarity with recondite investigations and theories , To advance
science and to philosophize on science were essentially distinct
activities. In the new mov ement | scientific epist emology is much
more intimately associated with science , J For developing the modern
theories of matter and radiation a definite epistomological outlook
has become a necessity; and it is the direct source of the most
far-reaching scientific advances,.

We have discovered that it is actually an aid in. the
se arch for kno wledge to unde rs tand the nature of the knowledge
which we seek .

Theoretical p h ysicists , through the inescapable demands
of their enm subject, have boen forced to become e pistemologists ,
just as pure mathematicians have been forced to become logicians .
The invasion of the epistemological branch of philosophy by physics
is exactly parallel to the invasion of the logical branch of philo-

/

-63-

sophy by mathematics. Pure mathematicians, having., learnt by expe-
rience that the obvious is difficult to prove — and not always
true — found it necessary to delve into the foundations of their
own processes of reasoning; in so doing they developed a powerful
technique which has been welcomed for the advancement of logic
generally. A similar pressure of necessity has caused physicists
uo enter into epistemology, rather against their will. Most of
U3 A __aj5j3lajji_m on of science , begin with an aversion to the philo-
sophic type of inquiry into the nature of things . Whether we are
persuaded that the nature of physical objects is obvious to com-
mon sense, or whether we are persuaded that it is inscrutable
beyond human understanding, we are inclined to dismiss the inquiry
as impractical and futile. But modern physics have not been able
to maintain this aloofness. There can be little doubt that its
advances, though applying primarily to the restricted field of
scientific epistemology, have a wider bearing, and offer an ef-
fective contribution to the philosophical outlook as a whole.

Formally we may still recognize a distinction between
science, as treating the content of knowledge, and scientific
epistemology, as treating the nature of knowledge 'of the physical
universe. But it is no longer a practical partition ; and to con-
form to the present situation scientific epistemology should bo
included in science, We do not dispute that it must also be included
in philosophy. It is a field in which philosophy and physics
overlap, ■ (122)

Scientists are becoming increasingly conscious of the
fact that (grha E) they get to know of reality is i nextricably bound up
with (th e way) they "get to know it , (and that as a consequence they cannot
be sure of what they knovf except by studyi ng, the wa y in which the a*-
get to_ kno w- it J To use the happy expression of Leon Brunschvicg, they
are no longer satisfied with giving an artificial communique of their
victories over reality, as was their wont in the past; they are find-
ing it necessary to give an account of their battles.

But philosophy has as much to draw from scientific epis-
temology as physics has — and more, For the philosopher few undertakings
are more rewarding than the study of the mystery of knowledge. And
of all the different types of knowledge none presents greater episte-
mological complexity than mathematical physics. In physico-mathema-
tical knowledge there are implications that are deep and far-reaching,
A false view of its nature leads inevitably to a false view of the
n ature of human kn owledge "in general or to a false view of the natur e
of roality7~or~to^ gth. It would be" interesting to point out the con-
nection between modern physical science and the many modern theories
of knowledge, but that would take us too far afield. We have already
alluded in a general v/ay to this connection in Cartesianism and

-64-

Kantianism, and this must suffice for the moment.

Because the true nature of physico-mathematical knowledge
has been generally misunderstood, ithasbeon almost universally sub-
stituted since the time of the Renaissance for the philosophy of nature .
J rf7 ™ e results ^^ been disastrous for both philosophy and physics,
Ouo of this substitution has arisen the great historical misunder- ..

(standing of the relation between Aristotelian and modem physics.
Looking back at the physics of Aristotle through the eyes" of modern
mathematical physics, and not taking the trouble to. find out what
Aristotle was actually talking about, scientists and philosophers
of science have become a prey to the fallacy of ignoratio elenchi .
They have not suspected that when Aristotle was talking about motion
his approach to the question was something entirely different from
that of Descartes. If this study should accomplish no other purpose
than to help to clear up this unfortunate misunderstanding, our ef-
forts will bo more than justified.

But even when mathematical physics has not been substi-
tuted for the philosophy of nature, the failure to grasp its true
opistemological character has led to abortive and extremely unhappy
attempts to integrate it directl y with philosophy. These attempts
have been numerous bo th inside and outside Scholastic circles . Before
the true relation between philosophy and science can be worked out,
an immense epistemological task of purification and clarification
of notions must be undertaken . It is hoped that this study will con-
tribute something to the furtherance of this task.

As we have said, the consequences of a false view of
the nature of mathematical physics are far-reaching. It would be easy
to show for example how it leads (and de facto has led) to a determin-
istic View of the whole of nature. In this connection Boutroux
writes:

Telle est la racine du determinisme moderne. Nous croy-

(ons que tout est determine necessairement, parce que nous oroy -
ons que to ut, en realite, est mathematique . Cette croyance est
le ressort, manifesto ou inapercu, de l 1 investigation scienti-
fique. (125P

But the implications are even deeper that this. In the
course of history the human mind has often been turned on the dialela '^Vt>
of materialism and idealism. It 'is significant that a false notion
of the nature of mathematical ph ysics leads to bath of these dia metr-
ically opposed extreme s..

The reason for this derives from the peculiar character of mathema-
tical science. As we shall see there is something necessarily materia}

-65-

abouu ^hematics HL thg_s gnae that it deals with quantity , which,
while it abstracts fronTsonsible matter does not abstract from intel-
ligible matter, and_even_int glligible matter imp lies homogeneity. In
so far as mathematics has reference to reality, that reality can be
iiathing_^bu^jra^terial. Hence any possible real mathematical order is
necessarily material. That_ig_why universal mathemat icism con lead
and has led^to nnteriajJJmTOn the other hand, mathematics is the"
mos-^.bstract of .all the sciences, - in a sense eve n more abstra ct than
mouaphysics, For mathematical entities are considered by the mathe^"
nvatician(in_j heir very state of abstraction! .and as a consequence
they are indifferent to real ity. Moreover, these mathematical entities
in their abstract state are prior to the sensible reality to which
we apply them. That is wh y universal mathematicism can lead and ha s
led to idealism . "* ~~ :

During the years when mechanism held complete sway over
mathematical physics the tendency' of mathematicism was towards mate-
rialism. In recent years, however, since the breakdown of classical
physics, the tendency has largely been towards idealism. Professor
Joad has described the dialectic by which mathematicism leads to
idealism:

But if the entities of which the universe is on a naively-
realistic view supposed to consist: substance and space-time ,
turn out to be mathematical , that is completely resolvable into
mathematical formulae, and if to be mathematical is to be men tal,
more will be implied "by the various statements as serting the mathe-
matical naturo of things than that the universe is describable
in terms of mathematics; it will be implied that the universe
somehow is mathematics. And, since mathematics is thought, to be
mathematical will also be to be mathematical thought. (124)

Of all the modern mathematical physicists who have been
drawn towards idealism, Sir James Jeans is perhaps the most outstanding
example:

The terrestrial pure mathematician does not concern him-
self with material substance, but with pure thought. His creations
are not only created by thoug ht but consist of thought , just as
the creations of the engineer consist of engines. And the concepts
which now prove to be fundamental to our understanding of nature...
seem to my mind to be structures of pure thought, incapable of
realisation in any sense which would properly be described as
material... The universe cannot admit of material representation,
and the reason, I think is that it has become .a more mental con-
cept. (125)

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And elsewhere he writes:

Broadly 'speaking, the too conjectures are those of the
idealist and the realist -- or, if wo prefer, the montalist and
materialist — view of nature. So far the pendulum • shows no signs
of swinging back, and the law and order which we find in the uni -
verse are moat_eag_il y described — and also, I think, most easily
explained ~ in the language of idealism. Thus , subject to the
. reservations already mentioned, we may say that our present-da y
soignee is_ favourable to idealism ., In brief, idealism has always
maintained that, as the beginning of the road by which we explore
nature is mental, the chances are that the end also will be mental
To this present-day science odds that, at the farthest point she
has so for reached, much, and' possibly all, that was not mental
has disappeared and nothing new has come in that is not mental.
Yet who shall say what we may find awaiting us round the next
corner? (126) „ ,

Yfe nust try to see whether it is necessary to choose
between materialism and idealism,,

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CHAPTER TWO

THE SPECIFICATION OP THE SCIENCES

.1. The Problem,

The expressions "mathematical physics" and "physico-ma-
thoma-cical science" immediately suggest an epistemological dualism
which implies both a distinction and a union.. And the crux of our
own problem lies in analysing accurately the nature of that distinction
and that union. In the present chapter w e shall endea vour to lay bare
the -basic (principles) which determine the [distinctioii ^bety reen math e-
matics and physics ; in chapter three we shall consider the principles
which govern the |unionjof the two, And the principles laid down in
these two chapters will serve as a foundation upon' which the entire
superstructure of the chapters which are to follow will be built;
they -will guide and shape the whole subsequent-analysis.

Our first concern, then, is to- see how physics and ma-
thematics are distinguished , from each other . The mere recognition
of the dualism implied in the expression "mathematical physics" does
not of itself predetermine the solution of our problem, Por a dualism
may be only nominal ; it may be only the superficial expression of
a basic identity. As a matter of fact, the dictionary of modern science
is filled with expressions which suggest epistemological dualism:
bio-chemistry, astro-physics, etc. And the very creation of these
apparently hybrid sciences seems to have come from a recognition of
a basic identity between the branches of knowledge joined together.
As science progresses, this basic identity seems to be growing in-
creasingly evident. Barrier after barrier between the sciences is
being broken down; there is steady progress towards epistemologica l
homo geneity . And on the face of things this- seems to hold for mathe-
matical physics as well as for the other hybrid sciences. Recent do -*
velopraents .seem to bo wearing pre tty thin the traditional distinction
betwee n physio s and mathematics . The most abstract conceptions of
pure mathematics are being "incarnated" in the physical universe;
the most concrete elements of the physical universe are finding a
mathematical explanation. And perhaps few would hesitate to deny that

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there is a greater dichotomy between mathematics and physics than
between biology and chemistry.

,. " . .? ! f P robl em 3 then, is to try to discover how deep this
dicho-oonry is becwcen physics and mathematics,, It is a problem which
liasinnc^-cablt; ramifications, and which cannot be dealt with adequately
in isoiauicn t^a irs epistemological context. In order to get at

one na .,urs of oho distinction between physics and mathematics we must
sec ,-m.v .toy in into the whole epistemological scheme of things,,
In other words > we are faced with the question of a classification
of the sciences t . And we mast explore 'this general question at least
to th/? extent in which it is necessary to throw light upon the spe-
cific problem we have in hand

It has often been remarked that the human mind has an
instinctive tendency towards monism, It is an extremely significant
tender..-.;-- and ■:.:-,.. which reveals the inner nature of ,the intellect,,
The history of philosophy has been a constant manifestation of this
tendoncv under a , great ^at-'.ity of ferns „ There have for example been
r:ou.at ,.oss (■■.■'/■■onpr.s at soino kinc; of on tological monism ,, But this is
not thi ^sp--ct of the tendency in which we are interested here; we
are concerned with -what, might be called epistemological monism : ) the
attempt to :cs : jVx'-fj all hjman knowledge to one homogeneous typej) _the
failu'..:-: to rncorinizc the radical heterogeneity of the ways in whic h
the hu.r^..-?. mind enters into contact with realit y* It would hardly bo
an ex£g;o:vation to say that one of the greatest intellectual evils
of modern times has been this persistent attempt to homogenize know -
ledge , /it is an evil wh ich has had far reaching consequences ., notabl y
in the fieJd of education J But these consequences are not particularly
re levan': hov-i; ' . '

In this connection^, positivism and scientisrn readily
come to mindo But even philosophical circles whijh have rejected
positivism and scientisrn (including the majority of modern scholastic
circles) have, Been affected by this evil in a number of ways,, Typioal
examples are: the iden tification of speculative and practical know -
ledge : the idantifioat ion_o f metap h ysics and the philosophy of nature ;
the - identif ication of dialectical knowledge and true scientific kno w-
ledge, ond. ti n.- 1 , identification of mathematical and, physical knowledg e.
This 1 last ■■nxartyla ij obviously the one which affects us most directly.
Bu J '. all the others have definite repercussions upon our problem as
we shall f -'.-•:• vbuallj- see It is worth while pointing out here that
the ur.if.''.»i--Acn of knowledge has historicall y been associated with
inath'ri!--.-'--: •■'■'■ ■"■'.,, And the reason is that in no science can this tendency
be i v.'.i-r.i.o' ' y.; far is in mathematics.

Now it is extremely significant to note that homogeneity

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is at once the source of unity and the source of multiplicity - in-
S o^ s^arTt^ fi^M^ ^^ting^ow n of ^an Sowlcdg o
uK^th^SS^ hasal most inevitably r^uTtid-Jn~ the breakin g
upo^he^ciunces into^^sgT ^iie^ami-bH nches. One has only .
Y Tm ^^ I ^ ¥I ^ S ^~^^^ ±onoe 3 attempte d by Bacon? Comte,

irorler^oouT'J^T 011 ^^ 1 ^' <*> ^mention onl y'q few?"
in ordci ,o see yio ^highly_arMtrag y = lhe distinctions (6g^e^the~
s.cignc^s.must be rrJ^e^^^jr^S^^^ m^TTW^ homo geneous *t
jpe^And because these distinctions are arbitrary, the advancement
of science has made short shift of many of them. That is why some
have come to the conclusion that all distinctions between sciences
arc purely capricious. And in this connection the following lines of
iiax Planck are significant:

■ . Rooked ai correctly, science is a self-contained unit y;
it is divided into various branches, but this division has no
( natural ) foundation la nd is due simply to the limitations of the
human mind which comp eT us to adopt a division of labour. ; Actually
there is a continous chain from physics and chemistry to biology
and anthr gp^logy ^andjhence to the social and intellectual soierio eg})
a chain which cannot be broken at any pom tc save caprici ously^r~( : 2T

In the sixteenth century two contemporary philosophers
I wrote on the question we are discussing. The one represented the birth
/ of a new philosophical movement; the other represented the end of

an old philosophical tradition that was passing away. The first was
^Rene Descartes, and the second was John of Saint Thomas. Descartes
was the principal source of what Maritain has justly called "the ra -
dica l levelli ng of the thin gs of the spirit " that is so characteristic
of moder n times j (sif In his famous page in the Regulae on the unity"
of knowledge, modern ep istemologic al monism received its first explicit
for mulati on o And the source of thiETformulation was the mathematization

(6f~nature"P~about which we spoke in Chapter One, (4) Around the time
that Descartes wrote this page in Regulae , John of St. Thomas wrote
an article on the unity and distinction of the sciences at the end
of his Ars Logica . (5) - an article which summed up and synthesized
with ad mir able clarity a nd precision all of the fundamental Thomistio
principles gov erning the classification of the sciences . Though it
must be admitted that in his philosophical writings ho neglected the
order of concretion , and that he seemed completely unaware of the
great scientific discoveries that were going on around him, no one
ever achieved a better exposition of the fundamental notions of science
and the principles which determine the unity and distinction of the
sciences. It is principally to him that i/e shall look for a guide
in our discussion of the present question. At the same time it must
be noted that ho merely s.yvbhesized principles already found in
Aristotle and St, Thomas; he in no way changed or added to these

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priiiciplos, as some have maintained.'

to point out t£t b the°re STS^^ ^" ^«*»i"> « 1b important
the Question of oJ=+L f ™? fundamentally distinct aspects to
too question of opMtgmolog ioal plura lism. For the problem may be

t^^?%^^? [ ^ S ^^~ Srii ^ rae ihe Polity of ?orrW
point of view S ^ n hG ^f ^ h ° ld ° f te TCali V, or from the
Point of view ox the plurality of the means of knowing employed by

^nTtl'T J J hQ intelli 8 ible secies. In other wordsthere are
cjroJigtangt^oblgga_gf_ttig L o ne and the Many . (6) Becaus^TSre

Bufft * ^ r l5 ^™T555E3ae f both aspects enter into ouJ problem.

the obio^f *o -, ° kGeP in "^ that a Polity on the part of
the objects does not necessarily imply a plurality on the part of
the means of knowing. In fact, in proportion as an intelligence is
more perfect, the plurality of its means of knowing decreases while
the distinctness with which it knows objective reality increases.,
ihe diyine intelligence sees the whole of reality exhaustively in
ios ultimate distinction in the one intelligible species which. is His
essence. At the other extreme of the scale of intelligence, the human
mnd_n|gdg_as many _intellig ible species as there are natures"to~b5
ffiffi' (If the human inlallacJi_wore_in_a_st ate of perfection, the problem
of_the_d istinction o f_the scigncg^_EQuld .bo easily solved : j there woul d - '
be as ma ny species of science as there species of things . Saint Thomas'
explains that in the infused knowledge of Christ there were as many

ysgec ies of science as there were species of things knovm by Him. (7)
But becauseof the imperfection of the human intellect, it is necessary
for it to know a plurality of objects which in themselves are specifi-
cally distinct in the light of a common scientific species. This com-
monness, however, is something quite, different from the commonness
of the intelligible species possessed by the higher intelligences
which enables them to grasp reality in its distinction . It is a com-
monness of potentiality which hides rather than reveals the distinction

,of reality.

In connection with the question of epistemological monism
mentioned above it seems necessary to point out here that if the monistic
tendency consists merely in an attempt to reduce the plurality of the
moans of knowing, as is done in the method of limits , it is a legitimate
and laudable thing. It is reprehensible, however, when it consists
in a reduction on the part of the objects .

These remarks should suffice to show that the question
of the distinction and specification of the sciences is an extremely
complicated thing, which depends essentially upon the nature of the V"
intellect in question. For God, for example, there is no speculative
science distinct from His one science which is wisdom, since Ho neces-
sarily must see all 'reality in terms of Himself, the First' Cause.

-71-

iwttnl ?J ' f C °^ Se ' that HQ fails t0 ^asp the ratio mo -

n ^ oxnnjplo whi ch, as we shall sae ^^rogg^ly, JT^Ttarml
ratio of all mturalJhinggTbut Ho sees it aufetio^ DeltatE ;

For all created intelligences there ' is a distinction of
speculative sciences even though all of them must remain essentially
subordinated to wisdom. And the nature of this distinction depends
upon the nauure of the intelligence in question. That is why there
is a plurality of sciences peculiar to the human intellect which,
unlike uhe angelic intellect whose knowledge is prior to things in
so far as it is derived from the species divinae rerun factivae , (8)
is dependent upon things for its knowledge. This dependence, plus
the fact that its object is necessarily material things, make human
knowledge esse ntially abstractive. And that is why the plurality of
the human speculative sciences is determined by abstraction .! No other
principle of division is nos sibleT}

But before we como to the question of how 'the speculative
sciences are distinguished by the different degrees of abstraction ,
it is necessary to go back further" in our analysis of the heterogeneity
of knowledge. For reasons which will become apparent Later, particu-
larly in Chapter IV, we must begin with the primordial distinction
between speculative and practical knowledge.

2, Speculative and Practical Knowledge .

The implications of this distinction are manifold, and
it would take us too far afield to consider even more important ones.
We shall content ourselves with a summary consideration of those im-
plications which have a particular relevance for the understanding
of mathematical physics, (9)

Briefly, then, speculative and practical knowledge differ
by their end, (10) The end of speculative knowledge is truth ; the
end of practical knowledge is an o peration , that is, a work to be
d one or made , (ll) When we say that the end of practical knowledge
is an operation, or a work to bo done or made, we mean an operation
or a work that is o utside the intellect . For as Saint Thomas points (12)
out, an operationliiay be either exterior or interior to the intellect,
■In the latter case the operation is a mere contemplation of truth,
land in this speculative knowledge consists. Moreover, within the intel-
lect there may be a kind of opus consisting in an ordering and a cons-
Vtruction. In this case v/e have an- art , but only a speculative art ,
and not a practical art, for the opuS remains interior to the mind.
Both logic and mathematics are arts of this kind. This distinction

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both G of ?hem h^ V ° ^ ? ract j cal <"* i*°f some importance, since
Stical plSsiSl V Pa t0 Play in the °on^otion of mathe- ,

+i <»,„ + ■;, ^f °^ cct of a11 Practical- knowledge, then, is some-
thing outside the limits of the intellect, (is) it is, in fact,
primarily and essentially tJie_^yecl_of_an_^petite,CfoLthe intel-

^Bit^Sn^SBJSSSS^S^^SS^eP gn^beogugeTIt"Bu biiilts itself i n
somgj my to an a pj3ejtjg^v^n~Ttou gh practical knowledge in itself "
does noo consist m a mere extrinsic submission ) . Hence ' iFf o ll owi
that practical knowledge has as its .object the good as good ( bonum
ut bonum), and not the good as true ( bonum ut verum ) which is the
ob.ject of speculative knnwTedgP. That is why in order to have true
practical knowledge' it is not sufficient that the object be in itself
an op erabil e, i.e. something that in itself is "makeable" ; it is neces-
sary that this object be considered precisely in ordine ad operationem ,
° r per modura operandi . (14) , Now whereas the object of speculative
knowledge is something within the intellect, and that of practical
knowledge something outside the intellect, if we consider the principles
of^these two .types of knowledge, the situation is exactly the reverse
(at least in so far as human laiowledge is concerned) , The principles
ofspjcula^yejcnowled ge are in thin gs, ( tind the movement is from thing s
to theiigS;y >th^~principles of practical knowledge are in the mind
and the direction is from mind to things . That is- why St. Thomas writes:
"Practicus intellectus est de his quorum principia^ sunt in nobis,
non quomodocumque, sed in quantum sunt per nos operabilia," (15)

Consequently the mind is the measure of the things of
which it has practical khowledge , Cwhe:reas it, is measured by the things ,
of which i t has speculative laiowledge , ) as St. Thomas . explains in the
following pas' sage t .'-,

Res aliter comparatur ad intcllectum practicum, aliter
ad speculativum. Intellectus enim practicus causat res , unde est
mensuratio rerum quae per ipsum fiunt; sed intellectus spfeculativus ,
quia accipit a rebus , ( est quodammodo motus ab ipsis robust et ita
res j jiensurant ipsum . Ex quo patet quod res naturales, ex quibus
intellectus noster scientiam accipit, mensurant intellectum nostrum,
ut dicitur X Metaphys. (com, 9) : , sed sunt mensuratae ab intellectu
divino; in quo sunt omnia creata, ■ sicut omnia artificiata in in-
tellectu artificis. Sic ergo intellectus divinus est mensurans
non mensuratus ; res autem naturales, mensurans et me nsurata; sed
intellectus noster est mensuratus , no n mensur ans quidem res na-
turales sed artificiales tantum. (16) ~~~~

Now there is an analytical connection and a direct pro- ■
portion between the operabilitas (the "makeableness") of a thing and

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wc arming STw^^ " 1IMSt b<3 noted i— lately that
In S scnoiil f' rialityl ' ^ its ^°adest significance,
^^^^^L3^SLC^E3SS&Jo j^ kind of potentiali ty, and
^^^to^Th^^ required fo"^

be not MentifJS 2th + ^actical knowledge is that its essence
g^g^gffif^jn ^ its exist aice . For the pr actical knowledge i s
knowledge or things to^groughTj&to existence. \That is wh y God

(except m the nemo that He is attainaWby intelligent creatur es
through practxeal knowledge). ( 17 ) As John of St. Thorns points
0Ut ' K^ ^^speculative abs tracts in some, way the exist ential (ab
exorcitio exIsTenfli), wherSaTthe ^ TO nti-^T^ sid^rs~its~obje' ct to
its exiatsntial _gtatg T ut stat sub exercit io existendi ' ). Yet it tm,, lfl
be highly ambiguous to say, as some authors have done, that speculative
knowledge has to do with the essential order, and practical knowledge
with the existential order. For there is an operabilitas in the es-

jsential order as well as in the existential order. All be ings vm&z"* <
potency in their essenc e, i.e. matter in the strict" sense of the terras
have anJjitrinsic_ontolo gical plastic ity, a "formability" which pure

\JTorms do not have. In all, material creatures, "formability" touches
the very substance. In their very essence is found the reason for
their intrinsic physical contingency.

Viewing the hierarchy of being dialectically, we may
say that in the measure in which Tire get farther and farther from pure
immateriality in which the essence is identified with existance, in
the measure in which we get deeper and deeper into materiality, the
closer we approach to pure operabilitas and hence the greater becomes

Vthe scope of practical knowledge, We are getting deeper and deeper
into contingency and hence farther and farther awa y from the necessar y.

( which is the object of spec ulative knowledge.) In this dialectical
process we start with the Being of which only speculative knowledge
is possible, and we tend towards a limit which wou ld be an objec t
that would bo purely practical . This object does not exist, nor can
it exist, but there is something like it in normal knowledge. Saint
Thomas points out that the study of morals' *not for the contemplation
of truth, (19)

It should bo pointed out, perhaps, that we. are considering
this descending scale from the point of view of natures, for if other
points of view were introduced, such as the large place that fortune
plays in human life, and the immense amount of contingency involved
in the supernatural order, what we have just said might be open to •
modification. Perhaps some might be tempted to take exception to the
last paragraph on the score that the ultimata elements might very
well prove to be few in number and highly determined in their cons-
titution. But even if this should prove to be true what wc have said

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and fcobilitS" Lx ^ th ^ ™ uld P^ess indefinite malleability
and formability ' and serviceability because of the fact that every-
thing in material creation would be made out of them„ -

^T a ^" 1 this has an ex treraely important bearing upon
the nature of physics, For the object of physics is down very far
111 the scale we have been considering. This is particularly true of
the part of physics which is far advanced towards concretion. And
the farther physics advances the deeper it gets into materiality,
That is why the things with which physics deals ore principally
operabilia, more operabilia than speculabilia . And as physics progres-

ses, the things with which it deals become less and less amenable to
speculative knowledge and more and more amenable to practical know-
ledgej)

Moreover , in order to possess fully the speculative
knowledge of which these~things are capable , it is necessary to' have
practical knowledge of them . For even though speculative knowledge
always remains something distinct from practical knowledge, in order
to have perfect / speculative , knowledge of things that are in their
vor y nature operabilia , it is necessary to have ,practical, knowledge
v of them,) And the more things are operabilia in their very nature,
the greater becomes the necessity of having -practical knowledge of,
them in order to possess wi th any kiiil of adequacy the speculative
knowledge t hat it is possibl e to have of them.

Now the difficulty is that this practical knowledge is
not open to us. For we cannot make natures. We can only imitate them
by making artificial things, (20) , Natures are, in fact, essentially
" rationes artis divinae , " as Saint Thomas points out in the second .
book of the Physic s, (21) In other words, art is essentially an
extrinsic p rinciple, ^a nd it is only in divine art that this extrinsic
principle can be the cause of the intrinsic principled The reason "
is that whereas all created art presupposes a subject, divine art
does not, and as a consequence it can reach the- very first principle
of the things it makes. -

But even though man cannot have a practical knowledge
of natures which alone would make it possible for him to have perfect
speculative knowledge of them, he can have practical knowledge in
re lation to natures , and by means of it acquire a more perfect spe-
culative knowledge of them. As a matter of fact, in order for man
to have a profound speculative knowledge of natural things in their
concretion it is necessary for him to have recourse to an immense
Mount of practical knowledge. He must operate upon nature with ins-
truments devised by himself . I And the deeper he plunges into concretion

-75-

^^^^^S^L^BOM ^mst those l^i^^ rAs^aj^^^rn,

n^S v-i TV iu f? clsol y because of the weakness of his specu-
Vlative knowledge that he must have recourse to practical knowledge.

Not only must physical construction enter into physics
| in an increasingly large measure as it advances, but mental construct-
T a ! ^l 1 '- , In . th e°iy-l3uildtog, which still falls within the genus
of_aro,C though 1,, be a speculatiye^ gTT t.he H M BT .t.^t. mm™. .,..-4+ -
wore, an ersatz logos which can never do more than connote objective
nature,. Moreover, in-order to mtlnnnl^Mmtim, ^Tr^v^^^ is
forced to borrow heavily from mathematics which is also a speculative
>>art.

Thus in a number of ways construction enters into the
I object of physics ~ enters into it so profoundly ' that it becomes
/impossible to distinguish between what is derived from nature and
Ijwhat^ comes from art. All this is necessary but it constitutes a danger.
For it is all too easy for man to come to look upon nature as a mere
malleable matter to be worked upon and used. Moreover, the knowledge
we acquire by having recourse to this construction makes possible
such extensive mastery over nature that the practical power that is
derived from this knowledge all too easily becomes confused with the
'purely speculative knowledge of nature which is. the basis of the prac-
tical kno wledge^) In other words there is the danger of confusing the
speculative knowledge we have of natural things with the knowledge
of what we can do with them,, or at least of subordinating the specu-
lative knowledge of nature to the practical knowledge we are able
to have in relation to it, in somewhat the same way as is found in
the case of the artist who is concerned with the nature of the material
he uses (onl y to the extent to which that is necessary for the, achieve -
ment of his work of ar t,) Then the -practical know le d ge is no longer .
the instrument of the speculative knowledge , 'but just the contrar y.
And oven when confusion between speculative and practical knowledge,
or the perversion of the right order that should exist between them
does not occur, there is at least the danger that, the abundant use
that we can make of nature might lead us to cease to wonder at nature ,
and without this wonderment, as Aristotle has pointed out, speculative
knowledge cannot thrive.

That the tendencies we have just mentioned have been
prevalent in modern times is all too evident. Already in Descartes
we find the following:

Mais sitot que j'ai. eu acquis quelques notions genera-
los touchant la physique, et que, commencant a les eprouve'r on

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ttl g L a f 0CUrGr Qutont ^'il est en nous le bien general de

de M D i S n ^ B! ° ar GUeS ra '° nt fait voir ^'il oBt possible
de parvenir a des connaissances qui soient fort utiles a la vie:
et dans les ecoles on ne peut trouver une pratique, par laquel-
lo, cormaissant la force et les actions du feu, de l'cau, de 1'alr,
des astres, des cxeux et de tous les autres corps qui nous envi-
ronment, aussi distinotemant que nous connaissons les divers me-
tiers de nos artisans, nous les pourrions employer en meme facon
a tous les usages auxquels ils sont propres, et ainsi nous rendre
comae m aitres et possesseurs de la nature . (22)

These tendencies have continued to grow since the time
of Descartes, and today it is not rare to find even in the writings
of those who have otherwise made valuable contributions to the phi-

/losophy of science passages in which the important distinction between
speculative and practical knowledge seems to have faded to a large

\ extent,, The following lines of F.C.S. Schiller are fairly typical:

The mental attitude which entertains hypothesise., feels
free... to rearrange the world at least in thought, to play with
it, and with itself. For hypothesis is a sort of game with reality,
akin to fancy and make-believe, fiction and poetry... It is by
this hypothesis - building habit that science touches poetry on
the one side, and action on the other : for it is akin to both.
The play of fancy and the constructive use of the imagination
reveal the creativeness of human intelligence; by their use the
scientist becomes a 'haker" like the poet . . . Yet on the other side,
this hypothetical attitude mediates between thought and action ,
bnd helps to break down the superficial distinction between the

/ theoretic and the practical,, J It "drives the scientist out of the

/purely receptive attitude, and makes him a doer . For to entertain
a hypothesis is to hold a mental content hypothetically, and this

I is (| o_ hold it experimentally^ ) which, again is to operate on it

\and to manipulate it . (23)

From many points of view it is in Marxism that the ten-
dencies of which we have been speaking have found their fullest ex-
pression,, Marx' eleventh thesis on Feuerbach states that "the philo-
sophers have only i nterpreted the world differently; the point is
to change it" At the heart of Marxism is a revolt against the humble
state of being measured by thin gs that is characteristic of specula-
tive Imov/leda oT and a desire to become their measure through practical,
knowledge^ There is a seeking to transform nature completely, to

-77-

reconstruct it to onp'q ™m -;,-,„„„ j -,.-,

to the exigencies o7cno^lSw°? ^*™**>*° subject it entirely
Dialectical Mn+P^iV ™ f prtua - s ° In ^-s Introduction to

ts this to —^^ ^^ 0on ". a **«** disciple of Mar.,

. Dialectical Materialism is surrounded by the glamour

So tire^entlo^hSh ^ 117 Strang<3 ' ^teriou/and startling,
taiown the SL™ St hl f n0W method 0f * hi nki»g becomes better
S?'»J^ ? UnlCnOV ' n Wl11 vanish » Jt ^ n **-seen that

tic'l tool ?t°h VXQ0G 0t te™^™> ^t a very prosaic and prac-
tical tool. It has more the functions of an axe than of a Chinese

vrvse , M

— > 4-u -, J N ° t thG mer ° mdersta nding, but an increased control of
^tno world, is the ultimate purpose of scientific method. (24)

But all this is an anticipation of what is to come in
subsequent chapters. Consequently, we must leave this point, and having
seen the nature of the distinction between speculative and practical
icnowledgo, v/e must pass on now to a consideration of the hierarchy
of speculative science. This will bring us directly to the central
point around which the whole of the present discussion is revolving;
tha nature of the distinction between physics and mathematics.

3. The Hierarchy of Speculative Science.

Science, writes Professor Urban, "is the most ambiguous
concept in the modern world," (25) In order to avoid confusion it
seems necessary to point immediately that at the beginning of this
discussion and until further notice we shall take the term "science"
in its strict Aristotelian sense. Both Aristotle and Saint Thomas
sometimes use the expression "scientific knowledge" in a fairly loose
fashion. Thus, in the Posterior Analytics (26) "quaelibet certitu-
dinalis cognitio" is called scientific knowledge. In the Summa (27)
St. Thomas sometimes uses the word "scire" in terms of knowledge of
particular contingent facts . But outside of a few exceptions of this
kind, "science" in the peripatetic tradition has consistently meant
a. knowledge that is universal and necessary j a knowledge that has
been arrived at by demonstration , and a knowledge that has been fixed
and determined in an intellectual habitus . (28)

Nov?, in ..ecming to grips wi th the problem of the distinct-
ion and classification of the sciences, it is extremely important to
discover the true criteria by which one typo of scientific knowledge
is distinguished from another. One cannot select these criteria in

-78-

llTX^fnJjV m °l° e±Cal confusi °*. What, then, will reveal
ObviouSv too n t, ,°f ^ ° b ^otive and necessary classification.
Obviously, the nature of knowledge itself.

tmn , nn1nnr pledge is essentially objective,, for, in Thomistic
cermanology, to know is to bo. the thing known in its very "otherness."
But ton knowledge, because of its limitations, is neve? completely
objective under every aspect. Potentiality always involves some kind
of subjectivity, and the intrinsic potentiality of man's- nature neces-
sarily limits the objectivity of his knowledge. Quidquid re citdtur
ad laodum recipient s recipitur; hence if the knowing facultyiiVery
imperfect, the objectivity of its knowledge, however true it may be,
must necessarily be very imperfect. It would seem to follow from this

■ that the segmentation of scientific knowledge into specifically distinct
types must be based on something which is fundamentally objective,

^but which has, at_Jhe__same_time, a subjective determination.

As we have already remarked, if human knowledge were
in a state of perfection the problem of the distinction of the sciences
would be simple, since there- would be as many species of science as
there are species of tilings., But because man is incapable of grasping
things ^ perfectly, it is necessary for him to know a plurality of objects
which in themselves are specifically distinct in the light of a common
scientific species. Now in order to grasp clearly the nature of this
common scientific species we .must introduce here the distinction
between "thing and "object". By 'thing" .we understand what is commonly
known as the material object of knowledge,, i.e. that which is known,
the res in se , considered purely in its entitative status. By "object"
we understand what is commonly known as the formal object of knowledge,
i.e. the particular determination or formality by which the cogni tive
power lays hold of the "thing " o For a thing can become the object
of knowledge only in so far as it is orientated to a cognitive power
in a certain determined way. Thus, an eye' con perceive a wall only
because the wall is orientated to the eye by means of its color.
jProm what has already been said about the nature of human knowledge
lit must be evident that the specification of scientific knowledge
/must come from reality, (29) not however in so'far as reality is a
Vlthing", but in so far as it is constituted as a scientific object. (30)

Consequently, whenever St. Thomas uses such expressions
as "scientiae secantur quemadmodum ot res," (31) he understands
"res" in the sense of formal object; for in the text just cited he
. immediately adds: "nam omnos habitus distinguuntur per obiecta, ex
quibus speciem habon t."

In relation to the formal object, Cajetan introduces
a further distinction which will be extremely useful for us, not
only for our present purpose, but also for the final explicit

-79-

f ormulation of the naturo n-p -r,u-,^A „ j., , . ,

we shall attempt in £ter X S TO ^T^l ^t^t I' hich
„ w +..,_ ,_■ , „„ p ^P^er Alii. (32) He points out that there

thin^t'elJ wS ,T ?? de ? t; ° ne is the f^nlity listing in the
thing i.self which^ S gtly^g rr nlllates the ant nf^ tinn.L by

™3.°^I h ^ th ° " thi »g" ^^e-^r^n^ne^te^oiniSve ?
power; the other is a formality whi ch actualizes the f irst i ft^my.
The concrete example usually given to illustrate this-diitS5H5n-
is borrowed from the reaLn of sense cognition: in visual cognition
there are two formalities:' the color existing in the wall, and the
ligh-o which plays upon the wall and actualizes its color.. By transpos-
ing -this example to the realm of intellectual cognition we discover
that tne second formality is a kind of objective spiritual light , (33)
which manifests and actualizes a determined f oruolity existing iri
the thing, which in turn renders the thing -intelligible by const ituting
^ as an objeoc. The first of these two formalities is know^in~Th5mistic
terminology as the " objectum forma le_auod" or the " ratio formalis
«uaa", or the " ratio formalis objecti ut res ." The second is known
as the "obje ctum formale quo," or the " ratio formalis sub qua ," or
the " ratio formalis objecti ut objectum . " This distinction may appear
extremely subtle, but Cajetan rightly insists upon its necessity:

Necessitas autem, qualitas et distinctio harum rationum
sumenda est ex distinctione duorum gonerum, in quibus oportet
locare objectum scientiae, Oportet .enim quod formaliter sit talis
res, taliter scibilis,, Et ideo oportet quod habeat et rationem
forraalem cohstituentem formaliter ipsam in tali esse r eali , et
rationem formalem constituentem formaliter ipsam in tali esse
scibili o (34)

Now, from what has been said thus far it should be evident
that the point of departure of the whole question of the specific
distinction of the sciences must be an attempt to discover in the
entire realm covered by scientific knowledge specifically distinct
"rationos forrnales sub quibus " For, as we have just seen, it is the
"ratio fomalis sub qua" that actualizes the ratio formalis quae .
In other words j what we are tryingto decide is whether or not
there are specifically distinct (way ^ . in which reality is scientifi -
cally knowablc , and it is precisely ratio formalis sub qu a which
constitutes reality |as) scientifically knowabl e, i,e, in esse soibili .
But where shall we turn to discover the specifically distinct- rationes
forrnales sub quibus by which one science will be distinguished from
another"Onoe again our answer will be found in the nature of know-
lodge in general, and the nature of scientific knowledge in particu-
lar,, (35)

The root of all knowledge is immateriality. (36)
This infeateriality is required first of all on the part of the knower
which, in order to be open to other forms besides its own .

-80-

It is also required oTJhftrfo^^

be known only in the mUsurc ^ ^-i L ^ " g ^T?!. f ° r a tMng can

that is t i in t ta , rs a S tt ^ l 1 } t M lil «sSf « S S.1SJ-.

s^by^ ob^Ltlnlhe m n SUffi ° ient -^-iality\f nof pSse -
sea oy the object m the state in which it is found in nature, the

if in th P W n?^ B ° ° f ^ de P endence of knowledge upon immateriality,
riaStv orfS „ speculative knowledge different levels of immate-
riality are discernable, there will be a stratification of the sciences
corresponding to, these different levels, Moreover, necessity pertaSs
co the essence of science, for no truly scientific knowledge is pos-
- i? ° f ^ngs in their contingency. (37) Consequently there will be as ma-
rg aillcra/it sciences as there are different types of necessity ; that
is to say, the sciences will be distinguished by the specifically '
different levels according to which the scientific object can be lifted
out of the flux of contingency ^ Hence St. Thomas co ncludes; "Et ideo
secundum ordinera reraotionis jeT a materia et a raotu^ scientiae specu la-
tivae distinguuntur. " (38) But the sciences will 'not be specified -
by the degree of immateriality and necessity of the object considered '
in its entitative state in such a way that the species of science
will correspond to the degrees of being. If this were the case, the
specification would be coming from the material object , which as we
have seen, is impossible. It is the de&ree of jjanateriality and nece s-
sityfarising out of th ejjm^jTn which th e ob ject is known by the intel -
^gcjf yEnat is the p rinci ple of specifi cation" ^ ~ *~- ~~

i Now the( 1nean^ by ■ which the intellect lifts its object

| out of , the opacity of matter and the flux of change is called (abstractiorS )
Hence it will be the specifically different | degrees of abstraction]
that will give us the rationes formales sub quibus we are looking
for, ( and these in turn will actualize in' the object the different
^rationes formales quae .) But before pursuing the discussion of the
diverse degrees of abstraction, it is necessary to point out that
we are concerned here not with total but with formal abstraction.
This distinction is of capital importance for the philosop h y of science ,
and no one has probed its profound implications with greater acuteness
(than Cajetan. (39) Since all positive abstractio n involves some
f kind of separation , ( the basis of this dual abstraction is a dual
composition; ) the composition of matter and form , Cand the composition
of_a _universnl whole and its subjective partsj ) Abstraction is called
forr.nl when it consists in disengaging a form from the matter in which
' it is concretized; it is called total when it consists in laying
hold of a universal whole (apart from the subjective parts in which

-81-

it is distributed } When a mathematician abstracts a certain quantita-
tive concept, such as the notion of line, from the sensible matter
in which it is concretized in the real world, he is practising formal
'abstraction, For "line" stands in relation to "sensible" as form to
matter n When, however, one abstracts the concept of animal from its
subjective parts, man and brute,, to consider it apart, he is using
v total abstraction.

In order to avoid confusion it is necessary to point
out that when we say that formal abstraction consists in abstracting
a formal element from its material concretion it is never a question
of abstracting the substantial form from the matter to whi c h it is
Sited, for as St, Thomas points out, f40) the \ interdependence) .
existing between a substantial . form and its corresponding matter is
such t hat one cannot be understood without the other ,- Thus, th e studen t
oflulture never abstracts the substantial form f rom its Cpatterj he _^n ^ fe
merel y presci nds from the (contingent) materiality proper to (j udividuals y ___■
fhlspolnTis_of_e xtremo importance for a prope r appreciation of the
rgpT^TTf'^vsics.rand it is usually misundersto od_by scholastic writers^
Similarly, the niathcmaTician does not abstract the substantial form,
but the accidental form of qu antity. The metaphysician lays hold of
substantial form only in so far as it is a co-principle of material
.beings)

There is a world of difference between the two intel-
lectual processes involved in formal and total abstraction. In the^
firlt case the Reparation results in a double concept each of. which
is complete by itself. The notion of line does not involve the notion
of sensible matter, nor the notion of sensible matter necessarily
involve the notion of line. Hence each can be perfectly conceived
in sepcration from the other. But in total abstraction only °ne com-
plete concept results: the idea of animal is conceivable withouo the
notion of either man or brute; but neither man nor brute is intelligible
without the notion of animal. Because formal abstraction reveals a
fSlthaf is purified of the T**^**^*^^^'
it gives. rise to .greater objective ,* f^f^Z'sl^i^of
greater intelligibility is oh ° ^^ much ^ intc iii g ible
the form. The notion of Jine, ^° r ^g e ' mtter , thon in its state
in its state °f ojatoaojjon from sens ibl ^ ^ ^ ^ ^.^
of concretion And let it ^ ™J - Qblen of mthoHat iool physics,
iroon the pivotal point of>how ^ ental reason why phys ic^in

^^Sl^^^^I^^^^^^-^^^r^^ot we

-82-

does not necessarily moan greater intelligibility for us. In fact,
there is ordinarily an inverse proportion between the two, as we shall
have occasion to point out in Chapter IV. We say "ordinarily", because
mathematical science presents a unique case which wo shall study in
Chapter VI. And this unique case will have an extremely important
role to play in the solution of our problem. '• ■ ■

From the point of view of actuality, the movement of
total abstraction is the reverse of that of formal abstraction,, For,
in ascending from brute and man to animal, and' from there to higher
genera in the Porphyrian tree, we are moving from what is more determined
and more actual, and hence more intelligible objectively, to what 1
is more confused^ more potential, and hence less intelligible objectively.
For the mind can abstract a universal whole from the subjective parts
of which it is predicable on ly by retreating from the actuality and
determination of these s ub jective parts into a state of_ greatgr_p_otent-
iality^ But it happens that in moving from what is less intelligible
to what is more intelligible objectivel y we arrive at what is more
intelligible for our imperfect intellects . The only kind of mind that
can be realized in a being composed of matter and form is one which
must acquire its knowledge through, experience, and which must, there-
fore, begin with pure noetic potentiality, a tabula rasa , and move
on gradually to greater and greater noetic actuality. That is ■ why
things are more intelligible for us in the potential and confused _
state of their universality, than in the state of concretion, It is
much easier for us to understand what a living being is than to under -
stand what a cow is. We shall discuss this important point in considcr-
■abl'~detail in Chapter IV, but it was necessary to bring it-ouo here
because, as we shall see in a few moments, a^umb er ^J^Z ? d d ow^t " '
^m-.» littin, in the abstract the distinction s we have laid do wn j
fallowed themselves to arrive at erroneous c^^' 1 *^* the
nature of science because of a confusion between the two kinds of
(intelligibility^ have just mentioned^

It should be clear from what has been said why the degrees
of abstraction which specify the scien <™. ^ "^r^'wi

F 55 * 8 the[ ^^^S^^^^isXllesl for us as our
to be made ^SrajTW^onsxder^ ^ betwQen positive

analysis proceeds. W ^ TO " a £^ aing total and formal abstraction
and negative abstraction, ^^^bsfraction. Negative abstraction
we have been dealing with jpgv£jy^^__ ^ ^^ science

toS^t^^^cesS/to deserve its nature brief^.

IS SOI

is extensive.

-83-

Ther.o are two distinct types of nagative abstraction.
The first type is called negative because it does not achieve a noetical
separation in the strict sense of the word. Just as a sense can pick
out a certain quality existing in an object, o,g. the color, and leave
aside all the other qualities coexisting with it, so the mind when
confronted by a plurality of formalities can concentrate its attention
on one of thou to the(n eglecr| )of all the others with which it is con-
nected. In thus concentrating its attention on one formality, the
Imin d. does not lift this formality out of its context , set it forth
IbyHLtsclf , and consider it formally as separated . | "5ence the term at
^ which it arrives remains tied to the context from which it has been
abstracted ^) That is why this type of abstraction does not achieve
even one complete and independent concept, and in this it differ s
from both formal and total abstractio n^ as is evident from what was
said above, Negative abstraction is like total abstraction in that
it arrives at a common notion, but it differs from it inthat this
cov.TKon notion is not considered in relation to its inferior s. (41)
It is like formal abstraction in that it lays hold of a certain form-
ality; tut it differs from it in that the separa tion is only negative,
and consequently it does not consider the formality f ormal ly <^g_sepa-
ratcd.

The second type of negative abstraction is that used
in lode (42) It gives rise to an object which, though related
to something in nature, does not have being in nature, but only in
the ramd, Positive abstraction always gives rise to an object thau
has boing in reality, even though, as in the case of mathematical
abstraction, the mind separates it from something to which it must
be united if it. is to have its being in reality. It is of great imr

Ld this s^S^S^iitiiejx^trsction, In mathemS.l abst race-

of t£ oSeot" whereas if negative abstraction the mind supplies the
very object. This type of negative abstraction P^ «? ^tant

am JO'.au.j- ui ,i^ ^nvmn-Kon of the univorsals that

mentioned above is employed m the formation oi ™ univ ersals

ion, • .

V ■ ' r.

„^^-;-Hnn tn -Dursue otir discussion oi
¥ e are now in a posit ^'° ^£ ht out vrith admirable

the dog^es of fcr^l f^^J^Z^ on the De Trinitate of

f ™- by Sain, .noma ^^i-^aTSo^ThSTpi-mphrafle his

Bofcr.hr.is, )^>f^^£e three kinds of mtter, ar.d_ consequently
treaonenx ci them., mere are ± h th0 ., lind liftg lts

three distinct levels m the pioccs& uy

-84-

scientific object out of the potentiality in which it is concretized.
First there is individual natter, i.e. the matter Y/hich sets individual
things off from each other with all the particular individualizing
characteristics proper to each. As long as these individualizing cha-
rac teristics are retained no science is possibl e , for: ornne individuum
j.ngf fabile , The reason is that a thing is intelligible only to the
extent to which it is in act. Matter is being in potency and every-
thing that is dependent upon it essentially and inseparable from it
is not intelligible in act. Hence it is from the knower that intel -
ligibility in act must come . Consequently the first step in the process
of scientific abstraction is to slough off these particular charac-
teristics and by so doing arrive at a specific intelligible essence
This first step is called physical abstraction . and it is used by
all the disciplines which study nature. The second kind of matter
is known as common sensible matter . By sensible matter is meant mat-
ter that is apprehensible by the senses , and hence something that
involves material qualities. This common sensible matter remains un-
touched hy the first degree of abstraction, for the biologist, for
example, studies flesh and blood, even though he does not study this
particular flesh and blood, the flesh and blood of Socrates, for exam-
ple^ (44) The second step in scientific abstraction consists in^
disengaging an intelligible form from this sensible matter. This is
known as mathematical abstractio n, for it is the abstraction employed,
by the mathematical sciences. In spite of its high degree of abstraction,
mathematics does not succeed in freeing itself of all matter, for
whatever is quantitative is necessarily material. But the matter which
it retains though apprehensible by the i ntellect is no longer agjgrg-
hensible by the senses, since all sensible qualities have been refined
lawayJ Hence it is kno'wn as in telligible matter . The last step in our
intellectual purification succeeds in freeing the scientific object
of this last vestige of matter and in setting it forth in its pure
intelligibility. This is known as metaphysical abstraction.

There is another qndj»rejorofauna way of presenting
these three degrees of abstraction: Some scien ti: fie °^°ts depend
upon sensible matter both for their being and for theijbeing known ,
that is to say, both for their objective existence outside the mmd
and for their subjective existence in the mind. As a conse quenc^
«"* «m neither exist nor he ^co nce^QLiS^^^L^^
of sensibiel^tter^ These con^tito^ the obpe^ of the ^oiplinoB
wHcTitudy-^tu^ All of the natural sciences study the ^ial_
cosmos in Its state of concretion in se "«*^ '^£ r fJ^ B *Z
it precisely from the point of view o^ggg ^t^^Sle
say, laUJ*sisfln^^

E^rJOnT^b^te mpted to °°^ ^lities invested in natter,
natter means, as we have s ^> ^?f ^ alitative determinations and
and physics seems to prescind from all quanta

-85«

to study the universe only in terras of the category of quantity.
The answer to this objection is, of course, that modern physics
is mathematical physics, and consequently not a pure natural science .
Other scientific objects depend upon sensible natter for their being,
but not for their "being known". That is to say, in order for them '
to exist outside the mind in the world of reality they must be in-
vested in sensible matter. But they are conceived and defined inde -
pendent ly of it * The notions of line, triangle, number three, etc,
contain no sensible matter, nor are they defined in terms of it}
yet if they are to exist at all in the objective world, they must
be concretized in it. These form the objects of the mathematical
sciences. Still other scientific objects, depend upon sensible catter
neither for their being, nor for their "being known % Not only are
they conceived and defined independently of all matter, but they
can exist in objective reality independently of all matter, either
because they necessarily do not exist in matte r, as for example

God and the Angels, or because they do not necessarily exist in

matter, ( as the concepts of substance, q uality, act and potency, etc ^
Here we have the objects of metaphysical science , (45)

St, Thomas points out that this threefold division
lis exhaustive. For the only other possible case that might be imagined
would bo that of objects that would be independent of sensible nat-
ter in their objective existence, but dependent upon it for_ their
Uubjective existence in the mind. Though completely immaterial in
their being, they would have to be materialized in order to be con-,
ceived and defined by the intellect. The inadmissibility of such
a case is evident, since it implies that the intellect is essentially
material and supposes the primacy of matter.. Moreover such a process
of materialization would be just the opposite of abstraction .

It is necessary to point out here in passing something
that will bo of consider^ significance for us ^-/ven * casual
consideration of the three degrees .of ^str^tion teings to ^gg
the fact that there is something peculiar about the uype fff™°>
ion used in the mathematical sciences. In it al one do^ we f i nd a

Sr^tion therf is^a correspondence be tween the way ^^
objectively and the way they arc [^^^^2^^'
In order for mthematical obje cts to e xist ^.^ ^

lect they must be immersed in ^g^^ £ lete inaepen aence
Vintellect they are conceived and f^^^s^^HoB) that
of it. Hence in this case abstraction iffi^^-i-^-^^ tM fl
is not found in th ^ther_degree,s. Later on ino^ ^ ^ ^^
dichotomy will throw a great deal of lifent up

-86-

r.iatical physics

This threefold level of formal abstraction provides .
us with the specifically different rationes formales sub quibus
tliat we set out to find. Yfe have three different grades of imraate-
riality, \thr ee different, ways of abstracting and defining the soient ~
ifio objefit.j In metaphysics everything is defined without relation"
to ivatter of any kind. In mathematics everything is defined in terms
of intelligible matter alone . In the study of nature everything
is defined in terms of sensible matter* Now these three ratione s
formales sub quibus in turn actualize and light up tlireespecifically
distinct rationas form ales quae : ( being ) in metaphysics; ^ juantilg ft
in mathematics ; ( mobility ) in the study of nature. The first ofthese
three objects is not of any special interest for our problem. We
shall remit the question of the second object to Chapter VT where
we shall discuss in some detail the nature .of .mathematical science.
Since we are particularly concerned v/ith physics, the scientific
object which has the greatest interest for us is the one that is born
of the first degree of abstraction. Thomists have traditionally
insisted that the proper object of the study of nature is ens mobile :
mobile being (46) For those who approach the question for the
first tiiTB it is noi>Wediately evident perhaps why this should
be so. There are a number of other ways of expressing the object
studied by natural science which would seem to suggest themselves
more spontaneously than "mobile being" } such as:' "natural body",
"natural substance", "sensible being", "physical body" , "natural bexng ,
etc. In fact some modern Thomis ts, have seen fit to substitute
"sensibl e being" for _bhe_t raditional "mobile b eing" . (47) This
action has been studiedldth great prof oundity and acuteness by
Cajetan and John of St. Thomas, (49) and though it would be too
long and tedious to summarize all of their arguments, there are
a few points which must be insisted upon with special emphasis.
The reason why mobilitas is taken aajhejorml^bjest of the study
of nature is thaTbette? than any other point of view that might
be selected it opens up the inner essence of natural things. In
De selected., xz opens uy mobility that nature can be
other words, it is g r-l y m Germs oi muuj-xx .j ^_^___^.

,ni W ) In his G^^^™Sm2±Zi> St. Thomas suggest an
intrinsic reason:

De his vero quae dependent a materia non solum
i 1 La P tian aeauitamrationem est Waturalis,
secundum esse l^^-SSSg--^-^^ materiam
quae Physica dicxtur, Eu quia o, * it subiectum
mobile est, consequens ^^.fl^rpMlosophia de
naturalis philosophiae . Naturaxis onu *

-87-

i naturalibus est; naturalis autem jaunt quorum principium
est natura;) natura autem est principium motus et quietis
in oo in quo est ; de his igitui- quae habent in so princi-
pium motua, est sciencia naturalis, (50)

The expression "sensible being" which some modern
Thomiats have attempted to substitute does not bring out the true
objective formality in terras of which nature .must be studied .
For, things in nature are not sensible for the separated substances, \A.fZ.
but only for usl) Hence "sensible" does not directly e xplain what == = _
things are in themselves, but only (How) they' are (known) by us . Of
course, every mobile being is at the same time a sensible being,
fo r there is an analytical connection between motion and sensible
matter in that both of them involve material potency , But sensibi-
lity does not explain the objective nature "of thing s;! it merely
explains how w e know them,, ) Mobility ,' on the other hand, is somethin g
objective ,! Even the se parated substances known natural things as
mobile bein gs,"; not, indeed, as we do., merely in terms of the general
formality of mobilit y, but in terms of the specific type of mobility
^proper to each ontol og ical species .

And just as no other word may be substituted for "mobile",
so no other expression can adequately take the place of "being":
not "substance", for that would exc l ude the consideration of accidents ;
not "body" , for as St, Thomas points out, (51) it pertains to the
science of physics to prove that all mobile beings are bodies , Cand
no science proves it s own subject .J John of St. Thomas clear]y indicates -
the positive reason why the expression must be "mobile being" : Motion ■
is rot defined in relation to substance or body, but in relation
to being , for it is: "actus (gntis) in potentia in quantum huiusmodi :

Fundamentum huius conclusions' sumitur ex his, quae
paulo ante sunt insinuata, quia videlicet (pj^pi2£_etjiteequata
passioT) quam physicus demonstrate de suo subiecto, est motus .
MolSrlutem non definitur explicando ordinem ad ■ corpus vel subs-
tantias, sed ad ens mobile; definitur omm, quod est actus
entis in potentia", ut patet in hoc tertio libro. Ergo foraalis
ratio subiecti physici non explicat rationem corporis. Nam quod
fomaliter est subiectum scientiae, explicatur etiam in fornix
aefinitione propriae passionis tamquam id, ad quod passio dicib
habitudinem. Ergo cum non explicetur in definitions motus ratio
naoixuaxnem, -t"B" notentia, non pertinet ad formalom rationem

corporis, sed ratio enuis m potem>j.a, x ^ ,,, '
■, . , . -n„ n i- -in r>p vorura sit, quod non sit; moDixe

-88-

extensurn, quod non esset corpus, adhuc videretur ab oculo, (52)

It is extremely important to insist upon the unityand indivisibi-
lity of the objuct of the study, of nature. The ca.rpooition fouad in the expres

I sion "cna i.iobile" is only yi.-rtbal . \It does not ii. i ply- a coupoaition
<5f _two objective formalities , the formality of being and the formality
o TT.mbilityg j Mobile being does not mean "being" with the addition of
a specific difference: "mobile". If this were true, ( philosophy of
nat ure would be a part of metaphysics or at least a science subaltern -

^atccTto it ^Both Oajotan- (53) and John of St, Thomas (M) lay
considerable stress upon this point, and we shall see its importance
in a few moments.

The assigning of "mobile being" as the object of the
science of nature gives rise to a difficulty, the solution of which
will enable us to penetrate more deeply into the nature of physical
science. Wo saia above that science is possible only in so far as
its object is lifted above the flux of change, for science is about
necessary and not contingent things. The etymological root of the
word episteme means firmness and stability. Consequently a science
of mobile being would seem to be a contradiction in terms.

, .„ de permutantc, idest de go quod movetur,,,
nihil verum dicitur inquahtum mutatur . Quod enim mutator
de albedine in nigredinem, non est album nee nigrum in-
quantum mutatur. Et ideo si natura rerun sensibilium semper
perrjutatur, et omnino, idest quantum ad omnia, ita quod
nihil in-ea est fixum, non est aliquid determmaca verum
diccre de ipsa, (55)

| In raising this question we are touching upon one of

the most persistent antinomies in the whole history of philosophy.

to reconcile the fluidity of naoue, clearly r ™^ * litu s Par-

... ,, .+,. ^ ar ,-ipnre. In the. doctrines Oi Iieiaciitus, rai
with the necessity ot science, ±u , ., .„ n „= „„<.„,,„ WPre i n
menides, Plato and their followers phi osophy and na ^ejere -

menides, Plato and tneir *££»— ^tLnes> J 7MjhjJ J .sophy_Jl

some measure y^""^ ^T^Z^^^t-m
suffered from the conflic t, ana at, uw> ___ __, — r^ VJ .

took the genius of Aristotl

to a philosop h y of nature^

are in a constant state of ij-ua, ^ J=""";"~ the is in nature a

in the midst of this fluidi *o ^ — ^ d ^lay hold of fcruu*

permanent, general s true cure ™

took the genius of A ristotle ^^^^^n£^t^il things
tg_gjhilosopJiZ_ofna^r|^_J^ZJ genera tion and corruption. Yet
are in a constant stace ot txux, oi & nature a

ioormanen-c, generax o»^ n v,nw

the process of abstra cti^njescrib^a^abOTe

,. , 1l]t ,ii itor cognosci possunt, Uno modo
■ Gontingentia du P ix °^°\_ io \ Q&0 seoU ndum quod in par.
secundum rationes umversales, alio

-89-

ticulari, Universalos quidcm igitur rationes contingentium
immutabiles sunt, ct secundum hoc do his demons trationcs dan-
tur et ad scientias demonstratives pertinet eorum cognitio,
Non enira scicntia naturalis solum est de rebus necessariis

(et incorruptibilibus, sed etiam de rebus corruptibilibus et
contingentibus „ Unde patet quod contingentia sic considerata
ad candeu partem animae intollectivao pertinent ad q uam et
nccossaria , quaia Philosophus vocat hie scientifioum. (57)

It is not necessary to transcend nature in order to
fin d imrau table types, Basic regulations in tho stream of phenomena
reveal tho fact that there are immutable types immanent in nature
itsel f^ It is only in their individual composite existenc e , not
in their universal essences that the things of nature are fluid.
As Aristotle points out in the eighth book of the Metaphysics , it
is only an individual house that is brought into existence, not
t the nature of house in general . In like manner when an individual .

man dies, the definition of man does not perish o "Etsi enim ista
sonsibilia corruptibiiia sint in particulari, in universali tamen
quamdam sempiternitatem habent ," ' (58)'. It is in this way that_
definitions of natural things are possible , (and wherever definitions
are possible, science is possible j ) These definitions gxve the
universal essences that are concretized in nature, shorn of their
individual matter (materia signata, in Thoraistic terminology) but
not of common .sensi ble matter ( materia non signata ) . Hence as we
have already pointed out, it is not a question of abstracting a
substantial form from its corresponding matter, for a form thus
abstracted would have no meaning. Now as St, Thomas points out, (59;
these abstract essences can be considered in two ways: first, in
their abstract state in which they exist in the mmd alone, and
in this way they are withouj L mattgr_(inaividual) and motion; secondly,
in relation to the mobil^lSteriirihings outside the mind from
which they have been abstracted, and in this way they are t he medium

b y which physical realiJ 3Li^JggI&? for things * re ^"T ^.f^ 3
oflteir~f^rZ~TS[s^Jlt^^ to have a science of mobile

being.

Nevertheless, it is important to point out that the

mobility of trthfngs^ich ^£^££^-1=

^inOo_ J2 ress_i|s_obJe^^^

todp, jh^ necessit ystorosto^a^Tha y cq ±ra£

to_do, the necessiystaros_^^ ■ Frea ter concretion, true

nature follows ^f^^^^S^^^^mi^^^BS

-90-

ledge, for reasons which will become apparent later.

In connection with the type of necessity found in
the study of nature the following lines of St. Thorns are signifi-
cant:

Modus autera demonstrationis est. divcrsus; quia quae-
dam demons trant l^gis_nocej,sarie f __siout raat heraaticae scientia e,
quaedara 'vero infirmius', idest nbn de necessitate; sicuTTscien-
tiae naturales, in quibus multae demonstrations sumuntur ex
his quae non scraper insunt, sed frequenter, (60)

Almost instinctively the "doxa" vri.ll attempt to erect
itself into an "episteme" ; the " modus infirmior demonstrandi " will
rea 9jL2HjLjri 3 i-^PE~i^9..fl-Hl £S,,^uro type of dem£nsjtration,(jbhe
scTence of "nature vri.ll _gegk_.to rid "it self ~oF"~the mobility to which
it is a prey ,,) And fehaj is why physics .vrijjf inevitably become^ jaathe-
naticalo ~™~ "~"

And now we are in a position to see how the degrees
of formal abstraction give us throe levels of immobility as well
as three levels of immateriality,,. The science of nature has to do
with objects which in their existence in reality are mobile,, and
which in their existence in the mind are from one point of view
mobile and from another immobile: mobile in the sense that they
are conceived of as mobile; immobile in the sense that they are
conceived in an 'immobile way %_^rtuo^f m a^str^%^xj^o^mA.rer-' _
sality_) Mathematical science deals with objects which have mobility
in their objective existence, but. absolute immobility in intellect.
Metaphysical science considers objects which are absolutely immobile
in both their objective and subjective existence.

In order to. round out our consideration of the hierarchy
of speculative science it is important to see the connection this
hierarchy has with both an objective stratification m the structure
of physical reality, (61) and a subjective stratification in ,he
cognitive powers, 62) Physical reality is constructed m such
a way that in it substance has a natural priority over ohe accidents
which inhere in it and determine it, But even among the accidents
quantity has a natural priority over the sensible qualities. Quantity
is, in fact, the first accident; of all the accidents it is the

closest to substance, for it is quantity which. .S^s tne^arg

of Material substanc<Sd gives it 4?*^3?J^e°g^^5g-

the substance, For example, a body can be determined by^a certain
this color, Hence sensible qualities ml,

-91-

thc__substanoo, ^!^.in_the_jiuantUy„ Only through it are they rooted
in the substoJicQo^Becauseof its closeness tojthe substance, quantity
•possesses a source of inteTliKiBiirty which" the" other "accidents ''
d o not have jjBvit at the same time it must be pointed out that from
another point of view it has less intelligibility than the, sensible
qualities, for these JLatter f o^ow^upj^jhe sub^-^antialj'|orm) _where ~

^^^^^LS^^^^^^^^^ 7 ^^^' ^ ^^ ^^^' ' fco 'fchi 3 paradox
later," for :V:-. has an important part to play in the solution of our,
problem,,

a
It

We find then, in the structure of physical reality
definite stratification ^substance, quantity,' sensible ...qualities ? J
xi is possible for the nrind to consider the essential determinations
of reality independently of any relation to its ' quantitative and
qualitative determinations. It is likewise possible for the mind
to consider reality in terns of its quantitative determinations
wi tl'.out .any rela tio n to its qualitative. relations. But the reverse
oi this process is not possible, Ji^.s _MTpossible ,_ ..for_jxangle, ^

(to conceive of quantity without substance ? )f : or quantity i s precisely
th e"order~o'f The" parts" of" the ' subs tance ; []an d order can not_be_con-
ceived of withou.t_the_^art£3At first glance this "point may seem
to TjTin~ conflict with what was said above about the nature of formal
abstraction, It was pointed out that total and formal abstraction
differ in that the latter results in two independent concepts. And
we added by way of oxamole that just as the concept of quantity
is independent of sensible natter , so the concept of sensible matter
is independent of quantity. But from what has just been said it
vailA seem that the concept of sensible natter cannot be independent
of the concept of quantity, The solution of this apparent | conflict
lies in this that there are Wo kindsof quantity: atos|raS3-55S2&-
matical quantity, and.concr^e_ m an,tity. The notion of sensible

.mtter is independent of the former, though no „ of the latter.

This distinction between abstract and concrete quantity

is of great importance for the question of ^^^ ± .^Tal-
n- ^ • -i,i „ j-^ i„v hold of the concrete quantitative ae
S-iJiJL'SSg^^a kind of native fraction
the roa.d is open to a^onfusion^n thi W^— g^
qr,_ntiiy and the way u xs conaido ^ Ag a lm . fcter of faot ,

i^^^.^^^^^£f^ aor ^ s ion > as we shall point
nrae authors have fallen into tnis cumu , . di „
out in a few moments. The consequences of this confusion c
sa 2 t:ro,, 3o For if ^^hemtical physics consisted og tur ^ b
of the concrete quantitative aeteminations existgg *

f ans of negative frt^^jg^^&rZt^
t J|- Z^^M to a r ata apart, to return to the physical world

-92-.

later with^a.ra^nality JN^damentally alien to_it, yet in a rayste-
rious way capable of being applied to " it'.™Thc mind would remin
enclosed within the physical world. This would change the whole
epistcmological character of mathematical physics.

Now the relation between this stratification and the
hierarchy of speculative science does not consist in this that na-
tural science studies the sensible qualities alone, mathematics
the concrete quantity as it is found in nature, and metaphysics
the substance of reality without any consideration of the accidents.
All three of thes e statements are false,. Rather, the connection
between the two hierarchies iSlai be' expressed in this way: because
of the logical priority existing in the objective structure of the
universe, it is "possible for the mind in its attempt to lay hold
of reality scientifically to take tharee specifically distinct steps:
first to prescjiid only from the individual characteristics and to
consider reality in terms of all its concrete determinations, in-
cluding the qualitative determinations of sensible matter; secondly
to prescind from all sensible qualities and to consider reality
in terms of its quantitative determinations alone (but here it must
be noted again that it is not concrete quantity that is being con-
sidered, but abstract quantity, for concrete . quantity is precisely
quantity concretized in sensible matter ~ here we have a key to
the Paradox just mentioned about the greater and lesser degree of
intelligibility possessed by quantity); thirdly, to prescind from
all matter and to consider being as such.

The hierarchy of speculative science also has an es-
sential connection with a hierarchy of cognitive powers. All know-
ledge begins in the external senses, but not all the knowledge _ term-
inates there. Likewise alltfche sciences considered £ro© the .point
of view of their origin have some kind of relation to the external
mmm, T^^side^Tfrom the, joint,, of .view. of_thgir, term, some
sciences are independent^ the external senses, and bear an essen-
tial relation to some other cognitive power. For example, our know-
ledge of God depends upon the external senses for its origin, since
the only way we can get to know God is through the nnoerial things
in the woria about us. But it does not terminate there, that is to
say, ir our conclusions about the nature of God we do not judge
that He is like -ho sensible things in the material cosmos,

the basis of St. Thorns' doctrine that natural

T'Vy : .'

Tir- q is trie uas-Ltj w »-»"• j -** . . n

Bcionc. ^rm^aies in the external senses , ™«f ^^^lone,
in _the Agination, ^^^^f^^JiM^^M^

m the i_>.--:err-an t-jcn.ses is clear. ^^ u^ ,.„„-». .^ Thomas

ions o:,c r.oco.aarily. .in3erms.of_ S ens.ible J; iatter, As St, Thomas

-93-

^

puts it, " qui _ sensuTii negligit in ^turalibusrlnciait in orrorem u ,(t63))
Hence all of its judgements' must "be verifiable" in "sensible" eii$erience^
It is to be noted that wo say " verifiabl e" and not "verified" in
sensible experience, for as we shall see later, it is only that
part of natural doctrine which is purely dialectical that Bust neces-
sarily be verified in sensible experience. We shall discuss this
question of the relation between the study of nature and sense
experience in Chapter IV.

The connection between mathematics and the imagination
is not so immediately evident perhaps. Since we have the intention
of considering this problem in some detail in Chapter VI we shall
content ourselves here with merely indicating the basis, of the con-
nection,. In the first place it is fairly clear that mathematical
science does not terminate in the external senses It is independent
of sensible matter in its conceptions and definitions » No mathema-
tician has ever seen in the world of sense a straight line, a per-
fect circle, or a line touching a sphere at only one pointo (64) . y
But that does not affect his science in any way* Yet, while inde-
pendent of sensible matter the mathematician still retains intel-

iligible matter, and it is because of this in telligible.,ipat.t.er., that.

| his sqience.TaistlterminatS^ For intelligible

! mtte'r signif ies(homogeneous exteriority) that is to say, the.J.iul-

I tiplication of th£;spr f°rR3Wou^

l \h6mogehertyo""This exteriority and multiplicity demands some kind
oFindivTduation, and it is precisely the i magination that Provides
this in dividuation rwhic h in physica-L tm ngs is provided by the jmat-
torOOf itself, the intellect has to do with pure form, separated
fro^matter. Hence if the intellect alone functioned in mathematics
we could not have the notion of homogeneous nult agjLgxjy. At iirst
glScTthis may seliTto give Hie to a serl °^ lffxcu n 1 ^ i f^ in _
is certain that God knows mathematics, and yet He is without imagxn-
uxtionl The difficulty vanishes, however,, when we take into account
the-vit difference between the human and the ^f^j^* t
Man's knowledge is posterior to things and his intellect is dependent
upon them and^asuLd^by them. All of his »^£^ P^^JL
are drawn from concrete material things. Con f ^f^^^f ^ bs _
are lifted out of concrete natter, there must be something to subs
titute for the individuation which *is matter provides. But God s

knowledge is prior to things, and His int f l^^^tion to pro-
by them! That is why He does not have need of imagination to pro

vide for individuation.

Th c connection ^^^etaphysical *f~* a f J^
lect is quite clear. We may arrive at the nfl£°£ exfern al senses
by means of material things g^^™ f^ge that Material
and the imagination. But m trie onu

-94-

things are like material things.

This point of Thomistic doctrine must be rightly under-
j stood if confusion is to be avoided. Even though only the study of
nature terminates in the senses in the way in which we have explained,
nil scie nce of reality rjust retain an CessontiaT) conne ction with t he
del iverances of the senses if it is to have any ^vaUdTty . That Is
to say, it must be able to bo resolved back to the sense experiences
^from which it took its rise, For abstraction does not consist in
burning bridges behind one. And this is true even of metaphysics,
as St, Thomas explains in the following lines:

Sed quia primum principiura nostrae cognitionis est
sensus oportet ad sensum quodaraodo re solvere omnia de quibus •
judicamus ; unde Philosophus dicit in III Caeli et Mundi quod
complimentum artis et naturae est res sensibilis yisibilis ex
qua debenus de aliis judicare; et similiter dicit iri VI Sthi-
corum (cap. VIII in fin,) quod sensus sunt extremi sicut in- •
tellectus principiorum; extrema appellans ilia in quae fit
resolutio judicantis, (65)

Taken in this sense , the principle of logic aljaosiiLiyism that no-
thing has mean ing except in the measure in which it is _capable_gf
verific ation in sense ex peri ence is quite acceptable /"and is ac tua l ly
realized ful l y in metaphysics ,") in spite of the violent opposition
Vto metaphysics on the part of the logical positivists.

Our discussion of the specification of the sciences
would not be adequate if we did not touch at least briefly upon
another point which emerges from a reading of the Phages in which
St, Thorns treats the problem. John of St. Thomas (66) calls our ,
attention to the fact that there are a number of texts in jhich
Aquinas seems to use other criteria for the distinction and speci-
fication of speculative science than the one .upon^^ have
based our entire discussion, namely the three degrees °f g™^
abstraction. Sometimes he finds the distinction upon a differ ence

ployed in th e ^^^^^^^iT^^loi^fTtrn^s
tifio. demonstrati on ; ( 68TWith «g£ of view are r a u _

gees on to show how all of ™°*V a * is £ 1Qrely rilaking explicit
cible to tho same thing. In Joxngso ne mntary on the

what is found in St. Thomas himself , tor in

tetaphysics (69) the ^^.SaS^XSedgo is precisely
already clearly indicated . Smco scien ^ ^^ ^

knowledge arrived at by demon *f ^'^n bG specifically different'
specifically different sciences theic via. *

-95-

•fcypGS of media used in the demonstrations by which they arrive at
their conclusions. Not; these media are the premises employed in
the scientific syllogism. These premises in turn are necessarily
definitions, and hence a specific difference of media reduces it -
s elf to a specific d i fference of definition . But a specifically
different type of definition can be had only by means of a speci-
fically different type of formal abstraction. Since immateriality
is the source of intelligibility, a specifically distinct level
of immateriality is at the root of the specifically distinct ways
the mind has of rendering reality intelligible, i e, of laying
hold of its essenoe, of setting forth its "quod quid est". But to
set forth the q uod quid est of a thing is to define. Hence the source
of the unity and. distinction of the sciences is the specific types
of immateriality. These types of immateriality result in different
types of definition. And this difference in definition gives rise
to a specific difference in the principles and media used -in scien-
Itific demonstrations. The difference in immateriality or intelligible^
lipht^|ound in the principles ) are communicated by means of the de~
Vnonstration to the scientific conclusions,

In introducing this question of the distinction of _
the speculative sciences, we said that we would adopt as our guide
the treatment of the problem given by John of St,. Thomas: At the
same time we noted that this treatment is merely a summary of the
doctrine of St, Thomas and Aristotle, and that it in no way adds
to it or modifies it in any respect. Perhaps the numerous references
of St, Thorns and Aristotle adduced in our discussion of the quest-
ion suffice to establish the truth of this assertion. But ^cause
the issue Is of some moment for our study, and because some contemp-
orary Thonists have thrown doubt upon it, we consider it worth while
to stop for a moment to consider the problem explicitly. (70)

It has been maintained that the doctrine of the three
degrees of formal abstraction taught by Caoetan and John of St.
Tho's is not found in St, Thomas himself. Aquinas, we are told,
taught that only r^thematical ^acti- xs J^^S^™.
and that thesjtudj_^fjiatorej^^

substance to this clai . In ^ f°™ of Boethiue, he seems to say
of his ^omnsnta^ostg DeTr^|t^_ ^ abgtraction of
that only in mathematical science ao w abgtr action found
a form from matter. And ^^^Sota abstractio non di-
in the study of nature, he adds^ J^i^^^^^arMculari^)
QiturCfo rmae a 1 mteri a - absplutej secn^n^^_______^ erent

In the next article, he explains WM speculative sciences

kinds of intellectual operation ^ ^ of nature is had "se-
and that the one that is proper to the stuay

-96-

Icundun oppositionem universalis a particulari, et haec competit
etiom physicae, et est communis omnibus scientiis, quia in ornni
scicntia praetermittitur quoa est per accidens, et accipitur quod
est per se„"

It is obvious that these texts must be interpreted
in the light of St. Thomas' general doctrine. And in the first place
it must be noted that if there is no formal abstraction of any kind
in the study of nature, it cannot be a science, for without formal
abstraction it cannot have a r atio formalis . Consequentl y, to hold
that St, Thomas and Aristotle in no way associated formal abstract-
io n with the study of n aturej^is_ equivalent to saying that for them
natural doctrine was not a true~"scienc e ) — which is patently absurd ,,
Moreover, there is a special reason why St. Thomas associates total
abstraction with the study of nature, for it is only in the things
of nature that there are individuals wh ich are not specie s, and
consequently it is only in natural doctrine ' that it is necessary ■
to begin by abstracting from individuals in order to get at the
object of science.

Many of those who deny formal abstraction to the study
of nature admit it for metaphysics. This admission should lead
them to recognize the fact that' when 'St. Thomas says that formal
abstraction is found only in mathematical science he is taking the
terra in a very special sense. As a matter of fact it is only mathe-
matics which considers forms that are separated from the sensible
matter to which they must be united if they are to exist. In other
words, there is formal abstraction in all of the three species of
speculative science, but over and above this there is in mathema-
tics a particular kind' of formal abstraction. The proper nature
of this type of abstraction will be analyzed in detail in Chapter VI.
When St, Thomas seems to restrict formal abstraction to mathemat ics
he warns us how this should be interpreted for he says: • ••£■«*- „
dicta abstracts non dicitur formae a materia jaWute." ""J™ 6 ^
that in the essences whioh constitute the object of .the * Judy of
nature there is common natter as well as form, but it is illegitimate
to use this as , foundation for a denial of formal abstention in
natural doctrine, for St, Thomas points out in ^~^n rlllttln
that even material essences can be considered ^ ° ^^»«
to the indivic^LJEiiii^ M which they haVe be abstractedo

And now we feel that enough has been said to bring

£ S° &£?%£ SEE » S "™ »> to °°" ala "

-97-

briefly some observations made by a contemporary Scholastic on the
Aristotelian doctrine of physical and mathematical abstraction in
so far as it applies to the problem of mathematical physics. In an
article to which wo have already made reference Professor Mansion
of Louvain has this to say:

Notons enfin quo los determinations quantitatives ne
sont pas plus independantes de 1' experience concrete et de la
realite existante que los autres attributs, — d'ordre quali-
tatif — appartenant au raonde des corps. Elles presentent seu-
lement cot avantage que, isolees par 1' abstraction, elles se
pretent mieux, — merveilleusement mieux, — a. une elaboration
1 conceptuelle ulterieure: cette elaboration, oeuvre de raison
tout a fait remarquable, a donne naissance, en effet, a des
disciplines independantes, construites suivant une rigueur lo-
giquc inegale ; Si 1'on voulait soumettre a un traitement sem-
blable un concept tel que- celui de chaleur, j'enterids le con-
cept- repond'ant de fagon immediate dans l'abstrait a no tre sen-
sation de chaud, nos speculations s • arreieraient court avant
d'etre arrivees fort loin, Cette notion, en effet, parait re-
fractaire a toute analyse un peu pouBseej elle est inapte a
entrer telle quelle dans une systematisation plus devcloppee,
ou seraient determines ses rapports avec des objets connexes,
tcls que le froid, etc. Ge n'est pourtant pas que nous ayoris
affaire ici a un concept abstrait a un moindre degre, que la
" notion de nombre par exeraple; mis simplement que nous somes
en presence d'un concept de contend different, moms accessi-
ble a notre intelligence humaine dans ses conditions actuel-
lcso, (71) .

This uassage is filled with ambiguities and contradict-
ions, in the SOT^^^g^^-^^^^

SiSSiS.IS^So 1 ^ " S Saliti^i^e.
rience and of existing reality -ban the J-g^.^&^tions
It is obvious that we get to f™™ 03 * ^ncretion through concrete
only by grasping them in their a tat ooi airoot3y givGn
experience. I* is likewise obvious ^.^ v f de termin ^ io ns ,
in existing r&j.lity along with the quaiixa-oiv

In this sense Mansion is justified in remarking:

,: + Zip, notes caracteristiques de l'objet physi-
toutcs (los notes caia . p£vrt ie originaire-

que et celles de l'objet ^ thel ^ iq ^U aW perception glo-

Lnt d_hmm|me^o^^

bale, et dans TSqTIeT^n-roT^ouvc los

tivos au memo titre que los autres, UV

-98-

But at the same time there is a sense in which it is
true to soy that they are more independent of concrete experience
and existing reality than the qualitative determinations. Because
of the hierarchical structure of physical reality ,__(V5) we got
to know the quantitative determinations only^ by~means of ) the qua-
litative do terminations o This does (noT| involve a process of illation ,
of cours e. It merely, means that all the proper objects of the senses
ai\) qualitative determinations, and that it is only through them
that these quantitative determinations can be grasped at alio More-
over, even though these quantitative determinations never exist
objectively except in the state of concretion with sensible matter,
they are, as we have seen, conceptually independe nt of this sensible
matter in the sense that quantity is the first accident arid the
subject jjf_ all the other accidents . That is why they can be lifted
out of it~and elaborated into a world apart — a world of knowledge
which does not have to terminate in the world of existing reality
as presented by concrete experience, but merely in the intuitive
imagination. Does not all this involve an independence of both con-,
crete experience and existing reality in which the qualitative de-
terminations have no share? Does not Mansion himself admit this
independence when he states that once isolated by abstraction these
quantitative determinations can be elaborated into "des disciplines
independantes"? Nor is there any force in Mansion's argument when _
he claims that Aristotle contradicts himself by postulating a special
degree of abstraction for mathematics and at the same time admitting
that m^.hnr^t.iP.nl beings are T& Sg aea.ipeoe& >£,
t hat Is to say T^tr^ted from the~elSemble perceptible to the senses,
Which constitutes the physical object, (74) How else could mathe-
matical beings have a special degree of abstraction except by being
abstracted from the physical objects presented by the senses?

After pointing out that the quantitative determinations
in their state of abstractive isolation lend .themselves readily
to a re^kable conceptual elaboration, j^^^XSL

^iTiSLr^i^-^.^^ Srlbjelfive^ -
as we have S pointed out, is ^sed^preexse *^ ± ^*g* ^veen
intelligibility. Moreover, to a f a ° ra the ot Lr sensible
the way the concept of heat is abst ^oted from abstracte a

qualities, and the way the concept of straigh * aoBtrine

from sensible matter is to vitia.e the whole 1 u ^ of heat

of abstraction. For the proces s £ "Jf £ g ]lot necessari ly positive
from among the other sensible qualities i^ batraotion . Actually it
abstraction at all, to say nothing of forma ^ f ^ .^

is merely a kind of. negative £;£*°J^_^ thin _ else that is
attention on one point while .f ff °™ S it J ive abstraction, there
\ connected with it. And even if ^ were y »

-99-

would still be a vast difference between it and the type of abstract-
ion proper to mathematics. Enough has been said to show that quan-
tity is in se more "abstractable" than the sensible qualities. The
former can be conceived without the latter, but not vice versa,
v/e can get at the quod quid est of a straight line, for example,
and define it, but it is imposs ible to give a proper definition
of heat or any of the sensible qualities . Perhaps we should mention
here something that will be discussed in a later context: it is
possible for the student of nature to consider quantitative deter-
ninations of the cosmos, but in his consideration they will always
be united with sensible qualities and connected with mobility ; it
also pertains to the metaphysician to study quantity, but only in
so fa r as it _ia _a principle of being . Both of these ways of consider-
ing the quantity of nature are vastly different from the way it is
considered by the mathematician in the second degree of abstraction.
The central error of this whole section of Mansion' s essay seems
to be a confusion between the way of grasping quantity that is proper
to the mathematician and the other ways in which it may be. laid
hold of by the mind. This is evident in the following lines:

En s'en tenant a. ce point de vue, on serait done au-
torise a af firmer qu'il y a moyen d'abstraire et d'isoler —
par la pensee seule,. bien entendu, — tel groupe particulier
de qualites sensibles, appartonant a 1'objet physique global,
— le chaud et lo'froid, par exemple, -- aussi bien quel'en-
serible des determinations quantitatives. On aurait ainsi un
objet plus abstrait, parce que plus simple, que si l'on rete-
nait tous les groupes de qualites sensibles analogues: on n' au-
rait pas pour autant un deere d' abstraction caracteristxque,
raais une meme abstraction poussee un peu plus loin, dans un
certain sens, choisi d'ailleurs de facon arbitraire. (75;

Arising out of this initial confusion is the confusion
beteeen the concrete, quantitative determinations as they «aat.^
nature and the.abstract quantity that is ^^^^^^'
Professor Mansion seems to hold th at tne °" - ^°^ qPn cdbles
what is known in T homisJ^tojjunglog y as the °™^ s ^ ™>

\he objects so strongly to Anstotxe s uxbw.
and intelligible matter:

et constituer ainsi le point ae ^. t fona amentalcmcnt,

Get obdet (mathamtique) est^ * * ^ S llobjet physiqU e ,
perceptible par les sens, tout aucau H

-loo-

et do maniere aussi directe. (76)

It is the same reason that leads him to write:

II y a plus, et cctte particularite ne manque pas de
saveur: le mouveraent d'apres lui eat caracteristique de l'ob-
jet physique; l'objet matheraatique en fait abstraction. Or le
mouveraent est aussi range parni les sensibles comrauns, mais,
en out re, c'est par la perception- du mouvement, que nous avons
celle de toua les autres, notarament les determinations quanti-
tatives, que re tient soul le mathematicien (De Aniraa T«. 1„
425,als - 19) (77)

I The basis of these difficulties vanishes when one points

out that Aristotle never held that the common sensibles constitute

l^the object of mathematics. As for the question of movement, it is
sufficient to remark that it falls under the common sensibles onl y
indireotl y 3 Cb ecause of the extension of space covered by the move -
ment q ) Movement in itself, i„e the act of being in potency in- so
far as it is in potency, is not a common sensible. The student of
nature considers it, not as a common sensible, but in its intrinsic
nature o

And thus St. Thomas writes:

Motus secundum naturan suara non pertinet ad genus
quantitatis, sed participat aliquid de natura quantitatis aliunde,
secundum quod divisio motus sumitur ex divisione spatii vel ex
divisione mobilis : let ideo considerare motum non pertinet ad,
nathe];naticum, ) sod tamen prxncxp ia mathematica ad motum appli-
cari possunT : et ideo secundum hoc quod principia quantitatis
ad motura applicantur, naturalis considerare debet d e divisione,,
et cont inui, et mot us. ut patet in VI Physicorum. Et in scien-
tHsfmeairs In ter mathematicam et naturalenp tractatur de men -
s uris mot uuin, sicut JiTscientiis de sphaera mota, et in astro-
,logia (78) X>e. fri w -^JX.

The last remark of Mansion, quoted above has _ no Parti-
cular relevance, for in the place indicated in the De Anima Aristotle
nerely states that sensibles are perceived on^y through an unmitat-
Uon of the sense,,

We have devoted considerable attention to those dif-
ficulties proposed by Professor Mansion not only because they serve
as an excellent back-drop against which to bring # out an f°*™
focus the fundamental notions we have been laboring ^° f °f^
in this chapter, but also because if left unsolved they inevitably

-101-

g i.ve rise to an entirely faulty view of Thomistic philosophy of
science in general, and of mathematical physics in particular. As
a mttcr of fact, they have led Professor Mansion to the fundament-
ally erroneous view of mathematical physics already pointed out
eavlier in this chapter - - that of considering it not as an inter-
pretation of physical nature in terras of higher science, but merely
as a study of the concrete quantitative determinations existing
,j.n the cosmos „ He writes:

Car, remarquons-le bien, s'il est question ici do scien-
ce ou de physique mathematisee, ce n'est pas qu'on ait substi-
tue, dans l'objot d' experience brut, a. des attributs qualita-
tifs, apparaissant corame tels dans la sensation, des entites
geometriques ou purement iaathematiqu.es; ces sciences nc sont
encore mathenatisees que parce que on a fait entrei dans la
construction scientif ique du phenomene la me sure exacte de ce
qui est deja donne comme quantitatif ou quantifie dans I'objet
d' experience lui-raeme. La part d' hypotheses geometriques qui
s'y ajoutent, par exemple en astronomie, pour importante qu'el-
le soit dans la construction systematique de la science, n'a
qu'un role secondaire e't simplement instrumental dnas la deter-
mination des lois quantitatives - - de forme mathematique- -
regissant les phenomenes etudies. Et de plus, h. oe stade de
Involution des sciences, les hypotheses utilisees ne sont,
par aillours, pas encore heterogenees en donnees empiriqucs,
dont on cherche a forrauler les lois. (79)

We shall analyze the falsity of this position later*,

In the difficulties enumerated above Professor Mansion
finds the reason why, according to hian, Aristotle cut himself off
from the study of mathematics and of mathematical physics. ™
then he draws his conclusion that in Aristotelianism no true science
of nvathematical physics ^^YS^^Sor^tic^vj^ssi^. We have
referred to this conclusiolH^Chapter F^nd perhaps enough has
already been said to call its validity into question.

4. Ultimate Specif icatig ru_ .

, j. u „-p ^ P hierarchy of speculative science
The ahove sketch of the hierarchy won P etween fl

tall serve to draw a clear cut lino oi a ' h f these scien cos

and mathematics and at the same time bo local ^e ^n

in the general field of pledge .But it "^^ ^^ to

for

the general field of ^owledge M £ mthaBiat ioal physics to
a true understanding of the. nature

-102-

pi-eas this question of epistemological pluralism a bit further.
The three degrees of formal abstraction provide us with the basic
structure of speculative science. But it may be asked whether they
give us the absolutely ultimate specification of the sciences. Is
it not conceivable that in the general framework provided by a certain
degree of abstraction, a plurality of more specific formalities might
bo discovered which would serve as the basis for a sharper and more
.ultimate specification of the sciences? In this case, the degrees
of abstraction would be a genus containing within it a number of
scientific species. To the question posed in this general fashion
the Thomists have traditionally given an affirmative answer,, And
.John of St, Thorns provides us with the reason, (80) Because abs-
traction is a kind of process or movement, there are in it two points
to be considered: the point of departure and the terminal point*
This point of departure is the materiality that is sloughed off; ,
and corresponding to the three types of matter there are three levels
of abstraction. The terminal point is the particular grade of im-
materiality, the specific spiritual mode, the- special typo of in-
tel ligibility that an object is brought tofwhen it is once cut free
of a certain level of materia lity^) It is not the mere leaving behind
of a certain general type of materiality that gives us the ultimate
specific difference of the sciences, but the particular mode of
intelligibility that is arrived at. For it is possible within one_
and the sane degree of abstraction to have an intrinsic differentiat-
ion consisting in a greater or lesser approach to immateriality.
In other words, once the mind has performed the initial abstraction
which gets rid of a certain general level of materiality, it may~ ^
havo the freedom to move to different points of termi nal abstraction.^
ThuTail of mathematics has the same general degree of abstraction: _
the leaving behind of sensible matter, Yet Thomists agree that within
this degree of abstraction two specifically distinct sciences are
ifound: geometry, which deals with continuous quan tity, and arithmetic
/which deals with discrete quant ity.Uliof_thi^thir_b^nches_of
mthemtics are nnt.h^ fW thSr~^laboration s, or_ap pendag e|, orcom-
Siiat io^or di ^cT^c a^ j™^™? "

scien ces,) The reason w frThqTaro apooIHoIHy^tinot is that
^iu5meilc achieves a closer approach to immateriality than S^uetry.
This can be brought out both by a proof and by a sign. The proof
consists in this that continuous quantity has mo re sub 3 oc^ y_ity
ondjnorejotentiality than discrete quantity. ^f^™^^,,
xVin^aitT^rin^ally Batter/ whereas number isja^eo^g^

iality for division. It is true THat^BoForS^o^^ra^.

is Taoii. % added toJffiwM^iS^S-ffirfia^

determined. Continuous quantiuy is some aims
ined.

-103-

Aristotlo brings out the distinction between arithmetic
imd geometry in the Posterior A nalytics:

A science such as arithmetic, which is not a science
of properties qua inhering in a substratum, is more exact than
and prior to a science like harmonics, which is a science of
■p roperties inhering in a substratum ; and similarly a science

) like arithmetic, whioh is constituted of fewer basic elements,
is more exact than and_prior_to ^geometry, ( which req uires addit-
ional elements ,^ )What I mean by 'additional elements' is this:
a unit is substance without position, while a point is substance

.• with position J [the latter contains an additional element » \ (8l)__^

It is clear that the distinction laid down here by '

Aristotle is based upon the greater immateriality of arithmetic.
In fact, as St, Thomas explains in his commentary on this passage,
the contrast brought out by Aristotle between geometry and arith-
metic is a contrast between matter and form: "alii autem duo modi
accipiuntur secundum quod forma est certior materia, utpote quia
form est princip iun co gnoscendi materi am. " (82) facjn'> 1-1, "& S~~-

A sign of the more abstract character of arithmetic
is found in the fact that it is far less dependent upon the. imaginat-
ion than geometry, Vfe can imagine any kind of a thing as a phantasm

I for number, as long as there is ho mogeneous pluralit y; but not any
kind of thing represents a circle, for example. Another sign con-
sists in this that by extension number can be /applied to spiritual

\ beings, whereas continuous quantity cannot.

Geometry still has something of the qualitative clinging
to it, even if it be only a^ajie£bi£nofquantita^^
OB_figurop Speakini of this distinc^nTito?e^ngeom^try and arith-
metic, Duhera writes: ,

Parrni les sciences, l'arithmetique seule, avec l'_al-
gebre, son prolongement , est pure de toute notion empruntee
I la categorie de la qualitej seule, alio est °°£ !™ ^T
deal que Descartes propose a la ^^^^^Jl^^ '
Des la Peonetrie, 1' esprit se heurte a 1' element qualitatii,
lies la goonewie, i <^i-^ n „treinte a la consideration des

car cette science demeure 'si astreinto a ^ f .„4.^ Buer

figures qu'elle no iP eut exercer 1-entendement ^ fatiguer
beaucuup 1' imagination.' - - 'Le scrupule ^P 8 *^^
i • -,. a +«wnr>c flp l'arithmetique en la geome-onu,

anciens d'user des tomes do J- ar« * VO yaient pas assea

qu& ne pouvait proceder que de ce qu lis " £ lt / et d , en _
clairement leur rapport, causait te ^ooup otscur ite,

Ibarras dans la faoon dont ils *'%***^'J*&«6trto la
cot embarras disparaitront si l'on chasse w>

-104-

notion qualitative de fome, de figure, pour.n'y conserver que
la notion quantitative de distance, que los equations qui re-
liant les uiiea aux autres les distances mutuelles dcs divers
points etudieso (83)

John of St, Thomas makes the following clear cut dis-
tinction between the two;

Sod Mathcraatica considerat proportiones et mensurasy
quae in quantitate discreta et continua ita variantur, quod
/ ad diversa principia reducuntur et a d diversam abstra ctionem
let modun definiendi, quia nensuratio^per magnitudineig ) nullo
r.iodo convenit cum modo nensurandi panes^nuraeratxonem a Heac enim
abstractiori modo procedit. quia maenitudo mensurat per modun a^ (> t» J/
continentis , uf locus , Inumerus per inte lleoturj numer ando^) ■ (84) __ ,

In the Ars Logica (85) he points out that geometry not only has
greater dependence upon place but also upon time It is not too
clear just what this dependence upon time consists in, but in all
probability he is referring to the generation of the figures in
geometry »

A further indication of the greater materiality of
geometry is found in the fact that some modern authors erroneously
believe that, at least in certain aspects, it is more truly a phy-
sical science than a pure mathematical science,, (86)

Telle etait deja 1'idee de Gauss, 'Nous devons admet-
trc humblement, ecrivait-il a l'astronome Bessel, que, le nom-
bre est uniquement le produit de notre esprit, l'^space, meme-
au P oint~dTvI[e~dc" notre esprit, constitue une realxtea laquel-
le nous ne pouvons ajriori dieter corapletement ses loxs„
Dedekind, dans la piFefe ds son fameux opuscule sur ^nature
du nombre a vivement insiste sur oetto idee de 1' aut °nome de
l'arithmetique a 1'egard du reel, Le nombre est toe ^nation
immediate des lois pures de la pensee • et < e^^ *f ?^~
dant des concepts de temps et d'espace j les nomb res sent des
creations librcs de 1« esprit tain, Us ? e ™* ^.^ t f W
saisir plus aisement et avec plus de precxs ion la dx ve rsiue^
des choses' (Was sind und was sollen dxe Zahlen? 52 ed„ Bruns

wick 1923, p 8 ill... .,„„„n-H- one 'le nombre est la plus

Mais Locke, deja, aufioaxt quo le ™ ( .

simple et la plus univorsolle d« touto s nos x ^ ^ g . Q _
Philosophique, II, Ch, XVI, no. ^' ^™ et i. a i geb re au
metric comma moins assuree que 1' aritlins ,xq.u e *
point de vue do la valour apodictxque de se a^mtxon a.
(Psyehologie, tr. Renouvier et Fillon, J^ris , *

-105-

, , .11 mS o^nr l h ° ! en<3ral scion ^fio framework which

maws all raaotor out of consideration, Thomists distinguish three
specifically distinct sciences: metaphysics, logic, and supernatural .
theology, and once again the distinction is based upon different
nodes of irxiateriality. Supernatural theology is distinguished from
the other two in that it enjoys the highest grade of immateriality
that any speculative science can have — that provided by the light
of revelation. Logic is distinguished from metaphysics in that its -
a bstraction ls jaurgly negative, that is to say, since the object "
° f lo S ic is not anythins _real,( lt has only a negative InmaterialityT )

Thus far all Thomists are in agreement. But when the
question is raised about the possibility of a plurality of sciences
within the first degree of abstraction, the issue becomes highly
controversial. The problem is whether the study of nature is spe-
cificall y one, or only genericall y one, [In its concrete form it
reduces itself to the problem of the kind'of distinction existin g
b£ige en_philosoB hy_of .nature, and ex perimental science,; Since this
question is of considerable importance for our purpose, we must
endeavour to give it a rather exact analysis.

Speaking in a general vtoy, we may say that until re-
Icent years Thomists recognized no formal distinction between the
/philosophy of nature and what has come to be known as 'science" ,—
/ at least no distinction of such a nature as to give rise to two
^specific sciences,, And this is of considerable significance, for
if there is anything that the medieval Thomists took pains to do
it was to introduce formal distinctions wherever there was an y basis
C2E_them This v/as particularly true in the realm of knowledge, (87)
But some modern Thomists, notably M. Maritain , while recognizing — U-a fl^y*
| the absence of any formal distinction between the philosophy of ae 0< UMv-

nature and "science" in the wri tings of Aristotle and the medieval
Thomists, believe that this v/as a serious error on their part - -

an error due to "intellectual precipitation" and an unwarranted

Viopjtir.Tism'Q (88) They have consequently seen fit to reject this
point of Thomistic doctrine, and have gone to great pains to ela -
borate an ep jistemological theory which attempts to set of rHqje_ p hi-
losoph y of nature and ex pe rimental science as tyro formally jUsjdnot

sciences! (89) While commending the motive behind this elaboration

" -"that of attempting to integrate Thomistic philosophy with mo-
dern achievements, we feel that it has resulted in a theory that
is in conflict with basic Thomistic epistemological principles,
to nuat try to see why this is so, and why these principles must
b e retained if modern experimental science is to have its true ex-
planation.

In order to set the question in a clearer light, it

-106-

Y ,il1. bo necessary go ,. lake several distinctions. In the first place,
it is evident that there is a specific difference between philoso-
phy of nature and mathematical physics, For as we have already sug-
gested, mathematical physics does not fall completely under the
first degree of abstraction. It is (a hybrid sciencc ) whose formal
eJLcnont_lsj3 prrowed from the second degree of abstraction. Hence
it is formally distinct from science that is of a purely physical
character. The whole quest ion at issue is whether there can exist
a plural ity of (specifically) d istinct science a rwhich fall c ompletely
Yathin_i}hLXirst_degr^ e of abstract ion^ In the second place, we
do not deny that there is a profound ) epistemologica ljdifference
between philosophy of nature and experimental science , In fact,
we shall lay considerable stress upon this difference in Chapters
IV and V, But, it is not a question of a difference between two
specifically distinct sciences of nature,, in the strict sense in
which science signifies [universal and necessary jud gements, ) Rather,
it is a distinction between a science of natu re (philosophy of
nature) and a purely(dialectical cont inuation )_of _that scie nce
(experiment al science ), We shall try to make it clear later that
experimental science is not scienss ( in the strict sense just defined ^)
None of its^udgejiicuxbsZaxe_uniae£sixl and necessory ;( ^they never go
Beyond a greater j ar— lQsaer_degrea_ of— prijbability^j Ojj ly the f aotg ,
j of science have certain ty,) And we s ha ll see that the g r eatest of
nioa5rn~scientists and "philosophers~of~science are in agreement on
this point J In other words, the reasoning used in experimental science
proceeda ^roiu hypothetical promises ) to Cprobable conclusions! ) It is
for this reason that we shall call this type of knowledge dialec-
tical knowledge. And in the future when we speak of experimental
science it must be understood that we are taking the term science
in the broad sense in which it signifies purely dialect ical know-
ledge The ambiguity of the word easily gives rise to confusion,
and lost some may suspect that it is merely this ambiguity that
is at the hasis of the difference between Maritain' s position and
ours, m shall quote the following linos from Yves .Simon, who is
recognized as Ithe most authentic inte r preter of M, Mantain J In
explaining MaritainTs philosophy of the sciences he writes:

, Whenever the mind seises an essence, a j^ioent^,
(albeit in the blind way .p ro per to the pe ring etical intellect^
a genuinely ■ ao ±stM£i^^^!^SES^JSSSS^^^^^ raal
Hnd"l^elsir7-f^rF^rbelng, however obscure may be th °W
it is graspel, constitutes a matter to which the mind can apply
the principles of scientific thought, that is, causal and ex
V planatory schemes, (90)

Because of theCes^itiall^dialectical character of
all experimenlrefeie, ifiTSft&A that there is no_p 21 s 2 bility

-107-

of .^plurali^^speoj^^^ distinct scicnces<xn the strict sense
gjlno word^gxjjan jhg_first-gegree of abstraction. But we do h oT~
intend to argue from this poinf~oT~vtew- horo. Rat her, we have in
wind to approach the problem from an entirely different angle. Our
position is that (evon if expe rimen tal aoi enceCwr^ science in the
strict sense of the term it would riot b ecfflpgjETI jE) distinct fro m
pMJgjopj^^_gJljiature,(b ut united with it to fornron e_indivisible
soience_of n aturg , j On^he other hand, if mathematical physics were
scionce~in the scrict sense of the term, it would bo ( f ormally dist -
incfr)from the science of nature ,

We can test settle the issue by first considering it
in a positive way before taking up the arguments of M. Maritain
and his followers, John of St. Thomas whose doctrine M. Maritain _

generally professes to follow, has written a special article to
show that_a plurality of sci ences ^in the fi rst degree of abstract- Cu*f. r 1 ^
ion")is_i ncoiapatible with basic Thoraistxc epistemologic al p rin&rples ,^ > ■£_■ I , «. -Z

(91) The clarity of the article is admirable, and we can do no _ _L

tetter than to summarize its content. The study of nature covers '

a broad field; it includes a number of branches which extend to

a great variety of things. Yet a close consideration of this study

reveals the fact that all of these branches roust of necessit y foil

under (5ne _ indivisible science ; > For (prescinding from the difference

be Ween dialectical and truly scientific knowledge, which John of

St Thomas does not con sider) the only fu ndamental difference between

these various parts of I natural doctrin e) is the dif ference between

ggnoxalliaLaiid_oon.cjetenesa . \Thi s difference cannot constitute a

I'ornal distinction be Ween sciences. ) F or as St. Thomas points ou t

on innumerable occasions (92 ) e very science necessarily begins _ % ^

.Yathj gencrfilit.i.BRniicLprog ressea to. greater and greater concreteness ^)
We have already"lndicated the reason for this: the human rnxiidtbegins
\n/thj2otency ,and moves on slowly to greater actuality. And on these
iBramir5bIe~occasions St, Thomas makes it very clear that the va-
rious branches of natural doctrine do not constxtute a varxety 01
sciences but only a difference of greater or .lesser concretion,
John of St. Thomas wisely points out that if the deference between
generality and concretion we^suff^n^toj^^ \

of sciences. fit would be impossible for a specifically distinct

sciences ^

"OT^^iolSiSr^EHaTnS^TEilOOS^

.y move fvom somg_

j^T-^J^^jT tv to greater cono rete-

ness.

Consequently, eve^scdence^^
I genu, (T^osoamfie^Is^i^^^^SS^:^^^^^
sSm^BoT^aF^TfiSii-^oies not have the full liberty of^e.
UffKSl^distinot sciences, they do not even ^°J^f ^°*f
Vlib^tTifCS: subaltemated_scienc5D because the difference wnicn

Hr

-108-

they add to the generic study is not accidental and extrinsic, but
intrinsic and essential. (93). As we have pointed out above, scien-
ces arc distinguished by thc (essentially dif f orentf prJn^Iges) which
they employ, for each soiencejliasjori nciples that are proper t o ±U)
Each science presses on towards its goai in the light of theie~pro^
per principles, and consequently as it moves from generality towards
greater concretion it cannot suddenly change its principles at a
certain point along the way, It is true that from a purely materia l
point of view new principles may be added. In this~"sense each new
natural species tliat the student of nature discovers in experience
becomes a n ew scie ntific princip le for hin < and the <Spu3Fc& of new
truths^) ButTTfc is obvious that in this context we are taking scien-
tif ic principles from the formal point of view which is determined
by the modu s def iniendi that is characteristic of them. In this
"sense, the principles of a science cannot change. No matter how
nanyjaew species the student of nature may discover (the y must all
be CflefinecD in t erms of sensible mattor) and considered from the point
of view of the ratio mobilitatis, It is evident that if the advent
of new principles from the material point of view were sufficient
to give origin to new sciences, ) there would be as many sciences
as there are natural elements or speciesO

Just three things can happen to a science as it moves
from generality to greater concretion. First, it may retain its
chararrtor of -strict science all the way, and then no profound epis-
ternologicai change takes place at any- points This is what happens
: : n the cone of ( geometry ^ which begins with axioms and postulates ■
of great generality, and which in pursuing its ambition to derive
all the implications latent in these axioms and postulates, remains
a strict science throughout. Secondly s it mayfat a certain point j)
lose its' character as a strict science land issue into dialectica l
knowledge ,) In this case the dialectical knowledge is a necess ary
continuation of the science as it moves towards concretion. It uses
the s.-,ue -principles >\but no.t_J 1 n^U Gh_a_Hay as t o arrive at strict
demon-; '.-.rationu „) ObvicW]3TEEIi - 35e¥~not give rise to a plurality
oF^^ffffRlraiy; it may_cal l ^^he help _^f_anoujsidescienge
in such a way that the "two" constitute a >S £tentia_media, In this last
case wo have the only way in which other principles besides the
ones y^t science started out wit h can be introduced. We do no

s~5 e W r.y other -ooaBJMl E^^STbo addw3 f*' Le * ™ a ffl*S
genez-.t o^iderations to our specific problem of uhe study of

nature,,

I This study begins with the consideration of ^"f

Dein, ,0 its WdSt generalities: what ^^^3^^^
are thoQonstituients of ^J^^^^^S^iZ^

-109-

this point the study moves gradually towards greater concretion, and the

nther .i^tural— fcgfiafa^ea are devoted, to following ont. this moyeraerfp

\le do not soe how at any point new principles oanbe'si^^^nijvEro-
duccd to transform the science into a different science, unless
they be brought in ab extrinseco . But if they are brought in ab
extrinseco, [the y necessarily give rise to an intermediary science.)
This is what actually happens in the study of nature when mathematic s
is_applied. But in this case we have a hybrid science composed of
elements fromtwo de grees of abstraction ; ! we do(n ot )have a pluralit y
o f sciences ^in_^e_£ix3-t-4egreg_ of abstraotiorial It is true that
as the study of nature progresses it eventually ^ issues into ; a purely
dialectical type- of knowledge./ But this does not give us a new science „ ;
Ifthat_d ialectical knowled ge ^ could be suddenly transformed into
stricti y"~scientific knovfI'edg e-) it would merely constitute a conti-
nuation of the (pne?) science of nature) in its movement towards co n-
cretion. ,

The obvious objection at this point is: what about
mathematics in v/hich you have two specifically distinct sciences
within the same degree of abstraction. And the a nswer is not dif-
ficult to find: There is no science of quantity (as such ji as there ,
is a science of mobile being (as~su cHP In other words a^ggneral science
of mathematics does not exist, nor can it exist . \If it did, geom etry
and ar ithmetic would not be specifically distinct, for as we poi nted
out above, the science which deals with the genus deals also with" - "
the species that fall under it. In other words, mathematics is not
the study of quantity ffrom the point of view of its essence -) nor
are geometry and arithmetic studies~~bf continuous and discrete quan-
tityefrora the point ofview of _ their essence, ; The study of quantity \ ^ „

and its^ip^cigs fro inl£te3oSiraf-V-iew_oJLgFsence is distinctly J

a metaphysical consideration, I For it pertains to metaphysics t o

colore the nature of all tfe_oa3eggg^<H°Ei tie ?°i nt °j £^ci

oT their essenc es i.e. i n so ISFiTthey are Cp rincipljssofJe^ngJ

Tl.iri^cludei^eVthe-Fategories that ^^J^V^^SL

a contradiction of what we said above about mot a P hysics preac inding

from all matter, for metaphysics considers and ^f^es «*** cate

gories noljWthe^oint. of view of their ^^^g^-j^ _ b-l!^-

f ar as theTj ^rtoStoles of being7 )This explain^ ^^^esT kJ> '° '

c^n-slxyT^Diriu^^ —

taphysici considerare." (g^^TSTTaWIrTEhe same lectio he

tes;

Ticet ad considerationem prince philosophiae pertineant

ea quae sSttepta^ ■^.gJ&SSffi^^^
mot^UonJffimenjBolumj^ sed_etian L dc^e^io^i_ ^_a,

sun^e^tKpphlKsp^husjers^ K^) #//6t.

M-7

me
va^ites

-110-

Sopho h ;4r" Ud (96)" Ge0metria aCCiplt ^est'nagnltuao aphl- -flM*.

„. * + Th L Cas ?. of the stua y of nature is ontirely different"
from that of mathematics. And it will sharpen the issue to present
it m the form of a disjunction. Either there is a specific science
of mobile being as such, or there is not. If there is not a special
science, then under what science does the study of mobile being
V-fall? Certainly not metaphysics, for mobile being isj iot a categor y
or a princi pal of bein g/^as quantity is^ On the otheFhand, if there
is a science of mobile being as suchTthen, everyth ing that falls
under the formality of mobility C foom the broadest generali ty to
the ultima te concretion^ will pertain to the same scie nce. 0ne~cannot -
begin the study of mobile being in its generalities and then some-
where along the road to concretion suddenly shift to other principles,
A particular, concrete (j^gg of movement is a concretion of movem ent
in general . But continuous quantityis not a contraction of discrete
quantity or vice versa. (In this case) thore ~ is something- enlirely
newo

This clarification of the difference between mathematics
and the study of nature will help to bring out the ambiguity in
the following statement of Maritain:

, , la difference entre la philosophie do la nature
et les sciences des phenomenes, soit empiriometriques soit em-
pirioschematiques, apparait comme beaucoup plus accusee que
la difference entre l'arithmetique et la geometric, lesquelles
etaient pour les scolastiques deux sciences specif iquement dis-,
tinctes. (97) "Dty^i K $<■*«•* , •>•>, o^ir2 .

J. •

Several distinctions are necessary here. There is a
greater distinction between the philosophy of nature and experimental
science in the sense that the former is strictly scientific know-
ledge, while the latter is only dialectical; whereas . jnjbg tg) geometry
and arithmetic t here is strict lv_scigntifio knowledge. On the other
hand, however, there is a greater difference between geometry and
arithmetic in the sense that they are two formally distinct sciences,
Cgagh jossessi ng its ow n proper principles ) Of course in the case
of the scienceslMcFMaritain calls empj£iofetric>here is a deeper
dichotomy separating them from philosophy of nature because_of_the
feet that the y constitute a hybrid science ,

/ As a confirmation of his position, Maritain writes:

"Jean do Saint-Thomas distingue ainsi la Philosophic naturelle et-
la medecine," (98) It seems o3j E Dst_inoreai!2le that this argument

Should bo adduced, especially sl^TthT^rd^insi" refers directly

-in-

to the linos immediate^ preceding wherein Maritain explains his
distinction between philosophy and experimental science. For John
of Saint Thomas, while admitting a distinction between medicine

lend philosophy of nature (which in his terminology, included the
entire study of mobile being) explicitly and in so many words rejects
this distmotion( as an argument for a. plurality of science 's) of mo-

Vbj le b eing? And the reason for this rejection ultimately "boiTs~down
to this that medicine and the_study of nature ore formally distinct
because medicine is not aCspeculatiyg) soienc e ^like the study of
nahu^but^a<^agtica l> science . For "while they both have the same

§rial)objoct:Q bodyj ) they have a distinct formal object in that
ral doctrine considersboaies as; mobile) and medicine considers
ra^uurabJOp, Even though the act of~curing takes place by means
otion, medicine does not consider its object in terms of , the
ality of motion, but in terms of curability.

St Thomas brings this point out with great precision
in his Commentary on the De Trinitate:

Quamvis enim corpus sanabile sit corpus naturale, •
non tamen est subjectum medecinae, prout est sanabile a natura ,
( sed prout est sanabile per artem} .. Et sic relinquitur quod

physica secundum se, et secundum omnes partes eius est speou-

lativa, [ quamvis aliquae operativae subalternentur ei.) (99) jL J /, •

It is precisely because medicine is a practical science
that John of St, Thomas writes: "magis concretive procedit magisque
ad singularia et proxim accedit." - (100) And v/hile experimental
science actually proceeds in a more concrete way than philosophy
of nature, and comes closer to singulars, no parity can be established
between it and meddcine, becausc fevg n though ) as experimental scienc e
progresses( 3.t takes on more and more the character of practical
knowledge) as_w g shall see .\_it_r emains <a sse.ntia.Hy?a s peculative
science^) If is difficult to see how a distinction between a specu-
lative and a practical science can afford any argument to prove
the existence of a plurality of speculative sciences in the first
\degree of abstraction.

-D r.

But it is tine now to consider briefly the positive
arguments of M. Maritain. (101) The basis of his distinction bet-
ween philosophy of nature and experimental science seems to consist
in this: The object of the study of nature is sensible being - -
ens sensibile, (102) This object presents a dualistic or_bigolar
2i^acte77^d it is in this dualism or bipolarity which gives rise
to Wo-Tastly diverse ways of studying nature. *<>r " is .f^^
to study sensible being in such a way that the emphasis is placed
upon "being" , and when this is done you have philosophy of nature.

-112-

rt is likewise possible u0 study sensible being in such a way that
the emphasis is put upon "sensible", and then you are in the realms
of experimental science. Out of this ( difference of accentua tion^
arise two diverse conceptual schemes, ^Urojayerselmgdes of~deri~
notion. The philosopher of nature defines his concepts liTterma of
intelligible being, the experimental scientist in terms of sense
phenomena. The one employs dianoetical intellection, which consists
in penetrating to the essenee of things. The other uses perinoetical
intellection which consists in grasping the essence only in a blind
and remote way in the phe no menal reg ulari ties themselves . The one
fr esolves its concepts; in an ascending~analysis which goes up to
intelligible being. The other resolves its concepts in a descending
analysis which' goes down to the sensible , the phenomenal ,, Hence
the one moves from the visible to. the invisible. The other from
^the visible to the visible.

Professor Simon, with his usual clarity, has attempted
to give an exact and concrete explanation of Maritain's ascending
and descending analysis:

Let us try a rigorous ascertainment of the meaning
of a word found in both philosophical and in positive contexts ,-
The example chosen may be very simple. To the question what
does the word "man" mean ? the ansv/er will be 'rational animal';
now, none of the elements of this definition presents a character
of irreductible ^clarity. Take one of them, for instance, animal.
What does the word mean? A correct definition would be: "a living
body endowed with sense knowledge" , and there are so many terms
which badly need clarification. .Take one of them, for instance,
'living', I would say that a body is a living one when it moves
itself, when it is the active origin of its own development.
If we go any step farther, we go bjyondjhe, l^its_of physical
thought. In order to render "Widea of life clearer, we would
hive To define it as self -actuation, The concept of self -actuation
does not imply any reference to, tj^propEr_principles of cor-
ruptible and observable Vtia&iiitte.&wtevb&i^oop^J
Its 'eleme"nTs''a?e'-rdenti'ty Md-causality. Identity is the first
propSFEy "^bTing7'"GausaliW'can be . analyaed into potency and
act. Identity, potency and act are so many concepts directly
reducible to that of being, which is, in an absolute sense, .
the first and most intelligible of all °°™ e P" Either
the ultimate term of the analysis, the notion which neither
needs to be nor can be defined and which does not admit of

^ b0y °1or'the zoologist, *»n. is a mammal <^f J^..
Primatc a ...How would he define such a M™fJ*^ fl ^roting
brate characterized by the presence of .spooial glands seciowng
urate onnracterizw, w ^. r i e fi ne d? In terms of color,
a liquid' called milk. How is milk d °™^ chemical compo-
taste, average density, biological function, 01

-113-

nents, etc.

Here the ultimate and undef inable element is some sense
datum; it is $hq_objcot jof jan_ intuition for which no logical
construction can be substituted anT'upon which all the logical
constructions of the science of nature finally rest, (103)

a duality of sciences ' in "5)jG'^tuJyf of "'mture ^ There are two main
differences be"tween"th"e "definition of "the philosopher of nature
and that of the experimental scientist. (104) Both of them, far
from constituting a specific distinction between sciences, absolu- .
■h eiv exclude the possibility of such a distinction . In the first
place, the definition of the philosopher is strictly scientific ,
whereas that of the zoologist is purely dialectical . Obviously,
if the definitions of experimental science are purely dialectical,
it cannot be a specifically distinct science, f^r_jth.e_aiiiTple_reasOTi
that it Ia nlA-a-acienge . The second difference between the tyro de-
finitions is one of generality and concreteness. Whereas the phi-
losopher of nature deals in broad generalities the experimental
scientist is far advanced along the road to concretion. In this
sense the former is far less immersed in the directly observable
than the latter. If this is what M. Maritain means by saying that
the one moves from the visible to the invisible, while the other
goes from the visible to the visible, he is correct; but besides
being an extremely ambiguous and confusing way of explaining the . •
situation. rit _-provides no foundation for a sp ecific distinction
\ between sciences P )

Because. the experimental scientist is deeply immersed
in concrete materiality, it is only natural that he will clarify
his definitions in terms of concrete, material observable, things.
If we asked St. Thomas to clarify his material ^"fion of a house
"a structure made of stones, cement, and wood' (105) he would un
doubtedly do so in terms of material observable things.

It should now be fairly clear that the difference in
^teriality between philosophy of nature and f^™^^'
upon which M. Maritain seems to ^^/is 9^0^ distinction i|_
not one that derives froj ^o^^^^^^^^^^,

, g^i^^rgS^^^^^^ ^o? sciences, abso-
This difference, far from constituting a auaii^

lutely excludes'the possiHLity of *"* * ^2'^ same acience
ready seen that the more particular must pertain
las. the more general.

t. • wi. if the main flifferenoe between
But it may be objeoted: if ™o ,.vx

-114-

•the definition of the philosopher and that of tho experimental scientist
consist in a question of generality and concretenoss, why should it
not be possible for the experimental scientist to clarify his defi-
nition by retreating into higher levels of generality and thus re-

ijoin tho philosopher, and why should it not he possible for -.the
philosopher to push ahead into concretion and rejoin the experimental
scientist o Our ansvrer is that not only is such a thin g possible ,

(b ut in a pertain sense absolutely neceasar y_.)Let us try to see why '

\this is so.

In the first place, it must be noted that the ascending
analysis attributed to the philosopher of nature is nothing but
ana scent of the Por phyrian tre e, (fa retreat into potentialit y^) that
is to say into generalities that become more vague and more empty.
The philosopher of nature may, indeed, make this ascent, provided
he does so in terms of mobility ^ But it is important for him to
realise that while this ascent is leading him in the direction of
that which is more knowable Quoad nos , it is. leading him farther
raid farther away from that which is more knowable in se . In other ■
words, by, the very fact that he is practising total abstraction
he is achieving greater intelligibility q uoad nos only , at the ex-
pense of sacrificing intelligibility ' in se , Now philosophy does
not consist merely in giving terms that are more knowable for u s,
(| ut in manifesting the natures of thing s as perfectly as possible^ )
It consists in getting at what is more knowable in se and not merely
what is more knowable quo ad nos . Definitions are supposed to manifest
things to us and this manifestation does not come from a retreat
into notions that become increasingly more, vague and empty. The
only way in which a philosopher can truly philosophize is, not by
retreating backward into potentiality, but by pressing forward into
fuller actuality. In no other way can he succeed in b ringing to
light I the proper natu res of things „j That is why, as we no ted above,
StTThSmSTIn all of his proemia to the natural works of Aristotle,
keeps insistin g that the p MjjjJehgr^f, nature must con^tantly_moyg

forward into .fuller concretion ^

With these remarks in mind let us return to the pas-
sage quoted above from to. Simon. In the . first place, it mu st_be
noted that Mr, Simon has chosen his examples with care, for apart
from the fact (over which we shall not linger that he has made
the philosopher explain the generic part of his f ^^f'/^^
the zoologist the specific part of his defi -t-n, h h- in select
ing the example of rational animal, ^^L^^^l-^^g-j^,,
As he .himsell suggests (106) ^JSH^lf^Eiom

this point of view it provides a k3,W of te ™"™ s thinps. This
Philosopher's quest to get at the proper natures of things,

-115-

ia far from saying, however, that thia movement towards concretion
has come to an end as far as the nature of man is concerned. For
both "animal" and ''rational'' are rather vague notions which must
be explored and concretized. Having determined that man is a rational
animal, the st udent of nature is forced to attempt to find out,
for example, what precise structure of bod y is proper to rational
an imalit y .( Tind this attempt will very speedil y bring ; him to the
definition given by the zoologist. ) But in order to bring out the
issue clearly let us use another example.

Le t ua ask the philosopher of nature to tell us what
a horse is. . And while we await the answer let us recall a remark
of Professor Simon: Philosophy of nature "does not reach its end
until it is able to answer the question 'What is the thing- under
consideration'?" (107) Where will the philosopher turn to tell
us what a horse is? Will he turn upwards in his ascending analysis?
If so, we are justified in becoming impatient and calling bin back,
for he is not telling us what a horse is; he is merely telling us
what al l animal s in general are . Is it not evident, that in order
to answer the question "what is a horse" he must move in exactly
the Opposite direction? It is useless to retreat into logical p o-
tentialit y; he must push forward along the road to concretion into
greater actuality j It'^rnay be that he will never be able to give
us a perfect answet,. : i,ut if he is true to his science that will
not keep him from aft endless striving to get at least a partial
answer, M. Maritain seems to admit the necessity of this movement
towards concretion in every science,, for he writes: "Toute science
allant d'ailleurs dans cet' ordre vers la plus grande determination,
exigeant que l'objet soit serre, pour ainsi dire, dans une notion
propre, et non pas enveloppe dans une notion commune plus ou moins
f lottanteT" "(108)-' <yk ( o,. p . j,<v , ™W) .Wi^ s<v * ■H^ ^ "^'« a V M ^ *"* iW

We know what reply this objection would receive: (l09).-<£~> A
thejphilosopher of nature must no t_attgm|3t_to answer such questions. >?■>"; ff

He must praotiiQHe ~spirit of p overty; |hejmist_r^4e_guil^gf _ j
the~e^agerated ogHmism(and philos^hj^al^jjmjeidalisjg) of the ancienc
TtoSitajHen^riel^e" ^I55tiolTi^FthaTHnd to the experimental"-
^ciSntisY who with his special science completes, the philosopher a
study of nature. And why? Because philosophy of nature is' wisdo m
(within the order of physicalrealitf> Or "toute sagesse est magna-
^iiVno^ ^barrasse ^a^TdelSirnTateriel des choaes, pauyre
done en ce sens, et litre, comme lea vrais magnanimes; et cette
sagease-li est obligee a la pauvrete; elle doit se re^^ooon,
mitre, elle doit^honore^e c onna trc , le - £. ^ er _ ^ ^ . U
pauvres sans pretend.ro epuiser le detail des phenom ^ J^ ^ ^

J-es cailloux du torrent." (.110; we iax-i. ou , __

argument. Strange mgtohlmity this, the renonclation of the too;;

-116-

l/UJi J )

ledge of things HL.theirjerg B grjBgoiflolfer. Par from being a pro-
perty of wisdom, such magnanimity is opposed to its true nature.
And if human wisdom cannot succeed in reaching things in thoir pro-
per.,' specificity, it. is not_because_U _i s vtisdom Cbut because it is
human and ther efore extremel yjjgporfectp But_ precisely becaus e it
is w isdom it must ever strive iiowards thelcn^wl edge_of_s pecific
natures o These last linos of Maritain are rather "hard on St. Thomas „
For let us recall that, he has already told us that the doctrine
of the ancient Thomists (St. Thomas included) which held that the
philosopher of nature should push f orward int o concretion was a
grave error,. If thon"^he reason why the -philosopher of nature must
abstain from concrete questions is that he is obliged to do sojby
the_ very fact tha tj philosophy of nature is wisdom , ( the conclusio n
is inevitable^ ) St. Thomas was unaware of the true nature of wisdom,, \
Ve prefer to believe that his ideas on the nature of wisdom were f~ f
more exact than those of H. Maritain,,

We admit that there is a sense in which it is true
to say that the philosopher of nature is brought up short before
such concrete questions. But the reason is not that he runs into

. another scienc e / Hrjut that~he runs out of science^But there is " no
reason why he should not prolong his study of nature dialectioa lly

( even when he is unable to do so scientifically .') And when this is
done the philosopher and the zoologist inevitably meet .

' If there were any valid reason why the philosopher
of nature should remain in his generalities and. feel satisfied with
his ascending analysis, it would have to bo because in fthig Vwgy
he could derive the greatest illumination concerning nature and
obtain the deepest insights into physical realit y j But this would
necessarily mean that the generalities wouM_contain all their .in-
feriors Actually and distinctly^ and that wh at is more knowable
for us would be at the an m e time more knowable secundum se . Not
a few modern scholastics, wfljTthejJjaisT^ir of pjgfundifrrin
dealing with these vague generalities which considered from the
pHnT"ofWiiw^rthe - proper natures of things that constitute „he _
goal of the science of nature, P rovide_the most mp^^m^^
knowledge it is possible jgji ave of the cosmos , seem to hold such
a^iiw~, at least impllcItl^AMfalii^^
to fall in the erj-oUf_j^ latonistS ( ^

^STi^^r^rS^F^S^rTm^ process of m ^ ^imng
■with the general notion of art is well known, (ill) In ^ las^
analysis this kind of philosophy of nature is nothin |^t Hegelian
ism, Iferl Marx's explanation of He B el on this pom, is extiemely
illuminating!

J

-117-

Quand, a partir des parames, des poires, des f raises,
des amendesreelles, je forme la representation generale: fruit,
quand je vais plus loin et que je mo figure que ma representa-
tion abstraitc;. le fruit, obtcnuo a partir dos fruits reels,
est une essence qui existe en dehors de raoi, est meme 1' essence
veritable de la poire, de la ponme, jo declare, -~ en termed
speculatifs - - que le Fruit est la 'substance" de la poire,
de la pomme, de l'araande, etc, Je dis done que 1'essentiel de
la poire, de la pomme ( pe n'est pas d'etre pomme ou poire^ ) L'es-
senti el de cos choses n'est pas leur "etre reel , tombant sous
les sens,|raais ]Jessence_de misrepresentation:,) le Fruit. Je
declare doncque~la poraae", la~poire, 1' amande, etc. sont de

simples modes modi du Fruit a Mon entendement fini, sou-

tenu par les sens, distingue sans doute une pomme d'une poire,
et une poire d'une amande, raais ma Raison speculative declar e
Cque cette distinction sensible est inessentielle et indifferen -
te o^) Elle voit dons la pomme la meme chose que dans la poire,
et dans la poire la meme chose que dans l'araande,' a savoir le
Fruito Les fruits reels partieuliers no sont plus que des ap -
parences du fruit, dont la veritable essence est la substance,
le fruit,,, Le Fx-uit n'est pas une essence sans vie, sans ca-
racteres distinctifs, sans mouvernent, raais une essence vivante,
distincte en soi, en mouvernent, Le caractere distinct des fruits
profanes ne releve aucunement de mon entendement sensible, mais
du Fruit lui-meme,. de la Raison speculative, Les fruits profa-
nes distincts sont des manifestations vivantes, : distinctes, '
du Fruit unique, ils sont des cristallisations qu'elabore le
Fruit lui-meme. Par exemple, dans la pomme, le Fruit se donne
^une apparence de pomme, dans la poire une apparence.de poire.
On ne doit done plus dire, comme du point de yue de la substan-
ce: la poire est le fruit, la ponme est le fruit, l'araande est
le fruit, mais Men plutot: le Fruit se presente comme pomme,
comme poire, comme amande, et les differences qui separent les
unes des autre s, la ponme, la poire, l'araande, sont les diffe-
rences meme du Fruit et olios font des fruits paruiculiers des
chalnons differents dans le processus vital du fruit. Le Fruit
n'est done plus uncuni^s^^ntem, sans distinction?, il
est l-unite'en to^Hi^riSnlEaliT^ue "totalite" des Fruits,
qui foment une succession, le fruit se presente co.^e une exis-
tence plus developpee, plus complement exprimee, jusguao
qu'il soit enfin "le resume" de tous les fruits en meme temps
quo leur unite vivante. (112)

We have quoted this long passage because it characte-
rs so well theTttitude of >-W modern scholas ics^ee^g
look upon the general as thg ^gjubBtoeo^sgff^L
c gictrn^e phenomenaT^Ii^iiEII^M^^^g^^^^

-118-

: ohxlosopher>ho must concentr&teJn^J}±1^2f!?^lT"r\ the profound
essences of things. We believe that the doctrine of Maritain' tends
to encourage this attitude. It does so in many ways: by insisting
upon ascending analysis and neglecting the movement towards concre-
tion? by describing experimental science as something which merel y
deals with phenomenal details -, without explaining that it is preci-
scly through experimental science that we are constantly carried
closer and closer to the proper natures of things which constitute
th e goal of the (whole) study of nature , closer and closer to the
raost profound knowledge that it is possible to have of the cosmos -
to the kind of knowledge that God has of nature ; etc. Maritain does,
indeed, point out the poorness of the knowledge provided by philo-
sophy of nature f but he does so in such a way as to make it appear
that the riches which it renounces are hardly woAth havin g. He com-'
pares the knowledge that experimental science gives with counting
the stones in a stream . St Thomas has already taken care of this
counting of stones when in explaining the opening lines of Aristo-
tle's Physics where we are told that in the study of nature the
raind must move in the direction of concretion by progressing from ,
imiversals to singulars. , he wrote:

Hie autem singularia dicit non ipsa individua, sed
species; quae sunt notiores secundum naturam , utpote perfectio-
resfexistentes et distinctan cognitionem habentes; genera vero
sunt prius nota quoad nos, utpote habentia cognitionem in po-
tentia et confusam. (113)

The same point is brought out by St. Thomas in the
Prooemium of his C ommentary on the Libri MeteoroloRicorum :

Unde manifestum est quod complementum soientiae requi-
rit quod non sistatur in comraunibus, sed proceda tur usque ad
species: individua enim non cadunt sub consideration artis;
non enim eoruin est intellectus, sed sensus.

But there is even a greater danger in Maritain' s doc-
trine that the one just mentioned. We belike that it .ends *o lead
tola confusion between philc^pJSLOJLiiature^nd^^hy^icgj m

/ 3pkc-^fltoitain's expliclt^fforts to keep the too distinct. (114)
The difficulty here arises from the initial error of. seeing in^the
object mobile being a dual or bipolar char acter whic h gives rise

Uo teo formalities! Earlier in this chapter we have re.ecte Mtos
error a*d pointed out that the great Thomists have ^aditional ±y
insisted that the dualism in the expression "mobile being ^ Purely
verbal, that it signifies one indivisible ^^^'{L^biect of
his two formalities, Maritain goes on to say .ha, the object of
philosophy of nature is mobile being or sensible being considered

-119-

precisely in so far as it is being. Now, as wo saw above, St, Thomas
j n his Conaont ary on thoJSj^h^o^-^hj^M^^jyjnR repeatedly
insists upon the fact that no other science can deal with any par-
ticular type- of being precisely in so fay as it is being ^except
notaphysicsj) And he says explicitly that this is true of" sensible
TjiingT "etiam de sensibilibus, inquantuin sunt entia, Philosophus
^perscrutatur,," (115) And the difficulty is only augmented ¥/hen
one constantly runs across such misleading; statements as the fol-
lowing: "oo^il faut dire que l'objet projre de la philosophie de
la nature ,,,,n' est constitue que par le ti'anscendantal otre en tant
qi^ determine et p articularise au monde corporel, mobile et sensi-
ble7"(116) "En realite elle (la philosophie de la nature) conside-
ro las choses corporelles et mobiles au print de vue du transcen -
d antal etre imbibe en olles " (117) " " " — ' "

And even if philosophy of nature could in this position
save itself from identification with metaphysics it would at best
have the appearqnee of an intermediary science subalternated to
i metaphysics « We do not accuse M„ Mori tain of holding this view, ^. .

but it is interesting to note that noise than one author who have ■ \'ffi ,V -^
followed in his wake have explicitly arrived at this conclusion. (118)
And a greater epistemological perversion couid hardly be imagined, (119;

But let us return to the definitions of the philosopher
and the zoologist,, From the foregoing it should now be clear why _^
the philosopher of nature must move forward towards concretion and
join the zoologist. But the question now suggest itself : can this
mooting be brought about by having the zoologist move backwards
as well as by having the philosopher move forwards? Once again the

I answer must be in the affirmative. If we ask a zoologist what a.
vertebrate is, he will probably answer: an animal with a spinal
column. By seeking for an explanation of "animal" we can make the
same ascent in the Porphyrian tree made by the philosopher. But
one will immediately be tempted to object: granted that such an
ascent is possible, why is it that it is never made. by ^e oxperx
mental scientist? Why is it that as Simon points ™»'V™> ea T
a way of explaining terms would ordinarily move a ^oologxst to ^

1 laughter? The reasons are not far to seek. Modern Wf ental soien
tists have chosen to ignore completely the higher f]ft°^ <g ne
rality in the science of nature, M^M^^S^^^
roly experimenfaljroE^Uons. Experiment al P- P °sx bions -yn

aro ^inited^ s concrete exp£r3£nce_alone 5 j Hence " •" houla

jEF^^aaSaT5^5E& ^gff^ s Tot nfc'sS for tLm
turn to concrete experience. While it is n experimental

to know philosophy of nature m °ff£ Q h ™° °to^dorstand the
scientists, such a knowledge would enable tnem

v ,f

V*

-120-

meaning of their science and the proper significance of the terms
ond propositions thoy employ. A zoologist with a knowledge of phi-
losophy of nature would have no difficulty in rakin g an ascendin g
amaysis_of_his terms and thus rejoin the definition of the philo-
sopher of nature. And in connection with the question why the zoolo-
gist ordinarily nates a descending rather than an ascending analysis
pm-haps this last remark should be made: experimental scientists
have understood far better than scholastic philosophers of nature
that the p ro per movement of the study of nature is f orward into
actuali ty ,(rathe_r than backward into potentiality,;

Before leaving this criticism of the doctrine of Ma-
ritain, we should like to put it to a final test. We are told that
dianoetical intellection is characteristic of the science of nature
which employs ascending analysis, while perinoetical intellection
is proper to the science which employs descending analysis. Let
us take the example of a definition of man in terms of the tongue
and the hands, Nov/- while most definitions in terms of the concrete _.

structure of the body are purely sy nthetic(and' hence dialectical ) U3

as in the case of the definition ofman as a mammal, it seems that
the definition in terms of the tongue and the hands is analytic, .
for there is a necessary connection between rational animality (which
in plies an animal that possesses both a speculative and a practical
intellect ) and__th ese tyro organs . If then one were to attempt to
resolve the concepts contained in this type of definition in which
direction would he turn? Tfould he not be laid 'to explain himself
in terms of concrete, material observable things? We are consequently ■
faced with this question: wh at kind of intellection do we find in
the proposition ,just mentioned ? Is it dianoetical? If so, why do
we have a descending rather than an ascending analysis? Is, it peri-
noetical? If so, \how explain that we h a ve an analyt ic proposition,;
for in nn"K7STiyHn ^positions t he~essenco is opened up and does
not remain covered over.

5. Natural Doctrine and Practical jfoowlgdgej

At this point it is necessary to introduce a problem
which arises out of the text of Aristotle, The solucion of ^is
problem will serve to clarify our conception of the ™^ g "f ural
doctrine and of its relations to the other branches of ^° A f>
The text we have in mind is found in the f -^^Lf co^ara"
book of the De_Partibus_Am£e^S. I* x f \f „,,° Im ot . of Aris-
«.vely littlT-alteHtiSn-haTbeen given by the canmentators of Aris

-121-

■toUe; yet it is prcgnait with profound implications, In _spite of
the _fact that m^all the other passages of his writings, where ha
c onsiders the nature of natural ddo lrino he classes it among the
s peculative sciences,, injb his particular text he seems to set it
in qp positxon to the S p eculative sciences *

The causes concerned in the generation of the works
of nature are f as we see, more than one. There is the final
cause and there is the motor cause a Now we must decide which
of these two causey comes first, which second, plainly, however,
that cause is the first Y/hich we call the final one. For this
is the Reason; and the Reason forms the starting point, alike
in the works of art and in the works of nature „ For consider
how the physician and how the builder sets about his work,. He
starts by forming for himself a definite picture, in the one
case perceptible to the mind, in the other to sense, of his end
- - the physician of health., the builder of a house - - and 1
this he holds forward as _the reason and explanation of each
subse quent step that he takes ,, and of his acting m this or
that way as the case may be c Now in the works of nature the
* good end and the final cause is still more, dominant than in
works of art such as these, nor is necessity a factor with the
same significance in them all; though almost all writers, while
they try to refer their origin to this cause, do so without
distinguishing the various senses in which the term necessity
qs.usedj) For there is absolute, necessity manifested in eternal
phenomena; and there is hypothetical necessity, manifested in
everything that is generated by nature as in everything that
is produced by art, be it a house or what it may. For if a house
or such final object is to be realized, it is necessary that
such and such material shall exist; and it is necessary that
first this and then that shall be produced, and firs, -.his and
then that set in motion, and so on in continuous succession,
until the end and final result is reached, for the sake "ion
each prior thing is produced and «iats.. As wixh these ^oduot-
ions of art, so also is It with the production^ of nature. The
mode of neoisaiJK, ^SSm^^SS^SS^^^^^^ Q _ A

^i^al-cienc^sT^^^^^^^iS^^^ ?°*-3S

retina.!, sciences: 01 wu^j^is^-ac — . v - -. — " • ..„ tVl „ form er

tne latcer tne st,u.j.- ^"fi ~.t^^uj^-~--~- — ■ . , . + ±. n \, p

previous production of tms ana „. igta or has been gene-

come into. existejice_o__{121iy

-122-

We have italicized the lines in this passage to which
xia wish to call particular attention, There can be no doubt the.t
in these lines physics is distinguished from speculative science.
And after all that was said above about the place it occupies in
the first degree of formal abstraction which distinguished the spe-
culative sciences, this presents us with a problem that must be
solved. Two possible interpretations of the passage just cited sug= ,
gest themselves: Natural doctrine is distinguished from the specu-
lative sciences either because it is essentially a practical science,
and consequently not speculative at all, or because though essen- ,
tially a speculative science, it has some chara cteristics in com-
non with prac tical knowled ge (tind insome measure falls short of ~
the pe rfection of speculative knowledge Q After all that has been
said thus far it must be evident that only the second interpretation
is acceptable. Natural doctrine must be essentiall y a speculative
science, because in it knowledge is sought for its own sake.

As our analysis proceeds we hope to make it clear in
how many ways natural doctrine comes close to practical knowledge,
and we do not wish to anticipate these developments here,, Yet it
vail be helpful, perhaps, to set down in skeletal fashion some of
the salient features of thb striking resemblance between the study
of natufce and practical saience.

In the passage cited above., Aristotle suggests the
basic reason for this resemblance. Like all the characteristics
of the study of nature, this resemblance derives from the fact that
the object of this study is mobile being,. Now mobile being^ means
not only being that is but. being_thatJecomeg. And the study _ which
deals with such a being precisely in. terms of its mobility will
deal with it not Der23 2 Jnite_bein£butjJ^^ AM

i r i^^e~ _ alTn SS5: aI^^

in nature something closely akin to what is found in art and pru-
dence; we find a becoming, a generation, a pr oduc tion, a ~ent
towards an end. And whenever there is an end, it ^W* aotsas prin
ciple, as Aristotle points out in the text ,n ^ °^ e ?' /"^J^
I it j. j.- • j. ^ \ +>,.,+ vjhirh is to be" While uhis cnaracxe

(the starting point is) that ™"J" ^j^.^ between them and
nstic of natural beings es tablj ^ e * a ^f^ ^ distinguishes
the things of art and prudence, it a. oho same ww- a

them from mathematical and metaphysical things. For, as ^we have se en,
the objects of both mathematics andmetaphysies are immobile, To
this it might be objected that there is a ind of Paction in
metaphysical beings, si^^an^els^^

But because it is merely a question of act i°» s > * the con _

touches only the accidental order. In ^ ^j the mtter

tatty, ^2^J^L^^ in them an

and priva^M^n^m^c^sienc^ of these beings,

-123-

intrinsio plasticity that makes them substantially formable. Thoy
are not merely called into existence f their gen^tiolTirThe torm-
inus of a lengthy process of composition ana formation in which
nature proceeds like art. In mathematics there is no formability.
It is true that there is a kind of construction in mathematical
science, but this does not involve movement or production in the
true sense of the word. And that is why the only kind of afct that
is possible in mathematics is speculative _ art.

Now we arc in a position to understand the pr"found
distinction which Aristotle introduces here between the "object of
natural doctrine and the objects of the other speculative sciences.
Since the objects of the other speculative sciences do not become,
they simply are. That is why Aristotle says that these sciences
have to do merely with tha t which is. But mobile being becomes.
And since all becoming, all movement |gets its whole specification
and determination from the terminus,) the science which studies such
a being will be engaged primarily not with that which is, but with
khat which will be , that is to say, the end,, which is first in in-

tention and last in execution. And this end is a good, and moves
as a good. All this reveals the fundamental role that finality plays
in the study of nature and in all practical science and explains
why Aristotle insists so strongly upon finality in nature in the
second book of his Phy_sics.

It ig __because of th isjl ependence upon the end that
existence plays a __p_a rt in the study of nature that it does not pla y
in mathematic s o r metaphysi cs which deal with esse,wces_ -IrJkJEE?-
"hat ia' similar ""to the_part~it pla ys in practical science; ) For in

Tj-s _.£> TI3 Ii.„~„ „«„ tm .montsi finfl in the order of intent-

the notion of 'Sr^TthsToaT^bwol^eots: end in the^ordSr of intent-
ion, i.e. end as a cause; and end in the order of execution, i.e.
lend as an effect. iHow it i s preois elg^g xistence whlch separate^
these two.) And it 3r^e^ause2ofmoj^g7^g°2™£i> that the Wo
SeraTSre unTEeaHW'gtt^ whjit_go|s,

.SQ5TT!KarirWT3-iriU merely concerned with the £E°iJ^
5itl£ mathematics and metaphysics are. And it is to be no.ed that
tte end involved in nature is the_very_form of mturalthings, and
consequently it is due to becorZnFthKt the very object of the study
of nature is constituted.

All this serves to bring out the striking r^gnce
between the study of nature and practical ^^^^^
mkes it clear that from this point of v f w ^ur|i^rincJ can^

^•s to say, becaus e 01 one naT.ui i_ uj. — ^ „ — e —

-124-

\/hat we havo been saying enables us to understand the
articular type 01 necessity that is found in the sciences of nature,
Since, as we have pointed out, all science deals with necessity,
(the n ature of the science is intrinsically determined b y~the~Tcina
of jicce3sitxJhalj.s_p ropog to it J Now there are two kinds of neces-
sity: absolute and hypothetical. As Aristotle explains at the end
of the second book of the Ehjrsios, (122) things which have their
necessi ty from a formal, material' or efficient cause en joy , absolute .
neces sity. On the other hand, the necessity which derives from the
final cause is only hypothetical. And hypothetical necessity con-
sists in this: if .a certain end is to be achieved, then such and
such means are necessary. But it does not -follow that given these
noans , (the end willnecessarily bo achieved^ For example , we may
soy that if a certain type of organism is to be generated, then
the conjunction of a sperm and an ovum is necessary. But it does ■
not follow from the fact of this conjunction that the organism will
necessarily be for the end' may fail to be achieved for some reason
or other o

In order to understand this point clearly we must have
recourse to a distinction made 'by Aristotle in the second book of
the Physics 8 (123) The end that is found in natural things, may
be considered in two ways, It may first of all be considered as a ^
principle of re asoning/ fand then it is taken as the (gjusg) if rom w hich
we nay demonstrate all the things that are n ecessary for; the end
B'be^ riaTZ zedTJIn^E'nis'sense we can reason from The end to the
Se^EsThaTarei necessary for the end, But_it mayj tlso be t aken_as
ia principle of( aotIo^ , that is to say aB the cause moving the agen t,
■In this sense" iTTiirapossible for demonstration to tactuall^ resc n
the end, that is to say, we cannot reason from the fact _ that the
I moans necessary for an end, are' given, that the end is going to be _
realized^)

In all of the speculative sciences besides the study
of nature absolute necessity is found, ^iJ5JB g^ t) g°* r g B g^
is onlvJEpottetic^necessity. Here we ^V^T Eludes 4oT
^rthrr^ETSTthTirlSH^l. And so Aristo ^^nonena- S
there is absolute necessity,, mnifosted in etei^^^e^ and
there is hypothetical necessity, manifested m ^^f^ that is
penerated bv nature as in everything that is produced by arv, be
fmo^ y -ofSt it may." V) ^^^^^W '

point of view of prjor_causes, whateve r "5g*Li-^j^

b^t^^^Sst^- -^5^^|^ a for the most
I^f^uTo-lSZ^riirTa^Fbe^h^acterized by wha, napp

part. And it is this that St. Thorns has in mind vh a

alread,y quoted (126 he points out that the sen

-125-

to0 A/2°!fe5fuM^^ becauae ..^^ demonstrationes
svamaxr ex his quae non semper insunt, 'sed frequenter," This dis-
tinguishes xt from the other speculative sciences whose demonstrat-
ions enjoy a greater necessity. At the sane time it reveals the
olose similarity wich practical knowledge, for as Aquinas points
out in the sane lo^feo 5 in_tte J aoral_s^ie il ces the "principia su-
V r jjntur_ex_his quae sunt ut in pluribus T" (127) ~

It is e-rf.dent, then, that in natural science demons-
tration cannot arrive at the ip„sum_esse_J^nis For example, in
the evolution of the cosmos, at no point was it possible to de-
monstrate- v/ith absolute necessity the future existence of any
particular natural species (128) - - even- though once the exist-
ence .of a certain species is given in nature it can be the prin-
cVciple of what had to be in order for it . to exist. In other words,
natural things are not knowable except in the order of existence;
that is to say, we cannot know them except by knoy/ing them as
^existing ;. This creates a great difference between the science
of nature and the. other speculative sciences,. We stand before the
universe as before a work of art in the process of being made.
We might have a geueral notion of what is to' come about, but as
long as we have no full share in the idea of the artist, we do
not know just what is to cone about or exactly how. Like praotical
knowledge , therefore , \ the study of nature has a close and neces-
sary relation with tho existential order, and(qonse g ueafly3vri.th
expe rience ,j This point will be developed at considerable length
in Chapter IV, and in connection with it, we shall discover an-
other closely related reason why^physics is associated with prao-
tical knowledge: it has to do with the objects that are for-aed
by divine art. This, is not true in the -sense-" of ^Metaphysics, for
angels are not Lrornec D in the line of (essence , Irijjathemticg eve ry-
thMigis_an^yJ^ical,

Besides being about things that are brought into
existence by composition, natural doctrine must itself engage in
eonposition. This is true not only in the construction of jheories,
hut already in the gathering of the various subjects considered.
The study of nature must be cbuilt^) out_of_bits_g arnered _ from
experience. And closely connected with this is another poino of
SHilEH^ with practical knowledge, namely i| s intimate r^i on
with^ingulars. The student of nature cannot deal purely with
SB^HaferSrfact, as he pursues his research^inohedir|c|i£n
of fuller c oncretio n, it soon becomes iaposs ^ lo /^.^*°^ Se
iu^ssfuliy abovo-thc realm of singulars to true universal itf,
and he is obliged J^^^u^^^

hyp othetical o on atrnot-^ lg^^^^g^;"^^^
necti on with t h^^elatio^Tbe^een naturar^o^rTne^nd singulars

-126-

it is vrowth while nooxng that in nature generation is always in
'oho singular. In mathouatios, on the contrary, it is possible
to have a quasi universal generation, e.g. the generation of a
line from a point. This makes it clear that the science of nature
tos_soncw hat the same cha ra oter of singularity as mc ral science .'
[in thegojbwoj^ias_jaono3s it possible to have history!! '

As the student gets deeper in the realrc of concrete
singularity his science becomes conditioned by a constantly increas-
ing multiplicity of elements. (129) In this it becomes renarka - •
bly simila r to moral science . And just as in the fief.d of concrete
taan actions the multiplicity of elements i s so greiLt \ thatgaotio§ )
rgmins poss ible onl y__b ecause man can ov erride thi s multipl icity
bya "de liberate act of~~bhe will J so in the parts of iiatm-aJTdoc-
^Hne~wHIch are deeply immersed in concretion, experience is con-
ditioned by such a multiplicity .'of elements that '(|cience]ji becomes
possibl e only because the scientist overrides this multiplicit y
by del iberate fiat .

All this makes it clear v/hy physical science ..is it
advances towards concretion soon issues into a purely dialectical
extension,. This happens both because of the materiality of natural
things and because of man's way of knoT/ing them. It is interesting
to note that if we consider the whole range of natural doctrine
from the highest generality to the ultimate concretion the p ort
vrtiich has a truly scientific charaoter is small indeed in coi ijpa-

riste, with the part whose character is. merely dialectical . Tt
is also interesting to point out that the passage of Aristotl 3
which we used to introduce this problem is taken from a treatise
which is already far along the road to concretion^ )

Now it is highly significant that no other speculative
science has such a dialectical extension. Theology, mathematics ■
logic, arithmetic and geometry can pursue their course m strict ly
scientific fashion. This does not mean, of course, that no probaLue

I factors enter into these studies. It means that in these sciences
'chcre~are no sections whose wh ole structure is dialectical. Ut
all the speculativT~sciences this is characteristic of the study

v of nature alone „

Bit at the some tine it is also ^actor^sticof
practical knowledge. In moral philosophy as soon a * J> ^ *™_
'lost general principles necessity likewise p^tersoutintp^ba
!>ility. (130) That is why St. Thomas often repeats that moral
plSTo SO phy(p?ocecdg) "figuralitor, ideatjreriai^ig. And the
closer thelS?al-philosopher draws to concretion, Jg^norma
tivo his science becomes. Nevertheless, the very nature his

-127-

acicnoc forces him co continue along this road, exploring the
re^a^o^ociolosj^Goonojn^s, etc., always pressing forward
towards greater concretion. Once again, as in the study of nature,
the part of the doctrine which enjoys strict scientific necessity
is suall indocd in comparison, with the part which possesses only
probability.

Our final point of comparison, between natural doc-
trine and practical knowledge brings us back to something consi-
dered at the beginning of this chapter. We saw that as the scientist
draws closer to the ultimate concretion, his attempts to lay bare
the secrets of nature make it increasingly necessary for him (to,
qp_e£ate_upon) nature , yto refashion it and reconstruct it. j ln this
wayjphysical scienc e gradually' takes on the aspects of an art .
At the same time man's practical power over nature increases.
And not only does his power increase, but at the sane time his
ars co npara tiva naturae , as in the case of the arts of medicine
and hybridization, for example, increases. And in this man knowingly
and through his skilful action pursues a terminus that in itself
is natural.

Those few ideas on the relation between the science
of nature and practical knowledge must suffice for the moment.
Later chapters will give them fuller embodiment. But it is worth
v.'hile pointing out hero what an important bearing all- this has
.upon the problem of mathematical physics. F or f ew jb hings could
( seen more diametrically opposed than mathematics and practi cal
knowledge^ Yet iTT s to this cosmos , which in" so many ways presents
such striking resemblances to the object of practical knowledge, ;
(that mathematics is ap plied ,)

6, Specification and Method. '

Prom this general consideration of the specification
of the sciences a conclusion must be Immediately drawn which is
of extreme importance for our purpose. It is this: the specif icat-
ion which sets off the various distinct sciences is neither arbi-
trary nor fluid; it is something very objective and d ^™;
As a consequence, each specifically distinct science has a specia^
character of its own which the other sciences- cannot share. Each
science has its own particular questions and its own pa rticular
answers; it has principles that are peculiar . oo it, it ha s itg

Thomas brings out this point m a geneiai way

-128-

on.theJ9eJMjjitato when, after explaining the distinction between
physics, mathematics and metaphysics, and pointing out how each
of these sciences terminates in a different cognitive power , he
concludes: " Et propter hoc peccant qui uniformiter in trib us spe-
culatiyaej partibus Jj^cedgr^nituntur^~(l53) As Maritaih has
renarked, these words should De written in letters of gold over
the doors of every university. (154)

In his Commentary on the Posterior Analytics , Aquinas
presses this point home with greater precision and greater insistence,
In commenting on Chapter XII he devotes a whole lectio (135) to
shewing that each science has its own particular type of questions
and answers and disputations . And he points out how this follows
from the very specific character of the science. For, as we have
seen, the sciences are specified by the type of -propositions, the y
us e as prin ci ples of their s yllogisms . But a scientific C^ies^xSi}
and a rc-jorItific( jproposTt3^ i are substantially the same, and - dif-
fer only in the node of expression ,. Since, therefore, each science
has its own particular type of principles, it will necessarily
have its own particular type of questions. And so Aquinas concludes:
Won ergo q uaellbet interrogatio est geometrica, vel medicinalisj
et sic de aliis seientiis." (136} Since an answer must be in
the saue genus as the question to which it replies, it follows
that each science has its own type of answers. And consequently
St. Thomas remarks: "Won contingit de quolibet interrogate- respon-
aero: sed solum de his quae sunt secundum pro prianLScientianu" (137)
It likewise follows that each science has- its own type of disputatr
ion, since dis putations jeroc^gdjby^gue stions and an swers. And in
order to press this general- point home , with, more precision he adds
to this lectio another lectio in which he shows that ea ch science .
ha s its own peculi arjtyj3g3_gf_dece ption, and i gnorance. (138}

But of even greater significance for our purpose is
his commentary on Chapter VII (139) wherein he proves that each
science demonstrates by BieansofJ : ts_own_proj P ^Pi2£2iPj^ ^
that .consequently, th^^^B^m^rM^BB-lS^^B^^^^-^^
toj£nggrge^TeJ^itog_ in_a nother scie nce. He writes:

In illis scientiis, quarum est diversum genus subiectum,
sicut in aritSeti" d-e Lt de ftumBris, et gc .ometria quae e S

de magnitudes, ^^o^^^^^f^lilTst
ex principiis unius scientiae, puta ^;.«Jg^ ' sunt subiec .

biecta alterius scientiae, sicut ad magnitudmcs, qua

ta geometriac. (140)

j-hr, ■m-inr'iules and the conclu-
And he goes on to give the reason: ubo P" n ^f oa for the

aions of a scientific syllogism uust be m the same g ,

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principles illuminate the conclusions; the latter are in fact pre-
containad in the former, , .

This doctrine taken as it stands here immediately gives
rise to serious epistemological difficulties. It seems to throw up
rigid and insurmountable b arriers betwe en the sciences in such a
way that one science cannot influence another, except perhaps in a
very extrinsic fashion. And has not modern science given the lie
to aw doctrine that would establish barriers of this kind? Must
ye conclude that it is illegitimate to ask geometrical questions
in terms of arithmetic or to seek to de mons trate geometrical pro -*
pcgMorib_J> y means of ar ithmetical principles; [If so, what about
analyJi^H^geometrjrTjAnd - - to come directly toThe issue with
Yfiiich. v/j •'..re concerned - - is it illegitimate to raise questions
about pbysics in terms of mathematics or to arrive at conclusio ns
about ( rlalv.'?^ ) through mathem atica l demonstrations? If so, what about
nathei i,: ! ;..c?.l physics? There~is not a modern scientist or philosopher
of science who would not immediately reject any doctrine which would
call i-ifco question the legitimacy of such procedures. And Emile
Meyerson terms the doctrine <taught by Aristotle in the chapter we
have been considering: si cnoquante pour le sentiment de l'homme
nodeme." (141)

Fortunately, there is no reason to take scandalo All
diffiouities vanish when the Chapter is read in its entirety in
the light of the commentary of St, Thomas, and in conjunction with '
Vo'j whole context, particularly Chapter XIII where Aristotle and
St. ■/'homes consider the problem of the subalternation of the sciences,
A:r hhis whole context must, of course, be integrated with their
other writings which treat of this question, notably the passage
from the second book of the Physics cited in Chapter I. This full
and integral reading not onl# dispels all difficulties but io leaves
us with a profound admiration, for Aristotle and Aquinas whose ana-
lysis remains accurate to this day.

In lectio 21 of the Posterior Analytics, after explain-
ing that each scie^~has its own particular quo bo ions, St. Thomas
goes on to give an example taken from geometry, &f >£**??$..
this example he brings in the case of the science of optics ygoh
is^ubalternated to geometry, mid he points out that ^ «>gita
r^e-l^ie^^caTlfestions^n W*%^^*™»*^

isjs ubo.l te rnatcd to ge ome^CM^Jig-^^^^^--^^^'-

And ho concludes: "

Bt quod dict^ OBt.^-^i^S'SKrStlo
te aliis sci/)ntiis: quia scilicet pioposioio, 1

dicitur proprie alicuius scientiae, ex qua demons tratur v__

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to ipsa scientia, vol in scientia ei suba ltemata.

In lectio 15, to the text cited above in which he aoys
that arithmetical demonstrations cannot be employed in geometry he
iixiediately appends this important qualification:

. ,„nisi f orte^ subieotum ) Unius scientiae contineatur
sub subiecto alterius, sicut si magnitudines contineantur sub
numeris (quod quidera qualiter contingat, scilicet subiectum
unius scientiae contineri sub subiecto alterius, posterius di-
oetur) 9 M agnitudines enim sub numeris non continentur , nisi
forte secund um q uod magnitudines nu meratae_sunt. (143)

In this passage written centuries before Descartes \
St, Thomas explicitly allows the possibility of a treatment of geo- J
noisy in terms of arithmetic » '

In giving the reason why demonstrations must be '.
in the same genus, St. Thomas takes pains to explain and qualify
his doctrine with great accuracy:

Quare raonifestum est quod neaesse est^aut esse sim-
pliciter idem genus, circa quod surauntur princxpia et conclu-
siones, et si c non est de scen sus, Cheque transitus de genere
in genusj )aut si debet demonstratio descendere ab uno genere
IrTallud^ oportet esse unum genus sic, ide gt quodammodq . Aliter
enira impossibile est quod demonstretur aliqua conclusioex a- ;
liquibus principiis, cum non sit idem genus vel simpliciter,
vel secundum quid. Sciendum est autem quod simpliciter idem
genus accipituV, quando ex parte subject ! n on sumitur aliqu a
differentia deterninans, quae sit Cextranea ) a natura illius ge-
n^Jr^icuT^Tquii^per principia verifi^ata de tnangulo pror
cedat ad demonstrandum aliquid circa isoscelera vel alxquam aliau
speciem trianguli. Secundum quid autem est unum genus, quando
assumitur circa subie^m aliqua differentia extranea a natura
illiua generis; sicut visuale est ^xtr^neu^generejangae
,(et sonus est extrangus a genere numeri...J ^-s^w

O^^TOiThulc- coniunxerSIi-quol dl Tf^ e q ^^ ae

sint circa diversa genera subiecta; ex necessitate, ^uitur
quod ex principiis unius scientiae non concludatur aliquid in
alia scientia, qu^e_j^i_sil(s^^a)p^sita. ..

I Et sLfeterTluod est ^^^ < r nt ^ v |^S

V /bare alia scientia, niEd^^una^cd^^

\jgSnica, JS^m^SS^^LSSJ^SS^SSSi' K '

A casual reading of these passages might give the im-

-131-

pi-ossion that St. Thomas contradicts himself. First he denies the
possibility of using the demonstrations of one science, such as
arithmetic, in another science, such as geometry. In the next breath
he seems to admit the possibility. There is no contradiction here.
He is merely trying to insist upon the fact that in order to unite
things correctly one must first distinguish them carefully, that
union without accurate distinction can only result in confusion.
He begins, therefore, by insisting upon the distinct character of
the sciences, each of which has its own peculiar mode of demonstrat-
ion From this he concludes that per se , that iq absolutely speak-:
ing, the demonstrations of one science cannot be applied promiscu-
ously to other sciences. Having laid down this basic principle he
goes on to explain that under certain conditions one science may
he brought to bear upon another, in t he measure in which one can
he to some ex tent integrated with the~~other Cj;hrou gh the process
'*' „ Vof subaltornationJ But in the union effected through this subaltera-
)ation\ neither of t he sciences los es its proper character 9 ) The union
of nathematics and physics does not mean that physics is mathematics,
\or that mathematics is physics. Saint Thomas is very careful to
keep before our minds the fact that the demonstration of a geome-
trical proposition through arithmetical principles is a process
that is essentially different from the demonstration of a geometrical
proposition through geometrical principles. All too many modem
scientists and philosophers of science have allowed themselves to
lose sight of this fact. That is why their union is a confusion.

And now, having seen the principles which govern the
( distincti on) of the sciences, we must turn our attention to the prop
blen of their Isubalternation. )

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CHAPTER_th^e

lo The Opecies of Subalternation.-.

In this question of subalternation we are touching
upon one of the most basic and pivotal notions in the philosophy
of science „ That is why it is imperative to handle it with as much
inoisiveness as possible. For the ancient Thomists subalternation
had a rather well defined meaning. But unfortunately not all modern
Thouists have kept its outline clear and sharp, nor have they taken
sufficient pains to keep distinct the various ways in which the
genera], notion of subalternation may be applied. The question has
been handled with considerable looseness and ambiguity, and the
result has been confusion. Let us try to circumscribe 'the meaning
of the word as closely as possible,

Subalternation is sometimes defined in terms of the
application of one science to another, or the dependence of one
science on another,, or the subordination of ' one science to another.
Its notion involves all of these things, but they do not adequately
explain its meaning. In the first place, not every case of the ap-
plication of one science to another is a case of subalternation,
For exar,iplc, in philosophy of science there is a kind of application
of nstaphysics to experimental science. But this does not involve
the subalternation of experimental science to metaphysics. The phi-
losophy of science is a purely metaphysical study, for, as we pointed
out in Chapter I, it pertains to wisdom to make a critique of the
nature of all the sciences including itself. ^ (l) Secondly, subal-
ternation is not coterrainus with dependence. For example, theology,
^ so far as it s-akos use of philosophy, i:iay in some senso be said
to be dependent upon it. But it is not subalternated to it. (2)
Thirdly, the notion of subordination is not sufficient to explain
the r loaning of subalternation. For, philosophy is subordinated to - 1 J

theology, but it is not subalternated to it, (3) yjforeover, _all_
Bjaotioal science is in s^~W~iubordinated to speculative science,
^tthiTsGSSSInation does not necessarily involve subalternatio n^
xt ia true that some practical sciences, such as medicine, agriculture,

-133-

One of the difficulties encountered in the problem
of subalternacion arises out of the fact that the tern is used in
a variety of Trays. Perhaps the best way to arrive at the positive
neanxng of the tern is to begin by considering the different ways
in which one science nay be subalternated to another. John of St.
Thorns distinguishes three types of subalternation. (4) One science
nay be subalternated to another either by reason of its end, or by :
reason of the principles it employs, or by reason of the subject
it considers. Let us consider briefly each of these types.

Subalternation that derives from an end pursued is,
as the very terms suggest, proper to the practical order; it is
found in the practical sciences and in the arts. When the end of
me science, though truly an end within its own order, is subordi-
nated to the end of a higher science in such a way that it is con- ' ~

trolled and directed by it, the first science is said to be subal- VV\<* \ ifa«-\ ^
ternated to the second,, Thus, for example, military science is subal - Ooii'tft^' sw'c.
ts^iated^o_po3i : ticDl_scJ ; eniceo It is important to note that the
first endTnusTTie truly an "end within a certain order, for if it
is only a neans, if the higher science uses it merely as an instr u-;
nent(bhcre :ui^oJreaJI^isj^inction _ of scienc es an d hence no subal -
'temntiohj) In this first type we are dealing with subalternation
in a very broad and improper sense. For, subalternation implies V ft ^, vt ,, ,4 H-.c
the dependence of one science upon another with respect to the rna- ) i
nifestation of truth, and very often when one science is subalter-
nated to another by reason of its end there is no dependence of
this Hind, but rather dependence, wi th respect to use 9 control, di-
!S2l^Sa^and__caxiand., - - something akin to what is found in the
interrelation of the virtues, as for example in the case of charity's
loourand over temperance. And this follows from the very nature of
the practical order, whose object is not the true as true, nor even
the good as truo, but the good as good. It is only in the speculative
order that subalternation in the proper sense of the term is found,
for the object of this order is always the true, and consequently ;
subalternation in this order involves a manifestation of truth.
v ''° are particularly interested in the subalternation of the specu-
lative sciences o

One speculative science may be subalternated to another
in two ways: either by reason of its principles alone or by reason
°f its subject. The first case is had when a lower science borrows
from a higher science the. principles necessary to iU«™*° its
°vm domin,(ahd thus becomes dependent upon^ But m order to have

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(subaltcrnation of this kind in the full sens e of thc tem the dG _
pendenoe must be|neces^ T _aM^ssentialJ tho.t is to say, the lower
science must be J^ms in per se evident principles within its own
Uomiii, and thus bo- forced to reach up to a higher science to have
its principles mde evident. This t yj3e_of_ dcpendence i a found in
the subalte rnation of supernatura l thcology_to_t he soiohce~of~the
blessed , 1 Theology does not rosolve its demonstrations into -o rinc-i.pleg
thatave peruse evident, ) For the theologian must accept his ^prin^
nvplgs on faith. But these principles accepted by faith have their
intrinsic evidence in a higher science — the science of the blessed
injjeayerio It is in this higher ncience that they find their mani-
festation and their proof » That is why theology is essentiall y subal-
ternated to the science of the blessed.

It is extremely important to insist upon the difference
between this kind of dependence and the kind of dependence that
philosophy of nature and the other sciences have upon metaphysics.
It is true that in some sense all the sciences receive their prin- ;
oiples from metaphysics, for as St» Thomas says, "ipsa (metaphysica)
laBgitur principia omnibus aliis scientiis," (5) Nevertheless , ;
( jhe lower sciences do not depend upon meta ph ysics for the evidence
of_thoi r principles „) They are -capable of resolving their demonstrat-
ions into per se evident principles' which are proper to them . They
do not have to turn to metaphysics to have the truth of their prin-r
oiples made manifest or proved. It is true that metaphysics explairis
the principles of the other sciences and defends them by a reduction
ad inpossibile, but it does not prove then in an a priori fashion.
The principles of the other sciences come under the influence of
those of metaphysics 'only in the sense that metaphysics is the most
universal and the most basic of all the sciences. And even though
it has becorecommon to authors to state that the principles of phi-
losophy of nature are (contr a ction?) of the principles of metap hysics
(e.go that tho principle of the composition of mobile being of matter
and form is a contraction of the division of being into potency
and act),<Jre feel_^t_such_£ta^^ the3 T e

is a world of difference between the i^ayiiiwhich the particular
Principles governing a certain f^p7|of friotio^ )are contractions of
toe general principles"" ^ motion T and the way in which the principles
of the philosophy of nature are" contractions of mataphysical prin-
ciples, For "as wo say in the two", in the latter case there is
| not newly a question of the application of the more general to
the more specific; ythg^^s^^u^stion^f two different|ger|^>
It is a serious errVlo~55nfute thc two types of dependence des-
cribed in these last two paragraphs.

It is true that the other sciences may ^netimes us e
'^•Physical principles in their demonstrations. It is likewise

-135-

ti-uo thao they may sometimes employ principles taken from the science
of logic, Bu-c this amounts to no more than on occasional borrowing
f r or.i these other sciences} It merely means the use of an extrinsic
proof o All thfs explains why the dependence ' of the other sciences' ^ .J 1
upon netaphysics and logic is not subalternation in the full sense c J)***^ ^
of the word a And if the tern subalternation is applied to this kind ^o^ .y
of dependence it should be made very clear that' it is only a question l^
of subalternation in a Very partial and limited sense, (6). '' ^

Mow for our purpose, it is not subalternation by reason
of principles alone that is of particular interest, but subalternation
by reason of the object. In this third type we have subalternation.
in the most perfect sense of the word, John of St, Thomas says:
"Tertius modus a ,, inducit propriissiraara subalternationem," (7)
n'e roust try to see why this is so

This third species of subalternation arises when the
object of one science \falls under the object of another science,)
But as ■we pointed out in Chapter I in our discussion of the fifteenth
leotio of the first book of the Posterior Analytics , one object
my fall under another in two ways, First of all, It may merely
be a question of a more specific object being contained. in a more
generic object, in the way in which, for example, animated mobile
being falls under mobile being. In this case it is evident that
there is no real distinction of science and hence no possibility of true

subalternatioiio Every science explains its, object by division :
as well as by definition, and consequently in order to have the
formal distinction of science that is required for subalternation,
it is not sufficient that one object add an essential specific dif-
ference to the other. And this explains why many of the apparently
hybrid sciences to which we alluded at the beginning of Chapter II
(e„g astro-physics, bio- chemistry, etc,) do not involve true subal-
ternation, since it is only aqugstjnn of the union of "too branches
of the same, science. There is, consequently, a world of difference
between the hybrid character of these sciences and that of raathe-
mtical physics in which physics is truly subalternated to mathe-
natics.

Because the subalternated science must be egr|n|ic
to the subalternating science, the difference wh ich the ob jeqt of the
one adds to the object of the other ffi^^^S|2^g^^-
An example will make this point clear. Let us taJte ™e y,
-tion of "line", We may add to ^a £ ion n ^ws.^

.<r>

nocion of "line". We nay acta vo ^ "~";~ n i. B tmight« and'
of all, we nay add the H2ES2LSe52*H 7 |2*£22£S2^ ^ra g (

"curved" , \^J^s^EM^^^^i^^^^^^4^ .
55_ d 'burved Idn^T^oin^f^nicFfSl under the g°^ic ° D £ ±

^IoTn^tMSWojHJLi*22^^ SX

-136-

which deals v/ith a certain genus necessarily deals with all the
p:toper species which fall under it. But it is also possible to add
to the notion of line the extrinsic and accidental difference "vi-
sual"? and thus arrive at a new object, "visual line". This new
notion is not a prope r speci es of the generic geometrical notion
of line T Hence it does not fall under, the science of geonetryln
the"sense of being a part of its object. In fact it constitutes a
new science, [the science of optical known to the ancients as "pers -
pectiya"° This new science, v/hile not falling under geometry in
the sense of being a part of it, does come under it in some wa y,
since the notion of line which is compounded with the notion of
visual to constitute its object is borrowed from geometry,, Tn other
words j (optics is subalternated to geometry by reason of its object ^

Perhaps another simple example will clinch the point
ve are trying to make. We may add to the generic arithmetical notion
of lumber the two proper essential differences "rational" and "ira
rational", and thus arrive at tw o num e rical species, both of which;
pertain essentially to the object of ""arithmetic. But we may also ,
add to the notion of: number the extrinsic and accidental notion of
sound and thus arrive at a compound object which constitutes a new

science, distinct from arithmetic, but subalternated to it the

s£ience_of_msic o

Now su balternation by reason of. the object aij jggg_in-
volves at the same time _gubalternation_by reason' of the principles..
ThiTihould^be fairly~evident from theTexamples just cited, F°r> _
since the formal object of the subalternated science is constituted
by the addition of an accidental difference to the object of Jie
sulalternating science, the subalternated science cannot treat its
object and prove its properties except by having recourse^ to T,he
conclusions of the subalternating science. But subalternacion by
reason of the principles does not always, involve suoalternation
by reason of the object. The contrast between the way theology is
subalternated to the science of the blessed and the way optics is
subalternated to geometry brings this point out wioh sufficient
"larity. As we saw, supernatural theology ™s, reach ^P* the bianco
of the blessed in order to find the evidence of its P™?£P les .

Nevertheless, i^^le^LJSS^m^^^^^^^.
^2Bd§Gtal djy^rejicejfcojhe_p^ecW .

ll^Tl^flcV^hT^y-iaSe-oble^^

lights: (the li^ht joLthaiSiW^^

Qb.ieot.-lh!hi-fIrIt-5aBe we have a slJ f ^ "°^°^f ^ confound
frora all sensible mttor. In the second case ^^ t ^l element
object made up of this staple notion plu^jmjicciaenx,

-137-

(vjhigh JJgpl yea .. scnsi blejaatter) There is an enon.ious difference
Ibewcen these two types of subalternation. In the first type, the
sU balte mated science remains a simple science . In the second type,
it becorjes a com plex science , fa h ybrid soience? )a scientia medi a,
(bnnau se its object is compounded of elements which involve two dif -i
I "oront levels °^ in telligibility g ) ~~ '

The three fundamental types of subalternation just
described are the only ones Mentioned by John of St, Thomas in the
article cited above. We may well wonder whether the list is exhaustive,
For St, Thomas in his Commentary on the De Trinitate (8) gives :
us a case of subalternation which does not seen to fall under any '.
of the three groups listed by his disciple We are referring to
the case already mentioned earlier in this chapter in which the
practical sciences of medicine, agriculture, etc, are subalternated
to the speculative science of nature. We pointed out that .this
subalternation does not arise merely from the subordination that
all practical science has to speculative science, (but from the spe-
cial ch aracter of the dependence which these few p rac tical sciences
havgjroo n the science of nature ) St, Thomas brings out the nature
of this special relation with great clarity and precision:

Quanvis eniin corpus sanabile sit corpus naturale, non
tamen est subjectum medicinae, prout est sanabile a natura,
sed prout est sanabile per artemj Sed quia in sanatione quae
fit 'par artem, ars est rainistra- naturae , quia ex aliqua naturali
virtu te sanitas~lperficitur auxilio artis, inde est quod propter
quid de operatione ariis oportet accipero ex proprietatibus
rerum naturaliun, ^propter haecj2e]lD£ina__£a]b^^
ggeT~eT~ ^^ratione~a1i ^^»_et_soj£nc ia de agriculture .,
eFomnijniiusmpdi^j Et sic relinquitur, quod.physica secundum
ieT^t secundum omnes partes eius est speculativa, quamvxs a-
iliquae operativae subalternentur, (9)

It does not seem possible to fit this type of subal-
ternation directly into any of the three groups described above. ■
It is not the case of subalternation by reason of the end, f ^ we
do net have one practical science subordinated to another P^^g_
science. No* is it question of subalternation ^ "ason ° f JJ° gin
ciples, for a practical science cannot receive ij^^erjxg^^s
fror, a'speculative science. Since the end of a pra ot ™f^™ c °
iLSot to know ^vhy" but "how " , ^^^^^^^^^T
iJ^BF^i^^T^r^e^^£^i^ts> Fin fg' Sect for
Vo^i^A^^rST^^S^^WrSSfon °£ n ^.* a ^ foments
clooents from a practical science cannot be ^^edwit f ~ e ,
^ken from a speculative science to oonstxtute the object of a sinp ,
unified science. As a matter of fact John of St. Thomas, alter

-138-

plaintag the three types of subalternation, explicitly denies that
nacioine is subalcernaoed to natural science: "Medicina (agit) do
corpora sanabili, et tamen non subalternatur Philos'opliiae, quae
agit de corpore," (10) From the context, however, it is evident
that he is merely denying the possibility of subalternation by roa-
so^oj^tho_oboeci» And even though the way in which medicine" and " '
agriculture are subalternated to natural science does not fit di-v ■
rectly into any of the three groups listed by John of St, Thorns,
it nay be reduced to a case of the second group, For while it is
true that a practical science cannot receive its principles from
a speculative science, the principles of medicine and agriculture
are completely determined by the principles of natural science be-
cause of the unique character of the relation existing between these

I sciences o Perhaps nowhere can the Aristotelian adage: Ars imitatur
natural a be applied wi th such fu'Xhess as here. In fact, the imitation
is so perfect that in a certain sense it results in on identification,
for in medicine and ag r icultur e , dthe works of art must be at the sa -

Vne~tis.re wor ks o T'n atureTX

It would seem that if the concept of subalternation
is conceived as embracing all of the various cases we have described,
it can hardly have a strict unity. Nevertheless, there are two kinds
of subalternation in which the concept is realized in its proper
and strict sense, and in which it has a definite unity, Tfe refer
to subalternation by reason of the principles in which ishere is an
essential relation of dependence between the subalternated science
and the subalternating science, that is to say, when the former
receives its prope r principles from the latter, and to subalternation
by reason of the object. When the ancient Thomists speak of subal-
temation, it is usually this strict and proper sense of the concept
that they have in mind., and it is in this, case that we shall speak
of it from now on.

And now, having reduced the notion to this definite
neaning, it remains for us to explain in what its essence consists.
But before pursuing this analysis it is worth while pausing a, this
point to remark that every effort should be made to maintain a clear
ou^istinction between the various kinds of subalternation we have
teen describing. As we pointed out at the opening of this chap oer,
, this toa not alway s been^oneJy_moden i Thomists,, We are being told
b7l^ori-tn^n - o^r'c'olv^mporary writer for example -cha-. P^° S °P^
of nature is a scientia media, born of a union of the first and
bhird degrees of "abHrlctiSnT or, even worse, arising out of the
^plication) of metaphysics to the data of empirical JW^'W

And we-Wsider it extremely "^^^HnsSt- as so" authors ftjA* &«.
qualifications and distinctions are .jado, go 1 ^sist, as V^-M^

do, that in modern tines mathematics has come to occupy the same W'»M^

-139-

position in relation to the experimental sciences that metaphysics
held for the ancient Thor.iists.

2. The Essence of Subalternation.

The intrinsic nature of subalternation follows from
the intrinsic nature of science itself. Science is a certain know-
ledge of things in their causes > and for the human intellect this
ueans knov/ledge arrived at by a process of demonstration. Now knowr
ledge that is arrived at by demonstration is never self-evident
knowledge o Conclusions do not have their evidence by themselves,
but from something else, namely from the immediately evident prin-
ciples from which they have been derived. That is why the intellect -
ual virtue__o f science is essentially dependent . upon another intel-
lectual virtue, known as the intelle ctus principiorum , which is
the habitus that enables the mind t o grasp . immediately th e truth
of self-evident principles. Now the, (essential difference ) between

(a subalternated science and a science that is not subalternated is
that the habitus of the latte r |is in Immediate continuity; with the
habitus principiorum , whereas the habitus . of the former is only
Mediately in continuity with it, through the habitus of a higher
science , (known as the subalternating science ) (12)

In other words } no science is a science in and by it-;
self, but in and by' its continuity with a superior habitus, for
without this continuity its conclusions cannot have the certitude
that is necessary for scientific knowledge. A science that is not
subalternated is a science that is in direct continuity with the
habit us principiorum from which it immediately derives the evidence
of its conclusions. On the other hand, a subalternated science is .
one that is in direct continuity with the habitus of a superior
science, and only throu gh jhig_habitu§ is it in continuity with
the h abitus prinoipiorum o

At this point it will be helpful to draw a contrast _
between the way supernatural theology is subalternated to the science
of the blessed and the way other sciences are subalternated - -
not because we are particulars interested in the + subal ^^°"
of theology, but because the contrast will serve to accentuate the
character!^ features that are found in the intermediary sciences
in general and in mathematical physics in partic «^r. In th e oub al
ternation found in all the other sciences besides teology,|g
ggxim te principles of j ^jsutoltemaj^
jjjon s^ratea by the iub^lt ernatlni^sciencgj;

-140-

o , » scientia subalternata non utitur principiis alia-
rutt sciontiaruivt, sed conclusionibus : assur.iit eniii principia
quae probantur a scientia superiori tamquara conclusiones, non
auten principiis suporioris scientiae utitur resolvendo usque
ad principia per se nota, (13)

When the subalternating science does not coexist in
■the .qn ne intellect along with the subalter'nated science , C thcsecon -
nlns ions are taken on faith. ) But this does not mean that in this
case the orinoiples of the subalternating science are taken on faith,
For the intellect which possesses the subaltemated science may
posset the principles of the subalternating science by means of
■the habitus principiorun ^ without possessing the habitus of the
subalternating science itself. In this connection John of St, Thomas
writes:

„„. in sciontiis naturalibus non potest verificari
quod ipsa principia per se nota ipso lumine principiorur.i in
superiori scientia, sint tantum credita, et non per se nota
in inferior!: quia quod est per se no turn limine principiorun,
orinibus est per se notun; at principia quanto sunt ,-superiora,
et ad sciention superiorun pertinent, tanta sunt raagis nota
omnibus propter suani universalitatera, (14)

This only refers, of course, to principles that are
self-evident, and not to the. postulates which a science nay take
as its principles. In this kind of subalternation th ere f^
points to be noticed about th £ _ 2 ra £ ^^c 2 vl^ f the subaLtern
ated science; first, thgjrjgg not evident; secondly, gey are me .
diate, that is to sayT^^ the fruit of demonstr ^™f™
pSTciples thaf arc evident.These two poin f^^^tkr.
for it is possible for principles noo ,o he eviaeno v ^_.

being mediate. And in this distinction we f ™\*J^™£n con-
ference between the kind of subalternation we have ^* c ™ con
Udering and the kind that is found in supernatural .neology.

The proper principles of theology orj J*^^
but not all of them are mediate «££%££ ^£s. Now as Cadetan
and others are truths consequent upon ™°se * inevidence and that
points out, (15) although both the « -evid ^^ ^

of nediacy are ordinarily considered 1»P^ tQ ±t ta a foroa i
subalternation in some way, the ' f°*££ P Hence , in order to have
Tray, and the latter only m a ^^^l^oesBB.vy that the proper
true subalternation it is not Qt £0 lut ety ^on^it ±g suffioi ent
principles of the subaltemated be qon ^ Thowas maintains

that they be not evident. In fact, JO ' u t conc iusions there

that in Theology's use of principles tha- ar

-141-

is a fuller kind of subalternation than that found in the natural
sciences where all the proper principles of the subalternated science
avo necessarily conclusions. For, whereas in the latter case, as :
x; a pointed out above, at least the principles from which the con-
clusions are drawn are evident, in the former case the fundamental
principles are in no way evident, (16)

But here it is important to distinguish between Wo
Jdnds of continuity, which for want of better terms we shall call
objective and subjective. When the continuity is considered from '
the point of view of the objects that the science is about it is
objective; when it is considered from the point of view of the scientist
it is subjective. Another way of expressing the sane idea is to
say that objective continuity is the continuity that a science has
b y its very essence , while subjective continuity is the continuity
that iT has because of its actual state . Wtien a subalternated science
is in its perfect state there is subjective as well as objective
continuity,, But when it is in an imperfect state, subjective con-
tinuity may be lacking. And here it must be pointed out in passing^
that when Thomists raise the . question about whether or not a certain
subalternated science is in continuity with the subalternating science,
it is to subjective continuity that they are referring, for, obviously,
there can be no question about objective continuity since it is a ^
necessary condition for the veiy possibility of subjective continuity.
But psrhaps the best way to explain this distinction is by means
of an example. The science of . optics necessarily has objective con-
tinuity with the science of geometry,, that is to say, its proxima-co
eyijiciples are geomstrical conc lusions , which in turn have their
evidence from their continuity with self-evident principles. But
from the point of view of the student of optics this continuity
my or may not exist. It exists jfJ iej : s_am& thel ^ tic:i - an . as . 1 wo1 - 1
as a student of o ptics ^t does not exist if the geometrical con-
clusions which he applies to his particular natter are merely ac-
cepted by him on the authority of a mathematician without their
intrinsic evidence being grasped. Prom this it follows ohat the
habitus of the proximate principles of a subalternated science is
;pqr_ae the habitus of the subalternating science. Per accidens , •
hovrever, it may be a matter of authority alone,

Tn this distinction of the two kinds of continuity
• We the sSuS n Tl P^to^ich £» g^-^gives

ZS^^^XJ^J^^ g Sf - - -e subal-
ternated science tD be a ^V^s ntceSarily obtain knowledge.
Beou not. For scientific knowledge is necessaiiy ^
And how can knowledge be certain if xt " reauo
ciples which are hold on authority and are not £er_s_

-142-

oiples? Docs not, St;, Thomas write: "quaecumque sciuntur propria
accepta scientia, cognoscuntur per relationcm in prim principia,
quae per so praesto sunt intelloctui," (18)

As wo have just said, the correct solution of this
problem lies in the distinction between subjective and objective
continuity. Even when subjective continuity is lacking, objective
continuity is always there, and that is sufficient to insure the
■truly scientific character of the subalternated science, For objec-
tive continuity means that the proper principles of the subalternated
science are do facto demonstrated in the subalternating science,
and thus there is the essential connection between the subalternated
science and self-evident principles which St, Thomas demands in •
the text just cited.

This problem has particular significance for the science
of theology, which, in this life, is based completely on faith.
But it also has relevance for the question in which we are interested.
For we can imagine the hypothetical case of a student of nature
who, though unacquanted with the pertinent mathematical demonstrations
that are presupposed, might accept the mathematical conclusions he
needs on authority and employ them in his interpretation of natura l
phenomena. The conclusions concerning nature that he would be able
to arrive at by using the borrowed mathematical conclusions as prin-
ciples would express ob jective truth,, even though they could not ■
to called scientific truths o n the pa rt of the 'student him self.

From this we nay conclude that a subalternated science .
is specifically the sane scientific habitus whether there is sub-
jective continuity with the subalternating science, or not. For even
when subjective continuity is lacking, the objective °°noinuity
establishes an essential relation between the subalternaced and
the subalternating science, It is this essential relation that de-
termines the nature of the subalterned habitus. And .his essential
relation demands completion by subjective oontinuit y. Jence, as.
long as subjecti ve continui ty is lacking the ha bi^us °* f* ^J
ternated science is in an imperfect state, Bu^n^s_a^yir|d,

JIjgB'icMoTr^The following lines ox So. Thomas wow &
this subtle point:

attingit ad rations* sciendi, nisi in qaan sci entiam

continuatur quodamnodo cum °°gnatione eius, qg ^ ^.^
subalternantem, Nihilominus taraen xm <^ xor con clusionibus,

de his, quae supponit, habere BOwnUmj. sed ^^ (±g)

quae ex principiis suppositis de neceasiut

-143-

At this point we must turn our attention to a highly
significant passage of John of St„ Thorns:

»;> non fadit subalternationem sirapliciter hoc quod
est nutuari aliqoud principium ad aliis sciontiis, ad proceden-
dum ex illo tamquam ex principio extraneo' et rautuato Ratio
est, quia subaltErnatio propria et simpliciter, requirit quod
aliqua scientia ex propriis principiis et intrinseois non pos-
sit resolvere in principia per se nota; sed pro evidentia suo-
run principiorun necessario debeat reourrere ad aliquam aliara
soientiam , quao talem evidential-.! faciat. Si autem utitur prin-
cipiis aliarun soientxarum tamquam oxtraneis et mutuatis, et ■
in illis solum rccurrit ad sciontiam extraneam pro illorum e-
videntia: non manot subalternata intrinsecej quia quantum ad
propria et intrinseca principia non accipit evidentiara ab alia
scientia 5 sed solum quoad principia extranea Et ex hoc judi-
canda est subalternatio propria et intrinseca: scilicet an in-
veniatur in principiis intrinsecis et propriis alicuius scien-
tiae, an solum in externis et rautuatisj (20)

These lines have two obvious references-, In. the first place they
refer to a point made by John of St„ Thomas in the Oui- sus Philosg -
phicus which we have discussed earlier, in this chapter? an occasional
and extrinsic borrowing of principles from other sciences, suoh
as metaphysics and logic, does not constitute subalternation in\. •
the strict sense of tho word. In the second place, they refer to
the immediate context in which the author shows that theology cannot
be subalternated to philosophy even though it uses philosophical ■
principles in its demonstrations, 'for fi:..-st of all it does not take
then as its own proper principles, and secondly it uses then only
after having judged" them in its own supernatural light and elevated
then in some way to its own level, and thus the whole essence of
the demonstration rests formally and ultimately upon the superna
tuTal principle.

But it is not particularly ^^Vre^alheTif ^
'references that we have introduced th is passage , her ^^it^
is because some of the statements in it « 1V ° "^ ??„ f ound ^
touches the very essence &f the type of subalternation found m

.mathematical physics,

,'_,,. „„ r,f Tnhn of St, Thomas suggests, the

As thiB passage of John ot ^ t ghall cal l

ancient Thomists do not seem to have co^er^^^---^^^^
fealectioal subalternation, thai is co say, au ^^ gcience

the subalternating science aoes not give to « principles

in ,an intrinsic and adequat ely the evidence of the pr^nc^ ^
tnaTo7e~propor to the subaltSrnated science

-144-

is not realized a sufficiently perfect continuity beWeen the two
disciplines m question to permit the formation of a science in
Uhe strict sense of the terra. Now this is the type of subalternation
that is actually found in mathematical physics. And that is why
vro nust develop this point a little further.

The raediaval Thomists recognized the existence of ma-
thematical physics, and they accurately analy g;d its nature as an
intermediary discipline that involves the fullest kind of subalter-
nation - - subalternation by reason of the ob.ject . The/ carefully .
distinguished this type of subalternation from that found in theo-
logy where the principles alone are involved„ Nevertheless, for
them there was a fundamental parity between these two types of su-
baltematioho Just as there was a perfect continuity between the

I principles of theology and those of the science of the blessed, so
there was a perfect continuity between the principles of Physics

! and those of mathematics - - at least sufficiently perfect to permit
Mathematical demonstrations to be applied adequately to physical

UihenomenaT)

We ore referring here to a point already mentioned
in Chapter I, where we explained that for Aristotle and the medieval
Thonists mathematical physics could constitute a science in the
strict sense of the terra because physical entities realized a suf-
ficiently perfect conformity with mathematical entities to allow;
for She former to be treated in terms of the latter in strictly
scientific fashion. The reason why they held this view was that
they were without refined experimental instruments, and had to de-
pend upon sense experience. Now rough sense experience is extremely
illusive. It often gives the impression that things in nature hav^
a perfection which as a natter of fact they lack. The sense of to^ch
say convey the notion that a surface is perfectly flat; the sense
of sight may give the impression that a physical sphere is aJ22£T
fectjrphere. Consequently, when there is noth ing .<^to go on bus
thiT^gTT experience one'is easily led to feel ^stifled in pos£
ting the hypothesis that physical lines and figures reasonably ap 7
VProach mathematical perfection.

The refinement of our modern toste^nts has e^i^ed

the ga p between physical ^^f^^/^^l^°l^ S
surenents are only approximative, For this re ason ti(jaQ ^

cessary to hold thatjaaJhemati^al.ph^i Sg if ^^ T^l^id~in~~^

jj_jj_ purely di alectica l,

,, .^oj^tpiv add that we are con-
But perhaps we should acedia cely a ^ ^

Bidering the question hero merely from trie poino

-145-

[lrao-Tledge of which the human intellect is capable in its present
state o For we see no reason to exclude a priori the possibility
of the existence in nature of entities whose perfections approaches

I mthemtical perfection sufficiently to allow for their being treated
in terns of mathematics in a striotl y_s.ci entifio wa y. We have no
nsans at our disposal to make it possible for us to arrive at this
perfection, but perhaps the knowledge of this perfection is possible
for the angelic intelligences, or even £or the human intelligence-
in a superior state. If this should be true, mathematics would be
able to provide a strictly scientific propter quid for natural phe-

■ nonena

But porhaps what we have just said about the opinion
of Aristotle and the medieval Thomists nay give rise to a problem,

]Por if they believed that there existed in nature entities whose i
perfection came reasonably close to mathematical perfection, why
did not such entities fall directly under the object of the study
of nature? Why was it necessary to study them in terras of raathe-
natios and construot the theory of a scion tia media ? Why was n6t
the so-called science of mathematical physics nothing but physics?
Does not this bring us back to something akin to the opinion of ;
Professor Mansion criticized in the last Chapter? The answer is
that even if the conformity between physical and mathematical en-
tities 'were perfect, physics would still have to be subalternated
to mathematics* For the concrete 'quantitative determinations of
nature, in so far as they remain attaohed to sensible qualities,
are not susceptible of the conceptual elaboration of which mthe-
mtical quantity is capable. Quantity is by its very nature more
abstract them the sensible qualities, and it has ios own reasons ■
prior to those of the sensible qualities, and this would neces-

Vsarily lead to subalternation»

A few general remarks remain to be made' in order to_
complete our consideration of the nature of su ^j^* 1 ™' ^Jf
first place, it should be evident from what has £^ady teen said
that a lower science must be subalternated to a higher science and
not vice versa, abste actiora et s^liciora

'Widerat, tontoS£ p'rincipia sunt magis appUoa^a aliis ^
"scientiis:'unde principia mathenaticae aunt applicabil ^lUtone
"bus, non autem e converse propter quod P^xoa 3 8 x supposition?
"mthemticae et non e conserve, ut patet m III Coeli. ■ W

-,„■■, n + tines use the principles of

fon-.nl, for the higher science in that case jj i ^ poste _
Pies of the lower in terras of its superior lxght. V ;

-146-

riwj£S&rtios» St Th ° rnas S ive s us an example in which a mathematical
proposition is demonstrated in physics &

Sunt enin quaedan propositions, quae non possunt probari nisi
per principia alterius scientiae; et ideo oportei quod in ilia
soiontia supponantur, licet probentur per principia alterius
scientiae Sicut a puncto ad punctum rectam lineam ducere, sup-
ponit georaetria et probat naturalisj ostandens quod inter quae-
libet duo puncta sit linea media. (23)

It should also be evident that the subalternated soience
and the subalternating science can coexist in the sane subject, that
is, in the sane intellect. In fact, this coexistence is the normal
case, for it is synonyispus with the subjective continuity we spoke
of above o One could not get very far in analytical geometry without
possessing the science of arithmetic and al gebra ,<j iQr in mathemat ioal
p hysics without a personal knowledge of mathematics » ) In the case
of theology this , coexistence of subjective, continuity v/ith the sub-
alternating science, is impossible in this life but it will be rea-r
lized in the next, for after death , | t he habitus of theology wil l ■■
p erdure „) efren thou gh faith has disa p peared .

The subalternated science and the subalternating science
my also coexist in the same object. That this is true of the mo»
terial object is obvious. It is also true of the formal object tfra-
tio^ojmlis^guae) but in that case there can be subalterhation
only by reason'""of the principles and not by reason of the object.
And here we touch upon one o& the fundamental differences between
the two kinds of subalternation Theology differs from the science
of the blessed only by its ratio fo_m ^is_sub_qua: it studies God
under a different light. But the ratio_f orjna lis ^ quae , that is the.
£tioW.etatis is the same. But in the intermediary science, not
only is "the ratio gornalis sub__ qua different ( a dif f erent type
of abstractionTbut also the ratxoJ^rmlis_quae s for it is a c»
Pound object arising out of the" addition of an extrinsic accidental
difference to the object of the subalternating science. And .in order
to understand what this involves we must ' now anatyae more closely
*e particular kind of subalternation found in the intermediary
science

3, Subalternati onjuid_JcienMsJfeaH^-

Let us begin our ^sis by "Tf^^S^W.
^quired in order for a^cientia_media to exist. We have aireaay

-147"

ohea upon sono of thorn.

In the first place, the object of the subalternated
science i.iust contract the object of the subalternating science ana
add something to it. This addition cannot be an essential, specific
difference, for otherwise there will be no formal distinction of ''
sciences, Neither can it be a property that flows essentially ftjmn
tho object of the subalternating science, for the- sane science which
deals with certain object deals' with all the essential properties
of it. Consequently, the addition must be an accidental difference
which makes the natter of the subalternated science extrinsic to .
that of the subalternating science. But not any kind of accidental
difference is sufficient to constitute a scientia media . For there
are soae accidental differences T?hioh are not the source of any
special scientific properties, and as a consequence they are incar
pable of constituting a new science. For example, there is no sci;-
entific fecondity in the addition of the notions of "hot" or "cold"
jto the mathematical notion of. "line". But there is great scientific
fecundity in the addition of the notion of "visual", as the science
of optics attests. In the sane way, the addition of the notion of
"visual" to the notion of number does not give rise to special sci-
entific properties, while the addition of the notion of "sound"
does, as is evident' in the science of nusic

It is important to understand accurately the acciden-
tal character of the difference that is added to the object of the
subalternating science This accidental character must not be con-
sidered from the point of view of the two sciences themselves, in
the sense of their being only an accidental difference between them.
As a natter of fact, there is a specific and essential difference
beWeen the subalternating and the subalternated sciences, Rather,
it nust be considered from the point of view of the being which
constitutes tho object of the sciences. In other words, to- use
scholastic terminology, the difference is accidental to^the object,
not in osse.scibili, but in esse rei. But, as has already been sug-
gastea/TiorwerFa'ccidental difference in esse rei is sufficient
to constitute a rd^ed science. It must be a difference of such a
nature that it gives rise to certain now scientific truths, «£
these truths must depend for their explanation upon the principles
borrowed from the subalternating science.

t ^ w?o tho relation between the two elements

In other words, thg_r eiaiiioii — j» tpTTla aiarv- science
that are combined to constitute the object of ^^^ e ^J^ io v
»»t be a natter-form relation. The *£*£^%*£?^ ^lowL
science plays the role of form, and the elemen; 6 f*™ ience

science plays the role of natter. For ^e subalternating soi
mist illuminate, determine and inform the Bubalternaw

V

-148-

This is what St Thorns has in mind when he writes:

Scientiae mediae, de quibus dictum est, communicant '
cur.i natural! secundum id quod est materiale in earum conside-
ratione, differunt autem secundum id quod in earun considera-
tione est formale, (24) Subjection inferiozis scientiae con- ,
paratur ad subjectum superioris, sicut materials ad formale, (25)

In every intermediary science, we have an application
of the object of a higher science to the object of a lower science,
When, for example, in physics, we speak of light being propagated
in ft straight^ line. I the line in ..question is neither ph ysical alone ,
,^y~7 Siheuatical alone „ ) It cannot be purely physical, for it is
conceived as being perfectl y straight . Nor can it be purely mathe-
mtical, for it is the physical entity of light that is being pro-
pagated. Consequently, it must be both physical and mathematical
at the same time, • ■ • :

But such a line does not exist as such in nature. It
exists only in the mind* It does not however exist in the mind me-
rely through a simple process of abstraction, ! Ra ther it is ' born j| _- [
there through an act of co mp osition on the part of the int ellect,,)
And "it is extremely important to grasp jshe difference between the
fconposite . character ^ of the notion of the physico-mathematical line,
and""the composite character of the notion of "rational animal",
for example. In the latter case, the composition is not treated by
the nind; it is merely discovered by, it. Th at is why it comes into
being through a simple process o f_abstraction° In the former case' _
the"ca^ol±tiSrii~ta=olted by the mind. It is a^ripriin the Ivantxan X
sense of the term. This is an important point to keep in mind. It-
vrill be of vital importance when in Chapter XII we_oorae^oj3sguBp
hov/mny concessions a realistic philosophy of- mth ematical physics
^ngEeTSTEhtianism. BuTl^st confusion arise it must be pointed.
STTiiSidiately thafevbn though treated by the mind, the "«i?n
tef,oen the two elements is not completely logical. They are brought
together by the mind - - but _for an ob je^tive_reason.

Now this composite character of the object of th^in-
temediary eciences gives rise te a serious ^ f ^"y^or John
of St. Thorns, (26) For an object that is ^f^^^ ^ ity ,
Wtlon of an accidental difference °anJiaye^y_an_a^id^|^ity,

^^Sis^Js^sJE^^^^^^^^^SS^ to have
Per accidens non datur scientia per se . " ^/^entally one,
enesaontial definition of a being thMis S" 3 ?. ^ i^ scxriethlrig
strictly one, and a being that is only accidentally ono

-149-

/consist of a genus and its specific difference. But the unity of
a soignee is deterr-tingd. by _the uni ty of its definition s, since, •
Jfv/e sow~Tn"Tihe lastChapter, dcHnTtions~are th e principle s of e -

Perhaps one night be tempted to think that this no
longer constitutes a real problem, once we have granted that the
intermediary sciences are not sciences in the strict sense of the \
otxI, b ut dialectics 0| We believe, however, that this would be on I
illegitimate inference * For though these sciences are dialectical \

thoy are not sophistical, and only sophistry deals with ens_p_er J

aooidenso Though they are not sciences in the strict sense of the
yor&jTthey must proceed a d nodtun scientiae » Consequently, the problem
is still relevant,,

John of St, Thomas solves this problem by pointing
out that a sci entia media does not have as its object simply and
directly the composite of the two elements considered as an.acci-,
dental being. Rather it considers directl y onljs one of the two e-
lenonts = - not absolutely and by itself, but in so far as it con-
notes the other and is modified and informed by it. For example,
the science of optics, as the very name implies^ has as its direct
object "the visual". However, it does not consider it independents
by itself, but in so far as it is determined by certain m-cheraatical
properties. And thus it is possible to consider a certain object
as being scientifically knowable £er_se, and as being the source
of certain necessary scientific truths, even though m °rdcr_to
be the source of those truths it requires the accide ^al addit ion
of an extraneous element. For there are a number of P*^^*"*
do not flow from' an object when it is merely cons idered absolu tely
by itself, but only when it is considered as determined ^ified
and informed by a certain element, which, though ^f^^ 1 "'
is absolutely necessary in order for these prop ertie s tc^aris e^
For example, .there are certain ^^^^ but as deter-
of sound when it is considered not by ^seli aion^ .

rUned by number. In other vrords, although th f J^^iaental, since
too elements is accidental, the °^.g^onxsno|^^ ,

by means of it certain ^^^.^^^^S^^^ *-

amlogy will add clarity to ^s point^Paternity ^^ fa „

cidental to man in the sense that not axx men fram

thera. Nevertheless, a number of ess enti g Proper bi ^

the notion of man when it is considered precise^ ^
notion of paternity, which do not arise when it
fepcndcntly of this determination.

Xt must be noted here in ?Z^t££<F^
because the mathematical element enters into

-150-

mtical physics bjLway_ofjaGr^_corw^^

rvvfci.cs in pbysics_is _ essentially funoti onoT lSia instru mental J

Now since the object of a mixed science is a composite
of elements taken fi-ou different levels of intelligibility, the
question arises whether the abstraction employed in it is dual,
or specifically one p John of St. Thomas explains that it is only
ono, ond that is a special intermediary abstraction that stands
jn between the two levels of intelligibility from which the elements
have been borrowed, and that participates in the nature of both.

Quod vero additur de Musica and aliis scientiis subal-*
tcrnis, respondetur in illis non esse duplicera abstractionem,
sed unicam, quatenus principia seperioris scientiae ex appli-
oatione ad talem materiam redduntur minus abstracta et conse-
quently? pertinentia ad diversam speciem in genere scibilis,
et ilia abstractio, quam induunt in tali materia, unica est,
et i deo aliquid parti cipant de_ut risque 1 (unica tamen abstrac-
tione^ sicut medium unum existens dicitur participare at> extre-
mis, (27)

The significance of the Thoinistic doctrine of scientia
gedia has not always been correctly understood. Thus, for example, ;
Professor Salman writes:

Quant aux sc ientiae mediae , dont on a d'ailleurs beaur-
coup oxagere 1' importance theorique, il ne faut y voir qu'un
simple accident historique, Quelques probleraes, plus faciles,
avaicnt recu des geometres grecs des solutions fort precises,

et dont lo caractcre nathematique etait des lors plus accuse.
On a done pu croire que la theorie des cordes vibrantes, lo.^
catoptrique, 1'astronomie, se distinguaient de quclque maniere
des autres parties moins evoluees de la physique. La differen-
ce n' etait oependant qu'apparente, comme on l'a souligne plus
taut en faisont valoir des elements mathematiques inplicites
/ des foroules rudimontaires du langage coramun. On remarquera _
d'ailleurs historiquement que ces sciences intermedins n*
terviennent jamais directemont dans la classification des sconces,
<a aiB sont seu lement ajouteoB^Jag-^SEgg^g^y^fg^P
ELLes ne deri^eTit^aT^n^Hitllo^allm^de la theoiie de s
degves d- abstraction, mis sont des donnees ^ fait, assea ge
nantes d'ailleurs, que le theoricien integre come il le pent
dans une synthase qui ne les iDrevoyait pas. \^)

Wo fail to see any foundation for the objection that_

^intermediary aoienoea do not ^J^^^£rS£ir,
cation of the sciences. By the very face that tnoy ar

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they obviously could not bo put directly into any one of the throe
general types of knowledge that ore based on the degrees of abstrac-
tion, I? * his is wha ' fc Professor Salman has in r.iind when he says
that they do -Hot derive normally from the thocry of the degrees
of abstraction, his observation is perfectly true. But then it is
an observation that is utterly lacking in significance. On the other
hand there is a sense in which it nust be said ;hat they derive
essentially from the degrees of abstraction. For it is only by seeing
these sciences precisely as intermediary sciences, that is, as com-
binations of too different levels of intelligibility which arise
out of two distinct kinds of abstraction that vn can understand
their true nature It is utterly impossible to grasp the meaning
of thcse__ soiences except in relation to the degrees of abstraction ,
THaifis why it :'..?, completely false to say that they are mere "don-
necs de fait" which the philosopher must force arbitrarily into
^synthesi s wbioh has no natural place for them^ ) Nor did Aristotle
or any of the yreat Thomists ever show any signs of the embarrassment
of which Profeasc-r Salman speaks,

Vfe feel that perhaps enough has already been said to :
show that the intermediary sciences were far from being 'a simple
histoi-ical accident", and that the difference between them and pure
natural science is essential and not merely apparent. The further ,
analysis which is to follow will add clarification and confirmation
to these points. Mathematical physics is specifically distinct from
pure natural science because it con .taius an essential element taKen
from the science of • mathematics . And yet the introduction of this
extrinsic element into experimental physics is necessary and not
narely arbitrary. The anoient Thomists recognized clearly hoth 01
these points o .

As for the remark that the theoretical importance of
the intermediary sciences has been greatly exaggerated - 7. ^eel
that the contrSy is the case. The great ^^f^^^Z
latent in this point of Thomistic doctrine and its relevance tor
nodern physics have scarcely been recognized,

•„i nhnricteristics of mathematical
To discover the special oharaot oris^ ^ ^^

Physics as a sciejvtia_media we must turn ™ Ag haa a i rea dy

of Aristotle and St, Thomas "sn^ioned in univ introduced

toon explained, the text from the P^sterio^AiSiffii--

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i„ connection with the discussion of the two types of demonstration:

aenons:oratio_juia, ^-e. demonstration which arrives only at the

existence of a fact without being able to give its proper reason

and cause, and demons traUo^ropJg iLg uid, i.e. demonstration which

gives the proper reason. After pointing out how these two types

of demonstration dxffer in the same science , Aristotle and St, Thomas

go on to explain how they differ in different sciences, and first

of nil in sciences which are subalternated one to the other. And

they state that in this latter case it pertains to the subalternating

science to know the pro pter quid , i.e. the proper reason, and to

ths subalternated science to know the quia , i.e. the sinple fact. J

Both Cajetan (29) and John of St, Thomas (30) insist that in

naking this statement Aristotle was speaking of something that is

special to the kind of subalternation found in mathematical physics

and not something -that is common to all types" of subalternation.

In order to understand why this is so we must try to
grasp the difference between a scientia propter quid and a scientia
quia, A scientia propter quid is a science that is explanatory in
the strict sense of the word, that is to say a science that assigns
the proper reason, for things. It is knowledge that is arrived at
by a propter quid demonstration, that is to say a demonstration
which proves that a property belongs to a subject because of its
ver y essence . A scientia quia is a science which arrives at the^
fact that certain things exist or happen in a certain way, but it^
cannot assign the proper reason for the fact. The d emonstratio ^ cu ia
which gives rise to this type of science may be one of three kinds.
In the first place, it may be an a priori demonstration, and then
it consists in pro ving an effect by its cause , | But in this case
^tj : s_alv7avs _a question of the remote and common cause ^) Secondly,
it may be ar TtTposTeriori demonstration, which, proves the cause
V the effect; and this may be either inductive such as is found
in the study of nature, or deductive, such -as is found xn natural
theology in the demonstration of God's existence. The last type ot
toonstratio quia is known as demonstrati o a simultaneo; it is used.
ttlhTaSHSfeion of a thing by__the existence of its corelative
Cor of somethin g that is distinct I jjnjj ^nOyjjy^ distxnc So^-ationis
v ratioc inantisj )

Since we are dealing. with the study of nature we are
interested in the type of sci^ntia^uia that arises from indu ctivo
^posteriori reasoning. ButTeTt confusion arise, it must ^ Pointed
ou'aKTin^thematical physics, it is not the f^ e °? *^°* #
(in the Aristotelian sense) that is subalternated to "^ematics.
^e first part of natural doctrine that is ^°™ a ^*^ ^fLmons-
nature doefnot enter into subalternation, " ^ red ^ ^ ™ s
Nations to its own self-evident principles. It uses induction,

ft. 8

-153-

to be sure, ^t ge of inductionj ^a rriyes at analyt ic and
^_^y^m^^mm^ti^ t iTiiTTS refore, a deducti ve
^as well as can inductive science. It is a scientia propter quid.

It is only the dialectical prolongation of philosophy
of nature, known as experimental science, that is subalternatad
to nathomatics. Tr.j s part of natural doctrine uses a type of i nduot-
inrMjiat arrives only__at_ synthetic propositions . There result I'aam"
this two important things to he noted about experimental science.
First, it pertains to the type of knowledge known as scientia quia .
It cannot arrive at a proper jgrppter quid. The best it can do is
^oc onstruct an initationT ) a substitute propter quidCb y means of
hygothesispaecondly, it is not even a scientia quia in the strict
sense of the word, for it does not Rive certain knowledg e iCb ut onl y
prabaMlxtyT) -

Now in these two characteristics we find two reasons
Tfhy experimental science inevitably reaches out to mathematics ,
For science is cortain knowledge of things in their causes, (31)
And in order to have science in . the full and perfect sense of the
irord, these causes must be the proper causes. That is why scientia
q uia is related to scientia propter quid , as an imperfect state '
of science to a perfect state. That is why all scientia quia aspires
to scienbia propter quid . Now experimental science is neither certain
knowledge, nor is it knowledge of things in their proper causes, ;
Hence it has a double reason for reaching out to a scientia propter
.quid, i,e„ mathematics, in order to obtain for itself at least a ;
substitute certitucle4nd a substitute propter- quid . That is why- the
subalternation of physics to mathematics is not an historical ac- ,
cident,.-, {lt is the result of a necessar y and inevitable scientific
tendencyjln this connection John of St. Thomas writes:

In illis scientiis subalternatis ipsi mathematics,
quae usque ad sensibilia excurrunt, pertinet scire scientia
quia eo quod res sensibiles per inductions attxngunt et usque
ad experientiam descendunt. Si autem ilia eadem, quae per ex-
perientiara cognoscunt, velint sci^e propter quid, necessario
debent uti principle traditis.a nathem tica seu a scientia
j uBaTEernante „J (,52)~

in subsequent discussions we shall adduce ^^T
dence to bring out the necessity Af the subalter nation of physics
to mathematics, but perhaps enough has already been said to show
hov, erroneous is the opinion of those modern sch f^^° S ^° f ^
that the grounding of physics on mathematics is a great and fatal
historical mistake, (33) M- > * • ^'^^ ' * j •

a t^ of qt Thomas points out, (34) when we say

w the propter quid of the natural phenomena, this ao

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ttat i* pertains to the subalternating acienoo to know the con,-
elusions ot tic subalternated science and to demonstrate them. This
would neo n that mxhoniatica wp uld descond to sensible matter , and
Hi order co do this it would havo to abandon its proper abstraction,
and thus cease to bo Mathematics. The expression merely means, as ■
Cajetan explains (35) that the subalternating science 'knows the
propter quid Cm an abstract and g eneral wavD and it is the subal-
ternated science which takes the. general principles that are given
to it and applies then to its own particular subject matter. This
is what Aristotle and St. Thomas have in mind when they point out
tha t the one who knows the reason does not have to know the £ act» (36)
It should bo obvious from what has been said that when Aristotle
and St. Thomas say that the subalternated science knows only the
quia 3 or the fact s this means by itself, independently of the subal-
tsmation to the higher science from which it receives its principles.
For, by virtue of its aubaX-be-rruvblon the subalternated science is
able to know the cause as well as the fact.

Just as it is possible to have subalternation in the
strict sense of the word without the two sciences being related
to each other in such a way that the one knows only the fact and
the other the reason for £he fact, so it is possible to have sciences
related in this way without being subalternated to each other, A-
ristotlo gives a single example of this taken from the science of
nedicine. (57) A physician may learn from experience that circular
wounds heal more slowly thaii other kinds of wounds; but it is geometry
nhich gives the (re^on^f or this : I the absence of angles ,J This,_how-;
ever , does not mean that medicine is subalterna ted to geometry.

Now St e Thomas makes it very clear that in mathematical
physics we really apply abstract mathematical entities to the phe-.
nonena of nature :

, Per-.-p.-'-otiva applicat ad lineam visualem ea quae demons-

/trantur a geoiuetria circa lineam abstracts^ et harmonica, i-^
/ dest musica. applicat ad sonos ea quae arithmetic^ °^iderap
circa proportions numororum.,. Perspectiva accipit ^neamabs-
\ tract am secundum quod .e^ J^onsideratio ne mathematici , et .
Vapplicat earn ad materiam sensibilem, (58)

When a physicist ^-^ ^Sf ^ight-
a straight line his ^lcu^tion P^cecds^rom Uth the mathematical
ness Of course, he is not properly concerneawx ,, tical

line, but with £ho physical line -J^-^gjL^|^|^

ling that is applied to iti * V! ^^SmScalentity that is
Uind thaTITTsl^tuOT^the abstract mathematical em> y

\ applied to nature.

-155-

n „ + „ J? aPP T+ C f 10n lsnot merel y thc reverse of nvxthe-
mtioal abstraction. It does not consist neroly in fitting back
into sensible natter what was lifted out of it by the second degree
of formal abstraction. For, as we shall see in Chapter VI, the abs-
traction that is found m mathematics is different from that found
in all the other sciences in this that we cannot go back to reality
from the abstract notions and find then realized there. There is
a world of difference between the abstract notion of man and the
abstract notion of straight line. In the first case, we can find
the notion of tail realized in the concrete. In the second case,
although we can find a line in nature, -we cannot find a perfectly
straight line.

Although we cannot pass from the world of mathematics
I to the world of physical reality by a process of direct concretion ,
which would simply' be the reverse" of abstraction, we can do so by
m pr ocess of extrinsic application ^ The fact that this is merely
^""application and not a dire6t realization shows that the mathe-
matical interpretation of nature is necessarily a scientia media .
It also shows that the propter quid which mathematics supplies to
the study of nature always remains in s ome_ sense extrinsic to nature .
This would be true even in the hypothetical case mentioned earlier
in this Chapter in i which a superior intelligence would find it pos-
sible to treat natural phenomena in terms of mathematics in a strictly
scientific way

For us the mathematical propter quid must also remain
extrinsic to nature in the sense of its being dialectical. The ina-
dequacy of all our measurements and the limitation of all our ex-
perience both with regard to space and time makes it necessary for
us to operate within an extremely restricted frame where no phenomena
can be sufficiently accounted for. Given this inadequacy of our
'Ksasureiaents arc"! experiments and the uncertainty of our reasoning,
the application of a mathematical proposition to a natural subject
nust be cons^reri as something essentially tentative. The mind i
ever goes beyour' the date of experience in this application, and
in so far as this application inevitably outreaches what is_ conveyed
to us by experience, the mind is out on its own, so to speak. As
a consequence, thcjub.iect formally attained tisnever^^vd|y^cgd
faXLthe mrt P l^o^^^onl tiilf^And-to the exoent in which
tWirfcrthTiSbjeot somethlni^oMng from reason alone, the
subject itself must be callecKkdialoctical entity.^

T4. • -i„n-,, therefore, that in mathematical physics

-156-

5h ononena vjas_fg und In Eucli dlan_geoivietry; in Einsteinian physics
the pro^e^ouxd for fthe same phenomeneQ is found in non Eu clidian

As we have seen, physics reaches up to mathematics
in an lill 1 ^- to osca ' pe " fane aial ectioal status imposed upon it b y
f^TnV^of true universal necessit y. But it is clear from what
j^fjust boon said that, because mathematics cannot provide an ex-
planation that will give universal necessity for the meaning of
nature; physics does not succed in escaping from its dialectical
status by becoming subaltermted to mathematics „ In fact, it becomes
doubly dialectical.

But for the present the important point is that phy-
sics, because of the opacity of the universe of matter, is forced
to go out into a new world to find light, and having found it in
the world of mathematics, it brings it back into the material worldo
As Cassirer has remarked, "that form of knowledge, whose task is
to describe the real and lay bare its finest threads, begins by
turning aside from this very reality and substituting for it the
sjnbols of number and magnitude," (39) It is a strange light that
ire bring back from our excursion into the world of mathematics,
for as we shall see, mathematical abstraction is in one sense richer
and in another senae poorer than any other type of scientific abs-
traction, In this connection it is important to note the exact form-r
ality of the expressions used by St, Thomas in his discussion of
the application of mathematics to physics: "Huiusmodi scientxae
irtuntur speciobus idest formalibus principiis, quae accipiunta
Sthemticis," This shows that the mathematical forms xn physics.
are sonething e--^.ontially alien to the physical world, .. and that
the role playco. r? rsathe'raatics is from this point, of vxow purely
jjjtam ontal o :

■■-, •- -oor "•■■■: oal Physics, then, we take a mathematical
line, for e*,- V '• " ",nS « * the physical line. In other words
» confer Si" V o r; . -' J g it were a straight ^ jj*^*^ '
Physios is e & ,c,,r,ially "a Iclenee of als_on. ^/^JnLd cannot
/-oduce into ; ,al,,re ±l^M!i^-SL3£S^Lf^^^^ ™
«&>t as such in roaii^T^SSE^Sn^^SS^
onoe agaiTit becomes e^ifenth^Tmuch Kantxnnxsn there
Vthenatical pliysics,

' *■ ,-Hth this insistence that what is applied
, In connection wibn tms ■"'"•'•"., Pn titv, we must

Mature is. actually the abstract niathemexcal ^^^
n sider for a mQmnt a . pogs ible totorpratetionrt^ ^.^

Posies vMoh at first glance appears highly plausxo ,

-157-

is
is

ftmdancntally erroneous. We refer to an interpretation which
jld consider v.ie so-called mathematical entities merely ideal-
nations or limx cases of physical entities. Experimental science
deals oonsuantly with idealizations and limit cases. When a physicist
speaks of the laws of gases he has' in mind a "perfect Gas" which ;
exists nowhere m nature. Does it not seem plausible that when he
speaks of a "perfectly straight line" he is likewise speaking merely
of nn idealisation of a s ensible line , that is to say, a sensible .
line pushed to its limit case? If this interpretation were correct,
m thenaUcal physics would not be a scientia media , for oust as <
the introduction of such idealizations and limit cases as "perfect:
gas", does not involve the application of a superior science, so ;
neither would the idealization of a sensible line. This would bring
us back to something similar to the doctrine of Professor Mansion
discussed in the last chapter.

Such an interpretation cannot be admitted. Idealizations
and linit cases are not the product of formal abstraction, but merel y
of neg ative abstraction . It is possible, of eourse, to push certain
physical entities to their! limit case and thus arrive at something'
which s ugerf icially resemble_s mathematical entities. It is likewise
possible to attempt to study nature in terms of these idealizations.
However necesfary negative abstraction of this kind may be, it re-
gains something common, and does not account for the peculiar in-
telligibility provided by the application of the positive abstraction
of mathematics , The great rational elaborations of mathematical
physics show that it is a specifically superior source of intelli-
gibility that has been introduced into nature which of itself is
less rational,,

It is true that the basic relations between variable
iaMiti§s out of which the mathematical physics is constructed
are given explicitly in a concrete quantitative determinations of
nature. But it is illegitimate to conclude from this, as. Professor
Renoirte seems to have done, that there is no subalternation of -
a lo-rer to a higher science involved. (40) For mathematical physics
is not a r.iere collection of concrete quantitative relations or of
concrete measure-numbers. It is essentially a mathematical elabo-
rationand interpretation of thesein itial data . And it is in this
eg Eor^o^anT3n T;orpretatio TrihaFtho subalternation consists.

, After explaining that the subalternation of physics

o. orthoptics consiste|in this that the former getsits propte£

f4, its cause and reason from the latter, Aristotle and St. Thorns
°" to explain tno particular nature of this cause. (41) Now
! "fr Jggptor quid which mathematics can give to the study of

R aW< I-.,,-.* cr—r-^r: — , . _~.- ,„t n„„ M iitv. For of all tne ioui ;

-158-

causes the only iype of cause that is found in Mathematics is the
foiw-l cause c i..:; mathematical world is a completely immobile world,
In it there is no becoming, and hence no subject, no agent, no pur-
„ os e, It is a world of -pure forms. And this gives U3 an insight
into the peculiar nature of mathematical physics. If it were purely
plril2SJJ wou ^— &y— ^° i-o solve things in terms of all the four
oa^SsTButbe cause it is formally mathematical it can see things
^fTx\ the light of formal causality. This is an extremely-, important
point, mid we shall return to develop it later. For the moment let
it suffice to bear in mind that -the cause whioh mathematics contri-
butes to physics is in the general line of formal causality, and
pertains in particular to the structural order.

Now since mathematical physics is an intermediary science
beteeen physics and mathematics, it is necessary to try to determine
to what extent it participates in both of these sciences , Does .it
participate in both of them in equal measure, so to speak, or dees
none of tho two predominate over the other? From what has been said
up to this point one might easily be led to deduce conflicting ans-
wers to this question. ,For in discussing the structure of a mixed
science we stated that an accidental element taken from the object
of the lower science is added to the object of the higher science. ■
FroQ this it would seem to follow that the most important element
in the object of mathematical physics is the element taken from
mther.iatics, and that the physical element is merely an accidental
addition to it. On the othei hand, when the question arose about
the kind of unity found in the object of on intermediary science .

ra said that the object that mathematical physics considers directly \
and per se is >the physical element, and the mathematical element I
is brought into the consideration in a kind of oblique fashion by /
ray of connotation,,

If we look for the solution of this antinomy in the
nritings of Aristotle and St. Thomas, our difficulty is ^vated.
For on the one hand, Aristotle seems to class the V^xao^^-
tieal sciences among the mathematical sciences, {4,2) ^eover,
w> read in Saint Thorns that these sciences are ' 'mag is ^™ ™?
themtiois, quia in eorum considerations id quod ^t physici. es^
luasi mturale; quod autem raathematici, quasi f orraa ^» p + wjinetis,
Joh* of St, ThomL says: "astrologus non agit de ^^f^S'
«t sunt entia mobilia, sed ut mensurabiles sunt eorum motus ^t se ,
«»ta varies aspectus diversam proportionem induunt g* J£«»
Partinet ad mathematician quam ad physiauro. W "n physical

teA we are told by St, Thomas that ^ ? se sciences ar ° ™' J£
ton raathemtical: "Huiusmodi autom scicntiae, ^°°* ^ hic a
«ter scientiam naturalem et mthematicam, tamen dicmi e

?lulo SO pho esse r^e^jiB^^SS^^!^^^^ *

-159-

denoninatur c J - .«r.r;cien habet _a_termino; unde, quia harura scientiarura
cons ideratio ' ;■ .: anatur ad natoriara naturalera ? licet per principia
a1 tJie)iatic^ Cl r - , ' > - :ad ^ t p !^gii-i ffl t naturales quara mathematieao ." (45)

There is a text in the Surma which, together with the
the conuentary of Cajetan, throws light upon this apparent paradox:

Quilibet habitus fomaliter quidem respicit medium,
per quod aliquid cognoscitur; materialiter au tern id, quod per'
radium cogrioscitur; et quia id quod est formale, potius est,
ideo illao scientiae quae ex principiis raathematicis concludunt
circa naterlan naturalera, magis cum rnathernaticis connumerantur,
utpote eis sirailiores, licet quantum ad raateriara magis conve-
niant cum naturalij et propter hoc dicitur in II Phys. quod
sunt raagls naturales, (46)

To this text Gajetan adds the following remarks:

In responsione ad tertium secundi articuli non dicitur quod
scientiae mediae sunt magis matheraaticae quara naturales: cum
falsum si":-,, absolute loquendo: quia sirnpliciter sunt scientiae
naturales* utpote non abstrahentes a materia sensibili; omnis
erdra scievtia non abstrahens a materia sensibili est naturalis,
ut pate-l- T\ Met. Sed dicitur quod connumeran tur magis cum m»'
thenatic l^: ? ut pote eis similiores . Et de connumeratione quidera
liquet, "quia cun geometria et mathematica scientiae nuraerantur
inter liborales artes. De simlitudine autera in nodo demonstran-
di nanifestura est, dura mensurando et quantificando conclusions
Jnonatrantur. Verura quia medium utrumque sapit extremum; ■««»;-
entiae istae ex parte formae ex nathdmtica vemunt et Pendent,
ex parte raateriae physicae sunt: sermgneg_Dgotg rum pie xnter -
VEretandi^unt, si quando ad alterum extremura nirais deolmant.

Perhaps a more sharply drawn distinction ^.^e
to dispel all confusion on this point. Prom the point or view ol
its rati o fomalis q uae, mathematical physics is ™rephysical ™"
mttoatica l; from t hTpoint of view of its £0i2^gl^L^>
it is more mathematical than physical, ^e mti£f^rmal^s_qyae ^

tho physical cc^id^rMjyLJSS^

"incd and HK>dil ; GdTyTX~CoTSSa^^^

Erectly, whereas the mathematical is ^"^^^Hs the

?nd obliquely. The terminus or end of mathema ^ c ^ /^ S ^^ atical .

pledge of nature. It is not theknowledg of the ^^^^.

™>rH that tho mathematical physicist is a™ 1 ™ * ; tal^tapter

?£ady presupposed) but of the physical world. As we saw ^ .

", SStaSticTdoes not terminate in ^nse^experienco,^ soien _

origin which it has in sense, experience is only r

-160-

wfJfl . Mathenv: ,'..,,.. physics, on the other hand, both originates ™fl
terminates an no so experience, even though, due to the rolepLved
b .athematics^here are introduced between the origin and the
tg^SBS >.Bny e^meiros which have no ^uSe^iarti^^^if^rience
■ 11 this explains why we speak of mathematical physics and no? S
phys ical mthemaoics And from this point of viev^ pnysS-^thS
m tieal science mayte numoered among the physical sciences. As
Cajetan points out in T;he passage just cited, mathematical physics
does not abstract from sensible matter, and judged by this crite-
rion it nay be said to be a natural science*

Yet it would be erroneous to conclude that physico-
mthemtical science is formally identified with pure natural science.
As a ratter of ft.ot it is distinguished from it specifically both
ty its £2&°_ f .?2Pi!ii3_gya£ and its ratio for ma lis sub qua . For in
so far as the ratio f ormalis quae is concerned, we have just seen
that, while the physical' is considered primarily and directly, it
is nevertheless^ considered only as connoting the mathematical and
as Modified by :;.t Wow this connotation and modification introduces
a profound change., As we pointed out in the last Chapter, the ratio
fis^ljsjiuae ci? all pure natural science ( Is mobility 7 ! This, how-
ever, cannot bt-. >;dd to be the ratio formalis quae of mathematical
physics,, for ax -»,-.'; shall explain later on, the introduction of ma-
thematics into pViyt;ics destroys all true mobility by the very fact
that ther e is ny ^ rne be coming intrinsic to mathematics , Movement
undoubtedly p?ayu "ITTarge part in mathematical physics, but it is
noveEient in the Cartesian sense, which is a state and a relation ,
and not a proca-jj and a becoming,, Mathematical physics does not
study the phys:V:al world as mobile, but as measurable ,, As John of
St Thomas says in. a text already quoted, "Astrologus non agit de
coelo et plane^is ut sunt entia mobilia, ged ut mensurabiles sunt
Mnnijnotus_ et aonundum varibs aspectus diversam proportionem in -
iSffifiFTluoTmalp.u pertinet ad mathematician quam ad physicura,' 1 ( 47)
let mathematical physics does not dispense completely with mobility,
For there is an essential relajsion between its formal object and
that of pure ra» tir-al science,, The extremely paradoxical character
°f mthemtiea.':. n\iysica has already been noted: in order to draw
ol osor to the av; -volute world condition it draws away from it' by
8°wg out into another world, that of mathematics. Applying this
™ the point uo:v.^ discussion, we may say that in order to under-
sea the mobi:.- -V f the cosmos it prescinds from it by introducing
athematios, R,.t. 'fche important point is J cftSo in prescinding from
p it is tend-:.,*- towards a more perfect understanding of it. The
jj-ra^of this r,^?.ds»icy would i^n^j^Tnntification of the formal
^Jta^TTtical ph-i/sic^itOgr^fju^mturar^ciej^.

|^Ukte_of_iQnflBn^on essential relation between the two formal

-161-

in mathematical posies thero is a triple dialectical
nove ,,ent ; First, there ls the movement fron the state of generalit
towards the ultimate concretion. Secondly, there is the movement
fron the acaco of probability towards the state of certitude. Both
of these dialectical movements are common to all experimental science,
UA thirdly, there is the movement proper to mathematical physics - -
the one we have oust explained. All of these three movements are
intimtefy hound together.

Physico-mathematical science is distinguished from
pure natural science not only by its ratio formalis quae , but also
by its ratio for malis sub qua . In fact, from the point of view of
this Latter ratio it is closer to mathematics than to physics, just
as fron the point of view of the former it is closer to physics
than to mathematics. Mathematical physics is formally mathematical.
It gets its pro pter quid from mathematics, and since the propter :
pd gives the reason and cause of the natural phenomena, it stands
in relation to the latter as form to matter. All this means that
aitheaatical physics proceed under the light of mathematical evi-
dence. This would seen to imply that the special type of abstraction
which constitutes its ratio formalis sub qua , and which, as we saw
above, stands in between matheinatical and physical abstraction and
shares in the character of both, is more mathematical than physical.
Though principally mathematical it is not, however, specifically
mthenatical, since it is' applied to a physical object in order
to constitute a new subject and new principles proper to a science
concerned with physical reality. In other words, though mathematical
physics is formally mathematical, it is not specifically mathema-
tical.

From what has just been said about the parts played
ty mathematics and physics, it should be clear that when we say
that mthenatical physics is formally mathematical and materially
Physical this does not mean that the formal object is mathematical-
am the material object is physical. For the objectum formale quod
fes to do with the physical world. Some modern soholastics seem '
4 « he confused on this point. (48) It should also be clear how:-
wipletely Aristotle is misrepresented by Professor Mansion when
ho writes:

On voit done comment, en ecartant de la physique , pour
Jos assignor au domains mathematique les sciences mentionnees
a 1' instant, Aristote a manque 1' occasion de traiter a fond,
suv des concrets parfaitement adaptes, le probleme de la dil-
ference entrc une etude philosophic^ et uno etude purement
scientific do telle ou telle portion du morale materiel. ^)

-162-

Aristotle in no way removed the physico-mathoaatical
sciences from the realm of physios. If he listed then among the
mthemtical sciences it was Merely because they are formally ma-
ther.ntical, iVnd he took pains to point out explicitly that while
they are closer to mathematics fror.i this point of view, they are
at the soko tine more natural than mathematical. In his mind they
cere, of course, specifically distinct from pure natural science,
but this did not remove them from the realm of physics, since their
nhnle raison d'etre was to get to know the physical universe.

At this point it is interesting to compare what has
been said thus far about the nature of mathematical' physics as a
joientjaj^dia , ("formally mathematical and materially physical^ -with
ted passages from Albert Einstein, one of which has already been
quoted. There is a remarkably close affinity between the ancient
Thoiaists taught about mathematical physics as formally mathematical
and what Einstein has to say in the following lines:

It is my conviction that pure mathematical construct-
ion enables us to discover the concepts and laws connecting
then which give us the key to the understanding of the pheno-
mena of Nature, Experience con of course guide us in our choice
of serviceable mathematical concepts; it cannot possibly be
tho source from which they are derived .; experience of course
remins the sole criterion of the serviceability of a mathema-
tical construction for physics, but the truly creative principle
resides in mathematics. (50)

In the same way the following passage seems an exact
confirmation of the Thoraistic doctrine that mathematical physics
is uaterially physicals

Pure logical thought cannot give us any knowledge con-
coming the world, of experience: all knowledge of reality be-r
gins in experience and ends in experience. The conclusion; abs-
tained by means of purely rational processes are, in so far .
as reality is concerned, entirely empty. (51) -

We are now in a position to understand with greater •
fatness a point to which some attention was given in Chapter I
Tfc wfer to the question of whether or not the role of ^'^
jnmttemtioal physics is purely instrumental. It should be evident
™nv,hat has been said that it cannot be purely ^ st ™^ n
*- sense of being a mere logici^iaoot^lA^E^Bi^^^ 6 ;^
-ither a logicall^rnor a language enters into the very ob

the

jeet

\tv

'-j-wier a logicaJ. 'GOOJ. nor a Aunis""a . ,. ,,

. of the science that enploya them. They remain essentially ex
' l "sic to that ob.iect. But in mathematical physics, an element

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of m thonatics enters in co combination with a physical element to
constitute the very object which specifies that science. And yet
because it does not enter into it directly, but in an oblique fashion
by vray of connotation, and because as a consequence the objectum
forvalej(uod> that is, the thing that mathematical physics is trying
to "get" to know, 'the thing that is the terminus and 'the end of the
whole science, is something of the physical world, and not the ma-
thematical world, we may say that in this sense the role of mathe-
mtics is purely functional. Mathematics is employed in physics
only as a means to get to know the physical universe. As Professor
Babin has pointed out, the physicist who loses sight of this purely
functional character cannot fail to pervert his science:

Parce que la fin du savoir physico-mathematique est
tout de m§ne la. nature sensible, le physicien-mathematicien,
a tendance mathenatisante , pervertit sa science, quand il se
deointeresse des choses naturelles elles-m&nes pour se complai--
re, corano dans un terme, dans l'o rdre et la beaute de son ob-
jet formel, done dans 1' aggregatum ut sic , on tant que celui-ci
est un compose accidentel | et ; oeu vre de sa raison ,j. et ; pur subs -
titut de la nature,, C'est un artiste egare ou fi-ustre, et qui
sofser t~de la nature corane d'une matiere ouvrable, Ce faisant,
il erige en fin ce qui est moyen seulement, et prgfere contem-
pler 1' oeuvre de sa iaison plutot que la nature, qui est 1' oeu-
vre de 1' intelligence divine. (52)

Enile Meyerson makes the following commentary on the;
pivotal text of the Pos terior Analytics in which Aristotle explains
his conception of mathematical physics as an intermediary science':

II y a evidemment", dans ce dernier morceau une sorte
de tendance panmathematique, laquelle n'a p as manq ue d'embar-;
rasser quelque peu l es commentateur s dont certains m§me ont
crTpolIv^ir^bs ^veVar u e le Ste gggEgf transgressantles ^regies
, o^^ra^ait^slSildileurs, P^^£S3^fcb3£npasse^_ici dmi
genre a un autre. (Note: Of. nolaSS^Ia-n^eTFJai^leSy-r
Sl^HilairiTLoiique d'Aristote, t. Ill, Paris, 1842, p._85.; . •
Mais si l'on fait abstraction de ces passages, qui senblent ;
plutot un heritage provenant des philosophes de * A ™'^,
la ponsee d'Aristote s'avere parfaitement °rientee dani le nfi
m sens que colle de Bosanquet, tout en etant en quelque sor^e
plus extreme que celle-ci. (53)

-r- ■ 4. ,„i,r difficult to find any trace of a tend-

lo is extremely diliicuj.0 w fln „.i-~i n e of rathema-
ency towards pannathematicism i* Aristotle s doct sutalternat ion,

txcal physics, On the contrary throug h hi s do cti their

h e kept thorn both distinct, while at the some uime

-164-

intimto relation. He never held that the whole of physics could
be subalternated to mathematics, to say nothing of the other sciences,
Uuch less did he ever attempt to erect the mathematical interpre-
tation of reality into a metaphysics. Nor have any of his great
connentators - - those who have understood his doctrine most cor-
rictty and given it most genuine and integral development - - ever
prvnifegt ed the sli ghtes t embarrassment over this text from the
Poster ior Analytic s , On the contrary they have t considered it. Ho
be in perfect harmony with all the epistemological principles of
,the Aristotelian synthesis.

There is no difficulty in admitting an influence of
the Academy upon this particular point of Aristotle's, doctrine,
Aristotle himself would certainly be the last one to deny his great
indebtedness to Plato, But it is not, as Meyerson suggest^ a hete -
rogeneou s bit of doctrine Cthat was accepted by a kind of strange
Conc ession to eclecticism^ ) Rather it is something th"at~has "been
purified of Platonis t exaggerations and brought into perfect line
with the whole body of Aristotelian epistemology. As for the charge
that this text represents a transgression of rules laid down by ;
Aristotle elsewhere - - we have already considered this point both
in this Chapter and in the last part of Chapter II, and there is ;
no need of reconsidering it here, .

These remarks conclude our explanation of the basic .
principles underlying the Thoraistic philosophy of mathematical phy-
sios, The chapters which are to follow will be an elaboration of :
those. As we" have seen, there are two pivotal points around which
these principles revolve: the nature of the distinction between :
physics and mathematics, and the nature of soientia media. The next
three Chapters will be a development of the first poxnt, and the
regaining Qhapters a development- of the second. The next two Chap-
ters will be devoted to an analysis of the science of nature, anj
the one following them to an analysis of the science of mathematics.
The study of scient ia media will fall naturally in two parts :_ firs „
we shall considSTthe way in which this intermediary science is,
constituted (chapters VIII and DC) , and secondly we shall analyse
the nature of the physico.nathematical world which results from
thjjijrediation (Chapters X to XIII) o

-165-

CHAPTER FOUR

COSMOS MID LOGOS

1, Movement Towards Concretions,

At the beginning of his Commentary on the E e Coelo
et Mimdo, St Thomas has this to say:

. ..Philosophus ostendit in scientiis.esse processum
'orainatun, prout proceditur a primis causis et principiis us-
que ad proximas causas, quae sunt elenenta constituentia essen-
tiam rei, Et hoc est rationabile: nam processus scientiarur^
est opus rationis, cuius propriura est ordinarej unde in onmi
opere rationis ordo aliquis inventitur, secundum quern procedx-
tur ab uno in aliud. Et hoc patet tan in ratione practxea, cuius
consideratio est circa ea quae nos facimus, quara xn ratxone _
speculativa, cuius consideratio est circa ea quae sunt aliunde
facta, (l)

It is proverbial that the most characteristic property
of msdom is order: sapientis est ordinare (2) ^V°^*Jg_
no W does the 'profound wisdom of Aristo.lO and St-Thoms manx^
fest itself with greater brilliance than by the order J^* « JJF 1
in their writings, This order is some xme^le ° a ^ ^Sng

SSe? tolf S S^lSTSTSS ^"pecial need of insis-
uLuea to it„ Ac o-pner -cxmos, ""~* effort is made to ex-

ting upon the right ordter to be f° llOT ^' ^Jf ^ teZ writings
Plain and justify the order adopted. And *™ e ^
ao Aristotle and St. Thorns lay such P artlc »^ *^ t rine. It
«ion of order as in their trea xses £*£% *£*%& bookg
w the first problem discussed ^ f ^xnn g subsequent trea-
f the Physics, and time after time ^° u ^ u . princi pies in-
tt*» It is brought back into focus, -f^il present!? attempt
volved in it are reconsidered. (3J As we aiio.

-166-

to mke clear, the history of philosophy, and the history of modern
thought in particular, have shown that this emphatic insistence
upon the correct order to be followed in the study of nature was
Vfar fron being gratuitous.

But if this question is to be put into proper perspec-
tive j v«3 must begin by recalling that there are two issues involved
in the general problem of scientific order. First, there is the
question of the right ordering of the different sciences among then-
selves, and this has been treated at some length in Chapter II.
Secondly, there is the question of the right ordering of the dif-
ferent parts of the sane science; this has been touched upon lightly
in Chapter II, but -we must now consider it in greater detail in
so far as it involves the study of nature,,

• St Thomas brings out this double movement of the scien-
tific mind in his Soramentary on t h e De Sensu et Sensato : (4)

Et sicut deversa genera scientiarum distinguuntur se-
cundum hoc quod res sunt diversimode a materia separabiles,
ita etiam in singulis scientiis, et praecipue in scientia na-
turali, distinguuntur partes scientiae secundum diversum sepa-
rationis et concretionis modum Et quia universalia sunt raagis
a materia gppn TO tn ^(jden) in scientia naturali ^ ab universalibus
adrainus j miyersalia proceditur a J

In other' words, both the ordering of the. different
sciences and the ordering of the parts of the same science are de-
temined by different degrees of mental separation, but m aacn
case a distinct type of separation is involved. In the case of the
ordering of the various sciences it is a question of separation +
from materiality according to different ^elsof formal abstraction,
and the natural movement of the mind is from the less abstract tc?
the more abstract In the ease of the ordering of ^e different
parts of the same science, it is a question ° f .^^^tZSZ
concreteness ^coTdi^Jo^MrsrmUm^Lt^^^^^,
And the naturSTI^^SSrSThr^aisftaii, the more abstract

the less abstract, (5)

/ -, „„,^ +hn+ -rvosress in science means

^^^^^onjy^p^o^gdth^p^glgjs^,^

^^rth^Tis-toliTle) ThiB re fers, of £*£'££, conorete
^ii^B3i5nnoTaiow existential reality. Oo g concreteness?
^ality better means to get to know it with ^^^^^^^^
Mathematics, precisely because it is a ! cle "^.„ n( , tnesSt but in the
f^^T^^^oBSS^^!^^^^^ towards
study of nature and in metaphysics the movemcn

-167-

fuller concre-cioru In metaphysics this movement is from the car-tmt-
nia entis up through the, realms of the created separated substance
impure Aoto In -one study of nature the movement towards concretion
carries the mind in some sense in the opposite direction - - into :
deeper immersion in matter

Porliaps at first sight al'.!. this may seem to he in di-
rect contradiction to the actual historical development of physics,
Bertrand Russell has claimed that "in proportion as physics increases
the scope and power of its methods, in thai same proportion it robs
its subjeot-hiatter of concretoness," ("l') Surely relativity phy- ' ;
sics and quantum physios arc immeasurably more abstract than anything
that the past centuries have produced,

I J : cannot be denied that progress in modern physics ;
has i.cant an increase in abstractness. But at the same time, it
has also meant an increase in concretei-oss,, atomic physics ;> for
exanple, in spite of its abstract constructions (or rather preci-
sely because of them - - as we shall explain in a moment) has brought
us into more intimate contact with concrete reality that we ever
were before „ There is nothing paradoxical in this double movement
towards concreteness and abstractness. It merely- reveals the fact
that modern physics is, not a pure physical science, but a scientist
raedia in which physics a science of the concrete is subalternated ;
to iriathema tics ,. a science of the abstract, (8) U ,^^° J '

In this Chapter we are concerned with the 'study of
nature in so far as it prescinds from subalternation to _ mathematics.
That is why the movement that must claim our attention in a parti-r
cular way is the one towards fuller concretion, Moreover, even i n:
rathematical p hysics, the movemeii t_towar ds abstractness is _ secondary
al^pr^Ty'fv^.+-i^ c ,i .g-inrv? i tgjwholgjourpose is to assist the
SSJgTlg^ ^ is of extreme importance

^"analyse the nature of~this latter movement,,

In the first Chapter of the first book of the Physios
Aristotle Y/rites:

The natural way of doing this is to start t ran the
things which are more knowable and obvious to us and proceed
towards those which are clearer and >-re knombla by nature,

wwarus xnose wnicn ur« ^o"-" - , , =

for the same things are not ' knowable ralatively to us and
•knowable' without qualification. So Kn the present "W
™ must follow this method and advance fr ™. ^* ^^S
cure by nature but clearer to us, Jowards^hat ^clear.^
and more knowable by nature. Now what is piain pr i nc i p les
at first is rather canfusedmsBes , thg_elem gnt3 and pnn cig

-168-

ofjwtaoh b^comcj^wn^oj^ater by analysis. iThus we must
adTOn^e_lroj^gener ali Gios ^tgjaiSImnSsTrfOT ^l^lTwhoTe
■that is best known to sense-perception, and a generality is
a kind of whole, comprehending many things within it, like
k partso (9}

It is clear from this capital text that for Aristotle
the basic order to be followed in the study of nature is one which
noves from the more confused to the more distinct, from the more '
universal to the more particular, from the more abstract to the
noro concrete o But he does not lay dorm this principle, which is
to serve as the guiding light throughout his long research into
nature, without seeking to give it full justification. And St. Thomas,
in his commentary on this passage, shows that this justification
can be cast in the form of a simple syllogism:

InnatiAm est nobis ut procedamus cognoscendo ab iis
quae sunt nobis magis nota, in ea quae sunt magis nota . naturae ;
sed ae quae sunt nobis magis nota, sunt confusa ,(;qua lia sunt
universaliaj) er go opportet nos ab universalibus ad s in gulaFia
procedere „ (To)

Each of the propositions in this syllogism deserves
attentive examination.

In the first place it is clear that in the pursuit W\_a. \0Y

of science we roust start with those things which are most knowable o

for us, and gradually pass on to those things which are less know?

able for us„ This principle is so obvious that it docs not need ;•
justification. But it so happens. that there is an inverse proportion
between the knowability that things have for us and the knowability

(that they have in se. And we do not have to seek very far to find/
the reason' for this. For, .^rn ce being and ontological truth are
convertible, things are objectively knowable according to the mea-

^s5e"of~perfection of being which they possess. And since things
We perfection of being to the extent in .which they are in act,
it follows that their objective knowability is determined by their
degree of actuality. That is why, if our intellects were in the
fultess of actuality, their order of knowing would ??"«^^
^e objective order of knowability. But it happens that they arc
far from possessing the fullness of actuality - - as f ar as it is
Possible for any intellect to be. As a matter of f act, they raist
*** the process of knowledge from noetic pure^oncy^^ ^^^
Sfej-asa - - and gradually move in the direction u •_

And that" is why the knowability of things for us ig^™^2-
Portion to the knowability of things in^eo In other words, tg
Collect mat acqjiir^Jmowledge, not^^S^or^^^^J^-^ >

-169-

but in^g^rmijy with ita jgotgngy. If it were to acquire knowledge
iHTonforoity with its act, it wouia suffice for it, to exist in
order for it to have knowledge in act. Hence the first object of
imowledge inust be that which is most in conformity with the intel-
lect's state of potentiality, (li)

In our discussion of the nature of abstraction in Chap-
tor II we pointed out that one of the differences between formal
and total abstraction emphasized by Gajetan consists in this that
as ye advance .in formal abstraction we are moving from what is more
knowable to us and less knowable in se to what is less knowable
to us and more knowable in_se, while an advance in total abstraction
ineans a movement in the opposite direction;- And this explains why'
in the ordering of the different sciences we must ascend the levels
of formal abstraction and advance from the less abstract to the
nore abstract, whereas in the ordering of the different pasts of
the same science we must descend the levels of total abstraction :
and pass from the more abstract to the less abs tract In both cases
we are raovdug from the more knowable for us towards the more know-

iable injse, that is to say, from potentiality to actuality. In the
first case it is a question of the potentiality of materiality;
in the second case it is a question of the logical potentiality
of universality o

And this brings us to an explanation of the minor of VWv'wOr

our syllogisn e It is fairly obvious why the mind, if it is , to f ol- -

low its natural movement of passing from potentiality to actuality,
nust begin with the more general and advance gradually in the di~;

Irection of the more particular. For unive rsals. contain their sub- ■'
jective parts on ly in a confused and indistinct wa y, that is to
say, in p otentialit y,, In other words* the universal stands in re-
lation~to~the particular as indetermination to determination, and

Whence as potency to act, ..

In connection with the conclusion of the _ syllogism C(?V\clt^i

it is necessary to note that the^exa^gji^nj^gulari a'' does not •

■£3pv to individuals but to species. We have already~brought out
tttilolSt'TiroSr^HtiHiiSTf Maritain in Chapter II. And perhaps
it is not superfluous to mention in passing that in this whole dis-
cussion Aristotle and St. Thorns are dealing only with intellectual
Pledge, for obviously a knowledge °f Particulars by the senses
is a prerequisite for the formation of universals by the mind,

Ihe terminus, then, towards which the whole study of
^•ture must ever move, is ultimate specific P^J 1 ^^^!
^ to lose itself in the infinite potentiality of ^^^
wetion - - de singulis non e^lBoiontto. It nuat begin with the

(A" 1

-170-

eaaldsration of mobile being in general and analy s> its structure
and properties; From there it nwst move towards the full and ade^
qur.teiaeteraiyiat^pfy^^ is .proper to each

naturaJUfieciess, This is a goal that actually transcSds~tWpOTers
oTtho human mind, as we shall explain more fully a little later; ■
but it provides a limit towards which natural science must ever
Vtend if it is to be true to its own intrinsic nature.

The study of mobile being, therefore^ is essentially
a science that must ever remain in the state of mobility ^ For though
from one point of view the generalities which constitute the first
port of the science o'f nature are the most satisfying to the mind,
since they are the truths that are most knowable for us,, and, as
wo shall presently see, the truths about which we can have the greatest
certitude , from another point of view they are the least satisfying.
For, by their very generality and: Vagueness^ they give us only a
superficial knowledge of nature; they provide orily a kind of intro-
duction to the study - material reality', in somewhat the same way
as the coiajunia entis in meta physics ( provide only an introduction
to the stud y of immaterial being ,J The true student of nature will
never be satisfied with the superficiality of this introduction.
He Tail want to come into more intimate contact with cosmic reality.
And in order to achieve this, he will never cease his efforts to
advance in the direction of fuller concretion, In his coranentary
on the Libri Meteorologieorum St. Thomas writes:

Sicut in rebus naturalibus nihil est perfectura dura
est in potential sed solum tunc simpliciter perfectum est, quan-
do est in ultimo actu; quando vero medio modo se habens fuerit •
inter puram potentiam et purum actum, tunc est quidem secundum
quid perfectura, non tamen simpliciter; sic et circa scientiara
accidit. Scientia quae autem habetur de re tantum in universali ,
(npnest scientia completa^ecundum ultimura a ctum,; sed est medio
aoao~~ie"15DenirTnl^ purig^ ac tum. Nam a-

liiiirioi^raliquid' in universali, scit quidem aliquid eorum
actu quae sunt in propria ratione eius: alia vero sciens in _
universali non scit actu, sed solum in potentia - - Unde_mani-
festum_est (quod nnmpl. ementum scientiae requ^iM ugdjiongisr
tatur in cor^nibusf sed procedatar_usaue_ad^speciosJ (,1^;

thoi

tatur_JjijcomnMnibiiis , jsg djprocedat

Aquinas points out elsewhere that n^l f o rms have
•ir very being" in concretion^ -£«£."(«) ^ V*«

«r very being"in concretions aa i^^^. \~-' -- ■ •

°Vcan come into intimate contact with then onty by delving deeper
and deeper into matter.. -

Perhaps this last point will present a difficulty to

-171-

nove

the m nd, For this delyuig into the depths of nature nay seen to
be leading us m the direction of greater objective unintelligible
lit y, whereas we stated a few moments ago that the movement towards
concretion means an advance towards things which -are uo.ve .Intolliirtbl.
inje 3 The solution of this difficulty is fairly simple: even though
the things of nature because of their materiality are less intel-
ligible in_ge lthan jjxiaterial things, - ) they are, nevertheless more
intelligible in_se in the state of concretion with natter than in
the st-ate of vague generality.

Having established the fact that natural science must
„„,„ , rom generality to concretion we must now consider the problem
of how this movement is carried out. This is a question of extreme
importance, for it has to do with what is perhaps the most misunder-
stood point cf the whole Thomistic philosophy of seience

It has become historical among historians and philo-
sophers of science to insist with great emphasis upon the completely
antithetical character of the scientific spirit of the Renaissance
in conparis.oiL. with the Aristotelianism that had dominated the pre-
ceding centuries <, We are told (almost invariably without any attemp t
at proof ) that Aristotle and his medieval followers had held that
the whole of cosmic reality could be deduced a priori from a few
general principles, and that it was only at the tine of the Renais-
sance that the essential role played by experience and induction
in the study of nature was first clearly recognized. This condem-
nation of Aristotelianism is so universal that it is found even
among those who have won for themselves considerable repute as his-
torians of science, Emile Meyerson, for example, tells us in more
than one place in his writings that, as Malebranche pointed out,
Aristotle's natural science was not physics but logic, that it was,
in fact, a panlo gicism similar to that of Hegel . The following pas-
sage from De 1' Explication dans les Scienc es is typical;

o.elle (la theorie d'Aristote) presente egalement
un essai de deduction globale de la nature. Comment s'opere
effectivement oette . deduction, par quelmoyen a l'aide des con-
cepts de matiere et de forme les phenomenes se constituent,
c'est ce que les manuels enseignent suffieamment pour que nous
puissions nous abstenir de l'exposer ioi. Contentons-nous de
relever que la deduction domine le systeme entxer. Tout doit
se ramener au syllogisme, et Aristote ne connait ^ Jura-
tion scientific^ que par le syllogisme cette ^'™^°^_
ocp» 1-a- justement tormLA Seller, etant ^ J^^s.
sion resultant des premisses gui sont el i^^f^ristotT
C'est au point <m2^L3^S^B^^^^^^ et tet t

-172-

X'^ression qu'en recoit un home eleve a 1'ecole de la scien-
ce moderne. Mais il est clair que, pour le maitre du paripate-
ticisr.ie, aussi bien que pour aes sectateurs de 1'antiquite et
du noyen age, les deux se confondent p uisque l a nature ne peut
lii^_aue_logi_que.... C'est la un etat'dTesprit qui, sans dout S,
paralt fort eloigne du n&tre. II n'est cependant pas impossible
de lui trouver un parellele a une epoque tres rapprochee de
nous,, Hegel, nous le verrons plus tard, a entrepris une tache

/sinon identique a celle que se proposaient les Ioniens ou Aris-
tote, du mo ins fort seriblable, en ce sens que, tout en ne pre-
trr.dant pas deduire la nature entiere, il croyait cependant
pouvoir recreer, par sa netaphysique, tout ce qu'il y avait

i,e:a elle d'essentiel, (14)

Later in Ihe same work Meyerson claims that Peripate-
tioisr.i v/as an even more extreme f om of panlogicism than Hegelianism,
since Hegel did not hold that the whole of natural science was de-
ducible whereas Aristotle did And he finds a reason for this dif-
ference in the fact that the graat advances made in experimental
science between the time of Aristotle and that of Hegel could not
help but influence tha latter, in spite of his "arrogance logique"
(15) Levelled against the decadent Scholastics of the late middle
ageo, or against the modern writers of Scholastic manuals (to which,
incidentally, Meyerson seems to have gone to find his "deduction
globale") this accusation has some justification. But applied to
Aristotle and St. Thomas it is nothing short of sheer calumny . We
do not hesitate to say that no system of philosophy is so diametri-
cally opposed to Rjxdpateticism as Hegelianism,

■In the first place, it is extremely interesting and
significant to note that in 'his commentary on the opening passages
of the Physics which we have been trying to analyse, St. Thomas
excludeFeliriTcitly-tlKLin^^

5SenT!mo^gH5ol [er7rM of sciengg. This

intirp^Eatlbn had alrea6\yT5elTp?oposed as far back as the tine
of Averroes. According to Averroes, when Aristotle speaks of the
novement from generalities to particularities heha^njiiind^ro-
?£ss of deduction or demonstra t^(^grebyJhe3aper^drawg

S^seallgjtiSrstrThW refutation of this interpretation
is precise and to the point:

Sciendum autem quod Commentator al J^W^lSS*
enim quod ibi, Innata autem est etc., vult °^ndere philoso
Phus modum demons trationis huius Bo«nta^, ^ aoxlioet de

luod ibi dicitur, intelligatur de proceasu in

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et non in decerminando. Cum autem diolt, Sunt autera nobis etc,
intendit^manifestare, secundum eura, quae sunt magis nota quoad
„os et minus nota secundum naturam, scilicet composita simpli-
ci bus,Q : ntemgens_com p_o S ita per confusa^ Ultimo autem concludit
quod procedendum est aElHiversalioribils' ad minus universalia,
(quMl- < M0M£"lJ^^ollariuiju;Unde, patet quod eius expositao non
est conveniens, quia non coniungit totum ad unam intentionem; '
et .quia Mo non intend it Philosophus ostendere modum demo nstra-
i tioni s huius aoientiae . hoc enim fap.iet in . mmf in i-j^ n S^7 n -
dur.i ordinem determinandi: iterum quia confusa non debent exponi
c-raposita, sed indistincta; non enim posset concludi aliquid
h:.-. aniversalibus, cum genera non componantur ex speciebus , (16)

The last lines of this passage which we have italicized
are extremely important. They show that for St Thomas absolute ly
n othing can be deduced from the generalities with which the stud y
of jiat ure begins , But in order to come to understand this point
as clearly as possible; it is necessary to analyse the nature of :
the universality that is found in the first part of the natural
doctrine „ . I

According , to St, Thomas (17) there are two kinds
of universality - - universality by predication and universality
ty causality,. As the name implies, uMversality by predication a-
rises from the possibility which a universal notion. has of being
predicated of a number of inferiors. It consists, therefore, in
pure generality, and as a consequence, the greater universality
of this type a notion possesses, the emptier, the more confused,
the raore ^determined it is. Because of this indetermination, _ no-
tions and princi ples which have mere universality of predication
(g5rSpe~3g urce3 of deduction; ? their emptiness renders them barren.
Universality of causality, on the other hand, arises from the ca-
pacity of producing a number of effects. Increase m universality
of thii~kind~mi5ni' an increase in "richness and fullness of being; _
it neons an increase in fecundity, since the effects actually derive
from the principle which possesses this universality as from a source,

The notion which possesses the greatest universality
/QLpredication is otiouiy the general and confused notion of ^being.
On th?TtheThand, the principle which possesses the gr eatestun i
versality 6f causality is the Subsisted Being, or God, That is
% n? greate^rTo^ouldbe made than to confuse the e two tojda
Vof universality,, And in this connection Proiessoi uv

II me semble que ^^^^^^Jt^Z
Phie la plus universeli^i^nL^PE?^^!,"^!- , ler
nous eVc-pluT-distSKt que le ^^^S^ L^corde,
abaoluraant, plus materialiste que le materials ,

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en effet, au premier connu, a 1'etre predicat le plus univer-
fsel, lo plus confus, le plus indetermine, le plus pauv^Tle
■ plus inevident en sol la place qui, dans notre philosophic"!
Vrcvient a Dieu. La position de Hegel est des lors inferleure,
meme a cello de David de Dinant, >qui stultissime posuit Deum
esse matenam primam/,. (l a , q „ 3, a. 8, c.) Car son principe
en soi premier a plus raison de natiere que la matiere physi-
que,, (18; r J

Now the generalities with which the study of nature
begins posses£{only universality of predication. Prom this point -
of view they are the .emptiest,, the most indetermined, the most con-
fused, the most superficial of all the truths that can he learned
about the cosmos. That ia why they cannot be sources of deduction,

There are some scientific first principles which have
not only universality of predication, but at the same time something
which may be compared with universality of causality. These are
found in mathematics, and that is why from a few primary axioms
and postulates a whole geometry can be rigorously deduced. There
is a world of difference between the principles from which mathe-
matics takes its start and the generalities which constitute the
beginning. of the science of nature Mathematics can progress by
sheer deduction; the science of nature cannot. Yet deduction is
sonething for which the mind instinctively reaches out, since through
it_raan can .. become prior to things arid in some sense the cause of
then. And that is one of the reasons why it is inevitable for the
science of nature to be subalternated to mathematics \ so that natur e
jaay be tr ansformed (to some extent at least ) into a deductive s ystem.J

But for the moment we are interested only in the way
in which the study of nature advances from generalities to fuller
concretion. Enough has been said to show that this cannot be accom-
plished by means 'of deduction. That leaves us with only one altern-
ative : lexperience andjnductionj It is important to come to see
that tte potentiality native to the intellect not, only demands that
we begin with generalities, but also that in attempting uo escap|
from these generalities W eJakg^very_^tgp iS c g .lete dependen ce
tjpon the data of experieSSeTXndThUs we are brought to ^ consider-
Sao^^F^hVp^rTtfet-induction and ^ience p^ in the ^°™f
tic philosophy of science. This . consideration will serve to cleai
up not only tne historical misunderstanding mentioned "-bove^but
also another misunderstanding closely associated with it the often
reiterated accusation that the generalities ^.^°^ l ^X
and St, Thomas proposed to begin the study of nature were nothing
but abortive and ill-founded hypotheses, (XV)

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.,S2_Ill25^iL2BiJ^SB2i. i ^i5a»

We Know of no better way of introducing this question
than by quotias a text of Aristotle which the historians of science
have consistently ignored-

Of things constituted by nature sone are ungenerated,
imperishable , T and eternal, while others are subject to generation
and decay c The fomer are excellent beyond compare and divine,
but less accessible to knowledge „ The evidence that might throw
light on then., as on x,he problem which we long to solve res pect-
ing them, is furnished but scantily, by sensationj whereas res-
pecting perishable id] ants and animals we have abundant infor-
mation, living as we do in their midst* and ample data may be
collected concerning their various kinds, if only we are willin g
to take sufficient pains o <> ,

Having already treated of the celestial world, as far
as our conjectures could reach,, we proceed to treat of animals,
without omitting/to the best of our ability, any_ member of the
kingdora . C however ignoble J ) For if some have no grace to charm
the sense, yet even those, by disclosing to intellectual per-
ception the artistic spirit that desi g ned the n, give immense
pleasure ~td all whe can trace links 6T~causation, and are in-
clined to philoso phy^ Indeed it would be strange if mimic re-
presentations of than were attractive, because they disclose _
the mimetic skill of the painter or the sculptor, and the ori-
ginal realities themselves were not interesting, to those at
any rate who have eyes to discern the reasons that determin ed
their formation. [W e therefore must not recoil with chil dish
ivijnMTTromlhe"^^

re^ln^FTStu^~ii!^el^^ «* strangers

whT^Eie-^o-Ti-iTMmT ; olI5rhi J -.i waring himself at the furnace
in the kitchen and hesitated to go in, is reported to have bidden
them not to be afraid to enter, as even m that kitchen divi-
nities were present, s^wg^jhouldjgnture on the stud yofeyery
kind of an^J^dthoot^i^^T^^ 011 f}* f 1 ^iT^f 1
^^^^^^:^^^^^te an^l Sngdom

VOrSOrC ™^f^^^A l7like dis-esteem the study

an unworthy task, he must hold in ii*e ^^ ^^

of nan. For no one can look at th « P"^ 1 £ _ _ ^ thout
- » blood, flesh, bones, vessels, and the nice
V^h_repugnanceg) (20) T>e Pwh'tw fl^. cA-r.

. _ " ~ We feel that this text ^rings into clear light the^
spirit of research and the respect tor cont./uu

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ArisweJ

„. ;le's study of nature. Nor mat it be looked upon as an ex-
ceptional and isolated passage that demands some ingenuity in order
to be reconciled with the accual practice and the epistemological
principles of the Stagin-oe. For other texts of like character could
easily he adduced; as for example the one found in the first book
of ^e.^jratmne_.et^c_r™p_tione, where he points out that the main
obstacle to the study of nature is_ insuf f iciency of experience and
that only those who live in great intimacy with natural phenomena

L can succeed in such a study, (21) As far as actual practice is
concerned., one has only to read the natural treatises that are far
a dvanced in the direction of concretion ., as for example, the Histo-
ria Aniralium and the De Partibus Anir.aliun, to see to what extremes

'he pushed the experimental method. It is said that Alexander the
&reat had thousands of men engaged in research in every part of
the worM that was then known in order to assist Aristotle in the

^writing of his Hist oria Animalium , (22) It is true that most of
his experimental reseacch is restricted to the field of biology,
but sufficient reasons have already been brought forward in Chapter
I to explain why this is so„

But the most important point in this discussion is
to chow that this experimental method follows logically and inevi-
tably from peripatetic epistemological principles. And in order
to do this we must return to what we saw in Chapter II about the
intrinsic nature of physical science

In discussing the distinction of the sciences we ex-
plained that natural doctrine differs from all the other sciences
by the fact that it does not abstract from natural matter ; and that
as a consequence all of its definitions must be formulated in terms
„of sensible matter. Pr opositions which prescind fr om sensible matter
(can have nothing more than a dialectical meaning in physics^) Wc
pointed out that St. Thomas drew from this the principle that un-
like mathematics and metaphysics, physics must not only begin in
sense experience, it must also terminate in it. Scientific conclu-
sions have no meaning in natural doctrine unless_the y aro verifiab le
to sense experience. And that is why Aquinas could write: qui sen-
aun negiiiirSHSturalibus incidit in errorem. Rt haoc sunt ratu-
ralia quae sunt concreta cum materia sensibili. (23 J It is onjy
'experience that can provide us with natural definitions.

All this evident^ ties up with the Peripatetic doc-
trin^of hylemoShisra! ^urallorms, which are the^b ec^of natural
science, have their very being "in concretise, ad ^°riam . And
this refers not merely to their existence,^uttothe^pr^gnce.J
It is extremely important to keei , in mind tfaata ™*%£f%^
not a quiddity. It is not knowable in itseii onu y

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dcntly of natter - - just as natter is no t knowable indc-oendentlv
of form. Even God does not know material foms except l7rSaSon
to ^tter, sj^c^^gpenfentj^y^f na tter a natur al L, is nothing.
As a consequence ,^the-p^rf^MSTcr^uin5okedg e of theso f 5rn^
depends upon the intimacy of our contact with sensible natter. And
thatj;!^!^-*^?^^ sul3scribe to

thejesanciple formulated by_Eadington:\ "Every item"ofl^si5aT know-
ledge must be an assertion o f what has been or would be thTreiult'
ofjgarr ying cut a s pe cified observatio nal procedure -,"7 T24p

There are many reasons why the whole study of nature
is couple bely dependent upon experience,, but in some respect the
nost profound reckon is the one hinted at by Aristotle in £he pas-
sage quoted above from the De Partibus Animalium; The material u-
niverse is a work of art,, And it is impossible to understand the
role played by experience in the Thonistic philosophy of science
except by corxaig to see the precise way in which art enters into
the • strvuiture of the cosmos 3

Towards the end of the long analysis of the meaning
of nature ' carried on in the second book of the Physics , St. Thomas
arrives at his weal known definition: "Natura nihil est aliud quam
ratio cuiusdau artis scilicet divinae, indita rebus, qua ipse res
novetur ad finem determinatum." (25) A nature is something essen-
tially rational; it is a divine logos » And this applies even to
the purely material principle out of which cosmic reality is cons-
tructed,. " (26) The whole purpose of the study of nature is to come
to know these divine logoi dn their' ultimate specific concretion. )

Nov/ at first glance, all this may seen to add up to
an argument against complete dependence upon experience rather than
one for it, For to say that the cosmos is constructed out of divine
logoi night seen to indicate that it is a perfectly logical and
perfectly rational system, and that it therefore lends itself more
to deduction than to induction. As a consequence Meyerson might
seem to be justified in writing: "La science d'Aristote etait non
pa 3 une physique . mis une logique . . Mais il est clair que, pour
le r^tre du peripateticisrae, aussi bien que pour ses sectateurs
^ 1'ontiquiJ et du moyen Sge, lggdguxse ^onfond ent puisque .la
nature ne^eut etre. que logiguo," ~TWT Moreover, *e ^terial
'^TCr^ wSTo-SSriTof "di^iM art, and yet the science which
Weals with it is not complete^ dependent upon experience.

,, . __ ttor f fact, however, there is a vast diffe-

»» f ^J%£ £,"££ r .7- * *-?£ "iTSS SL.

-178-

verse that is free of matter. And it iq ^ tw. ,1. r •_■.
^ili* that. -the conplete d^^^^e^iSStS 1^1

Immaterial forms are fashioned by divine art, but only
with respect to their existence. This does not mean that their es-
sence is m no way formed by Godj it merely means that this format-
ion consists only in bringing the form into existence. Because of
their simplicity, immaterial forms have no plasticity intrinsic to
their very essenoe, and consequently within this realm of essence
the art that produces them canno t (Compo se) Material forms, on the
other hand, are fashioned by divine artTnot only with respect to
their existence, but also with respect to their essence. The very
fact that they .are not pure forms, that in their very essence there
is a principle of indeterMnation that is susceptible of an infinite
variety of determinations, gives them an intrinsic malleability that
leaves free scope for composition, This principle of inde termination,
this source of plasticity, is obviously prime matter s which is in
potency to all forms a And all this brings us back to something we
saw in Chapter II in connection with the similarity between the stu-
dy of nature and practical knowledge : as we ascend the, hierarch y
of beings th e operabilitas of things increases

But perhaps we can give clearer outline to this point
by having recourse to a rather crude illustration, drawn from the
realm of mathematics. Eetween any two given numbers in the series
of integral numbers there is only a finite multiplicity of numbers.
And the numbers in this multiplicity are already predetermined.
In order to actualize them a simple process of designation is suf-
ficient. But between any two points in a continuum there is an in-
finity of points, and these points are not predetermined. In order
to actualize a certain magnitude a simple process of election is
not sufficient,. There is required a previous process of determinat-
ion by which the magnitude in question is carved out, so to speak,
of the potentiality of the continuum,

T» somewhat the same way, we may say that between any
too given angelic species in the hierarchy of the separated substan-
ces orlv a finite number of species is possible. This is not a li-
mitaticn of God's power to imitate. His essence m ^f^ial forms
since JUBt , a there is no superior lindt * ttoB««s of ntegra^
numbers so there is no superior lunio to the niciarciiy * ,

substances which God can' create, Bu^gtween^Jwo given materia.1

number of other species J^jaos^l^Im^ei lax ^ > qinrole

Process of election by which existence is given ^ ^

i^al forms are not predetermined; if tney were, e

-179-

not be pure potentiality - - there would bo a latitatio format
That is why previous to the process of eleotior^hflrlsifce
is given to them there must be a process f coroposition by which
their very essence is formed. In other words., the production of
iraiaserxal forms merely consists in giving existence to essences
already predetermined in the divine exemplary ideas; there is no
composition in these exemplary ideas themselves „ But in the case of
material beings there is compos ition| in the ver y exemplary ideas
acc ording to which they are pro ducedn_) . "

' In the mathematical world nothing is formed in the
true sense of the word; nothing depends upon art in the sense of
depending upon free determination for in mathematics all things
a re analytical. And if mathematics is called art, it is only on
the sense of its being a sp eculative art, like logic. In the mete-
physical world there is formation by art in the sense of dependence
upon free determination, but only with respect to existence But in
the physical world there is formation both in the realm of existence
and of essence o The material universe is essentially plastic.

That is why there is no way of arriving at a more
profound view of the cosmos than by seeing it as a work of art.
In spite of his tendency to look upon the universe as essentially
mathematical, Sir James Jeans touched upon this truth when he wrote:
'To my mind, the lav/s which nature obeys are less suggestive of those
•which a machine obeys in its motion than those which a musician
obeys in writing a fugue, or a poet in composing a sonnet." (28)
But in order to understand just how completely and essentially the
cosmos is a work of art it is necessary to recall that because of
its transcendental freedom, divine art is not tied down to the yias
(^terminates that are characteristic of human art._ In this respect
divine art is similar to prudence which proceeds (per) vias determinan-
daso Divine art can dominate contingency in a way that completely
transcends human art; it can order it with infinite finesse. In facx,
divine art shines nowhere with greater brilliance than in the realm
of indeterminism and chance. And in the Thomistic view of thmgs,
the physical universe is essentially immersed in contingency, simply
became it is essentially material. That is why the divine *.ogos
that is found everywhere in the cosmos is not the perfectly analyti-
cal rationally that is found in the mathematical world, nor the
type of rationality that is found in the metaphysical world li-^
essentially an artistic logo_s - - ratio, artis divinae - - which orders

¥ Aristotle, St. Thomas and Cajetan on the part that contingency

-180-

aiid chance play in the universe to apnreoi/i+o «,„<■! •■

charge „ The Peripatetic and the SnlSt+f • fals "y of ^is

antipodean. (29) Spxnozistic universes are completely

All this helps us to understand the part that experience
plays m natural science. For as we saw in Chapter II, L L^udy
of nature we stand before the universe as before a work of art
There is no way of telling a_priori what an artist is going to do.
One has no wait to see what he actually accomplishes. Nor is it pos-
sible oo deduce from the first general outlines the particular details
thai, will eventually enrich the composition. The only way in which
a_priori knowledge can be had of a work of art is for the artist to
reveal what he intends to do. Something of this nature has actually
oociK-ed in the case of the angels, into whose intellects God infused
the intelligible species of all the things which were to cone frcm
His creative art. But for us whose knowledge is posterior to things,
the only way in which we can get to know nature is by experience.
It is true that given the subject of a certain work of art sons vague
generalities may immediately be known about it. Given, for example, <
the fact that an architect is going to build a house, there are some
general things common to all structures which serve aa, shelters that
we can immediately know about it. These do not depend ohe free dis-
position of the artist. But as soon as we wish to come down to parti-
cularities we become dependent upon the free will of the artist. For
{ there is an infinity of ways of making a house. In somewhat the same
vray, given the idea of a material universe, there are some things
that we can immediately knov/ about it. We can know, for example, that
nan must exist in it, since man is the rais on d'etre of the v/hole -
universe. But there is an infinity of ways in which the material uni-
verse in its evolution may prepare for the final production of man.
From the beginning the cosmos has been a continual process of form-
ation and artistic composition. That is T/hy there is a great deal
of truth in Plato's idea of the' demiurge which constantly works the
world „ And the only way to discover the actual line of species that has
!ed up to man is by natura l histor y, as St. Augustine has pointed out, (30)
This brings us back to the profound significance of the "erit" in the
Passage of Aristotle quoted in connection with, the question of the
relation between physics and practical knowledge. Natural things are
no t knowable except in the order_o f existence . The only way to get
toTSo^hem~ia^blTknowiHg _ Them assisting* that is to say by_expe-
dSSpo. As we remarked in Chapter II, the study of nature, because
of its likeness to practical knowledge, must be built up out of bits •
generated from experience. This constitutes a great difference between
the science of nature and the other sciences.

it

. There is, then, great wisdom in Aristotle's remark Jbhat
is noble to soil one's hands in experiments because by so doing

-181-

one gets to know the art of Him who made all things. There is all
the difference xn the world between a^^turaldst^anda peripatetic.
The former merely delves deeper and de^FInto^to-^ curity of ma t-
ter. His knowledge is something like the cognitio nocturna of the
fallen angels, because it is not referred to^dTSltlASreas the
end of his study is night, the end of the study of the peripatetic
is light - - the light of divine intelligence, for the deeper he delves
into matter the closer he is coming to divine art, since he is getting
into more intimate contact with things in their plasticity. The far-
ther advanced science gets towards concretion, the more it gets into
/the realm where divine art composes more than anywhere else.

That is why every true Thomist has a profound respect
for experience. For it takes the place of the infusion of the angelic
s pecies ; it gives a share in the scientia visionis of God, And the
farther advanced the student of nature gets in experience, the more
his knowledge becomes like that of the angels which depends directly
upon the divine species - - the more he participates in the scientia
visionis of God And in this connection it is interesting to note
that if the term of this increase in experience could be realized,
if the ultimate concretion could be reached, \there would be a com plete
destructi on of experience, ) for there would be perf oct a priori know-
ledge. This is just one instance of a very significant truth which
we shall examine in some detail in Chapter XI, namely that if the
term of the ten&enoy of experimental science could' be reached there
would -be a contradictio n, "L 1 esprit huraain est absurde par oe qu'il
cherchej il est grand par ce qu'il trbuve," (31)

The conclusion that this discussion imposes upon us
is that every part of the study of nature is dependent upon experience,
but not in the same' degree, the generalities with which this study
begins are not a priori hypotheses, as so many critics of Peripate-
tic-ism are inclined to think. They are truths that are drawn from _
experience. But precisely because they are so general and superficial,
and because they are the truths that are the most proportionate to
our minds, they do not iom^Aa^e^tJ £S l^_e^eri-e^o;[x^^^^

f^^F^S^i ve at the -gelSraTlSture of moti on, l^f^^L
[simple experience with any kind of motion, such as the fall of a leaf,
the movement of a finger, or a change of color in the sky is ^° 1Qnt <
for everything that can be known about the general nature f™*^
Vis contained perfectly and completely in any one of these examples.
But in order to get at the nature of the_spe^ial^p^f^giity
fet is proper to a particular Jioturaljpecies Li^fJ^^g

^^-srvr^ssr^sss^^m^ SreiTtive slJI?pllcity

upon experience increases. And it is puti"*^

-182-

of the experience that is required for the generalities which mark
the beginning ox the study of nature, and the comparative ease with
which the mind disengages them that have led to the erroneous opinion ..
that they are nothing but abortive, hastily formed and ill-formed
generalizations* , (32)

But perhaps at, this point one might be tempted to ob-
ject: did not Aristotle frequently have recourse to hypotheses that
were not fully founded in reality? Assuredly - and so has every other
scientist worthy of the name who has. really understood the nature
of science, from Thales to Einstein. And this applies even to Newton,
in spite, of his well-known dictum: Hypotheses non fingo, Newton me-
rely failed to grasp the full significance cf the method which he
put to such good advantage, Hyp otheses, as we shall bring out presentl y s
are of the very essence of the 'study of nature And to admit, tha t
Aristotle had recourse to them is simply to say tha* while on the
one hand he had., ho part,, in the apriorism of Descartes who spurned
sense experience and .wished ' to deduce more geometri co even such spe-
cific elements in. natuije as ."the heavens, the. stars, . the earth, and
on the earth: water., iron and minerals," (33) on the , other hand
he was far from falling into the naive empiricism of Francis B acon,
Although both Descartes and Bacon are counted among the principal
founders of modern' science, it, is certain that modern science has
sprung neither' from, the rejection of experience of the one, nor the
rejection of. hypotheses of the other, but from a union of hypotheses
land experience, such as is found in the doctrine of Aristotle.

'But were not the hypotheses of Aristotle hastily formed?
The answer is yes and n6. For in a' sense all good scientific hypotheses
are hastily formed, ' Of their very nature, they must anticipace reality;
they must, reach beyond' the: actual deliverances of experience. From
this point ofview a! scientist who is too cautious is apoor scientist.
It is true that as we look back now from the vantage poim, f™W-
centuries of scientific progress some of the ^theses of Aristotle
look extremely' precipitant. But, as we suggested in Chapter _I, is
it so certain that when as many centuries of progress have pa ssed
over the hypotheses of Einstein they will not appe ar oust as pre
cipitant a^the Aristotelian hypotheses look to us *°^ h ^
lowing well-known passage of Poincareis extremely relevant

■ • Chaque : si|cle ^^^^^^^^^
d' avoir generalise trop vite f J^^^it SO urire; sans
pitie des Ioniens; Descartes a son tour nous i
aucun doute nos fUa riront demons wg*"^ \ sulte jus _
Mais alors ne pouvons-nous aiier J raillerie s
qu'au bout? N'es^ce pas le ^°^'^ P ^ntenter de 1'expe-
que nous prevoyons? Ne pouvons-nous nous

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nenco touce nue? Hon, cola est impossible; ce serait mecon-
naitre completement le veritable caracterc de la science, Le
| savant doit ordonner; on fait la soience avec des faits oonrne
June maison avec des pierres; mis une accumulation de faits

n'est pas plus une science qu'un tas de pierres n'est une mai-
Uon " (54)

In connection with this question of hypotheses one
often encounters the charge that the Peripatetics were notoriously
guilty of arbitrarily and artificially forcing facts to fit into
preconceived theoretical frames We do not believe that this charge
is justified. For, in the first place, it is something that was
explicitly and strenuously combatted by Aristotle, In the second
book of the De Goelo, for example, he writes:

In fact their (the Pythagoreans 1 ) explanation of
the observations is not consistent with the observations . And
the reason is that their ultimate principles are wrongly shsu-
medj they had certain predetermined views ,C an fl w ere resolved
to bring~ eve^bhin g~ljito line with themaoj But they, owing to
their love for their principles, fall into the attitude of men
who undertake the defence of a position in argument. In the
confidence that the principles are true they are ready to ac-
cept any consequence of their application,, As thou gh some prin-
I cj ples did not require to be ju d ged from their re suit s ,<3nd
/parti cularly from, their fin al issue. ) And that issue, which m
the case of productive knowledge is the producij in the knowledge
of nature is the unimpeachable evidence of the senses as to
i ^eac h fact) (35)

Moreover, a number of cases could be cited in which
the great respect they had for sense experience led them to formu-
late points of doctrine that could only with some difficulty be
harmonized with their fundamental principles. An °>^ ^°J ££
mediately suggests itself is that of the doctrine of incorruptible
natter. Becaufe sense experience revealed no other ^ange^he
heavenly bodies. eoccepjLlo^j^tion, they. were led to ™* *£>%£?
that these fcodi^riiSrirTnOSSG^ incormptible and thai con
sequently the I^ime_JBMErw^

fe l rirIi ^F 3 ^^ has shown

ion of prime matter. In fact even ™aay> *"■ intrinsic
that the celestial bodies are susce p tibl * °J f °^f™% re do
changes as terrestrial bodies, ^J^^l^TtntZu^le
not think it possible to prove apodic^caiay x re conciliat-
matter oannpt exist somewhere i*/ the °° a ™ S ; h f * ripate tics had
ion demands considerable ingenuity, and if ohe per P

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had less respoot for sense experience it would have been a good
deal easier go arrive ajjxiori at the conclusion that the celestial
bodies were capable of intrinsic mutations;,

Another example of this kind is found in the doctrine
of spontaneous generation. This doctrine was formulated because
sense experience revealed the generation of living beings out of
putrefying matter, and at the time there were no adequate' means
for detecting the fact that eggs had previously been laid in the
^decaying mass, Here again we have a doctrine which was adopted in
order to save sense experience even though it could only with con- .
sidei-able difficult y be r econciled w i th the b asic principl e of the -*
essentia l difference betw een livin g and non I JA^gjmttsr,

. One of the most common objections brought to bear
against peripatetics is, that they failed, to recognize the hypothe-
tical character of their hypotheses, that, they consistently mistook
them for certain principles c In order to assess the justice of this
charge we mist consider a few texts „ Speaking of the theory of the
incorruptibility of the matter of celestial bodies., Aristotle re-
marks: i' ■'■.'■ '

The mere evidence of the senses is enough to con-
vince us of this, at_least w ith human certainly 9 For in the
whole range of time past, so far as our inherited records reach,
no change appears to have taken place either in the whole scheme
of the . outermost heaven or in any of its proper parts, (36)

Coranenting on this text s ..St. Thomas has the following to say:

Secundum, signum ponit ibi.; AccMit_autenJioc_et_per
sign-m etc,: quod quldem accipitur(ab experientia) longi tempo-
"risTEt dicit quod id quod probatum est per rational! ec per
concern ojpinionem/'aooU±t f idest consequitur sufficienter;
non ojiiaem siinplioiteTrSSa. sicut. potest dlo ^^°3^^
nem ael humnom f idem, idest quantum homines possunt testificare
do hiT^e-pSvo-tempore et a remotis viderunt. . , gcjpi

tatio deprehendacurS sicut tiansr depre henditur trans-

ditur On ^-^f^gSs^te:^ &01b breviorem vitam
wata.txo cams P vel alicuiuo <*-.» caelum sit

\ d^ndora c iuo transmutationemo \o()

-185-

•wri'ces:

In the second book of the' same vrork, Aristotle

Duabusautem dubitationibus entibus, de quibus meri-
to utxque quxlxbet dubitabit, tentandum dicere quod vldetur;
dxgnum esse reputantes promptitudinem raagis impuiari vereoun-
diae quau audaciae, si quis, propter philisophiam stare, et
parvas suffxcientias diligit, do quibus raaximas habemua dubi-
tationes, (28)

St, Shoraas commentary on this passage is extremely enlightening:

Qicit er ?o primo quod, cum circa stellas sint

duae dubitatior.es de quibus rationabiliter quilibet potest
dubitare, tentare deberaus dicere circa istas dubitationes id
quod nobis videturj ita scilicet quod nos reputemus dignum es3e
quod prompitudo hcmiriis considerantis huiusmodi quaestiones
oagis dobeat imputari ,ye£22!£^iaes idest honestati vel modes-
tiae.- q;.1£31 audaoiae % , idest pr-aesumptionif si tamen ilie qui
Ir.i.iuiiraodi duiPcat-iones considerate diligat etiam parvas suf-
ficientias, i<,e c parum sufficientas raidonesj. ad inveniendum
de lllis rebus, de quibus habemus maximas dubitationes; et hoc
• propter desideriimi quod quis habet ad philosophiamj ut scili-
cet eius principia stent, idest f irma permaneant . ,

■Illoruim- (ludoxi, Aristotelis, et Ftolemai) tamen sup_r
T jositiones quas odinveneru nt,, non est neaessarium esse veras:
licet enimp T,p. : llbus aup-oositionibu.s factis „ra pParentia aalva -

i rentm-^non tarnen oportet dicere ha s_su EPQsitione s esse veras -,
quio forte socuridura aliquem alium modura, nondum ab hominibus
compreherisum,; apparentia circa stellas salvantur, Aristoteles,
tamen, utitur huiusmodi suppositionibus quantum ad qualxtatem
motuum, tamquani verido (39)

Another very significant text is found in the Suraa:

Di-ioadum quod aliquo.m rem dupliciter inducitur ratio,
Uao irodo k -1-obP.ndum sufficienter aliquam radicem; sxcut xn
rv3ienvla r fl .turali inducito ratio sufficiens ad probandum quod
mot is «o^i se^e- sit uniforms .velocitatxs. Alxo modo xndu-
cSS ratio, Se"non sufficienter probet radium, geOjgg

sa lva.ri pogse njTi (40)

Relieve that these texts, which were completely

-186-

'ignored by historians until several of them were brought to light
by Pierre Duhem, (41) establish beyond a doubt the fact that A-
ristotle and Saint Thomas were acquainted with the hypothetical

k method employed by modern science.

It would be interesting to examine each
of them in detail. But for our purpose a summary conclusion will
suffice,, lie believe that they make it abundantly clear that the
peripatetics had accurate knowledge of the hypothetical method that
has become the very soul of modern science,, The fact that in indi-
vidual cases they have erroneously believe! that they had apodictic
arguments in favour of certain propositions when such arguments
did not exist, doe s_not in no way invalidate this claim . As is e-
yident from these texts, the position of Aristotle in this matte r
ig_ le_s s unambig uous than that Of St, Thomas „ But there is ample
reason for believing that even the former had great diffidence a-
bout the truth of the theories he proposed, that he attributed to
them the certitude that is necessary for working hypotheses, that
he posited them as if they were true, in order to save the phenomena.
But whatever may be thought about the position of Aristotle, there
can be no doubt about the position of Aquinas, In the passages just
cited from him there is an accurate description of the hypothetical
V method used in modern science.

It is not without interest to note ! that the theories
to w hich Acuinas attributed onl y probability were precisely jhosg
u~p^nliich_?iItSOS

What y ihfie^Tode rn critics TaiTtol^e is that }^* ^™> y
saved the phenomena that were known at that time jus. as successfully
"the theSies of classical ^^^^^S™^^™
known during the seventeenth and eighteenth ce nturxes - -^u st^as
successfully as the theories, of Einstein save Oie phenomena that
successruxxy as tno _ significant that nowhere do we

are known today. It is extreme^ a Jf credited with being the founding
find in the writing of those who ar * ^ediv Galileo,

fathers of modern science, such as °°^^ n ' oJ Jhejrue_meihod
anything that comes fig_olgJ§-J -Mff^ g^ qulna; . .^ nPjTt^T

that Copernicus in his Com^tar3£^i2^____ postulates:

lestiumseems to posit^his ^^^^f But late? in his
ffsilSbis aliquae petitiones . • • °°£ e J£ attitude is far less
De Revo lutionibu^Cael£SJibu^^bri^ ^ osiander brought

reserved. In his introduction to tn £ mthod? „ Neque onim

out with great accuracy the true sci ^ veris imiles quidem;

necesse est eas hypotheses esse ^° r > tionibus congruentem e-
sed sufficit hoc unum, s \ f i° u1 ^ °^ r Vwith such a doctrine: »Je
xhibeant." But Kepler would have no part witn s

-187-

'n'hesite pas a declarer que tout ce que Copernic a amasse a poste-
riori et prouye par 1' observation, tout cela pourrait, sans nulle
entrave, etre deraontz-e _a .priori, au moyen d'axiomes geometriques,
,au point de ravxr lo temoignage d'Aristote, s'il vivait," (42)
Galiloo distinguished between the point of view of astronomy in
which the hypotheses, have no other sanction except conformity with
experience, and that of philosophy of nature which bears upon the
objective nature of things c But if we are to believe Duhem (43)
this was a purely theoretical distinction formulated to avoid the
censures of ecclesiastical authority, and Galileo accorded full
certitude to all o f his theories In any case there can be no doubt
that throughout the reign of classical physics full, certitude was
universally attributed to doctrines which were in reality only hy-
pothetical,. And if today the hypothetical character of sciences
has become generally recognized, it is undoubtedly due in large
measure to .the rude awakening occasioned by the downfall of Newtonian
physics o St. Thomas did not need such an awakening . [ In spite of the
fact that the p h ysical theories he held saved the phenomena, known
a t the time as successfully as modern theories save the phenomena
known now , he was sagacious enough to recognize their hypothetical
character J

But even more important than the consideration of
the texts cited above is the consideration of the certitude that
the propositions of experimental science enjoy de jure in the Pe-
ripatetic philosophy of soience. And this requires an analysis of
the relation between certitude and experience in the study of nature.
Before embarking upon this analysis, however, 'at least passing at-
tention must be paid to one last objection that is frequently pro-
posed against the position we have been maintening with regard to
the importance of the role- of experience in the Thomistic philosophy
of sciences It is this: If according to Thomism experience plays
such an indispensable role in the study of nature, and particularly
in that part of it which is to sooe degrees advanced m the direction
of concretion, why' is it that St. Thorns and the medieval ?<*ooMen
were so KotoriOusly remiss in the actual practice ^experimentation.
We do not hesitate to' grant the premises upon which .his objection
is based. Aristotle was, as we have already pointed out, a ^
experimental But St. Thomas and the medievalists, with^a few notable
exceptions, such as St. Albert the Great, were not. Jere was^ how
ever, a reason for this. Thg ^aievaligts^r^ primari^ theologies .
This'does not mean that th3re were n^Tartte^ txma great phi
losophers, nor that theology dictated to their Philosophy xn the .
manner usually described by his torians. « .^p^he probSf
interest in philosophy was conoent ^f^ f£ problems that had
that had a bearing upon theology and upon the P^ ± _

the greatest significance for human Jjg.^^ot sense of the
marily interested in science in the run anu ^

-188-

vrord, that is to say, science in which there is certitude, and as
we shall see m a few moments, experimental science does not give
true certitude,,

Whatever may have been the actual practice of St„ Tho-
mas and his followers, the only important point is that in princip le
according to the Thomistie philosophy of science, the student of
nature must, if he is to realize his purpose s he carried constantly
forward toward fuller concretion,, and this advance demands an ever
^increasing dependence upon experience Here we run across a remar-
kably striking paradox,, Auguste Comte r the father of Positivism,
denied the necessity oncL validity of extended experimentation,, He .
rejected, for example, what hs called the abuse of extended micros-
copic research. (44) Nowhere do we, find anything of this sort
in the doctrine of Aristotle o? St, Thomas i which, if we are to
believe critics, was so thoroughly antipositivistic On the contrary
the very principles cf this doctrine demand unceaaing experiment-
ation and recourse to the most refined instruments of research a-
vailablea It may readily be admitted that neither Aristotle nor
St Thomas ever anticipated the perfectibility of our means of ob-
servation and experimentation that modern progress has repealed,
and that as a consequence some of the positions assumed by them
were far more provisory than they suspected „ But the fact remains
that their conception of natural science demands a conformity of
observation which must constantly increase both in breadth and in
depth,

5 Experie nce ■ and Certitude

, Let us begin, our analysis of this problem by consider-
ing the following text of Aristotle:

The sconce which is knowledge at once of the fact
and of the reasoned fact, not of the fact by xtsolf wxthout
the reasoned ^t is^^ore^act^the P-^ence
Asexence such as arxuhmct.xc, v,n . ^ r

tie, S ua inhering n a *uh tratum xs more^ ^ ^ g
■m a science like harmonic a , ™^ , science like arith-
inhering in a substratum ; ^« Ufl ™^ io elements, is more

erect than zrA ?rior to *%^'j£%^ B , is this: a unit
elements . What i mean by_ ^dit.onax ^ subst ance
is substance without position, while a y

-189-

with position] the latter contains an additional element, (45)

In this passage Aristotle brings out the throe basic principles
^oh_doterming_thfl ;L rgla tlve certitude found in the~ioli^ ogT Al-
though in writing this passage ho did not have explicitly in mind
the point which is of interest to us here, we may apply these prin-
ciples to our purpose j whiih is to shoy/ that in the measure in which
the study of nature becomes increasingly dependent upon experience,
its certitude decreases,,

The first principle laid down by Aristotle is this: ( )

a science which not only gives us facts (the _quia) but also the V_

reasons for the facts ( the propter q uid) is more "certain than a
science which provides only the facts v/ithout the reason for them,.
Now as increasing 'experience carries us forward towards fuller con-
cretion, the abundance of facts continually grows, but at the same
time it becorces constantly more difficult to disengage the prop_ter
quid to explain these facts a And the reason for this is fairly evident:
"the 'more we advance, the more we approach things -under the aspect
in ^7hich they depend conpletely upon the practical knowledge of
God, and scientia vis ionis, which involves something that is out-
side the realm of knowledge, namely the divine free will. (46)
It is precisely because it eventually becomes impossible to disco-
ver a proper prop ter quid in the parts of natural doctrine that
are advanced towards concretion that it becomes necessary to reach
up to mathematics to find a substitute propter qu id through_a_p_ro-
cess of su baltemat ion, That is""anotner way of scaring that as we
eraBriTfroia the- parFof the study of nature that is most conforma-
ble to our minds it becomes necessary to substitute the science
that of all- the sciences is most in harmony with the human intel-
lect, (47)

, The second principle of Aristotle consists in this (o

that a science which deals with a subject is less certain than a ^_

.science which does not. In his commentary on this P^f » f "J™
am explains what Aristotle means by the term "subject s Et acci^
pitur hie subaectur^p^5Ste£^^2SSiMlig-f ^"^J^jjT '

acced "ur^^taTr^^ than

as a science which deals with sensible matter is loss^ ^ ^

one that does not, so that part of -He syuoy ^ ^ t ^

porience has carried deeply ^^^ -ncSe sensible
that part which is not so completely tuimersea -u

\ natter. ^~

V/M.Qh has to do with fewer elements is more certai

-190-

!« . &

■Ak-

v/hich the elements are more numerous. This has a direct application
to our problem. For increasing experience carries the study of na-
ture forward from generality to greater specificity, in such a way
that the proper distinctions of things gradually emerge This is

Ivrhy the fartner tne study advances the greater becomes the need
for more p^tigular(an d consequently ) more numerous principl es „ /For
tfe jaroper differences of the natural species cannot be deduced "

\f ron each other} ) as we have already pointed out,, Hence the necessity
of as many principles as there are natural species to be known
It may be said that the nur.iber of principles in experimental science
tends towards inf i nity . Each natural species is a primary datura
and the source of a number of original p ro positions „ And the multi-
tude cf possible natural species is infinite c It is true that the
theories of evolution will attempt to reduce this great variety
to a basic unity, but these theories presuppose experience with
the original variety and must succeed in leading back to it„

, From all this it follows that there is an inverse

proportio n between the dependence of natural science upon experience
land the degree of certitude that is possible in it, That is why
the prudent student of nature will corur.it himself less categorically
and with greater reserve and with more abundant qualifications the
raore he advances towards concretion,, As Aristotle points out, "since
the truth seems to be lite the proverbial '"door, which no one can
fail to. hit, in this respect it must be easy, but the fact that
we can have a whole truth and not the particular part we aim at .
shows the difficulty of it." (49) And it is for this reason that
the universal propositions advanced in the more concrete parts of
natural doctrine do not L enjo X Jrue_^ertitude» Nor is it any cause
for wonder that in~a^cielTcT^hIchluial^ith mobile being, ceroi-
tudo so quickly fades into mere probability. (50) But it is ne-
cessary to try to analyze this question more accurately, BLJS§
general t.ro positions which the_j mnd^ir3t_d^eng % ges from .its ex -
Eorie^ilrifh cosmic re^ ^^^^M^^^^^

a nd predicate, \me^^~^^S^^^r^^^^i

jn^sitioiroT. t^arHsnE^^^^JS^ffitoine

but also the £ropter_guict r _That is ^^.^^^^^,0^
Yidchjire ja ad^p_suj^^

,the Biy^csjmd^ is direct co r-

in jL he_strict Ji ^^ ^ for

impendence between the oiarity *£ to w hat is found in theology
us and their certitude, m cont * as Y °^for us have greater cer-

Uose principles though extremely obscure f^hav J

Uitude than principles which have greater oxori „

-191-

But as natural science advances towards concretion and dependence
upon experience increases,, ana lytical relations become less and
loM^pparent . \Prop_os itions become more and more exn eritne-nfaTTl
There ultimately comes a point (and it is very quickly reached)
at which the propositions are purely experimental, that is to say,
thgyjnerely Cfor-'Julato ^v vhat experience presents to the senses , From
that point" f orward no true scientific knowledge in the strict sense
of the word is possible The propositions give only the _quia and
not the propt er qu id,., In other words they are not analytic7~but
.purely synthetic,. It is true, as we shall try to bring out presently)
(that the mind will not rest satisfied with this pura synthesis^ )
it will try to triuinph over it yby the projection of its own sub- -
TRct ive logos) by the creation of a 'pro pter quid " , in such a way
that Cin a sense) it will be able to arrive at synthetic a priori
judgments o | But- in the last analysis the propositions remain s ynthe-
tic and n ever become analytic ) At this" juncture we have arrived
at the frontiers of philosophy and experimental science

John of St. Thomas has brought out this point with
considerable precisions

Non est idem propositio per so nota quod intuitiva
sivo per experientiam sensuum . nota 9 quia quod sensu cognosoi-
tui-j non cognoscitur O)_propositi0i> ) sed ut simplex ob jeotun ^
a pprehensurft;,) neque ex sola explicatione terminorum innotescit,
sed quia experientia externa atting itur Et sic nivem esse al-
ban, licet in sensu sit per experientiam notum, in intellects
taraen non e st_pro positio nota ex ^gxminis^Bor_se_cCTmeja3,Csgd
potius in materia contig entl<,>> (5l)

Even though all experience that has ever been had with snow _ has
presented^ vrtiitia, this experience does not prove that it is
contradictory for snow not to be white. It remains possible, of
course, that there is some incompatibility be two en the essence of
snow and any other color, and further experience vail render this
possibilit/increasingly probable. But of itself experience^ will
never traiform this probability in o certi tude Nor do-it do
any good to have recourse -oo the principle ua ™ 1ri H«ie is vuwuest-
^Pl-lbus comes from -bure *or theugh^his P-n-ple^unqu

ion ably valid, it does no, s ^f* ^g^ the wh iteness of
« involved, in other- words, the re f^ e> mt is it

snow is obwasly ajsign ^ "S.it derives frol^S
coming from the nat ure of_thg_snow? ^rnaps

Planet. mm^ve_^jmff-J^f^ff ^ t ^ t it rerS SZT&K
vely Bir,^ 2 _OTOcesB_aBjho_H°^2i3^-g^ ^ a# It becomes
possible" to brace l^rii^H^badc to its source.

-192-

apparent, then, that the proposition "snow is white" is not neces-
sary and universal at the same tire In so far as it is proposed
as necessary, it is not universal, hut restricted to the snow that
has been re* thus far in experience In so far as it is proposed
as universal it is not necessary, A3 a consequence, it cannot he

/a scientific proposition which must he "both universal and necessa-
ry, Hence it is evident that the universalization that is effected
•ineY perimenbal. science is purely functional ,, That is to say,, when
propositions ore universalized without evidence s there must be a

^functional reason for doing so. In other vro.vds, when we act "as
if" this does not mean essentially that in so doing we nay be right,
but rather that in so doing we nay get somewhere

It is clear, then, that the propositions of experimental
science remain completely tied down to experience „ It can never
truly abstract from expe.rience because experience is never complete.
This means that they can never effectively rise above the realm
of singularity. In this sense all experimental seience is essentiall y
n ominaliatioj That is why experimental science must never remain
in a state of becoming, And we mean by this something over and a-
bove the progress that is characteristic of all human science. We
mean that the very genesis of the (-concepts ) employed in experimental
science is never terminated. There most toe a cons trait recourse in
the psrt'of the intellect to sense experience which is xraraersed.
in contingency and the flux of time. And this flux and contingency
will ever remain refractory to complete abstraction, I„ will always
be possible that further experience my change to a greater or less
degree of concepts already f orred, (or^tj^ast the relaxions bet - _ ^^

£oFT3eTColHHckTSs pointed out, (52; history ^ por u^*^^ j

essence of a X perirental ^gte5Sg> wtoMQ T EH ^^SStSlv^-
s-cien^eTlFli-itrl^^sTTf the word arg^y_ac^dentally M^

P licated..in history. And in- this connection it is "*«g^§*°

rational ^ ste « th ^\f J^y the methods that are proper to
it could never be known as ^uch by ^^ Qnly te a ^

Suc^lSit^ct^xS^Snoience could constant^ approach
without reaching.

timo of Hume, Yfc believe that much 01 ,t oglecte d. And perhaps

because a few basic distinctions ^™o been n ^ oiting the following
the best way to embark upon this question y

in the study of
over which

SnSicarte^of^hn^of St, Thomas

-193-

0mm. s. nostra Spoculatio dependet ab inductione sicut
dependet ^ a sensu et experientia; unde si propositiones univer-
sales alicuius scientiae non sunt ita abstractae et communes
quod ex quoeurague individuo manifestari possit ipsarura voritas,
sed ex pluriura numeratione et experientia pendcat, sicut scien-
tiae naturales, non sunt ita certae sicut aliae scientiae abs-
tractions eb ccroraunicresj, ut metaphysica et mathematicae. quo-
rum principia in uno individuo habent totara certitudinem ut:
(q uodlibet est vel nonestl) (53)

When John of St Thomas says that all of our specu-
lation depends upon induction just as it depends upon the senses
and experience, he is evidently taking the term in a rather broad
sense,, in a sense in which it is coterminous with any deliverance
of sense experience to the intellect. But under this generic notion
it is possible to distinguish three types of induction „ In the first
place, induction may be understood to mean the abstraction of uni-
versal concepts from singular objects. Taken in this sense, it is
found in all of the sciences and in all intellectual activity.

Secondly, it may signify the arrival at analytic
propositions from sense experience, and here it must be noted that
the terra "analytic propositions" is not taken in the superficial
sense in which it is understood by Kant, It means all. propositions
in which the predicate is for any reason necessary (and therefore
universally) connected with the subject. Since all sciences m the
strict sense of the word must begin with necessary principles, and
since all of our knowledge is: dram from sense experience, this
type o-f induction is found in all of the disciplines which are truly
sciences,, that 'iB to say in mathematics, in metaphysics, and in
philosophy of nature. The. way in which this induction takes place
is nob in every respect the same, for all the sciences. Magics
presents an especially particular case about which much has been
wcitten in recent years. It is not our purpose ;°.^k »p°n ™is
que-tion here and it is sufficient to point out thao even mathe
Tticar-princlpS, ^ spite of their intuitive and J£E*S f^

is ^^B^^^^^^^^^^\eing can be dra^ from
^^^SS^^Hi^i^SJiir^^eoE^^ being not because

sense experience for they ^ realised in ^ of nature

«J3_§2S§^ J G*iiL]^^ from, ex-

analytic principles g^^ing raobiJ.et^ing . are e nunciate d in

f ,ienoe, and unlike J^^^ST^'o^T^S^
terns, of sensible matter^) And in ^ p^erience to the univers-
^S^^SmS^^^ff^^^Z logically invalid,
ality and necessity of analytic principle

-194-

sinpty bggause tho 'basis of the universality ana necessity is not
the_fact76he subject ana preaicate are united in experience, but
^tefact tliat the mind can see that the predicate -pertains to the
vflrTTn ature of the subject , For example, the principle that the
whole is greater than any of its parts is drawn from experience
in which concrete wholes are presentea as greater than concrete
parts, but the universality ana necessity of the principle is foun-
aea on the analytical nexus which the mina aiscovers between the
subject ana the preaicate.

Perhaps the passage quo tea above from John of St,
Thomas nay give rise to aoubt about the possibility of such analy-
tic principles, in philosophy of nature, for at first. glance he nay
seen to restrict them to metaphysics ana mathematics, A more care-
ful reading of the text, however, suggests another interpretation,.
In comparison with all the propositions found in natural science,
the number of truly analytical propositions is almost infinitesi-
iPlly small, and that is why synthetic propositions may be consi-
derea as characteristic of the stuay of nature. Moreover, even the
few analytical propositions that are found in philosophy of nature,
though fully certain i n_ themselves , are less certain in comparison
with metaphysical and mathematical principles because of the mate-
iriality involved in them.

The third type of induction is the one that is of (^3

special interest for us. It is the type that is characteristic of v
experimental science, and it takes the form of an illation in which
tte mind progresses from a multiplicity of f^*^ never
to a judgment which is proposed as universal, but wtaehon never
be anything more than teMivelyjmiversal ^ cause the nexus of the
judgment is" basea -el^UEonl^g^-^^^
aTmrehension o f a necessary connection _b£t^ens uD,iec_G j ^

anything more than probable. It is true ™ increase to the

are multiplied the probability may m some °£^"£ w ^ the

,extent of reaching pracjtical^ certitude, mz oontention that

infinite limit of theoretical certitude. It is our con

f experimental scien-STiTlSde up °ompl^ely of this pr ^ ^^

h ^h^o^Lnrsni understanding arise, it mast be noted
- _.;. _ of the word s ) But le f,:'" refers only to tmiversal_prcpo-
teUately that mSJSE^^J^^^j^^^JnB^^
sj : tig 2a (andj^grg_is_j^

titude of fa^ts_e^tablHhed^Lg2 ^rim ^vre raol propositions

is~that" science iTc^itituted essentially
and not of singular facts.

The type of induction we have just described is Known

-195-

M ascen^g induction. There is also a corresponding descending
process xn which the mind passes from a universal proposition to
singulars o Thxs descending induction is often confused with deduction.
There is, however, a vast difference "between the Wo, for like as-
cending induction, descendin g _induction lacks a true middle tern , (55)
This descending induction is also used extensively in experimental
science, For since the universal proposition arrived at by ascending
inductions only tentative it must bo continually submitted to 'fur-"
, ther experience for verification, and it is by a process of descend-
ing induction that this submission takes place. It remains true,
of course, that deduction plays an important role in physics, but
that is principally bacause of the introduction of mathematics
which is a true deductive science.

The most important point which emerges from this dis-
cussion is the clear cut distinction between the second and third
types of induction Most of the difficulty that has arisen about
the nature of induction has resulted from a confusion of these two.
Until fairly recently it was customary to identify the third type
with the second in the sense that the induction of experimental
science was believed to give absolute certitude,, Until the downfall
of classical physics., nothing seemed more certain than Newtonian
science. But since this downfall occured it has beoo:,is customary .
to identify the second with the third and to extend the lack of
certainty that is characteristic of experimental science to_oll
science, (56) and in dged_to all human knowledge .

This distinction is important because upon it is based
the distinction between philosophy and experimental science, as
has already been suggested. The principles of the philosophy of
nature are drawn from experience by induction, but because they
are analytic, it is possible to infer from tter 'l con ^"°f J^_
are certain. If the inference is good the co "°^f ? n ^™^ eB
sarily true, Thes^onc^sdonsjm^st^^

in the way alriiayTxplo.ined in Chapter 1±. *?™™XrSr ^
that the/have to be submitted to sense experience for further vo
rificat/on - - since they are already -ces g ly true,, In^xper^
mental science, on the other hand, the princip inferred from

rience are only probable. Certain conclusi « u 5onf £ Tnot
thon » ]2iiJ?ve iL j^the_tofe^^

nej^sar^lytaue. That is why ^ ™ al science is,
S^omroIIeTby further ^ 1 . 0noo « JSETita orig in and i n
consequently, aouWy^xperirae^l v^^—-^-^^ th±s
itsj^mnus^lts principles ^.^Viseagagement ; the principles
'dr^Eiir^i/notconsist in a ^^f^J^ aoh ieved. The.
remain tied down to the actual exp^J- into expel - ie nce
conclusions of oxperimenta^scicnc^j^l^-Ea-^

-196-

/agoino Philosophy of nature on the other hand is experimental only
in its origin and even hero it transcends experience in the sense
* tat th e nexus of ms propositions is not based upon legpr^i mm „
That is why, m opposition to the term " expe riren tal^li my Fe

\ called "rational", ' J

And now, having arrived at this important distinction
beLnreen philosophy and experimental science, we must pause to exa-
nine its nature in some detail.

4, Philosop h y and Experimental Science .

It has become customary for modern writers to point
out that in the writings of Aristotle no distinction between phi-
losophy and experimental science is encountered,. The inference that
one is invited to draw from this observation is either that Aris-^
totle was unacquainted with experimental science (57) or that W«m»' *
he erred in falling to recognize that these two types of natural
doctrine are formally and specifically distinct sciences in the~^ (

^strict sense of the. word o (58) Perhaps enough has already been — VW
said to show that the basic structure of modern experimental science
is clearly and accurately outlined in the writings of Aristotle,
And in C hapter II we poin ted out why^A^^otlef ailed to recogniz e
the B^rmlaS~s ge^ific7distinction upon which so ' much st ress__has _
jgnn^ni^j^n™^^ neithe r exig-ca

nor_can exists )

Does this mean that Aristotle recognized no distinct-
ion between the two. parts of natural doctrine that have become known
as philosophy of nature and experimental science? In the f irst book
of De Partite Animalium we ran across the fol1 7 ™ ^sage: "
J^T^Sm^rSrKSk^Tot what mode of nee essity are we sp eaking
^ when we say this. For it can be neither of those two ™j£*£oh
\S are set forth in the^h^.o^^Mcal.^atiseSc" (59) These tew

lines make it ^rfeAriatot]* recognized a d «*^£™j£* re<m
the parts of natural doctrine that ore advanced J? tt» direo tion
of concretion and those which deal with. generalities . T°^c latter
he applied the tern "philosophical" ^V^^Sca^Yef later
is t£t the former are in some sense not philo op cal. ^later^
on in the same work he tells us that it P^«^° * at first

to handle the subject of this *^£' ft <f {J^ closer examinat-
glance to constitute a paradox, Ys * ^lioiS suggest the correct
ion will reveal that these two texts i.npli ^^^ g | h sug g est
solution of the problem of .philosophy and science, iney gb

-197™

both the precise way in which the two parts of natural doctrire
are distinct an&t he way in which they must he kept united.

In the first place, let us recall that the tern "phi-
losophy" had for the ancients a much broader meaning than the one
it now enjoys. It was, in fact,' coterminous with all human science
taken in the strict sense of the 'word (with the exception of theo-
logy for the medievalists) o Consequently, when Aristotle says that
the more abstract parts of natural doctrine are philosophical whereas the
nore concrete parts are not, he is simply saying that the former
are strictly scientific and the latter are not. And this is preci-
sely the conclusion to which our analysis have already led us In
Chapter II we demonstrated the impossibility of more than one true
sci ence in the first degree of abstrac tion. And earlier in this
Chapter we saw that because of the type of induction employed by
experimental science, it can never effectively rise above singularity
to the point of achieving true .universal and necessary propositions,
We saw that whereas in philosophy of nature the nexus of the pro-
positions is strictly formal and analytic., in experimental science
the nexus is material and synthetic. There are, of course, Wo types
of material and synthetic nexus. There is 9 first of all, the comp-
letely material and synthetic nexus found in such propositions as: •
"this table is white". In this case we know that the nexus is me-
rely material and synthetic because we have seen tables which are
not white But in the case of the propositions of experimental science
to ar-3 not sure that the nex-js_jsj^HS?Li25JSa£il- and synthetic.
In fact' we'tentatlveTylSdvFat something more than that. That
is to say, there is a movement away from pure maierialiiy and pure
synthesis tow^s formality and analysis. Nevertheless this remains
/a purely dialectical lixiit that never can be reached, In ot^mlJ*
whereas in philosophy of nature we get at both the ^ ^ d «*
propter, quid, in experimental science we get only the ^a. *f .
Vrrol^he-st content/with the mere quda , There is a constant stri-
ving towards the discovery of a propterjuido Th f h ^ o ^ ggg nd s
(by raans of ) hypothesis But the validity of every hypothesis depends
dfe-^S^S^taT^onf irmation, ani this ^'f^f^^*"
ion give/S only an ^ ^enta^osi ^££^££3 .

set out upon an infinite series of mterpiays concepts

of experimental science e/e.c ro.iKu.11 * u „tion can never reach

W ard perfectible . ^"^S^J^^^jnatorfi*
experience in such a way as 00 ^^^^S^M^t^Ting
a true universal^ ) exper:.^enoal B02B ?^^r° ^^^ a t it nor at its
toward, formal abstraction never ^^ o ^ental science seems
o.rtitude. The perfect certitud e t ^^^ ^ illusion deriv-
to possess^, iu the Oast af^axs, nothin & fl r

Uir from a cortitade that it is possibxo. to

-198-

object or a group of singular objects.

Since, then, experimental science aoes not arrive
a^JtHaJ!o^^Li!£5±J25iio2j ^ cannot be a science in the strict
sense of the word. And if it was experimental science that Locke
had in mind when he said that natural philosophy is not capable
of being nB.de a science, he was quite correct,, (6l) As has already
been stated, experimental science belongs to a type of knowledge
which must be termed 'dialectical," We shall devote the whole of
Chapter V to an analysis of the meaning of this term, and for the
moment it is sufficient to have pointed out in a general way the
nature of experimental science in order to make evident the precise
v/ay in which it is distinguished from philosophy of nature. It should

/be apparent from what has been said that the frontiers between phi-
losophy and science are some thing definite and clear cut and not the
nebulous thing that so much of the discussion of the. question has

q.nde theny )just as soon as the study of nature has arrived at the
point at which the nexus of its propositions depend only upon
experience, the frontiers between philosophy and experimental science
have been reached. And it oust be said in passing that if the, reason
why the term "experimental" is applied to science is not that the
propositions are purely experimental , (w e knew of no definite and
absolute meaning that c an_be_ attributed to it.j

At this juncture it is necessary to consider in some
detail the distinction between philosophy and experimental science
traditionally proposed by scholastic manuals: experimental science
studies reality in terms of its proximate causes, whereas philosophy
studies it in terms of its ultimate causes., Wg_belieye that in this
fl-igt.W^nr. th,»re is an extremely pernicious fflbig^c™£iL*gg
i^awS^y^hnlftJ ^ion of tte relation between philos ophy and
Icie^jFolPfch^xpWsion HJltimte cause" may De taron to raan
too-different things. It may, first of all mean the principles which
enjoy tru? universality of causality, and not merely *™sality
of predication. These causes can be arrived at as such and inan
absolute fashion onl£jy_ neans o L^^^^J^^^?
causes „ Thus it ii^iiblo to demonstrate in the Degim that
b^Tis tto last end of all the natural ^ OMS '^ e ^ 2
that this gives us, though certain, i^2|E2S^scure^|_con
fused I "-ho theories of ovoluti on are3n_attojpt_^odMsip^tetnis
Lc^. o| ^nc_oneorics ^.--fp-^^^pq^-^ r^aer of oori cretion o

But the egression "^SvoSS^ P^tloT
to mean the principles which have °^™sality p i ^

that is to say, those encountered in the first par

-199-

trine. 'Those causes may be called ultimate only in the sense that
they ere the farthest renoved from, what ..constitutes the essential- '
. and primary, object of the study of nature^;- the knowledge of -things
in their 2Fopjer causes.- They are not ultimate in the sense of bein g ■
th^jteminus.- towards whi ch the whole study of nature is orientated .
In fact, they are at, the- opposite~'-cxtreme. That is to say, far from
being- the ultimate, causes, they are- the very first causes which
the mind lays Hold.. of in its initial contact with nature. Nor are
they ultimate inothe -sense' of being-'- the most profound causes- in.
the true sense of the word. For 'the' most profound knowledge that
one" can have of nature is to .know -natural things in their proper
causes ; and the -..causes of Yfhich we are -speaking are the most common
that it is possible to discover,, -That is why from this point of.
view they •provide. us with, the most superficial knowledge that it
is possible ■ to have ■ of the ' ' cosmos . And it can be considered the
nost profound -knowledge, only by confusing the study of nature wi th
the type of knowledge that is had in raatherjatics where the most ...
known for- us is- also the most, known in. s.e, (62)

-The., .following passage from the second book of the -
Physics brings out what Saint Thomas understood by profound, cau-
se:

- ...in naturalibus oportet semper supremaracausam
uniuscuiusque requirere-, sicut contingit in artificialibus.
Ut ail quaeremus quare homo aedificat, respondetur, quia est
aedificatorj et similiter si quaeramus quare est aedxfxcator,
respondetur, -quia- habet artem aedificativum; et hie statur ,
(quia haecest prima ' causa (jtH^gine) Et ideo oporcet xn
Irebus feturalibus pWedere^Sgug^TJa usam aupremaq . Et -hoc
ideo est,- quia.eff/ctus nascitur nisi scxatur causar unde sx
alicuius- effectus/causa-'sit etiam.alterxus causae e«Jctu s ,
eoiri non P oteri/nisi ; causa.eius.sciatur; et sxc quousque per
^veniatur tad'-'prirn tun' causa l-- - (65),

It is fairly clear from f^^X^^^^^

cause which gives us Jta .most &™™* _ l _tbecausewhich.

^^^j^^£^^f^^2B of_the_eftect^

*u"+ ->-hP -rp'ioritv of the mode rn Scholastic s

have confused the two' me anangs_ofu^^^ misunder-

^^hl^c^rfu^Tha^^ Pr om it

standing- about -the true character of the stuoy ho]£lsUcs

has cono HHcAJ^e^LSS^^^ J a ? n reality constitute ■
have assumeTlFdeliinF^ith thxngs which in leality

-200-

the most indetermined and confused knowledge that it is possible
to have of nature. Prom it, too, has come a view which' when analysed
i can hardly be distinguished from Hegelian idealism. Yfe have in mind
the notion that by means of the most general considerations possible
one succeeds in grasping the very substance of things. Scholastic
manuals give the impression that in the De Anima, for example, one
grasps the very essence of the soul, and that the study of bees,
and birds and horses has to do only with accidental modalities o f
thes ubstanee 'of the brute animal .) If this were true , the general
woul d, be identified with substance / as in the doctrin e of Hegel,J
( , and~theijp _ecies would be o nly' a kind of phenomenal mode, ^ or ulterior
^ESoration of the substance , which is not of interest to the phi-
losopher whose task is to get at the profound essence of things.
In other words, what is the most clear and the most knowable for
us woi;ld be the essential substance oi Lthing s, that is the mosT
clear and knowable in se. Early in this Chapter we have seen that
this is diametrically opposed to Aristotelian and Thomistic doctrine.

From this same conflusion has arisen a false view of
the order in which nature should be studied, Instead of following
th- ^raditJonal Aristotelian and Thomistic order which begins_witb,
a.-.-ifiralltiea^ and moves on to wards fuller congretiong in such a way
t^F^eriTneirtal science iTSol£nga^^3S^S2^aL^J!£*!S?'
most modern scholastics have made the philosophy of mtwc £nox_~
tentton of experimental scienoe(in^uch_aj«ay th ^ ^ f ™^^?
^^i^i^^r^^i^:^^^^)^^ dependence is

v,a-: uorA "Under no consideration must it be thought the Philosophy

v-and .".iiemistry, .. ; . ,•■;:..• ' •

*, ■- -A4-, *f thP relation be twe on' science and phi-

data are not P^°^ t the'hy^otheses used by experimental
nnnt and observation, but tne^nyy e f tBt (65)

\ science for the coordination and descnp

4.^ + if 'this 'ware the true relation

between philosophy and science to .

-201-

Tfintlo al than the latter ,, (66)

In some quarters the anteriority of philosophy of
nature to experimental science is recognized in one fashion or
another, but then philosophy often 11000033 nothing but a highly
theoretical vanguard of science born of hasty generalization which
science .gradually su pplants by its constant progress. "The increas-
ing independence of natural scientific branches from philosophy
from Aristotle's tine to the present," writes Pascual Jordan, "has
simultaneously also . emptied ..philosophy of its original content and
t problems. " (67) '

Sons modern Thomists, while not making the dependence
of experimental science upon philosophy complete and absolute, con-
sider it nevertheless to be so essential that the constant progress
of experimental science makes every treatise of the philosophy of

.nature extremely short lived.- Thus Maritain says: "Je pense qu'un
traite de philosophie de la nature., au maximum peutvivre une vie
d'homme, cinquante ans, soixante-dix ans, si autem in potenta tibus,
octogent a anni - - et encore a. condition d'etre periodiqueipent re-

iSs"a jcjurTTTs'JTOposer qu' il ait des editions successives; parce

que ce traite de" philosophie de la nature doit^necessairementavoir

un contact intine avec les sciences des phenomenes, et ces sciences ^

so renouvellent beaucoup plus rapidement que la philosophie. {bti)

We cannot subscribe to such an opinlai. We believe thao a treatise

of philosophy of nature, if it is good when first mitten, can live

faAeyond the life of a nan. WeJ^iej^ati^

without any substantial_^ha nge. In everything that is ess ntial,

the treatises of Aris totle^nd St. Thorns upon those parts of na

tural doctrine which are now known as philosophy ot ™*f e f !

eight books of the Physics and the three books of the ffe Anima

IL S oust as alive tofy^ when they were first ^ tton^AUt
many modern Thomists think that they have genu 1 although

the perennial vitality of Thomismwhen they claim that alth ougn
the writings of Aristotle and Aquinas - P^ical ub °°^
obsolete, their metaphysics and moral Pg gg mke such

alive, l^is^afe_tg_sa y tha^iost ^ l^ff^^^p^gcTSnd
statements have never tak ent^_^^-^- 1 ^ r ^ cT f T ^ ing
the DeAnina a olose and in teljjjsgntjgaaag* extreme^

would-riv^al ^^EK^^^^^^^^^^TSEE^

minor details ItoTt^ je^ig^^g^a^^ and there-
T „ „-i,wi„. 4-^ g » t.-rPn l^eTl^ri'iiisntlSll y anterior , -—3—- -

is simple: these treatises are^ssaaa^ j alrelSTTx-

^r S J^ V e I1 ^n L ^ > _^2m^^^{]^ f the nature
plained, in order to arrive at the <ge^g___^ noe Qf ^ ^
of motion Aristotle needed only tne sii £ ^ tota ii y contame

of a snow flake. The .generic » a .^° °* n _ ed frD m it. If his analy
in this one instance and couH be diseng-g

-202-

/ of this generic nature was correct, and we believe it was, then
.his definition of motion' vd.ll'- ever remain unaffected by the innume-
rable highly complicated experiments- subsequently made to 'determine
^ the -nature ; of motion in a more s pecific way; -And the sane' holds
true of all. the; fundamental propositions of the philosophy of na-
ture o . (69) ,.■■'''■•'

This brings us to .the consideration of an objection
that has' frequently been brought to bear against, the view we have
been upholding in relation to the question of philosophy and science,
It has been formulated by Prof essor Alexander in the following
. terms: " ■-...''

' Mr„ Adler defines philosophy as a body of logical

•. ' " ' conclusions' drawn from common sense .observations, 'and science
' as a body, of conclusions drawn from specific observations ob-
tained by specific investigative methods, I agree with Mr. A-
■ dler's definition of science but not with his definition of
philosophy, Mr, Adler. reduces philosophy to reasoning about
:::v\&squaie (common, sense) observations, science representing
. at i;).ie' aasce time reasoning about more' adequate 1 observations
obtained by -refined and improved, methods of investigation. And
yet," -in order- to save the medieval hegemony of philosophy, with
' o peculiar' twist of' reasoning, Mr. Adler tries- to subordinate
" soierice -^ - that' is to say conclusions dravm from iiaproved

dbiSrvations -' - to philosophy, which according to- his own de-
' fjnition consists of conclusions from inadequate observations.
If Adler' s ^fini tion of philoso phy is correct(5)hilg^ojjhy_^uld.
bVdis ^arfeTiriAe _p^oporttorlo which scientific- knowledge^
progresses by the :^ ^Bte^i3^J^^r^s_^ cx :lteo\m^ m S
•'• v^-fcsaH ggBS figZ^^igi^" Ad^r kuaself^peoks
■' the death sentence of ; p hilosophy. (70)

. Let us suppose that the ten, "philosophy'' ^f.^f^^^Lion"
mists understand by philosophy .of nature, and ^J^4£™?£ n
'•conion' sense observations" peans the *?^ e ^°*££^ tf™*
that is the paint of departure of the first s P -ulations o^h ^

mind, about nature l\^SS^i^^^snaLB^S^

■ servation is completely Made ^ uat ? ^p-i^-r-^

. probjsris, But no one^^f^aj^t^^ is adequate for
a purpose. Our position is that common speola li Z ed observat-

ion, generic problems, and that only J^^ obgorvat ion from
lion is 'adequate for specific J^ ™^^ completely inade-
which is derived the generic notion °^° len con ccming the
quate for the solution of a. ^« e ° a S n l, let us say. But

Inspiratory tubes of a fJ^^f^liXf notion found in
at the sai-.ie tino knowledge, of the exaoT; . ...-._,.

-203-

| a particular type of respiratory tubes is wholly unnecessary for
I a determination of the generic nature of notion,

Doctor Alexander's objection with regard to the su-
bordination of the experimental sciences to philosophy recalls what
was said in Chapter II in connection v/ith our analysis of the twen-
ty-fifth lectio of St. Thomas* Commentary on the Posterior Analytics ,
This subordination does not r.iean subalternation in the strict sense
of the word. From this point of view the experimental sciences are
completely independent of philosophy, It can only mean a subordi-
nation arising from an order in which one mo/es from the more ge-
neric to the more specific, that is to say a dependence of the more
particular upon the more general , We feel that enough has already
already been said to make it clear that this dependence does not
mean that the more general knowledge acquired in the philosophy
of nature predetermines the solution of the more particular problems
of the experimental sciences. Nevertheless, the anterior parts of
->natural doctrine have a definite influence upon the posterior parts.
For '-tbs definitions arrived at in the philosophy of nature become
/ ne theolog ical princip lesc to guide the construction of hy potheses
in-the experimental scie ncesp to tapose limits upon them , landtp
gag^r^Th^Tn by ^lan^yH STfro. criticized7y rhu¥, for exam-
p le, the d efinition of' intelle"otin the Pe Anima becomes a netho-
Vdological principle for experimental psychology. This role of Phi-
losophy of nature is not a restriction upon the experimental sciences .
Rather it. frees them from becoming enmeshed m false and useless
hypotheses.

This discussion of the subordination of the experi-
mental sciences to the philosophy of ^^f^sts an^ortont
question: Is it necessary or ^^J^JZS^S^S, of
scientists to be acquainted with philosophy 01 n-
no better answer to this question than the one found m
lowing passage of Professor De Konmck:

N „3st-il Pas -i f^es^eineurs physicien. s mo^
dernes ^ozent ^gu premie tou Wes^ ^ t „

les prennl-zvs pax ties ele J- pi f 6 t la definition du mou-

ila meiltoxvu physiciens s'xIb aa ^"™ . flt efl te ce differen ce
vement, -csuo^a^mp^s^^

. suppose vne^e^^^^^^^^—^j^^^^^TL-e-n^
d¥ irp^HS? A cela on peut ^P™^ ^ ^^ Leg ^^
g -Hn-i5SiPil meilleur macon b^ us philos0 p hi(1 ues '

ges des savants modernes sur les aspec ^ ^3 du mcon
de leur science, nmtrent suffx s™t ^ ^ f<jnt violen _
qui veut f^ _V ^^B^^^^^ com!xiaS o^o si
ce a 1'orare qu'Tlnous fauc suivx

-204-

nous voulona en arriver a voir la partie dans son ordre au tout,
lis ont iiega/xge les considerations lo giquoment anterieuros_ a
co3J£s_ae_J : Gur_p^pre_suaet, negligence qui se fait sentir quand
ils veulcnt sortir.de celui-ci, Faire violence [ a 1' ordre , - ) ne
fut-ce qu'a celui qui nous est impose par la nature, in&ne de
1' intelligence huraaine, c'est faire violence a la sagesse, a
la science de la nature en tant qu'elle est philosophique , " (7l)

The greatest mistake of the modern students of nature is that they
have insisted on starting in midstream. The most fundamental and
most basic questions have been ignored,, Having started midway, and
pursuing their progress into deeper concretion, they have thought
that they could ultimately find the solution of the fundamental
questions;, But the progress of the study of nature does not move
in a circle j it moves in a straight line And one has only to con-
sider the answers that scientists have brought forward to such fun-
damental questions as: "v/hat is life", to be convinced of this.
Because the simple baoic questions have been ignored, modern text
books are filled wi th phrases and expressions which are utterly
devoid of any definite meaning. They have rwch to say., for example,
about "animal behavior" without ever having raised or solved the
simple question: what is an animal in g eneral , And all this brings
hone to us once again the utter futility of the efforts of modern
scholastics to prove or disprove the doctrine of hylemorphism by
means of chemistry and physics. The substantial composition of mo-
bile being is a fundamental question that is anterior to,. -and there-
fore independent of all of the findings of modern experimental
\science.

The o-perteiental sciences are, then, dependent in _
some way upon philosophy of nature. But f«*r £^7££.
we may say that P^-o^f na ure is J^^^SSe

SSSlSTS-tK Sh tstre knowable g^*^™*
the more abstract parts of .natural doctrine are suto ^ ^
the more concrete parts. MJStgBS y " ^""g \lf UTrenC That is
concrete part. r,ho abstract parts find their ^ ±:L ^ sf ,? ed ^ th
why the true , h ,l oa c,pher ^"KrftofsrtW
the common general ^ths ab ^l^^^^^rsSSfTE^T^s-
t^ili^js™^ and are conse-

titu^o-o^^^S^ction Jo ^e study concrete parts which

^^^\S^^^L°E2S^p^ n f^^ X ^ n e V er lose sight of

.fc^^ WlS^ jHgj^pter^L^^

S^aMrf^o^^^^^-SoSlfatf ; ^ mil simply
of a naivo optimism., or of ™ ^ na] -.^ sn that is intrinsic to
be obedient to the impetus oi tno ay

-205-

the very study of nature. For the end towards which all the expe-
rimental sciences strive isj ^the same time , the end towards which

the philosoph y of nature strives ,T) And here TO^e--touchlni~iroon

the profound wisdom contained in the two texts from the DeParti-
bus Ani malium which seemed at first sight to constitute a paradox.
On the one hand, the concrete parts of natural doctrine are distin-
guished from the more abstract parts by the fact that the latter
are philosophical, that is to say truly scientific. But at the some
tine the philosopher of nature must study the concrete parts as
well as the abstract parts, since the latter are a prolongation
and a necessary fulfillment of the former,, The following lines of
Sir Arthur Eddington are relevant here:

Not so very long ago the subject now called physics
was known as 'natural philosophy 1 . The physicist is by origin
(a philosopher who has specialized in a particular direction^
But ho is not the only victim of specialization,) By the break-
ing away of physics the main body of philosophy suffered an
amputation,) (72)

Perhaps we can sum up this discussion of the relation
between philosophy of nature and the experimental sciences by drawing
the following contrast between them,- The former is of greater in-
trinsic importance than the latter for three reasons, First it pro-
vides us with the knowledge of nature that is most in conformity
with the human intellect. It is significant that in modern times
the mind in its dealings with nature has almost universally rejected
the object that is most proportionate to it. But perhaps one might
be tempted to object that experience shows that the experimental
sciences are more easily accessible to a greater number than phi-
losophy of nature. The answer to this objection has already been •
suggested earlier in this Chapter. In speaking of the rf^ive
"taowability" of the different parts of natural ^inewehave
in mind only intellectual knowledge. In the measure in wh **££*
knowledge enters into the discussion, it is ev ^nt that concr ate
singular sensible objects are the most ^i^ taowab^ *f ^ *°_
far as the experimental sciences enjoy a close proxi mi^ to^ensi
ble singula objects they possess a facil %W£™£ ^measure
philosophy of nature. It mat be no ted , »°^ , ^

in which physics- is mathematicized it P^ c ^ believc tbat

Uat is the most I^^£^,^S^*. of physics
these two facts explain the c^orat *ve
and the extreise attraction which it exercises uv

Secondly, the philosophy of nature provides us with
truly scientific knowledge. St, Thomas writes.

-206-

Illi qui sciunt causan et propter quid, scientiores
sunt et aapientiores illis qui ignorant causam, sed solum sciunt
quia 3 Exporti auten sciunt quia, sed nesciunt propter quid, (73)

It reiwv->!3 possible to have scientific certitude as long as the
nind renr.ms in generalities. That is why the wiseman in the realm
of nature must be humble. To reject certitude in these tilings is
a kind of pride . Thirdly, the philosophy of nature has as its ob-
ject tho most noble, thing existing in nature, the focal. point of
Vthe whole of material creation - the spiritual soul of nan,,

On the other hand, the experimental sciences are more
iinportant than philosophy of nature in the sense that they cone
closer to the realization of the goal of the whole study pf nature
"=• — the knowledge of things in their proper causes. (74) From
this poi.iv of view they provide, as v/e noted in Chapter II, a type
of knowledge that is closer to the knowledge that God has of the
Cosmos rh::r: the knowledge found in philosophy of nature ,

5„ The Interrogation of Nature ,

V/e have seen that nature may be defined in terns of
la ratio indita rebus . It is this intelligence, this logos realized
I lnlEteHSn555irEhat makes the science of the cosmos P^xble. _
And the goal of this science is to capture this ratio xn some partial
Wvav at least to bring into contact with the ratio of man. We haye
\vay a-c least, to uixiit j.h „,-„„n w difficult as experience carries
seen that this becomes .increasingly ditlicuiT; as -<£P ,

the mind forward into deeper concretion. N ^ a ^ars les s a nd
less rational, less and less homogeneous ^f^e intellect. It con

tinually throws up greater ^^^rillS in whi^t^con-
engage the objective -logos from the ^™riai y thi for the rjind
cretized. And there ultlmteg ^"^^unon Lture the ra-
to do if it is to continue its tas.c t o ^ . ti ^ lo of tha
tiona^y_whiciit^cks, to °*£^ f * °°f ££ it. This process
cosmos by injecting "%°™ "g^gSe! In tne m^homa^zation
of rationalization eventually terminals speculative^ sci-

of nStSeT^^iSh the most ^^fj^^, The P intell e ct finds,
ences become subalternated to the most r. x enough for it> s0 it sub .

for example, that the visual line « r to the introduction

stitutes the mathematical line, w.* ^tionalization takes place,
of mathematics an extensive process of rationa
Ho must now try to ana^e this process.

-207-'

• +11 n ! ? st place ' " is important to recall that
experimental knowledge- is essentially imperfect, for it implies
physical passivity. To have an experience means to become subject
to something, and in the case of sense experience is always a quest<
ion of becoming entitatively subject to material things which phy-
sically affect the sense organs. That is why man cannot be satis-
fied with purely experimental knowledge. By the very fact that
knowledge is vital it is opposed to passivity, and by the fact r -^^-
that it is intentional it is opposed to the purely physical, (75)
That is why the mind is impelled to go "beyond" experience, to an-
ticipate it by searching for the reason of what is presented in
experience. The more the science of nature approaches concretion
the more experience gets the upper hand, so to speak. The intel-
lect cannot accept this state of affairs. It must try to rationa-
lize experience and thus get the upper hand itself. For the intel-
lect can never rest in pure givenness; it has, as Meyerson says,
"une repugnance irremediable ,,devant tout donne," (76) It can-
not be content with a more quia ; it must search for the propter
quid o It cannot remain imprisoned within singularity; it must strive
to achieve universality. It cannot rest satisfied with purely syn-
thetic judgments; it must find a way of making them a priori , /aid
when nature does not provide what it seeks, it vra.ll reconstruct
nature in such a way as to make it render what it wants , or at
least in such a way as to allow the mind to give itself what it
wants. All this explains why as soon as the propositions of the
study of nature start to be purely experimental there begins a
gigantic task of reconstruction of nature. And the greater the
part that experience plays in this study, the greater, must be the
part that the mind plays , Science becomes a mixture of fact and
fiction, and as fact increases so does fiction. As Duhem has re-
raarlBdf Le developperaent de la Physique provoque une lutte conti-
nuelle entre 'la nature qui ne se lasse pas de fournir' et la rai-
son qui ne veut pas 'se lasser de concevoir'." (77) Tfe must now ■
try to point out the most salient features of this rationalization
of experience.

This is far from being an easy task. For not only
do the objective and subjective logos ultimately teoons so inex-
tricably fused that it is impost to draw the line between then,
hut it is also ^possible to find an absolute ^^^ ?°^rocess
the introduction^ the ^^^KTtSS IhffirsTslep
is essentially circular. It might De suggest

in the rationalization of experience ^^^^f^^^
beginning of a scientific experiment the scientist ^ 3^3 —^

of the t ele im ^th a i_ai 2 ^^ th * who le

them in-iiiiclally chosen conditions w^gfggEnGafer
experiment is an ^^X^^LSSH^S^^JSSS^^l^

-208-

(ba suggested tint the second step consists in an intellectual fil-
tration and purification of the elements entering into the

expe-
.ve

riment m such a way that they becone idealizations which b,„
k no "exact counterparts in experience. There can be no doubt that
experimental science deals with idealized entities of this kind,
such as perfect ga ses.^ jaovement without fric tion^ absolutely ri gid
hodiesj perfect levers J~perfecTly geom etrical "cr ystals . absolutely
Ipure metals, perfect fluidity, perfect elasticity, etc (78)
And all this represents a projection of thought into the cosmo s.
But the nature of this projection must bo rightly understood, For
at first glance it might seem that all that is involved here is
Ithe substitution of limiting cases)for the brute phenomena that
/are directly, perceptible. If this were true, we could, as Cassirer ^ !

'has pointed out, "attempt to do justice to thi s method b y_.a,simple- S (^>v
^ .extension of the positivistic schema,"") (79)""~As a natter of fact, i

howevor, the problem is much more complicated than that. And an
attempt to unravel it will immediately show that in the process
of rationalization there is a good deal prior to the steps mention-
ed a moment ago

This brings us to the central point of our present
discussion,, And we know of no better way of coming to grips with
it than by considering a passage from Kant's Critique of Pure
Reason : (80)

Mathematics and physics are Wo 'types of theoretical
knowledge which must determine a priori their object': The first
in an absolute way; the second at least in part, and to the
extent to which the other sources of knowledge besides the
reason allow it to do so.

After attempting to show that mathematics is a completely^ prior^
science and that it has made true progress only since mathematicians
have come. to realize this, he goes on to consider the a .priori
character of physics:

When Galileo rolled balls down ^°} ±ne ^ J ^ m
with an acceleration determined and chosen j* h^self ^when
Torricelli attributed to the air a ^Mf^he computed
as equal to the weight of a known column of™ er, or^to
later Stahl transformed metals into ^> ^ ertain eleKfflhts ,
turn into a metal, by se P^^sicists! ?hey understood
then a new light dawned for all g^^s y r^ ^^^
that reason discovers on^_what^pr^__- th prlnclpXes ^ loh
to its o wn desig ns; it wust taxe i« „ nn _ tnnt i aws , and force
deTe^mln^Tts-judiments ^^^^^eato"^
natur^_jtoj^p^ndjto_it^ques3£ns ,

-209-

be conduced by nature as though by a string: for otherwise
our observations made at random ^(withoutanyplan traced

Ibef^rehandjj yould neyer _lead to a necSiia r v Jaw, wh^.h t. hP rea-
son nevertheless looks for and denands. The reason must present
xtself before nature, holding in one hand its principles which
alone are able to give the concordant phenomena the authority
of law, and in the other hand it must hold the experiment such
as it has planned according to the same principles. Reason
demands to be informed not as a school boy, who is bound to
speak only v.'hat pleases the teacher, but as a judge on his
bench, who constrains the witnesses to answer the questions

Vput to them. Physics, therefore, is indebted to the happy re-
volution which has been introduced into its method by this
simple notion that it must seek for (and not imag ine ) in nature ,
in accordance with the ideas which the reason itself brings
to it, what the reason ought to learn of nature, about which
it can never learn any thing simply by itself. It is thus that
physios has been able to unter for the first time upon the
sure road of scionce^after groping a long for so many centu ries , )

The gist of this passage may be summed up by saying
that according to Kant experimental physics owes its emancipation
and its progress to the fact that it proceeds to a certain extent
in an a priori fashion by posing questions which anticipate expe-
ience and predeterminate it.

This doctrine has in recent times been applied to
biology by an ardent disciple of Kant, J. von Uexkull:

Natural science falls into two parts, doctrine and
research. The doctrine consists of dogmatic assertions, which',
contain a definite statement concerning Nature, The foms these
assertions take often suggest that they are based on the au-
thority of Nature herself. This is a mistake, for Nature reports
no doctrines: she rarely ! exibits changes) m her phenomena,
Yfe may so en B j £ y_thes^^tonjes C t hat they ap pear_asanswers

. to our queHi^Tlf^^^to get a right "^^f^'g^
SfargoBition-oTsoieHoe vis-a-vis of Nature, ™ oust transform
each of the statements into a question, ^ .™ ™* f B °£_£
ves for the changes in natural phenomena ^/™ t °^^

I have used for evidence for their answer. Instigation cannot
proceed otherwise than by making _a ^position (hyp «^es)
in its questions, a Bupjo^^on^whicht^^

•, , . n • • z — mf7S^iii+-jrrr> te recognition ot xne aiibwcx
is already implicit, The ^^„ ^ * „ s soon as the invest-
a^ThVle-tUnTuTof a toa^Joltom 3 sufficient

igator has discovered in Nature !<^^ e f positive or ne-
number of phenorena that he can interpret as posi

\i

-210-

gative on the lines of J^H^J^Trvtjl^ig'-

_ The sole authority for a doctrine is not Nature,
but the liwostigator, who lias himsel f_a nswered hi s own quest-

lOtto (81) iVKrtM Uiolo^,, fr6( a te. ■ ~

We do not subscribe to all of the implications of
the doctrine found in these two passages. Nevertheless, we believe
that the central idea running through then is essentially correct.

/Kant was right in holding that if experimental science is to have
any significance it cannot rest satisfied with the purely synthetic

^character of experimental propositions. The i.iird must introduce
an a priori element into then. And this introduction does not take .
place only after the p _r_ocess of expe rimentation has been acconplished ,
It is sornething that is effected during the process itself. The
nind must anticipate experience and by this anticipation predeter-
mine the experimental process, Kant was wrong in believing that
Newtonian physics was definitive, and that as a consequence the
a priori elenent introduced by the. mind was something absolute
and necessary Let us examine each of these two points in v turn . (82)

We have already suggested that modern science is
far from being an outgrowth of the naive empiricism of Francis
Bacon whose ideal it was to have experimentation carried on without
any preconceived ideas. In this connection Poincare writes:

On dit souvent qu'il faut experimenter sans idee
preconcue, Cela n'est pas possible; non seulement ce serait
rendre toute experience sterile, mais on le voudrait qu'on
ne le pourrait pas, Chacun porte en soi sa conception du monde
(don^lnepejut_se_j3j^^ C 83 )

Perhaps the first author in modern times to bring
out with great clarity and emphasis the importance of preconceived
ideas in scientific experimentation was Claude Bernard. In his
classic work, Introduction a 1 ' etude de la Medecine.Exp^gggfa^^
he says:

' II n'est pas possible d'instituer une experience
sans une idee preconcue; instituer ."^^SEH^^g-^

en vue d'une idee precon^ie, peu ^^\f*^ (c"st)
. -1 . „i,,a mi noins bien deiinie... v u <"*>"/

plus ou moms vague, plus - »°^ ou le ^^mo^ens

fl'idee qui constxtue,.. Xe poin x en egt

de tout raisonnenent scientific^, et c es * 1Iin „

. generalement le but, dans 1'aspiration de 1 esprit

-211-

oonnu... Sans cola on ne pourrait qu'e'tf&sser des observations
aterxleso (84;

This opinion of Claude. Bernard has become universally
accepted among the best noderh scientists and philosophers of science,
Innumerable authorities besides the ones already cited could he
brought . forward to attest to this universal acceptance, (85) -
It has bec.or.-e increasingly clear that, as Meyerson says, "toute
experience n'.est et ne pout etre qu'une experience de pensee, " (86)
And these authorities have been unanimous in attributing the whole
fecundity of experimental science to the projection of an a priori
idea into experimentation,, Y/ithout this projection experimentation
could render only pure data without any unified significance. And
these data could lead to nothing be yond themselves . They would
bo utterly sterile, unable to carry tho mind forward in any defi-
nite direction,, It is from tho a priori idea that science derives
its essential dynanisn, (87)

But it is important to see in what precise way this
projection of the a priori into experimentation is effected. The
texts cited above have already suggested that it is brought about
essentially by the way in which the experimenter interrogates na-
ture. Every, experiment is in fact a very definite question which
the experimenter puts to nature. And the results have no meaning
except in so far as they are the answer to this definite question.
That is why these results are already predetermined by the expe-
rimnter. The whole pattern of the experiment, the\ selection of.
the elements) that are to enter into it, the |structure of the ins-
truments ) that are to be employed, the precise! character of ever y
action I that carries the experiment forward.-- - all these are pre-
dSSSSdned by the precise question that is mthe mnd of the ex-
perimnter. And this question has no meaning in relation to the
very complicated 'theoretical background which forms its context.
Max Planck has brought out this point with his usual clarity.

' Therefore from the results that are given by expe-
rimental meaSLnts we must choose those^ich -ll^ve^
a practical bearing on the object J« ^' phYsical u -
particulor attempt at discovering * teln q uestion whic h

Inverse ^gresentej^Eeoial^ ^£^£5^-3 ^^^r^

we put^toj^ature, ^^fS^in fhe light of which it
unless you have a reasonable tneory i theore-

lis3X)In other words one ^t ^^^Wtjajhe

itioal. hypothesis m one's mind g-^aj^TroTiSTRoBpehB
test of research ngagura^. i"^ ^meaning in the light

\^^S?SInm^^J^%^S, very often the

lof one theory but not m ^nai; oj-

-212-

(07

. / significance of a question changes when the theory in the light
{ of which it is asked has already changed, (88)

But it is necessary to try aid analyse more accura-
tely the character of the questions that it is possible to put
to nature in experimental sciences. There are in fact two concei-
vable ways in which a question nay be posed. In- the first place
it is possible to ask a question which demands in an absolute fa-
shion what the nature of a thing is, for example; "what is nan?"
Such a question can never be answered by either "yes" "or "no".
The answer roust be "rational animl" or "feathorless biped" or
sone thing similar. And the reason is that such a question does
no t contain an hypothesis ,, But there is another type of question
which does contain an hypothesis, for example: "is the definition
of ran: featherless biped?" In this case the hypothesis involved
constitutes a suggestion to which one is forced to answer by either
y ^yes" or "no " 7) This suggestion is already in some sense a prede-
j termination of the answer. And it is clear that in posing a quest-
lion of this second type the mind is taking the initiative and an-
ticipating nature, ■ .

Now it is only q uestions containing an implicit hy -
pothesis (that are used in experimental science ij) As Meyerson has
remarked, "11 est parfai tenant impossible d'arracher a la nature
ses secrets en l'interrogeant directementi" (89) And because
it becomes increasingly difficult to induce nature to yieldup^
its secrets^ as progress is' made towards fuller concretion, it is
necessary that the questions posed by the scientist become_jjicreas-
ingl y artificial and_ hypothetical. Scientific method has often
Seen compared to the msthods employed in tracking down criminals.
Now the criminal which is nature will never answer a direct quest-
ion. And as a result the scientific detective never, succeeds in
pinning this criminal down in an absolute and definitive fashion.
For there is this difference between nature and ordinary criminals
that when the f ormei- answers "yes" it does not necessarily mean
"yes" in an absolute way. That is to say, when the hypothesis of
the scientist's question is verified in experience, this does not
raon that the hypothesis is necessarily true - - "quia forte se-
cundumali caiem_alium modum appa.r_en tj £ _s^v i mtur. J^" ™* fo1
Cw-fraTThisTTiowever, that v-onHJilEkulXii-^onTpletely oorroot
in maintaining that "the sole authority for a do °^° ^ ™* ™
ture, but thfinvestigator, who has himself answer ed his owr queso
ion," For though it be true that mture-s ana^ra.are gJffiLJff
tent predetermined by the questions footed bythe ^vcstigat or,
Jey" L not comEletely determined thereby It ^\£/™£*oa t
that nature has something to do with ^ £™r, ^ the moa-

the whole dialectical process of interrogation it remains

-213-

S ui-e to which the scientist must ever seek to conform Mraelf .

_ Even among those who readily admit that hypothesis
ploys a major role in experimental science the notion is often
current that hypothesis is always something posterior to experi-
mentation and merely superimposed upon it, in such a way that it
remains a comparativel y easy_task_to disting uish the factual ele-
m ents deriving from experience from the hypothetical elements con-
tributed^ by the mind . We feel that enough "has already been said
to~show that this is false. Hypothesis must anticipate experience
and predetermine it. And this predetermination is such that, in
the more complicated experimental processes at least, it is impos-
sible to distinguish sharply between the subjective and the objec-
tive logos. The analysis which is to follow will serve to bring
out this truth with greater evidence.

6. Operationalism .

Tn order to come to understand more fully the way
in which the subjective logos is projected into nature in the pro-
cedure of experimental science, it is necessary to examine closely
the precise character, of a scientific experiment. (90) During
the reign of classical physics, it was generally believed that
a scientific experiment was essentially la revelation of a propert y
that existed as such in obj e ctive reality. j it was taken for granted
that the whole experimental procedure was merely a means by which

I the scientist was able to disengage a definite feature that was
embedded in the absolute world condition. Contemporary physics
has shown how naive this view was. In fact, we are touching here
the very heart of tfe. profound difference between Newtonian physics

^and Relativity and Quantum physics.

' We have already laid considerable insistence upon
the purely experimental character of the def talt "" 8 .J^*^-
the 'structured experimental science. We have .seen *at experi
rental science ngver^eally_su r.ceeds in di se " ^ f^^".^ S e ™ ^
that it never rial^Ti^ above the reato of s£^ity -%£
consequence, the definitions of e ^ rment ^ *°^e^ 111 thZ

} B ^^^^^^^^^^^S eSmenTof control
ion, that is to say, observation into wnicn n
or artificial construction has been introduced.

-214-

. . But the true well spring of science, .and particularly

of physios, is not this ordinary observation. By the very fact
that the scientist is unable to really disengage essences from
it and thus rise to true universality and necessity, it appears
as a frustration to the r.iind. For this reason the student of nature
cannot rest satisfied with it. If nature, will not yield up its
secrets of its own accord, it must be forced to do so. That is
why he finds it necessary( to operate upon nature^ to bring it under
hisjc uidance and control , ) _to Manipulate, it in ways dictated by
his _preconceived. ideas .J All this is known as a scientific experi-
ment.

An experiment has often been defined as controlled
sense perception. But it should be clear fron what has just been
said that it is a good deal more than that. It is, in fact, a re-
construction of nature. Because the routes provided by nature are
not sufficient to enable the scientist to arrive at his goal, it
is necessary for him to construct an artificial detour. This de-
tour carries him closer to his goal than he would have been able
to get without it, but it does not do so in the way conceived by
the classical physicists. For the detour is inseparable from the
goal. And this brings us to an extremely significant paradoxto
which we shall return more than once in this study: scientific
nethod carries us closer to nature only at the expense of carrying
Vus farther away from it. (91)

And what happens to the scientific definitions in
this process? The reconstruction of nature effected by the scientist
enables the r.iind to penetrate more deeply in its reaning, but uhis
penetration never arrives at a point at which the mind is able
to rise above purely experimental propositions which are of the
very essence of, experimental science. In fact, as we have just
suggested, from one point of view the very reconstru ction makes
it even less possible to escape from them. *7™ ^
down to experience, bound down to a mere formulation of what is
PresentedTy experience But^now ^hat *e present ^ S ™ a
has become something different. JLLiU^HP^f ^I ^p^+^t h imself

merits have no meaning excg pJLJJL- terms °, M "- ^n ^, ^iw r^rftTnh
Iby^ich ^heFKr^lFol ^a. They depend upon e very ele non t
/SKtira-Efto the experiment: upon what h. do °*>™ ™ t

he does it, all the concrete -^^^^wlxactly what
Vgtc) Arid because it is impossible tor "£• „ erat ion. he is never
M* doing H}^^^^f^^£i^^e^evt^y

able to rise above the se ^^^^^gentraUzatip^lSTSs
Vr.igan s of provisional and _dialectioai_ggii .

-215-

| amounts to saying that the definitions of experimental science
derive their significance from the series of operations employed
\ in the experiments wh ich led to their formulation „| That is to sa y,
|(lihe_only_vjny_to define, physical q uantities is by an enume ration
I pf_all _the conc rete o peration s b y which these physical quantities
have come to be known,") And every attempt to analyse the c meaning-,
oflfoe definitions of experimental science must necessarily end
in the me re [designation ) of a concrete series of operations performed
Ovith a concrete set of instruments,, (92) There must be a reductio
ad materiam sensibilem indiyidua lera The more experimental science
attempts to achiove the natural desire of the intellect to rise
above the senses and the pure givenness of experience, the more
it is obliged to fall back upon them.

In order to be convinced that all the definitions
with which physical science deals are (essentially ) o perational one
has only to open a book of physics and read the definitions of
the fundamental quantities which constitute the science. Mass,
force, temperature,, electricity, magnetism, light, sound, energy,.

entropy, atomic and molecular properties, etc, : all without

exception are defined in terms of definite physical_op_erations
pe rformed with definite physical instruments . And we must be cons-
tantly on guard against the natural tendency to hypostatize terms

I which designates no more than experimental processes . The way in
which scientific progress forces physics to introduce progressive
modifications into the definitions of its fundamental quantities
should be a constant warning that these quantities are not real,

^ ontological propertie s

As we have suggested, the realization of (gi g oper a-

tJ : onal^^te JL oiV^^
ttSl^Flo^^rthe^uSe^ortetneGn oias^al^nd contemporary

physics! One has only to read Einstein Jo^v^edrf this.^
The relation to the central problem of ™e wno <± k t

tivity - - that of Bl^ta»i2^^ l g^ 1 & ™^ a iol 2 t7
the quarter: what meaning can s ^f^g n ^Ion of si*Staniity
And his. answer is always the ^f^/g^t designates a series

,can have waning for a P^ c ^ t °^ £ ^uJ in thlTco^crite
2f^E§SSttonB^fj3saaa8SgnJ that can o^ ^ eyents

and that will make it possible to a f*!fSL w wflnoip ie, it Hereby
lare simultaneous or not. HavtoS P 031 ^*™ ^determine stoltaneity
remains, for hto to show that ^ery attempt ^^^^rfioular
by means of concrete operations ^f^^n^STbe^oniiflelFea
observer, and that consequently 3X ™^ U f ^ Bve n ts themselve s,
by a physicist SS^n^ab^o^pro^X^-^^^ fnr a¥ Th^stand
(feut as something heljongiJigj£_i!E2| 7 ^-^^
Vindi cation to a g lyeTr^segg^I^gJl^- —— — ~

-216-

veiooijy. We shall return to this question again in Chapter VIII »
For the moment it is important to note that operational definitions
maintain a vital union between experience and theory. No matter
how far the experimenter and the theorist rray go, each in his own
direction, they will always be sure of remaining in contact with
each other, a s long as their definitions are operational . (93)

It is worth while pointing out in passing the sirni-
larity between this principle of operationalisn and the fundamental
thesis of logical empiricism: a proposition has meaning only i£
it st ates the. means for its verification . This thesis is acceptable
in slTfar as it applies to experimental" science; the error of the
logical empiricists • is to have extended it to all knowledge. (94)

This whole question of operationalisra has been summed
up by Sis Arthur Eddingtun in The Mathematical Theory of Relati-
vity;:

To find out any physical quantity we perform certain
practical operations followed by calculations; the operations
are called experi^nts or observations according as the con-
ditions are more or less closely under our control. The phy
sical quantity so discovered is primary the result of the
operations and the calculations; it is, so to speak, amanu
fLtured article^ - - ^^^X^^^^%Zl^
KSes^ S5SS ^~ - S-rSea S

he would see his ^nuf-*-f J* ^g* ^t he can Sy x .
tinct feature of the picture g finding ^

unit measuring-rods in a line Dew ^ distance between
nufactured the quantity x which .^.^g^ x . is some thing
the points; but; he. believes^hat this dxsta^ ^ _ ^ ^ ^.^

' already existing in the pictured g ^ existillg

measuring rods... . ; , . . . . distinction between physical
Having regard to this ais. def . ne a pVysl _
quantities and world-conditions, we shax ^ vovl ^ ic ^ e

cal quantity asjhoug ^^^^Sjuanti^^^li^
which had to be^Suihtout. ^ff^SJ&^^ichji^

b y the. serie S _^^EE^-°-g^^^ L ----'^ . .

the result .7. . k the physicist what °°» ce P^ on

" We do not need to asK to ±- ^^^giength, and

he attaches to • length' ^n^^SSSSS^jSrior^. (95)
form our definition according

-217-

The epistemological implications of this principle
of operationalism are far reaching. They may, perhaps, be summed
up by saying that the physicist is never confronted with a pure
objects The fundamental quantities, such as length, mass, energy,
potential, etc, out of which the whole structure of physics is
erected are not thing s or natures or propertie s or features of
the absolute world condition. They are articles manufactured by
the subjec t o They are synthetic products. They are not things of
nature, but things fabricated in order to explain nature. As Pro-
fessor Petit has remarked, "le faire est au coeur du connaltre
experimental" . (96) In other words, in the experimental sciences,
specul ative knowledge can reach out towards its object only by
giving way in sons raeasure to p racti cal knowled ge.

All this, however, does not favor the idealistic
position. For the operations which constitute a scientific expe-
riment are physical, and they are performed upon objective physical
nature. As a consequence, the results, while not purely objective,
are not purely subjective. They are a composite of the objective
and- foe subjective. But it is extremely important- to recognize
the part played by the subjective element. As we shall have occas-
ion to point out in a future Chapteij it is only by acknowledging
the role of the subjective in experimental science that we can
k become truly objective.

It should be clear from what has been said thus far
about operational character of experimentation that the aub a e ct ive
enters into science in two ways. In the first plac f^e is a
rental intrusion through hypothesis and ^e^^.Jf.^^^s
all of the operations and the whole structure of the ^trumsnts

employed aredetermined \r i ^°°SS.l2 , I SS l Sen
are, in fact nothing ^jmajerialxgedjhegi^. This^
^velo^e^T^He-iprse^^Tttas C^pte^ -U subject ope „
there is a physical intrusion in the sense x ^

rates physically upon nature through J»L^ las ^
on by physical instruments _ constructed of °°PP e ^ ^cdj^ter--
aluminum and silk, etc. This obviously res ults * n^g^^^_.
action between the object and tte au^ot, ^°^ e state \ f
sible for the subject to get at tne o j ion ^ our d i gcU ssion

objectivity. We intend to return to ™" * vm but perhaps
of the limitations of measurement in <•£? the fol i ow i n g lines

at this point it will be worth whileto q significance

from Heisenberg, who has done so .aioh to bring
of ihis interaction.:

Particularly <^^«^^T^
follow is the interaction between observe

-218-

sical physical theories it has always been assured either that
this interaction is negligibly small, or else that its effect
can be eliminated from the result by calculations based on
•control 1 experiments. This assumption is not permissable in
atomic physics; the interaction between observer and object
causes uncontrollable and large changes in the system being
observed, because of the discontinuous changes characteristic

Iof atomic processes. The immediate consequence of this circums-
tance is that in general every experiment performed to deter-
mine, some numerical quantity renders the knowledge of others
illusory, ( since the uncontrollable perturbation of the observed
s yste m alte rs _ the v alues~of previously determined quantities^
If this perturbationTJe^lTollowed by its quantitative details,
it appears that in many cases it is impossible to obtain an .
exact determination of the simultaneous values of two variable s,
but rather that there is a lower limit to the accuracy with
which they can be known. (97)

Unitl rather recently it was customary to contrast
fthe method of introspection employed in experimental psychology
with the methods used in the other experimental sciences by point-
ing out that in the case of introspection the intrusion of the
subject makes it impossible to arrive at the object in its pure
state of objectivity. And it was more or less taken for granted
that this pure objectivity was attained in the other experimental
sciences, Neils Bohr, however, has shown that this pure objectivity
is a mere illusion and that throughout physics there is an intrus-
ion of the subject comparable to that found in the method of in-
trospection. One of the reasons why scientists ^come easily ■sus-
ceptible to this illusion is that, as Duhem has brought out so
fully and so accurately, 98) they tend to substitute in their
mind an idealized instrument, a kind of '^ the f^ r ra °g f °\
the actual physical instrument employed For a «**££« *£*
certain breadth, for example, is substituted a goui definit e
without breadth for a steel magnetic needle which has a definite
magnitude and which is unable to move without frict "^"^
tuted an infinitely aaoll horizontal magnetic axis which move
around a vertical axis without fric ion ^ c In J^f.^^
a tendency to go even beyond this g^^^T^^^
com P 2£tely,(tp_a±£2iuie_Jo_ iL ttt^^

5ojnlti^e f°S5l€7AiSThri55i^-rw all * thingg j^
to the nature of the intellect _gua intexxec
pendently of physical means.

Perhaps the «»t fZ^^tlV^on-
be drawn from this discussion of opera h ±t doe3

ality enters into experir^ntal science m way

;ains

-219-

not enter Into any other science. It is true that irrational ele-
ments enter into all the sciences in one way or another, but in
all the other sciences these elements rema i n extrinsic to the form-
nTjty_gf the concepJs_jhat_are_prg per to these sciences . But he-
cause the very notions out of which experimental science is cons-
tructed remain inseparable from the physical, material operations
by which they are formed, -that is to say, because a mere series
of ph ysical operations (~ plays the role that essences play in phi-
losophi cal knowledge )) there is a profound element of irration ality
intri nsic to these notions . And it is all too easy to lose sigEF
of this fact simply because of the operational clarity that these
notions possess ,

7, Laws and Theories.

Mic4iw/

\r yvww.

But science is not made up merely of isolated notions.
It is a highly coordinated and unified system. And this coordinat-
ion and unification is brought about chiefly through the formulat-
ion of laws and theories. To this formulation we must now turn
our attention. Since we shall have to return to this question later
when we come to consider the mathematical transformation of phy-
sical science, we shall content ourselves here with a brief out-
line of the structure of the physical laws and theories and with
a summary discussion of their epistemological significance, in
such a w that the central thought we have been pur suing, ™^
the wo.l gotion of the ^ subject Av^l^go^nto^ffixture , will be rounded
out and fully crystallized.

Unitv is a condition of intelligibility, for pure
unity is a ggm — . i , . the j^^

diversity is essentially "mtxonal. (99^ That is^ ^ &
in its efforts to rationalize nature cannot; r b
nero collection or tabulation of Phenomena. As we ^^
Chapter VIII, the process f-^^^g^sStial^
is already a_miif3cation, f or ^f^ r ^ f a standa rd) But this
in reducing a multjgj^cijyCtot ne uni ^ ^- r^^-^^axA' a desire

,iniHaTimIHomonT3T5otV?^t^ g 1 ^ ^ as de-

fer rationality. It has an ^f^^l^^otU^^^^
sely as possible to .the higher forms o ^ piuralitj^of^spe-

i ncreasing pl urality^o^jtang^Jg^-SH^^

Wes.Olt instinctively tends to £ 1S ° } ' Qf events whioh

inraefiniteC^^MonLMt^^.^ 1 ^;^ t £ e development of
reveal themselves in experiment. That is w ^ ^ Qne hand>

science- manifests too .paradoxical tendenci

-220-

T /e have seen that the movement towards concretion is a movement
towards greater multiplicity, since it approaches things in their
proper scientific nature, ThjgJg_a_jgndency_tOT7ards_ a plur alistic
universe ,, On the other hand, the mind instinctively seeks to reduce
this multiplicity to an ever more perfect unity, and the terminus
of this movement is a completely monistic univers e . | The .amazing
thing is that t hese two contrary movements, far from being irr e-
concilabl e, are actually coo perative J (Tool The early part of
this Chapter was devoted to a consideration of the movement towards
pluralism. Now, before bringing this Chapter to a close we roust
discuss the tendency towards monism,, This tendency is carried for -
ward principa lly by means of laws and~~~theories .

Nov? nature lends itself admirably to this tendency
of the mind. For the <§vents which present themselves in experience
are not mere desperate phenomena. They reveal themselves as belong-
ing to a pattern, For nature is defined preci sely in terms of those
things_which_hagpen, "ut in pluribus." (lOlX This natural "order
ancTregularity make s~xt~~possible for the mind to establish legalit y
among phenomena, and this is the first step in the movement of
the mind towards a more perfect unification than that found in
the reduction of phenomena to a standard.

But are physical laws a_ mere reflectio n of the order
and regularity of nature? Classical physicists seem to have been
persuaded that they are. All the best modern epistemologists, how-
ever, are agreed that is very far from being the case. And we feel
that enough has already been said to show why this is so.

For in the first place, it is clear from our discus-
sion of the nature of the propositions of ^rimental science
that the universality and necessity which are found xn. Pg s ^
haws, and which are of the very essence of all law, can be nothing
Wt a gift of mind to nature. Nor is this f ^J^°^' g S
mind bestows it only that it may ^ car ^hv^ TJ^e et-
, towards which it is striding. ^^^^^lolefu^
sential^y functional. That is why f ^^fScti'on of In ab-
as some thing fixed and static, as a tinisnea
Uolute order existing in nature.

But there is ^Zt^T^TfT^s^f '
For, as we have just seen, the ^^ed>re not obqectiye^ntities ,
o ii j L of^^h(phy^i£a^l^ operations

They are artfcl^ilSiuTa^tured by ne a j both j^theses

Won nature, (102) Into ^V'^hfre^tant g^ ^ ^ ^
andjhysical action. .That is ^ JJ^V subjective logos that
ing except in terms of the proo^

-221-

all this entails Moreover, in the highly complex structure that
is physical science, laws do not have a completely independent
and absolute meaning in their own right. Their meaning' is deri-
ved from the ir context , fwhich is a closely woven pattern of mutuall y int-
erdepend ent laws and theorie gjJTn this connection, Professor Camp-
bell writes: "Nous remarquons d'abord que les termes ne sont pas
habitUQllement des jugeraents simples et imv.iediats sur les sensa-
tions, mais des collections complexes de tels ju gements.' Dans la
plupart des lois, ces collections sont telles que les lois ne sont
vraies que si d'autres lois le son t. Elles en dependent a la fois
pour leur sens et pour leur verite. Ce caracterc de dependence
mutuelle est tres important pour nos recherches, (103) The si-
gnificance of laws also depends upon the particular theory into
whose structure they are fitted, in such a way that if the theory
s changes the significance of the law changes. Duhem writes: "selon
que 1'on adopte une theorie ou une autre, les mots mime qui f igu-
i-ent dans l'enonce d'une loi de physique changent de sens, en sor-
te que la loi peut etre acceptee par un physicien qui admst telle
theorie et rejetee par un autre physicien qui admet telle autre
theorie," (104) The difference of meaning attached to the law
of gravitation in Newtonian and in Einsteinian physics is a case
in point „

It is evident, then, that there is a vast difference
between the objective laws of reality and the laws of P^sical
science, Eddington has brought out this difference m the follow-
ing terms: . '

St'TcSSS intend^them to ^,^^85*
I will discriminate ^^^^^yJ^r^S that
and 'laws of Nature' , Law of Nature will na emmating

the term was originally ^tended to bear a personify

from the world-principle -J^^teStofore a regularity
as Nature. Law of nature ^ q ^ t ^" pledge, irrespect-
which we have ^^"^t^of nfture is Xtever would
ive of its source. In short a -^ " ^hvsical practice,
be designated by that ^£,™\?^ture is a law of

It will be ^^"^^ognLed laws of nature are
the objective universe. But aJJ. r B bal ™ ra dox that no
subjective. We have thus f ao ^f w *ture.|Effoctiv^ly_Jhe_Je iE s

known law of nature Jsala^V^ 5 - ' '

have becorneiS^al^xclusivey open ing. A law of

~~ I^iiTHI5TKaT^h av e let ^^ necessa rily if

Nature is a law of "^° ^ Susies. This brings me
it already is) ^f^*** S *e aS reason to believe that if
to a furthe~quostion. Have we any

-222-

a law of Nature - - a generalization about the objective world
- - were to become known to us, it would be accepted by current

' physics as a law of nature? I think it would only be accepted
if it confomea to the pattern of physical lave that we are
accustomed to . But this pattern is the pattern of subjective

i law. We shall try later to show by episteraological study

how the pattern has grown out of the subjective aspeot of phy-
sical knowledge. The pattern is the very hall-mark of subject-
ivity o Any expectation that we may have formed that the object-
ive laws of Nature, when they are discovered, will conform
to the same pattern is quite unreasonable. (105)

In order to be convinced that physical laws are ideal
constructions of the mind it is sufficient to analyse any one of
them accurately. This analysis will reveal the utter impossibility
of their being realized as such in nature. And this is true of
even the most fundamental laws which have cone to be considered
as the principles of the whole structure of physical science.
The^rincipJ^_rfj£e^iaj's^_casiLJiL£2^ b '' ( i06 ) l^The^ verif i-
c^tion of this law in_ natu re would involve jt contradiction;) For
g^^^^hoVto^FTm^vl^^rb^ay preserve^rEIT^ilinear and
uniform motion unless influenced by another body, itwouldbene-
cessary to' have only^ne body in exi stence - - andjhen allmotaon
wo^MSSonSli y since bodies otm.ggTO- onjonrgjataon^
to^tter^io^ver the exact verification of the principle would

is ii^ortortlo^ote-tMTlSwrrf this kind. beconBcp^e|tioss
^hich serve I todefing ) ^J^-^I}S^J!$^^^^§M~^-' )

yc «m-To^h e ,S71 i1^^

^tii^SaFSrunif-om motion a scientist ^t^ ^^
that the law of inertia had been violated, v/ ^gfejgy^ n

^^vini^bWrSnikTmnn^TThe law wmc ^

SSSSTreTZHoh between the .W* <* £ il^coeffiaientjf
temperature is transformed into (i^^^^^en^-orth^sTress
y^a^e X p_ansion J the ^ which st «*£* J ^ int0 a definition

in an elastic body upon the strain is w ^ light tra _

of el^tic^onstant. First the law is esta ^ hea0T ^ s the
vels in a straight line, and then the pax ^ ^ ^ Lq RQy
definition and the norm of * 3 X^V„ hlea a prendre les choses

, could write: "Les lois sont inv ^i£° X ^ ti £ uen t le critlre mSme
en toute rigueur . . ., P^ce % £ s Rhodes qu'il faudrait uti-
auquel on juge les apparences .etx ^ pr( s oision BO it sus-

. liner pour les soumettre a ^ ^'^g^le." (108)

^ceptible de depasser toute liuite assign

-223-

l im -j Ifa

It is necessary to conclude,, then, that physical
laws are not found, - - they are made. They do not exist before
they are formulated by the mind. This does not mean that they are
purely fictitious. They have a basis .in reality in the sense that
■they are suggested by experience. The law of inertia, for example,
was f ormulated only after it had been suggested in countless ways
by nature. Moreover, the terra of the process which constructs phy-
sical lows is always the true, -objective laws of nature. And that
is something which those who insist upon the subjective character
of scientific laws usually forget. Nevertheless, it remains true
that only a suggestion of these laws is actually found in reality.

(That_is_ whv there i s something essentially Platonic about them .
That is why Kant -was in this respect correct in making the mind
the lawgiver of nature „ [ For scientific laws come- from real ity only
ma teriall y; formally__ they are from the mindu ) The essence of scien-
tific knowledge is made up of a kind of( _ noetic hylem orp hisip in
which the matter presented by reality is formalized by the mind.
In all of the laws of experimental science, as Eddington writes:
."the mind has by its selective power fitted the processes of Mature
/into a frame of law or a pattern largely of its own choosing; and
f in the discovery of this system of law the mind may be regarded
Vasjregaining fron Nature that which the mind ha s pu t into Matur e^)

(fray ~~

The establishment of legality among phenomena was
for Corate the ultimate terminus of the scientific movement. But
in this respect as in many others Comte failed, to seize upon the
true spirit that animates scientific endeavor. As Einstein and
Infold have pointed out, "la science n'est pas une ooUeotxon de
lois... Elle est une ■creation de 1- esprit humin au noye n ^ idees

at de ^.ffi^^^rffiK

£ s^.^£&^?~ -r r thSs'for-

se just^ent seulement si, ^^*$%^g%BS,
mentun tel lien. " (110; >> US J .JT. . n unification _achieyed
Hity-I^illi^t to rise ab ove ^f^gffiShilfto.

fiHitrreEHon in the multiplicity at P n Synthesis which

impellsTT^rio further and ^^^f ^This higher
establishes rjlaUons_jmJihej^ The kinetic

synthesis is achieved by means of a P^icai^ to J s ynthetize the
m^> of gases, for example, f k ^ X y Avogadro . By means of this
CgwFof Mariotte, of Gay-Lussac, and *_ A Y^ thetize the laws
principle of gravitation Newton^ was able ? mlng the tid es.

•arrived at by Kepler and Galileo ana

Without theory the movement of the scientific mind

-224-

H.

would be essentially frustrated. For the two essential properties
of nature are universality and necessit y. By means of laws the
mind is able to rise above the singu larity of ph enomena and arrive
at a kind of universality. But this Cuniversality) is lacking in
(Necessity) That is to say, even when, laws have been formulated
there is nothing intrinsic to them (which shows that could not have
been otherwise,) In other words, propositions which merely state
an fassociation between the values of one variable and the values
o f another variab le ja re not logically necessar y. For example, an
increasing temperature is associated in a determined way with in-
creasing volume but there is nothing in this law which shows that
jthe reversec lnight not ha ve_.he.en the cas e^> The mind cannot rest
satisfied with this contingency; it must strive to reduce it to
some kind of necessity by finding a reason which explains why in-
creasing temperature is associated with increasing volume. This
is accomplished by the construction of a theory which postulates
the existence of (u nobserved entities whose hypothetical behavio r
will explain the observed phenomena^ ) Thus physical^tEeory )bgcomes
a~sub3tl tu te for the analytical oharacter that the propositions
of expe rimental science lack .

In other words, science, as we saw in Chapter II,
is a knowledge of things in their causes arrived at by demonstration,
But without the ories experimental science is unable to discover
the causes oFthillws Cit formulates^ norcan_it_deduce these laws,
ThatTs~why it is 5nTy by having recour!e~Eo theories that the
scientific mind can realize its ideal of i» tion ^ : iB:ui 8. na * u ^S

fiJatltelStSair terminus towSSI which all science moves m the

**i°n of this ide-a3Tw^uldlEiiS-that the whole of ™^° J ^^^
be deduced from one s^s^-^SSBT'^J^^^^ ^I^^

oTiatu^e-iould mean its comple te dest ruct ^ J^ noted ^

another example of a phenomenon which ^^^^^Osctence
to which more attention wall be given ^ • -< o^Tts'ldiil^iil
tends towards a contradiction. ^reali.aUon^of ^ ^ ^
ever remain a mere dj^aBOjtn^aix^; — , -nerfect deductibility

to reveal irrationaTilSSStrEo^eveno its pen

- „,_ +pnflq towards monism while it
To say that science tends Jona^ ^^ yniversal-

moves towards pluralism is to s %.^ nnoretioni> But it is important
ity while it moves towards SP 8 ?"*^ —U-gr-^ tends is not the
to note that the universality Awards wh ich re ^ ^^

same kind of universality from which .^^ ^^ pointed out ,
/cowards specific concretion. For as

-225-

this latter universality is a raere univgr Ba i ita3 in prae aicando .
3?}HkiJLB°WJgBglL2Jgg^^ seek3 t o

achieve in its construction of theories is a universality which
will permit deduction, And that is why it instinctively reaches
out to mathematics whose principles are not only universal inprae-
laicando , but also in causando . And this explains why Descartes'
[attempt at the global deduction of nature" by means of mathematics
was much more intelligent than Hegel's attempt to arriye at the
^same goal by means of logical categories.

It is in the construction of theories that the mind
finds the fullest scope for the projection of its subjective logos
into nature, For to a far greater extent than in the case of laws,
physical theories are not so much a gift of nature to the mind
as a gift of mind to nature. They are fictitious constructions
freely chosen by the subject. (112) It is true that these cons-
tructions must be made to conform with reality. Nevertheless, this
conformity is not a logical proof of the objective truth of the
theory concerned, for: ex falso qoudlibet . In other words, one
cannot conclude to the truth of a theory from its perfect and cons-
tant verification in reality without falling into the logical fal-
vlacy of affirming the consequent. (113)

It is true that deduction from a theory can lead
I to the experimental discovery of a fact. For example, the. law of
gravitation as conceiv ed by the theory of Relativity led to the
discovery of The fact that in the neighborhood or ponderable bodies
\a path of light undergoes considerable deviation. This fact is
true, but the truth does not derive from the logical discourse
which first suggested it. Rather, it derives formally from the
experience by which it was actually discovered. And this brings
/us once" again to the essential reason why experimental sciences
are experimental* their truth is in experience onty; the logical
discourse is onlyan instrument, and even the ^J u :£°» ot ^ a
discourse is only instrunBntal in the sense that it leads to or

ThTw^Telictrostatics, for example, can ^ e ^^ e * °
cessfully by a number of different Tories, such as the theory
of two electric fluids, or the theory °^^ sx ^ff ct ^ s ^ protons,
theory of discrete smallest charges, "f^'^° ™ of Xch
The corpuscular and undulatory ^^Vc^sgcal example of the
have been successively "verified", are a ciassi

same thing,

,•<■ ^irlv prevalent that physical
The impression is fairoy prova

-226-

theones are founded directly upon facts. That is, however, an
inexact way of representing them. They are not founded directly
upon experience, rather they seek to _ posit a point of departure
from^wtogJLjexpjrigjice^may ^e arrive"d~a\ " that is to any, from which

relations may be logically deduced (w hich. will be equivalenOo

5hos e derived from experience^ ) " '

It must not be thought, however, that theories may
be constructed in a purely arbitrary fashion. There are certain
criteria which must guide the mind in this construction. And the
three most ijnportarit of these criteria may be deduced from the
foregoing analysis of the nature of physical theory. First, because
every theory is an attempt to arrive at the most perfect unity
possible, the one which has the greatest \ logical simplicity) will (Q
be preferred to all others. Secondly, because every theory is an
attempt to make nature deducible, the one which has the most per-
feotl co nformity with reality ) must be chosen. Thirdly, because the (V)
ideal of science is a merely dialectical limit towards which it
must ever tend, that theory will be preferred which has the great-
est (fecundity,j that is to sa y, cwhich is most significantly sug - (^

gestive~of new experience^ ) This last point means that a good the-
ory"! 3 one whicfi reaches beyond itself ; if it does not give rise
to new problems which it cannot adequately solve, it is not truly
scientific, A good theory must not only solve problems} it must
create them, for otherwise science will become static and sterile,
(114) The new experiments suggested by a theory will at once in-
crease the multiplicity of data an^preMi^J^^fTlf^^T^
unity .(-that is to say,_J^r_ajnor^p^rfe^tJheory>) (115; That
lilriy agoST'theofylnuBt contain thTiiSraTta own destruction
within its bosom. For a theory that explains everything explains
nothing, Newton's theory was good, not only because ^explained
many things, but because it brought to light *^^*^ t ^
unable to explain, "Crises" are essential for ^/^J^f 1 !^
science, and if contradictions did not ?™*^£ t ^° ^?W
become stagnant, (116) But it is significant that no matter ha,
mny contradictions may arise in the face of one theory, it^s
not abandons ed until another theory is ^ *? *^^; B |S tni

constant interaction of objective and subject "eJjOgo^
is this interaction that we must now attempt to anaJyse
bringing this Chapter to a close.

this

-227-

8. Objective and Subjective Logos.

If there is any conclusion which emerges from the
preceding discussion it is that the evolution of science is essen-
tially a creative evolution. The mind does not merely discover
nature; it constructs it to its own likeness arid image. And it
is only by no doing that it is able to discover it. (117) But
because this construction is never free from its relation to dis-
covery, it is not a pure creation, but a re-creation . The mind can
progress in production only by becoming increasingly dependent
upon induction; it can perfect its construction, only by perfecting
the instruction it receives from nature, (118) It can advance
only by keeping up an incessant dialogue with reality . 1 It canno t
reason without experimenting, nor can it ex periment ^without rea-
soning ^ This is not, "however, a circle without any definite direct-
ion For the reasoning is always orientated towards reality.

In other words, experimental science must be at once
synthetic and a priori . And it is only by maintaining a proper
balance between these two elements that the extremes of idealism
or empiricism can be avoided. All this may be summed up J>y saying
that experimental science is (a mixture o f science and _ art ,)_jing.
for this ^^njj,^ neither a science nor an art m the fni l
Si SrortHe-^Srl Ttod there is perhaps aTbetter way of getting
grit' s nature as a quasi science than by analysing the way m which
v art enters into it ,

ttousselot is correct in maintaining that in the e-
pistemology o/st ^o^the sciences in th. ; modern senseof the
term arejkther^rt^than sciences. (119) An d it is highly si
gnificeSTlhaTaTttoToienoa of that part of ^^ W £an as
I we saw above, ca™oJjDe_iefined_i^^

formed in such a way that there is no moi ^ %

ling it perhaps,' than by viewing it «a» jjt. ^J^ ^ ^_
a remarkable paralle^etween the way in wh ^^ soiences

ture, and the way in which it enters in created

of nature. As we pointed out ^^^"ivirl art penetrate
reality is a work of art, but "^^^i^smos . In the same
so deeply into reality than ^ J^ ^. f for n0 other reason than

"ay, ^^saJs^^^^^rsssrsr^^^

that_they all employJfS^^ " tural soiences. And it is ex-
so deiply-l[s^in-tfie~55^Hifental «™™-

hremly important to see why this is so.

/

(

-228-

Logio reaches farther down into- the structure of
the soiencea than might at first be supposed. It has to do even

with the first operation of the mind. One might perhaps be tempted
to doubt this statement. For logic has to do with an ordering of
thought, and since simple apprehension grasps things in an abso-

| lute fashion, it may be difficult to see how the mind can intro-
duce order in relation to this first operation, as it does in the

i construction of propositions and syllogism, Nevertheless, as John
of St, Thomas says, "pr ima apprehensio absolut e et per se_perti-
n et ad log icam." (120) As is evident from thlTCategories 6f~A-
ristotle, a, certain distinguishing and ordering of terms is neces-
sary prior to their( oonst ruct ion into ) propositions . In this way,
art surrounds the terms in ail th e sc ience s from the very beginning,
But the vital point is that in all the vital sciences besides the
experimental sciences this art merely surrounds the terms - - it
does not posit them . Only in the experimental sciences are the
very terms themselves artefaota . The student of nature fabricates

I the very stuff out of which the whole universe of physical science

(is constructed. To use scholastic terminology, the objects are
never a pure quod ; they are always a mixture of a .guod and a quo.
The quod and the~quo constitute an accidental unity and are con-

i sidered ad modum unius .

This penetration of art into the very essence of
experimental science is continued throughout its w ^%^^^
As we saw in our discussion of laws and theories, the grm .of ex
perinatal science proceeds not only from the object, but also
from the subject, (121) ?^JS}^9S^^L^^j^^ lo
tructed byje i ^j2f_j^,(tor^^

of the_logic they employ. But in the expe rime structure,

art employed becomes an essential W**^*^ independently
That is why they aro not sciences in their own r g ^

of the dialectics they use. They^ re J^™ * ^| e the re-
be clarified in the next Chapter wten J e °°^ c g s^
Nation between experinental science and dialectics.

Another way in which art Pirates £* £^£3
essence of physics ia.f** » xt s gggf^ How aeep this_
(which is at oncea_ao^noejx^a^VJ^^^- 1 r^-r^ ti ^ te union exist-
^vSS^^S^^STBSr^^^^^^^s. The mind,
ing between subalternating and suoax^ . ai soovers great

which finds it necessary ^^t^va^t co^tructibilijyof
scope for its artistic impulae iAS^^^^^j^-to a si-
mathematlos. In this connection a*"" 1 gt Thoraas says that

gnlf icIHFtext in the D^Jrinitate in wn

-229-

logic, mathematics and mathematical physics "inter coetaras scien-
tias artes aicuntur quiaj ion solum habent _ cognitionem f ^d_opus
aliquodj) q uod est immediate ipsius rationis , ut constructionem,
syllogismum, et orationem formare, numerare, mensurare, melodias
formare, cursus siderum computare," (122)

It is interesting and instructive to try to deter-
mine the nature of the art which enters into experimental science.
A moment's reflection will reveal the extreme complexity of its
charaotero For, in the first place, it is at once both speculative

I and practical,, In so far as it involves the use of dialectics and
mathematics;, it is speculative; in. so far as it involves a physi-
cal operation performed upon nature, it is praotical. In the se-
cond place, it has the characteristics which are proper both to
fine art and to useful art. The fine arts are essentially arts
of imitation. But as St. Thomas points out, (123) an imitation
is not a mere similitude, that is to say a materially exact copy.
It is the expression of an original by an intellect, and this means
that the original has passed through an intellect, and in passing
has acquired something of the order and light that are proper to
the intellect. Andthe 1S ir S oae_a£_^J l ti S B_ S ^(^m0 E i^^^
art;) is to .nttta~WoH g^jJiLggggjrcyJgtt^^

s^iel^jThe-Wkc^rSniferse constructed by the scientist is
STSdtetion of the real world. IXis^tan^xa^py^mgeJ.
of it. For the intellect has contributed much to this ^^on,
AMln.this imitation „yS_the_wo^

it really_is. Our knowledge of. material .thin gs is Jf^han the
thinis-themielves; ^te^noe u^est ShosfSund L
forms that are found in the nund are better tnan u _

But precisely because ^ ^V^'ST
They are worse because experimental «»«"£ ™ for ^ is to i ead
a science. That is why the whole purpose ofjhese t ^

to the WLe'dge of the forms existin g ™™*^ ^ ^^.any-
perfect the constructions of ^^ as ( ^° a ^ y ±a to sa y, the
thing more than mere scaffoldings, V ' purely f unctiona l,
art that is found in experimental ^^^jsESTSSSTSxTto

Cand from this poJr rt-^^Sg-^M t "i^ria nga Kst which they are
^iSTlEHTmita^^ outlines at least,

erected for they must take fj^"' ect does not consist in

Nevertheless their ^ost todamental aspec ^ ^^ ^ reaoh

this but rather in the fact that they ar

the house,

-230-

in the arts - - the distinction between t hose which, cooperate with
nature and those which do not . In the latter case there is a pro-
jection into matter of a form which is independent of the natural
fo rm that is native to_ thejgitter. In the former case there is

Canj3xtrinsio assistance ; Droug ht to hear yfco enable the natural form

lt o_achieve its end more fuLjy.) It would seem that the art which
enters into experimental sciences participates in both of these

Icategories, For in so far as it is purely functional, in so far
as its purpose is "to induce nature to yield up its logo i, it is
an art coo perating with nature . But in so far as the projection
of the subjective logoi is not a purely extrinsic assistance, as
is true j for example, of the use of logic in the sciences ; in so
far as this projection results in the construction of a physical
universe that is in a sense distinct from the absolute world con-

Vdition, it shares' in some way in the second category,

A number of recent authors have insisted upon the
fact that modern scientific progress has meant a gradual emanci-
pation of science from the profound anthropomorphism that was cha-
racteristic of the. views of nature current in past centuries, (125)
And the truth of this can hardly be doubted. Yet if the foregoing
discussion of the projection of the subjective logos into nature
means anything at all, it must mean that from another point of
view modern science is immeasurably more anthropomorphic than an-
cient science, For allart , as Bacon taa jcemri^jAjagLS&f
to nature, ThisTi-jSitlSother of the innumerable Paradoxes that
5nTl*55£tantly encounters in attempting to analyse the nature of
experimental science: modem science ^ a ! °»*^£f£ %?
cisely because it is more anthropomorphic; xn other words it is
more objective precisely because it is ™ re ,s^ctxve ^ specific
'example of this is found in the ^ematization of na to. *
mttemtization is in a sense anthropomorphic for it °^
viewing nature in tern* of ^^ ^Z^lll that delivers
\to the humam mind. And yet it is ^ ^atn sub j eot ivity

us from the anthropomorphism J^riv \ out this paradox^

of sense perceptions, Ernest Cassirer i*x
of modern science:

Physical thought. igZl^gSF^t&Sr"
in pure objectivity »^f ^ f^^^'its own principle,
necessarily expresses itself , "^ isra . f all our con-

Here is revealed again that ^^° P the ^ sdora of old age,
cepts of nature to ^^^Z' oTnXcT U still only anthrc-
loved to point, -AH P 1 - 10 ^^ SThlmself, imparts to
pomorphism, i.e. . . »"} & L^ unity, draws it into his unity,
everything that he is not, this un^ *» ^asure, calculate,

Imakes it one with himself .... ve can

~231«

our

weigh, etc., nature as much as we will, it is still only UUi .
measure and weight, as nan is the measure~^flarihiig^^Only
after our preceding considerations, this 'anthropomorphism'
i tself is not to b e understood in a limited psychological way
l(but in a universal,_gij^y £ al_and_teanscendental senseljPlanck

points out, as the characteristic of the evolution of the sys-
tefmof theoretical physics, a progressive emancipation from
anthropomorphic elements, which has as its goal the greatest
possible separation of the system of physics from the individual
personality of the physicist.. But into this objective' system,
free from all tte. accidents of the individual standpoint and
individual personality, there enter those universal conditions
of system, on which depends the peculiarity of the physical
way of formulating problems. The sensuous immediacy and part-
icularity of the particular .perceptual, qualities are excluded,,
but this exclusion is possible only through the concepts of ;
space and time, number and magnitude. In them physics deter'^'
mines the most general content of reality, since they specify
the direction of physical thought as such, as it were the
form of the original physical apperception. (126)

1

As Cassirer suggests, one of the fundamental differences between
the anthropomorphism of past centuries and the anthropomorphism _
of modern science is that the former tended to be individualistic,
whereas the latter tends to rise above the restrictions of indi-
vidual sensuous perceptions and of the interpretations proper Jta
particular groups. There is some truth in Claude Bernard -S remark,
"Si l'art c'est moi, la science c'est nous." Yet of the artof
which we have been speaking, it may. be said: "C'est nous." Ahd the
reason is that this art is at the same . ttao a science.

All this explains the spell that mathematical physics
has succeeded tfputting^pon the human ^^^fS?^
For in it nan can be at once_ both the ho mo J ffa ang andjhe g2
faber, (127) The mind is allowed to indulge _in "^^J?*™

TatiSn in the realn that * ^£^£5^ £» contraction

mathematics, and this speculation is inseparao

in which tte intellect posits its own ob^ot. At the sam

this speculation brings it closer to the °^<^ ^

to it - - the essence of mt ^f .^"f^ity and nriLeaUlUy
ledge of material things reveals the P^xc .y ^

that is native to them and thus gives to the mind
refashion nature according to its own designs.

x- u 4- QO n crvpat intellectual danger,
But this spell --^f^/J^cientism which will
For not only will man fall W'' attention, in such a

rake mathematical physics absorb his wno

-232-

wiythatux the speculative intellect(wii^) will be dethroned
by(science) and not by science in the full sense of the word but
by mere dialectical prolongation of science; and in the practical
intellect, (£rudgncS>ri.l l be dethroned b y (art) and not by the highest
form of art but by technoloR ioal art - - not only will he fall a
prey to this form of intellectual suicide, but because by nature
he is more a being of action than of contemplation^ more an artisan
than a philosopher, ( he will be tempted to make all science a kind
.of ar-Q That is to say, he will become so fascinated by the pro-

jection of his own subjective logos into nature that he will sever
this projection from its complete orientation to the objective
logos and make it an end in itself, Bergson has characterized this-
tendency in the foil oaring terms 5

Nous ne dirons peut-Stre pas homo sapiens mais ho->
mo faber. En definitive, 1' intelligence envisagee dans ce qui
paralt £tre la demarche originelle, est la faculte de fabri-
quer des 1 objets artifioiels, en particulier des outils affai-
re des outils, et d'eh verifier indefiniment la fabrication,
"Son objet n'est pas , . . de nous reveler le fond des choses,
m is de fournir le rneilleur moyeh d'agir sur elleg ." "Quel
est 1' objet essentiel de la science? C'est d'accroitre notre
I influence sur les choses, (128) M-Ck*-

Ye have seen that experirrental science is more a
priori than the disciplines that are sciences - in the strict sense
oTThe" word precisely because it is less a priori. That is to say,
in the latter case thgonnections of things are independent of
the experience in which they are first recognized, _ and in this
sense they are a priori . It is precise^ because ?hat is lacking
in exper4ntal-stiSn^ that a substitute ^rips must be intro
duced^But this ^priori of expe rental ^^Sgigg^g^

in .sdngjSrBeSoVge th^prancipl^of^gerig^^xp^^ ^

HoTma^ifist, it ^re3^ confirms the nmnifestati tal science

has made to itself. That is why the ^* e £££ £ camp tel i teB
becomes the creator of the universe, as Professor o y

remarked:

Un Newton, un Faraday £ Maxwel ^-ncoivent^e^
.theorie, et la vio ?'^pte ^aur^ou^^ ils \ r eent
predites. Par la puissance de leui XT s deg <jreature8

la structure durable du monde. lis * e ^" f t deg aem . il s
c Aos, enctoinees pa^es lo- du tem^ ^ ^ ^ ^

sont les creatures qui entanten
flots leur obeissent, (i-^)

-233-

When this creative element- is made an end in itself,
the mind becomes utterly free, and the measure of all things. In
this connection the following lines of Abel~Iey are extremely
pertinent:

The present era announces a new liberation, as pro-
found perhaps as the two previous ones. It aims at these im-
mutable s> these mathematico-physical absolutes. There is no
longer a tool that serves the intellect, except the intellect
itself in its inventive omnipotence a The universaliaation of
the hypothetico-deductive method, in its broadest signification,
is the logical illustration of it . .It renews itself by chang-
ing, whenever necessary, even its very foundations. Logic,
a collection of rational formulae, appears no longer as an
architectural conception constructed once and for all into
an unchangeable unity resting on an eternal foundation. Thought
must constantly be ready to build on new foundations, or to
modify the arrangement of the edifice, and consequently to
complete, to adjust, and to renew its tools. (130)

This tendency has been extremely prolific and extre-
jmly virulent in recent years, (131) Qneof_it g results has been
the instrumentalismof^oJm_Dewey. The fo3Io5Eg passage, which
is typical of his thought shows how the creative element has been
made the whole raison d'etre of all scientific enddavor,( now science
Vhas been transformed into artj ) it*rcv;c».«- & w^bm, />/>■ ^~>, Si"?.

If Greek philosophy was correct in thinking of know-

worker - - as we significantly say ^ his ^

a^-^rth to the dilettante who onjjoys «»wbu
bors. But if modern tendencies are ^stified xnpu g
and elation first, then the ^^f^ then be seen
should be avowed and carried through. -^-^ that th
thot_soienoe_M_anj^OE|^^ pxactice_and

^^rjg^tiHotioTT/orth toa ^ g /4f 2 S^cticTthat arejiotj.n-
theory,<but betv^enjhosemodes ioi 2__— ^j^ggj—j and_those
*^mSrIt^D^^^^^fSffW|efc4tion dawns,
wHjr^_far^Fjn3pC^a^f;J'2 the mode of activity
iFwiU be iT^oTiSnplace .S|pJ^ e f Mediately enjoyed

that i^^^ 5 ^^ 2 ^S«£S^£t!SS' "* ttet

posses

-234-

,;;W

' science ' is properly a _handraaiaen that conducts natural event s
t o this happy issue. Thus would disappear the separations that
trouble present thinking: division of everything into nature
and experience » o:f experience into practice and theory, art
^ond sdience, of art into useful and fine, menial and free, (132)

Enough has been said to show that there is a sense
in which the whole structure of experimental science is instrumen-
tal and functional, but as we shall point out in a f ew moments
fit is so primarily in relation to contemplatio n,) to t he apprehension
Qf ~tHe~ob~jectiv e_lggos of nature . 1 Dewey segregates this instrumental
and fu nctional charact er and destroys its essentia l orientation.J

But the tendency to exact the projection of the sub-
jective logos has led man far beyond this form of instrumentalisin.
lit has led him to conceive the mind, as a kind of Platonic demiurge
whose sole purpose in to work the vrorld, to fashion it according
-to its own designs. Nature bec OTag^jiiergjy_jJdj^ofnj^^
the nrH-. of mnn:m. is viewed only in term s of its plasticr^
E^ir^thinFin'nature that does notytola itself up ^malleable
matter for the free play of human art isjKglectedCox_its_existence
£ denieS-Sl thec^^distn^ctions ■^^J^^^Lt
^TBlSsticity andieT^Ehem up a^_natures_rin their own right) must

Man substitutes himself for God.

We believe that this is the profound aij^ioan^

of the Marxist'philosophy of ^^£^T***^ 5
the whole Marxist system. Marx writes La |u ™™£ Ieflt
la pensee humaine peut comperter une veri objotav J
j.j «,i™i™w rnixs pratique. •*> ^° u . , »■■■.,._ •

la pensee humaine peut °^ e ^ une ^ st dans la pratique que

une question theorique mais pratique. L ^ dirQ sa re _

1-homme doit prouver la verite de sa ,5^* ^ n , ont fait

alite, sa puissance, son --ae- 5 a. nte ?S«5ito.B. QrO^aeJtJS

qu' interpreter le monde de din erenow

1© transformer^ " (133)

1 1 torched the oore of Marxist philo-
Bertrand Russell touched tn ^ acc0 rding to

sophy when he wrote: "Roughly sp ^gS, ^ of mchlnor y: it
Marx, is to be thought of as ^ naturally ^ ^ . ts

has a raw material giving PP^tunity 1° ^ en ™* toS s Y? ceeded
pleted form it is a ^/^nation which re S is^tos__creation
in breaking dovm every dot ermin £«£ ™ Hj^KFg^bSuThmielf ,
of .the cosmos, he will at last be^ ^ ^ ^ let loose
his own true sun." ^°>

-235-

upon the world a more frightful philosophy, nor one that is more
pregnant with fearful consequences.

Prom many points of view this doctrine is hut the
logical outcome of the general trend that modern thought has taken
since the time of the Renaissance. In every order there has teen
a tendency to construct rather than to accept . And in the last

I analysis \this revolt against mere givenness) is nothing hut a re-
volt against the finiteness of the human mind . As great an autho-
rity as Ernst Cassirer assures us that at the time of the Renais-
sance all the properties that the Doity had formerly claimed for

^itself (were made the attributes of the human soulT )

In so far as all this affects the philosophy of science,
it is clear that the error of the moderns has heen to divorce the

(projection of the subjective logos into nature from ±ts _essential
orientation to the objective logos. The subject becomes the mea-
sure of the object only in order that the object may in a more^
perfect way become its measure. (136) Kant was correct in point-
ing out that in the construction of hypotheses we anticipate ex-
perience. But even before we give our assent to an hypothesis we
have already admitted an objective criterion by which it is msa -
sured, namely objective truth. For an hypothesis must be likely,
that is to say, have at least the appearance of truth. We are not
the ones who create this likeness to truth. Moreover the only rea-
son we posit an hypothesis is to he lp us to know obj e ctive truth ,
(and wes ubmit it to experienc es to the determining me asure of
TtTSSrth. The modemi see in the power to f ^^f^^ 3
a manifestation of the supreme excellence of man. Undoubtedly,
it is better to be able to construct h3T°theses than J° j^T *°
remain in the state of pure passivity. But an the last anal ysis
the necessity of having' recourse to hypothesis in °^* ^_
mture spjring3_from the^extremJjmrtec^on^ Jhe human mt gi

lectp

vet the modern e^ion^^c^cti^ genius

of man in experimental science is but ^-^r--r^^^- ^~

found truthf For we have °^."***?t£££ n £ becoming
science. means that man' s knowledge of th * ™^er ^ ^

at the same time njor^ob^ot^^jmdj^e^^^^ to th±a

/interesting to note here in P-^f^^f J^ow God the
is found in geology m which th e no«r ^ |^^ which is in
greater becomes our recourse ™ *^ Jj^fl^mHim. Now if the

U sense getting us farther ^ frthor ^ ny ^^ ^ ^^

limit towards which e ^ e ™ nente „ r ^r be complete^ objective,
man's knowledge of the universe wuw completely a projection

but at the same time the universe wouia

-236-

of the subject, Ma^pejmlatiy^knov ^dge of natu ro would te one
with his practical knowlid||. \Natureang art would b e -Ta5ntiHe1Q
In other words, .Bffi^ouldJ^gTsurW^^
in Dante's remark: "Si che vostr' arte a Dio quasi e nipote."

Perhaps to move •
His laughter at their quaint opinions wide
Hereafter, when they come to raodel heaven
And calculate the stars; how they will wield
The night frame: how build, unbuild, contrive
'So save appearances.

- - - Paradise Lost,

-237-

CHAPTER FIVE

EXPERIMENTAL SCIENCE AMD DIALECTICS

1. The Problem .

In the first book of the Topics Aristotle tells us
that in seeking to discover the nature of an art it is advisable
to begin by consulting those who are experts in that art. -No one
who attempts to follow this advice with respect to the nature of
experimental science can fail to be struck by a remarkable unani-
mity in the opinions, of those who in recent years have achieved ^
the greatest renown as scientists. Experimental science is consis-
tently described by them as a discourse in which from freely chosen
suppositions (certain conclusions are inferred} ? And in this hypo-
: Ehetical character attributed to experimental science two parti-
cular points are generally stressed; l) it is, at least to some
extent, a priori knowledge; . 2) it never goes beyond probable know-
Vledge „

In the foregoing pages some passages have already
been cited which show that this represents the opinion which the
most eminent modern scientists have of their own art. I™™g e
texts of similar character could easily be adduced from _tho wri-
tings of @U ch experts as DeBroglie, Le Roy, Po "^.J^g"'
Planck, etc, etc. Perhaps the following linos of Sir Jeans will
serve as a typical example:

.we have seen that efforts to — fu-
ture of reality are- necessarily doomed to failure, *>J**
if we are to progress further it ^^Xafprinciple of
objective and ^^^Z^ZT^^f^t tnemselves.
which we have not so far made use. i descr i-bed as pro-
The first is the principle of what Leibni ^^ ^^age,
bable reasoning; vre give up the qu alternatives before
and concent r -atl on that one of the various^ ^ ^ ^
us which seems to be most probably true, mt ^ ^^
decide which of the alternatives i most like y ^^
This question has been much discussea o , v ^ rQ ^ on
by H. Jeffreys, For our purpose it is suit

-238-

*w TtS tn Cr i+ Gd a ?. the ^^lioifa D^falnte, this assorts
that of the Uo alternatives the stapler is likely to he nearer

x „ * n . real science also a hypothesis can never be proved
orue. If it is negative by future observations we shall know
i t is wrong , but if future observations confirm it we shall
never be able to say it^ia^rigjitj since it will always be
at the viiorcy of s'HIlfurther observations. A science which
confines itself to correlating the phenomena can never learn
anything about the reality underlying • the phenomena, while a
science which goes further than this> and introduces hypotheses
about reality, can never acquire certain knowledge of a posi-
tive kind about reality} in whatever way we proceed, this is
forever denied us. (l)

We cannot claim to have discerned more than a very
faint glimmer of light at the best; perhaps it was wholly il-
lusory, for certainly we had to strain our eyes very hard to
see anything at all. So that our main contention can hardly
be that the science of today has a pronouncement to make, pe-
rhaps it ought to be that science should leave off to make
pronouncements: the river of knowledge has too often turned
back on itself,

Many would hold that, from the broad philosophical
standpoint , the outstanding achievement of twentieth-century
physics is not the theo r y of relativity with its-welding to-.
gether of space and time, or the theory of quanta with its ,
present apparent negation ofthe_law s of causation , or the _
dliiSotiSTof the atol^SiTrtteTSultant discovery that things
are not what they seem; it is the ■ geggraLre^gnj^onJhat
wea^jToJ^jretJji^contac^^ ^'

This attitude, which Bertrand Russell characterizes
as "humble and stammering", (3) is a f ar cry f rom the proud dog-
matism of the classical physicists whose ^f^J^^^
wards experimmtal science had been summed up in Desc ^es dx ctum
that thSe who wish to find the true road in s £"^*£ „£•
cupy themselves with any object about which tney
tilude e q ual to that found in the frustrations of o^ttaet^
and geometry. (4) If this new att ^ « °°£ ft ^ iscovery of
is surely right in suggesting that it ^prosen ^.^

far greater import than the '^^ a ^^ Lt ^ SSS S^L^
itself, \Por tt^tmmr^^Bj^mm^^^^--^^. of th f s
ter r^anTlSr^y X^^pTg^^Bu^ int ia not that
W attitude must be cliSly^og^ aeo. ^ * eri nental science
scientists have come to realize that n with abgo lute cer-

knows nothing tjia^j^jmiversal^ndneces^ so ience lfl such

titude, but mtTSTteFthTlStar^f experin

-239-

that it can never arrxve at certain knowledge. In other words,
the expression which Bail du Bois-Reymond made so famous must he
applied to the very essence of experimental science: " Ignorabimus" ,

This new attitude raises a crucial problem for those
who wish to establish the relevance of ancient epistemological
| schemes with modern saience. In fact, the majority of contemporary
v/ritors both Scholastic and non-Scholastic seem to hold that this
new attitude is incompatible with the epistemology of the ancient
Peripatetics, The Scholastics see in this incompatibility a proof
that the new attitude is false. The non-Scholastics see in it a
proof that the old conception was only a provisional stage in the
evolution towards modern thought. Both of these positions have
consequences of great import We believe that in the last analysis
the first is a denial of experimental science and the second a
denial _o f_ philosop hy.

Sir Arthur Eddington has crystallized the issue in
the following terms $

In view of the closer contact which now exists bet-
ween science and philosophy, I Y/ould like to raise one question
which effects our cooperation. A -feature of science is its
prog ressiv e approach^ truth. Is there anything corresponding
to~~this in~philosophy? Does philosophy recognize and give ap-
propriate status to that which is not pure truth but is on

the way to truth, . . , ,, „_„„„,,.! „

It is essential that philosophers should recognize
that in dealing with the scientific conception of the universe
they are dealing with a slowty evolving scheme. I do not mean
simply that they should use it with caution because of its
lack of finality; my.point.is that ojjehiouWjrogr^ is
not furnished in the same Itaes as SJE^E^^^^'^
The scientific aim is necessarily s»^™!F fc f ™" if
philosophic aim, and I am not willing to concede that it
a less worthy aim, (5)

to a vehicule of progress which is nuo Does
as a mansion of residence? In the se cond P^> s ^ proach to
the philosophy of science ™ 00 f^° ftte v£y essence o£ experi-
truth which for Eddington constitutes w ^ itg mt j aning? Ge-
neral science, and does it admit its jc ^ affirmative
nuJ^ieThomisticjhiL^^
answer to both of~~these questions . Ana .

-240-

the explanation of this answer must be sought for in the field •
of dialectics, • ' ■ •

In so far as the first question is concerned it must
be pointed out that Aristotle and St, Thomas in the most explicit
fashion "recognize and give appropriate status to that which is
not pure truth but is on the way to truth," And they do so not
merely by granting; this "vohicule of progress" an insignificant
place within the realm' of philosophy^ but b y admitt ing that it
m at »ake up the major portion of. every philosophical treatise
(even of that which constitutes the verysoul of all philosophy^ )" -
Instaphysics, At the end of the first lesson of his Comme ntary on
the Third Boo k of the M etaphysics Aquinas writes; "Dialecticam
\lisputationem posuit quasi partes principales huius scientiae," (6)

But, it is evidently in the second sense that Edding-
ton wishes his query to be 1 understood. And here we come upon some-
thing quite different from the case just considered,, Dialectics
as a vehicule of progress roust constitute the major portion of.
every philosophical treatise because the arrival at philosophical
truth usually entails a long journey for the hunan mind. Neverthe-
less in philosophy there is an arrival, there is a mansion of re-
sidence furnished on different lines from the vehicule of progress,
and the long journey is caused only by the limitations of the hu-
man intellect But in experimental science there is no arrival, ■
"iEOTeTTnoTianBion, of residence; one is committed to remain for-
ever in the vehicule of progress. And the(iiSsg)for_the endless
journey is not merely the l:h-jit ationg_of the human mind^butjbhe •
very nat ure of the object studied.^ )

We must try to see why this is so. And our first
concern will be to examSe the nature of this vehicule, of pro-
gress<,' . .'•. ' .■,' , '- ' . ■

' 2„ The Katur ^ "f Dialectics.^

■j.'™, ™ the posterior Analytics, ( 7 )
««. *. , . Inhi % C r e SrencfbeLen metaphysics, logic

St, Thomas brings out the diiiereno
and dialectics:

, „n. ratione dialectica est
Sciendum tarnen est quod alia lation

-241-

de conmunibus et logica et philosophia prima. Philosophia pri-
m eniia est de communibus, quia eius cosidoratio est circa
ipsas res oorarjunes, scilicet circa ens et partes et passiones
entia. Et quia circa omnia quae in rebus sunt habet nogotiari
ratio , |logioa_ajrfaera_est_ae_ operationibus ration is ; ) logica e~
tiara erTF"de his7"quae comauhia sunt omnibus, idest de inten-
tionibus rationis, quae_a a onnes res se habent . Non autem i-
te, quod logica sit de ipsis rebus communibus, sicut de subiec-
tie* Considerat enim logica, sicut subiecta, syllogismura, e_
nunciationem, praedlcatur.i, aut aliquid huiusmodi. Pars autem.
logicae, quae demons tratiira. eat, etsi circa comraunea intenti-
ones versetur docendo, tamen usus demons trativae sclentiae
non est in procedendo ex his communibus',) intentionibus ad a-
liquid ostendendum de rebus, quae sunt subiecta aliarura sci-
entiarura, Sed hoc dialectica f acit P quia ex communibus inten-
tionibus procedit arguendo dialecticus ad ea quae sunt ali a-
inara soien t iaruii iji sive sint propria sive sint coramunia, maxims
^taraen ad corxiunia, Sicut argumentatur quod odiura est in con-
cupiscibili, in qua est amor, ex hoc quod contraria sunt cir-
ca idem, Est ergo dialectica de communibus non solum quia per-
tractat intentiones communes rationis, quod est commune toti
logicae, sed e tiara quia circa commuhia rerura argumentatur.
Quaecuraque autem scientia argumentatur circa coramunia rerura
oportet quod arguraentatur circa principia coramunia, quia Ve-
ritas principiorura comauniun est manifesta ex cogmtione ter-
rainorun communium, ut entis et non entis, totius et partis,
et similiura.

The terra "dialectics" has come .to possess a number
of meanings, but its most fundamental meaning ^^^LSiof
all. others can be reduced is indicated in this ^'^f 1 ^
consists InCStooEEllcation^nn ensrationis to ^g|- f ^

is to say, i^isTgggJiP^lBf %^^L ^Sr words,
the modus intelligendi moves towards the Hoauajg|. ^- "' °^ ct3 j

\<iiir^tteri^^
' conclusions vrtiich regard^ rgality.

This point is brought out ^h^» ^at« ^
by St, Thomas when in his nnmientary on ■ ffi J 1 ™"^ _ u „| | — r )r-
Metaphyslcs he distinguisheTtetween the dialectician

losopher:"

i, -invi^em. Philosophus quidem a
Differunt autem ab ^vavaa, rn v £ rtutis oat con-
dialectico secundum potestatem. lima' ^ , ^ Rntlol> Ph iio S ophus

sideratio philosophi ^^^^^ditrd^rao^to^^T 35 ^
enii de praedict dgcgroaunibug Vg^^A^r r t c 5 gnoscitivus
51als-^sAabeYe-^iSSi^3i-Hii2^5^ s '

-242-

eorum per certitudmem. Mara certa cognitio sive scientia eat
leffeotus dem o ns trationis,) Dialectics autem circa omnia prae-
dicta proceait ex probabilibus ; unde non facit scientiara, sed
quamdam opinionem, Et hoc ideo est - quia ens es t duplex : | ens
scilicet rationia^_^ns_KktumejEns"autera rationis dicitur
proprie do illis intentionibus, quas ratio adinvenit in rebus
consideratis; sicut intentio generis, speciei et sirailium,
quae quideni non inveniuntur in rerura natura, sed considerati-
onera rationis consequuntur, Et huiusmodi, scilicet ens rati-
onis^ est proprie subiectum logicae. Huiusmodi autem intenti-
ones intelligibiles ? entibus naturae aequiparantur, eo quod
omnia entia naturae sub consideratione rationis cadunt, Et
ideo subiectum logicae ad omnia se extendit, de quibus ens
naturae praedicatur, Unde concludit quod subjectura logicae
aequiparatur subiecto philosophiae t , quod est ens naturae, Phi-
losophus igitur ex principiis ipsius\procedit ad probandum
ea quae sunt consideranda^circa huiusmodX coramunia accidentia
entisj )Dialecticus autem proceditCadJeg^gnsjjgranda ex inten-
tionibus rationis, fcuae sun t extranea (a_rajaira rerura^ Et ideo
dicitur, quod dialectica esTtentativa, quia tentare proprium
est (ex principiis extraneis procederep (8)

It is" clear then, that dialectic involves a process
| which begins with a construct and. hence ab extrinseois . That is
wh y there is Ca movemini 5 in .dialectics - - dialectica est tentative;
the mind attempts to pass from the extrinsic to the intrinsic,
[tron logical construction to reality. But as is evident from the two
texts of St. Thomas just cited, there are more than one kind of
construct from which the mind coy attempt to reach reality. A close
reading of these texts and of other passages in which Aristotle,
Saint Thomas, and their medieval comn-mtators ai ^s the mture
of dialectics reveals that Ith^j^cognim- t.hree distinct types ,
^dialectical reasoning^) The first type ™f «^,

that iTtTsaF, terSl^hlcTriiinlfy 8e ^„^„^woa|ln

Pie of this is'found in the seventh book ox ^jggggl^ .

^*i2lL^LJaetaj2h^sician^ , ^

5rogtH^^3r^dlcated(^ndwhich^

The ^TSF5e4raElS^^ takes

eroployed are not proper to the so lence in . ^ ^ ^ mB

place, (but are common to sey ergL^SS2S|^|_. not form ed_by_tho

<$

z

ff^Qfwhich; the priaiciples^re ASSI^S^-J ^iS~ES^J^m^
njnd, IbutJh^priaicjjDi^itoemsel^^^

§iffiS3mMj : ss^metb±ng = Jh^ genus,

on'J^TWThe-IBgicTan that angel and ," 1 ^ 1 „ enua ,/-they canhaye
for when things do not sliarejai_a_mturax_g _,^ —

-243-

I nnhr_ a logical genus in cornm onT) Ah example of this type of dialec-
tical reasoning is suggested by Saint Thorns in the passage from
the Pos terior Analytics cited above : from the coirmon principle
that contraries are in the sane category, one concludes that hatred
]T<vriv iins to the concupiscible ap jetitg^ be cause it is the contrary

\ofjSv<l]) (10) The third type of dialectics consists in reasoning 3

from principles which are only probable but which are accepted
as if they were certain. It might not be immediately apparent why
pHncTDjnSsoTTTJhis Kind can be considered logical, and how reasoning
based on them can realize the property of dialectics insisted upon
by Saint Thomas, namely, that it be in intentionibus, ex extraneis .
The answer is this: syllogistic form necessarily require^ univer-
sality's and when there is mere universality ut nmic , that is to
say a universality that is not seen in things , (but is supplie d
tentatively by the mind p there is obviously a formation b y the
nindo (11 )

Whenever conclusions- are drawn from any of these
three types of principles they are purely dialectical. For conclu-
sions must be considered formally jjLJ he light of the principles
by which they are illuminated. TM £ J : s_trug_eyen when only one
of the premises is dialectical ,1^^ Way sorne what analogous to
thTc^ of reasoninFwhichi; foully geological even when only
one of the premises is a datura of faith and the other ^^ .
sical). And in , all re^ soninj^jMsJcind .l^^^g^fl
is alwJ3telab3MOOSIII)TpFl^/hy, if, a _we sh all try ,

tT^ri^rlSSntaricIe^cTisC^^ ^|

necessary to conclude that the • hg^gBS^li^^^r- •

tain matter pertains to a habitus f ^rjtaanth e scienc
.atter, it is obviously necessary to have -»* «™ otloB . It
ratter concerned in order to be able w i ub ^ the faot that>

is also worth while here calling the avs g Qf aialeotios

speaking formally, the abstraction used in ^ lfl reduc tiv e

is that of logic (i^'^J^^^^^^^^a^Jhej!^^

^^^^^^^^^^^

■ How since all of those thre ^^^
reasoning are a functioning of a "^ c 6^cirSa7TheylSS^from

scientific habitus pro per j£jgS3S_-— —^r^tif io reasoning.
thls-^oinT^^wbedrstinguished from a ^ j^f ied

Yet from another point of view the fust £ £P soi ^ nt

with scientific reasoning. ^ *^ lons tration, and it is evident
reasoning is that it is a strici. aeIT1 onstration.
that only the third type is lacicms _ .

-244-

„ „-n A ^^^y of Ringing out this point is by saying

to t while all dialectics consists in an attempt to get at reality
from a logical construct that is extrinsic to it, this construct •
rJa y be extrinsic in two distinct ways. It may first of all he ex-
trinsic from the point of view of truth, and then the reasoning
is r.ierely probable and does not give strict scientific certitude.
Secondly, it may be extrinsic from the point of view of what is
specifically proper to the reality concerned, and then the reasoning
my give strict scientific certitude. Since a failure to grasp
this important distinction may easily give rise to confusion about
the way in which dialectics is employed in the study of nature,
it is important to try to make it as explicit as possible. And .
we can best achieve this by considering the question in terms of
definitions „

Definitions may be considered in two ways: either
(rarely as definitions^ or as principles of reasoning . \ Taken by
themselves , definitions^ are not pr oposition s; they "do not involve
predication.J "Hence they cannot be either true or false, but only
good or bad. Now definitions may be either intrinsic or extrinsic.
They are intrinsic (or proper) when ihey define things in terms
of T/hat constitutes them intrinsicall y; they are extrinsic (.ot _
a^E55aa3JCwhen they de fine things in terms of something extrin -
sic to theiOArTag^xample of this distinction is found in the
teo definitions of substance. The proper definition of substance
is: that whose nature it is to exist in itself and not in another
v as in a subject. The dialectical definition is in terms of sose-
/( thing extrinsic to substance, namely predicat ion: substance is
that of which everything is predicated and which is predicated
of nothing. In this distinction we. have the explanation ot the
contrast which Aristotle draws between the physician and the dia-
lectician at the beginning of the De Anima:

nif ferenter autera definiet physicus et dialectics
unumquod^fSo^rut iram quid est ^c ,gde -«e -
titum reeontristationis, -J.^^f^Horum 'autem alius
fervorem sanguinis aut call ^-° X ™* °peciem et rationem. Ra-
quidem assignat n^xteriara, alius verospeci ^

tio quidem enim heac species rei. Necesse
in materia huiusmodi, (12/

essentially to mobile beiugs, all ffl^S-— F l h nhiugs of nature
in terms of it. That is why any definition & tfl to def5£ e

v/hich does not include sensible mattex, w . x _._..^ nnfl nrotier*

which does not include sensible mattex, » inslc and proper}

few in terms of the f qr nalong } <3^^^^=^5^^S^3V^ry
since it does not touch cosmic reality

-245-

being o It can be nothing but extrinsic and dialectical, for the
foi-ns of natural things can exist independently of sensible matter
only in the mind; the very quod quid est of these forms demands
nattero

Definitions however may not only be considered in
themselves, but also in relation to the thing defined i In this
senss they arel virtual propositions) and can becone principles of

Sens J uii^j t^.j.^ i .^»»^ ^x^ |J ^aj.uj.u i j 0fuIJ u. Kio-LL ut;uumj }jjl xixuj.jjq.tj a Ui

syl logisms , as St„ Thomas points out in the Posteriora Analytics :
Principium autein syllogismi dici potest non solum propositio, sed
etiam definiiio,, Vel potest dici quod licet definitio in se nofo
sit propositio in actu, est tanen in virtute propositio quia co~
gnita definitione, apparet def initionen de subjeoto vere praedi-
cariV^) (I 3 ) Considered in this wa y, (definitions may be either
scientific or diale ctical, ) They are scientific If" the connection
with the thing defined is necessar y, in other words if they are .
vjrtiial -pro positions that are true Q They are dialectical if the
connection is not necessary, in other words if they are virtual
propositions that are ' kb rely probable. (14) It is clear, Then,
that definitions can be truly scientific and at the same time dia-
lectical in the first sense of the term. It is likewise clear that
they can be truly physical and natural, and at the same time
dialectical in the second sense. Hence it is extremely ruapor tent
to keep distinct these too ways in which the term "dialectics
is employed by Aristotle and St. Thomas in relation to natural
V doctrine a ■ •

And now, having made these necessary distinct ions
between the various meanings of dialectics, ™ ?*?}*% *££
in what sense experimental science ^^g^^^Sdent
Prom all that was said in the last . G |fP ter " „" -, „ cienoe is
that the most fundamental way in which ^^^To St
dialectical consist in_thi§ ^SL^^^^^r^Ti^^^
atjtejruth^bailnature ^yjrea^of^^^^^-^^^^
^Me~?S^5S jColSeo^ntly in this Chapt er we sha^ ^ ^ ^
upon the meaning of dialectics in which « ns ™ true denons _

strictly scientific, that is to say, to wnas ^

tration, and leave the consideration of °*££ g^ sense , dia-
sics is dialectical to ^ture contexts • TaKen ^^ ^

lectics is defined by Aristotle at the opening ntari de

of the Topics as : " mathodus per quam .P° 3 ^ . Imputations
onni pro^STto problemate ex probabiiiDus ^-^ notion thttt
sustinentes nihil dicamus repugnans. ,^ oua i y that of proba-
msfc be analysed in this definition. is obvio
bility. „. .,

two kinds of probability: real and dialec

There are

=■246"

ticalo The former belongs to objective reality independently of
knowledge, and it arises from the indeterminism of nature. The
existence of chance in nature means that there are some future
events which are not completely predetermined in their causes,
These events are not necessary, and hence are at b<igt only proba-
ble. Only conjectural knowledge can be had of them, (15) Even
the most perfect created intelligence is unable to foresee them
with certitude. Of course. a created intelligence can judge with
certitude of the present probability Of the future, and in this
sense real probability can be the foundation of a true proposition.
But the truth of . the future event does not follow from the truth
of the present probability. Dialectical probability is not found,
as real probability is>upon an indeteimination inherent in things,
but upon an in deterraination proper to the intellect which must
jjorcfrom^o^ncjXLto^ct, And it is with this type "of probability
that we are concerned in the dialectics of experimental science,

Aristotle defines dialectical probability in the
following terms: Probabilia autem sunt, quae videntur omnibus vel
plerisque vel sapientibus, atque his vol omnibus vel plerisque
vel maxime notis et Claris," (16) The important word in this
definition is "videntur". Probability must be defined in terns
of appearances. As Aristotle points out in the fourth book of the
Topics, (17) I the probable is not a speoieg^f being^ Itmust

thTlikiHe-ss of being - - th^rwhichgppearsto_bg Just as being

, gives rise to tru«i **££*>£ S^lfS^S ^totle
^rice to the likeness of truth. That is W •"» fl ~ "tr„th" ( 1Q )
"defines probability as that which is ^«to he truth. JM
grobabili^ means verisxnalitu|e In otter words, JU . g the

is the adequation of the mind with what is, so y exp iains

adequation of the mind with what appears to be Ana^ ^^^
,why, as Aristotle suggests ^V J^lflr^ruth and take delight
impetus which moves the mind to seeic f\ tg likeness and take

in it, likewise moves the mind to seek at letely sa tis-

delight in it, even though this delight is ^ ^^ ^.^
Uying, In his commentary on the iogioa ^

. j, 4.1 = = Maleoticara distingui a Philo-
Respondet Aristotle sDialeotic^^^ ^^ omnes

sophia per hoc, quod licet aiaie hilosophus sc ientif icus ,

res et circa omnia problemata, sic * llogophug e nim non
adhuc different in modo ° on3 f^ a ". ' h ounl a secundum verita-
est contentus apparentia, sed exam iag causas reruns

tern, ac quaerit, propria pri ncipm e I ^ ^antia veri
dialectics e converse contentus est qu ^^ ^

et procedit ex comnunibus et probabi
opinionera, ( 20)

-247-

A first reading of Aristotle's definition cited a-
bove may make one v/onder why in it he gives so much attention to
the various kinds of knowers. But from what has just been said
i t should be clear that probability must necessar ily be defined
in jceras of the knower a nd noTT^rTerms of the thin g known ,~Tn
oTher words, it is essentially related to appearances and" hence
to the apprehension of the knower and not to objective reality, (21)

The judgement which is the subject of the qualifi-
cation "probable" is known as opinion . Just as a truly scientific)
judgement is necessarily true, so an opiniative judgement is ne-
oessarily probable. Opinion is opposed to certitude as indeteroi-
nation to determination. And the indeterraination which is proper •
to opinion is in the mind arid not in things, (22) In other words,
the object of opinion considered formally as such exists only in
the apprehension, (23) By the inde termination found in opinion
the mind is opposed to reality as logical being is opposed to real
being. In other words the mind interposes itself so to speak bet-
ween itself and reality. And the attempt to arrive at reality from
this state of indetermination will be a dialectical process.

There was profound wisdom in the recognition by the
analon-k Grooks of the fact that at least much of the atudy of na-
ture was merely doxa and not episteme in the strict sense of the
word. For a study"which can never rise above the appearances pre-
sented by experience except by having recourse to hypotheses which
are nearer more than probable and whose, sole purpose is to save
the phenomena", can never rise above the state of opinion* °f"
never become a science in the strict sense of the word. In this
connection St Thomas writes:

Ita et in processu rationis, qui non est cum .om-

nimoda cer^Sin^ gralus "^^^S^gua-
magis et minus ad perfectam ^titudinam ac ced it u^
modi enim processum, quandoque ^ide^etsi non f
fit teran fides vel opinio propter P^ oba ^ iter aeclinat
num, ex quibus proceditur: quia ra ^ foraidin e alterius,
in unam partem contradict ionis, ^ ti Nam sy iiogismus

et ad hoc ordinatur Toxica, siye 12£±|^- ^ Arlst oteles in
dialecticus ex probabilibus est de quo agit
libro Topicorum , (24)

But before turning to °°^ ^ ^ JerSSta?
dialectics of prebable reasoning is emp ^ accurate ^ lts
science, we must try to de taB» J t f ron wha t has al-

precise nature. It should ^ e .^^ what the schoolmen terned

ready been saic^hat it pertains

-248-

loRica_utens, as opposed to logiog docens which merely gives the
Jules for the application of scientific principles that are already
given and which does not enter in the very construction of these
principles o But the term logioa utens is employed in a variety
of ways, and John of St. Thomas has brought out lvith great clarity
the sense in which it must he understood here:

Tertius usus Logicae est ipsi specialissimus, qua-
tenus praebet usum in aliis scientiis seu materiis probabili-
ter disputandi sine hoc, quod procedatur demonstrative et re-
solutive usque ad prima principia, Et tunc proprie dicitur
Logica utens, ut distinguitur a demonstrante et docente, eo
quod demonstrans non praecise utitur discursu sistendo in eo
sed pervenit resolvendo usque ad prima principia, quae discur-
su non probantur, sed sunt terminus discursus. Utens suteia
discursu, sed non demonstrans, ita utitur et sis tit in discur-
so, quod non pervenit ad terminum discursus, qui est resplu-
tio usque ad prima principia, et hoc pertinet ad procesun dis-
putativum seu tentativum, quando inquirendo, non autem resol-
vendo proceditur. Et ita vocatur probabilis processus, quia
non cum certitudine ultiraae resolutionis usque ad principia
fit. Hie est actus Logicae utentis, et sic explicat ilium D,.
Thomas opusc 70, q,6„ art, 1 dicens

Logicam utentem esse, quae utitur discursu, sed non
v termino discursus, qui terrainatur in principia per se nota,
A ubi oassat usus rationis discurentis ... , mTV!in+lir nir „

Logica utens tertio modo accepta solum versqeur oir
ca partem topicam et sophisticam, id est processu non resolu-
tivl sed probabili seu probative et gsput atxvo^ Jt^talis
usus fiat in aliis scientiis ex pnncipiis ^ ^ -e-tinet

ad Logicam solum directive, si auo i lura ai _

ipsius logicae talis ^^f^^^SCsecundarius
rectiva, sed elicitive erit a Logica, quasi

et imperfectus . . o , ,,(.,, Thomas opusc, 70 cit (

Expresses autem ^g^^Jvroaenas
q, 6, art, 1., um docet, 'quod ££& eaenao , ultimas

rationalis ex termino, m quod si *™itio perducere debet,
autem terminus, ad quern rationis ^ resolven do iudicamus;
est intellects P^P 1 ^ 1 ^ d £ionstratio. Quando autem
quod quidem quando fit, dic ^ , ' termi num non X :>reducit,
inquisitio rationis usque ad uxto sc ilicet quarenti

raed sistitur in ipsa inquiaxti one, q 1±B prooessus dis-

adhuc manet via ad utruifll^t, sic ^ q procedi poteg t

tinguitur contra demons trativum.Lt p^a^ilibus pare-

rationaLiliter in qualibet s<^ntia, ° t hio l ea t alius modus,
tur via ad necessarias °? nol ^^i!s, non ut est docens, sed
qoud Logica utitur in alias scientiis,

ut utens „ Sic D, Thomas

-249-

• t ** l ^ C2SXt P raeben ao principia propria tali
dxscursui et disputation!, ellcitive totum illun discursum
produced Logica, quia non solo praebet malum disputondi, sed
etian materials seu principia, (25)

In order to understand that passage correctly it
is necessary to recall the distinction -made above between the two
v/ays in which the extrinsic character of dialectics can he under-
stood, V/hen John of St. Thomas suggests that the use of dialectics
which he terns directivus does not provide the principles for the
process of reasoning, but merely the modus disputondi he obviously
has in mind the meaning of extrinsic in which it signifies some-
thing exterior to the matter that is specifically proper to the
science involved, as in the case of the definition of anger in
terms of fom alone, or of substance in terms of predication.
For if axtrinsic were understood in the other sense, then even
the dialectics of probable reasoning must be said to provide the
principle Sp

In any case, it is in the use of logic which John
Of St„ Thomas calls directivus that we are now particularly inte-
rested. Later we shall have occasion to see that mathematical phy-
sics also involves a use of logic that is simUar to what he terms
elicitivus.l in so ^^ ^^n^mm^S-m^^^^^^^^
phenonena j-nj rerras of l o gical construc ts^)

' It is clear that a study which remains within the
dialectical discourse just described without ever being able to

emerge from it can never be, a .f ^ViSht sScelt never achieves
word. It is not a science m its own right, ^"JS?"^ soienc Q
strict demonstration. Nor can it be_consideredg_^g|C|l_sci^
since the logic involved is not loSiE^£2H ^Hlfafhysics is
lowing passage from St. Thorns' CoB mentary on the l femgny__.

relevant here:

L icet autem dioatur, quod P«^ ? ^T '
"non autem dialectica et sophistica, nun - Dialeotica enta p0 -
"quin dialectica et sophistica smt scion ^ gecundun quod est
"test considerari secundum quod, esi; uu , ^ cons iderationem de
"utens. Secundum quidem(^odes^docen^^ ^ procedi possi t

"istis intentionibus, initituens noaxm *. aMliter stendendasj
"ad conclusiones in singulis Bowntiw v ^ BO ientia.lUtens

"et hoc demonstrative facit, et se °™; itur ad CO ncludendum_ali-
"vero)est secundum quod modoadiuncta u t a nodo so ientiae,

"quid probabiliter in si^ 13 ' ^,^i oa ; l^a P*™* est &°° e ™
"Et similiter dicendum est de Bophxstxoa^ q^.^ ^^ arcy endi
"tradit per necessarias et demonstrates

-250-

apperentero Secundum vero quod eat uteris deficit a processus
verae argumentations. Sed in parte logicae quae dicitur de-
monstraoiva, solum doctrina pertinefc ad logicara, usus vero
ad philosophiam et ad alias particulars scientiae quae sunt
de rebus naturae,, Et hoc ideo, quia usus demons trativae con-
sistit in utendo principiis rerura, de quibiis ,fit demonstratio,
quae ad scientias regales pertinet, non utendo intentionibus
logicis, Et sic apparot, quod quaedam partes logicae habent
ipsam scientiam et doctrinara et usum, sicut dialectica tenta-
tiva et sophistica; quaedam autem doctrinam et non usuum, si-
cut demons trativa, (26)

Prom all that has been said thus far it follows that
the meaning'v/hich the term "knowledge" has for us when applied
, to experimental science coincides exactly with the sense in which
\it is understood by Sir Arthur Eddington:

Some writers restrict the term 'Knowledge' to things
of which we are quite certain; others recognisa knowledge of
varying degrees of uncertainty. This is. one of the cormion am-
biguities of speech as to which no one is entitled to dictate,
and an author can only state which usage he has himself chosen
to follow* If 'to know' means 'to be quite certain of, the-
term is of little use to those who wish to be undograatic, I
therefore prefer the broader meaning; and my own usage vail
recognize uncert ain knowledg e, (27)

Enough has been said to show that if we wish to dis-
cover the principles which reveal the true nature of experimental
science it is to the Topics especially that we must turn. And_it

of-logic has been aESsTexcHilOTTgted to ™ __ on -^r
bsripT^tios And we believe thrf there ^ tholr negleot of
ween the scholastics' neglect of diale cues disrega rd

gasana pt towards oonoretto njn^iejtuglgn^ug.. in *

for tte taportance of dialectics goes baclc as iar as
Thomas hiraself :

ad materia* logicam seu ad P 08 ^"^^ ord inatur. (28)
xiRB in demonstrations, ad quara _P- * topioan quae agit

Quae ento pertinent ad J?™\ bl 4 Elencho rum qui
de probabilibus, et quae P ^™ in pra esenti, quia non
aguat de parte sophistica, omitt unwr ^ et lde0 so lum

agunt- de certa et perfecta ^^T^tici ad Aristotle. (20
libri Priorura et Posteriori vocantur amoy

-251-

At thetime that these lines were written the modern
development of experimental science was already underway. Without
realizing it, men like Galileo had already discovered in dialectics
a potent intellectual instrunsnt for the advancenent of the study
of nature in the direction of concretion,, It remains for us- to see
just how this dialectical instrument is employed by experimental
* science „

3,, Dialectics and Experimental Science,

As we have already explained, the propositions that
are proper to experimental science are devoid of intrinsic and ob-
jective universality. But because the intellect cannot remain im-
prisoned in singularity, the scientist is lead to confer universa-
lity upon them ab extrinseco . In order to get at the reason for
the regularity appearing in nature, the scientist is lead to act
as if these (-proportions") were universal. In so doing he is applying
ttelrtocipliTaidto^T WSrliSoile-ln the Topics: "4f ecumque
in omnibus aut in plurims apparent, sumenda-iSHTqua ^ P^ncipia
et probabiles theses," (50 In this way he, uses the P™le
dicide omni to the sense in which it is employed in the Pripra
where it is not restricted to science in the strict sense of the
tern, but is ooimon to both science and dialectics*

Ad quod sciendum est quod ^^-S'^S^i-

■undtur, ad'dit supra ^Sk^^J^iT^SSS)
orum. Nam in libro Priqrum accipitur £ci a , - ^^— r

prout utitur eo et aj jggoJa °M °*. g^ ^^icatoB insit
pTSs ponitur to ^^^^^^^ubiecto! Hoc autem contin-
cuillbet eorum quae .contira ntursuo ^ di ci de ontt il dialec-
git vel ^S^^^^XS^^^^TS^o solum
ticuaj vel simpliciter evsa"
utitur demonstrator, (31;

/ we have already ^e^^^^^^L

'which are prosed by the sadist ^^.^^^^ beyond

are purely i f5nctional. Their position ^ i^^f^esearch. In o-

But as we explained in

-252-

versalized propositions do not satisfy the mind, for they do not
"save the phenomena", That_isJo^ay,( j h ey merely state the connec-
tion between subject and predi cate without giving the reason for
iU) Consequently, the mind is lead to~feaoh out for the propier "
quid tby constructing hypotheseiVhioh vri.ll give a. provisfonaTex-
planation^of the experinental^aropositions . In other words, purely
experimental propositions contain an implicit problem, and in order
to solve this problem we transform propositions into questions
which anticipate experience. In connection with this use of hypo-
thesis it is worth while pointing out, lest confusion arise, that
the term "hypothesis" ( suppositio ) usually meant for Aristotle and
St Thomas something quite different, from the sense in which it
is now understood,, It did not mean something that was lacking in
certainty, and that as a consequence could not be demonstrated.
On the contrary, it meant something that was absolutely certain,
but that was accepted without demonstration either because of its
self -evidence or because of its demonstration in another science,
or at least because of its acceptance by the adversary or the dis-
ciple with whom he who used it had to deal, (32) It is clear,
however, from the passages cited in the last Chapter from the De
Coelo etc, with regard to the planetary systems that the. ancients
alscTrecognized the use of hypothesis in the modern sense of the
terra) Taken in this sense it means, as we have already suggested,
proposition or a group of prrreositionsfposed by the mind)in_order
to save sensible phenomena l by offering a provisional expEgaggn
ol^thT^oFtehinT-e^rke^^

go es beyond probability,- ^ ^Z^JSBSSj^^J^^^^
guess" - - an antioipated-lKanfersrS problem. ^"g* ^St-
the-product of the creative imgination and of sci ^" °^* ruot
Ion. From hypotheses of this kind.posited as premses, g§L^*

and, thus explain it. It is clear J^* *^ h X nlind attempts to
dialectical: they are constructions by whicn Tine-
arrive at the nature of reality,

, -l ,„ut -i<* similar to the truth

The scientist accepts wha^ is^in^ ^ ^^

as if it were, the truth and uses it as a p ^ ^. ^ ^^ whl(jh

In doing so he is following the natural «P fo the trut h when

upon whatis^simiia^ ^ ^tiplyjdtb

lerative ,

as we saw above must seize upon whas is ms tjr^Piy_4ih-

Ut cannot have the ^^J^^^^Md^^^^^^^SS^

ou t end his conjecture s^
ftmctioTr.l, insl jSi jerriirtX val"°

-c^- tigon_^ghihg eJ.

Les theories n-ont pas pog^f ^ellfde ct
veritable nature des choses; . • ' lenoe nous fait connai-

ordonner les lois physiques que 1 JfP r< 5 e llement: o'est

tre na „ Peu nous importe que l'ethei exi

-253-

IV affaire des metaphysiciens ; l'essentiel pour nous, c'est que
tout se passefoom me s'il existait T)et que cette hypothese est
\ commode pour 1' explication des pfienomenes," (33)

Ever remaining within the realm of the conjecturable, '
the experimental scientist must carry on a methodical interrogation
of nature which never has any final issue. The(art)which guides
this methodical interrogation is dialectics ,,

The mind is therefore free in the construction of
these hypotheses. We have already quoted several passages from Ein-
stein which show that the premises of experimental science are free
inventions, creations. This freedom is not absolute, to he sure,
for the dialectics of experimental science must always he kept in
tow, so to speak, by constant recourse to experience. Nevertheless
there is liberty and creativity in this dialectics. "The scientist
is free to choose between contrary and contradictory hypotheses
the one which seems to serve his purpose best at the moment. He
is, for example, free to choose between the opposing corpuscular
and wave theories of light the hypothesis which gives him the greater
help in achieving his task. All this recalls what St. Thorns has
to say about the dialectician:

Secundo, ibij Diabetica etc., ponit differentia*
inter dialS^'propLitionem et demons trativamdx c ens quod

cum propositio accipiat alteram V^™™™^lTe^f^
tica indifferenter accipit quaecumque earum. fobet enm vgn

proposxtxo accxpxt alteram part demon3 trandunu Unde e -

habet demonstrator vxam, msx ad v CO ntradictionis ,

tiam-semper proponendo _aooi P it yeram ■& ± demonstrat

Propter hoc etiam non interrogat, sea B umw,.«i
quasi notum. (34)

Because these ^«^^^^£Sr
ble, experimental science must never ., thls characteristic

its conclusions but its very P^ 3 *"™?',- his C _pmnentarv_pnjhe

of dialectics, as St. Thomas points out m

Posterior Analytics: .

lectica enlm norLBoJAaii^i^^^TeriHtMrogat, sea -
de vrer^BaiBrKr^^r^f^^l principia probata: sed
V surait quasi per se nota, vex p

-254-

intcrrogat tantum de conclusion. Sed cum earn demonstraverit,
utitur ea, ut propositions, ad aliara conclusionem demonstran-
dam (.35;

This brings out the difference in the way dialectics
is employed by philosophy and by experimental science. In philoso-
phy it is used roerely( as an instrument to search out principlij )
which, when found , [ jjipose themselves upon the mind by their certi -
tude o) In experimental science, dialectics is employed not merely
in the search for principles but in the very choice and positin g
of the principle s ,o (36) This ties up with what we. saw in Chapter
IV about the difference between the Thomistic and the Kantian mean-
V ing of a priori.

In all this we have the reason why experimental science

is essentially variable and transitory a vehicule of progress

ani not a mansion of residence. And in this connection De Broglie
writes:

II ne faut pas's' etonner si souvent la decouverte
d'un ordre nouveau de phenomenes vient renwerser comme un cha-
teau de cartes nos plus belles theories, car la richesse de
la nature depasse tou jours nos imaginations, Les savants sont
bien feB&s de vouloir reconstruire par la pensee des portions
de l'univers: la grande merveille e'est qu'ils y ont parfois
reussi, (37)

As Dotterer has remrted, "the firs t principles of
the sciences roust be regarded asjeost^^i^aji^S^^-
in which all science isJ^und^TonJ^ithg ^^™^^ nte i
cTa^Bel^dl^o^^

science that he made doubt the grea ^™tal , ? t S pSlosophical
great experimental principle, therefore, is ao , *

doubt which leaves the mind its ^f^^^SstStor in phy-
which come the most valuable qualities m an aiwes*igax

siology and in med icine," (39)

„ ,fl,nnres bv a gradual rationa-
• Experimental science ad yances by^ continua i reor-
lization of irrational elements; but mis ^ emplojed an a

ganization of its rational system. J*™ 1 in ope n to revision,

the corpus of doctrine achieved must ever ^ aeyelop ia by a con-
The only way that experimental science passing through

tinual process of subst itution. I* can gl0W
crises and revolutions. (40)

i«l« up tta structure- of experMent.l scion

-255-

wlia-t St Albert the Great calls " interrogatio consensus in probabile" „

Sed dialectlca propositio est interrogatio consensus
in probabile, nee consensus requireretur si probari' non debe-
ret: manifeste autem falsum probari non potest, et manifeste
verum non indiget probari, sed ad altorius alicuius assumitur
probationenio

In diffiniendo ergo propositionera dialecticam secun-
dum potissimum suum statum dicimus, quod propositio dialectica
est interrogatio probabilis, ita quod probabilis sit genitivi
casus j hoc est, interrogatio de probabili, quod est materia
propositionis dialecticae. In probabili enira (quia ponitur in
-judicio eius cui proponitur, utrura sic videatur vel non) opor-
tet quaerere respondentis judicium et consensum, antequam pro-
cedere possit opponens. Sic ergo dialectica propositio inter-
rogatio est probabilis. Et.hac ratione etiara Boetius m diffi-
nitione syllogism! dicit, quod est oratio in qua quibusdam _
positis et concessis, respiciens. ad propositionera syllogism!
dialectic!. Cuius causa est, quod probabile de se non habeo
sufficientem causara consequentiae vel inferentiae, et causam
inferentiae sufficientem accepit a oonoessxone reaponden xs.
Heac igitur est tota diffinitio propositionis dialecticae. (41)

Sir Jaras Jeans' has brought out the dialectical cha-
racter of the scientist's interrogation of nature:

• ,,+ i-?Vp pverv other, amounts an ef-
Such an «^r^ n i' IgL^SL question can never
feet to asking a question of nature .f£* A ^ ,

be - - 'Is hypotheses A true? but Jf™ phenomeri^which

Nature may answerour ^^f^ showing us a phVno-
is inconsistent with °^r_ hypothesis or^ * ^^ she ^
msimwhioh is not inconsistent ™" it;l one phenomenon is

never show us a V^omnonj^oYi^^.^^^^^^^
enoughJo_disjrowjLjOT2«^ n ^4ricTSnTii^5iniewr^I5im
HciVjr^vl3g)l ; S r ^i^on : f flirect facts of observation,

tol^o^anything for °°!^"£j^ L bui lding up hypotheses,
Beyond this, , he can only P"***** * than its predecessor, but
each of which covers more phenomena tna ^^ ^^ in
each of which may have to give V m ^ rep i ac i ng a hypo-

due course. Strictly speak ing, the ^ (4Jj)

thesis by a' data to certainty never

t (45) (3heart_oI_in-
As von Uexkull *» P^J^ oJ:^ISU^T
tgr rogation of natu re? ^"-5^-gg^Sen o ugh has been said to
t rfpertaent a ls cISntist We feel tha ^ the "logica inter

show that this art is substantially t » tiva „ of the ancients,

rogativa", '.entativa" , ^is tg ^ ^ ^ as exM ^
(\,e, the dialectics_of_j££_^L-----' y

-256-

significant to note the similarity between the following passage
of von Uexkull and the lines quoted earlier in this Chapter from
St, Thomas' Commentary on the Fourth hook of the Metaphysics:
"aialecticus autei.i procedit ad ea consideranda ex intentionibus
rationis, quae sunt extranea a natura rerum, Et ideo dicitur, quod
dialectica est tentativa, quia tentare proprium est ex prinoipiis
extraneis procedere„"

In the present book 1 have endeavoured to frame the
theoretical considerations concerning biology, in such a way
that there can be no longer any doubt that, in their very na -
ture , biological doctrines always remain unsolved problems*

In nature everything is certain ; (^ in science everythin g
is problematical^ ) Science can fulfil fEs purpose only if it
be built up like a scaffolding against the wall of a house.
Its purpose is to. insure the workman a firm support everywhere , -
so that he may get to any point without losing a general survey
of the whole, Accordingly > it is of the first importance that
the structure of the scaffolding be built in such a way as to
afford this comprehensive viewj and it must never be forgotten
that the scaffoldin g does not itself pertain to Nature, but
is always some thing(extraneous~p (45)

The comparison of nature with a scaffolding, which
had already been employed by Goethe, is, as we su gge at e ^ ^ ™e
Last chapter, very exact. It brings out the fact that ^mental
science is essentially a logical construction which ^ reuses
in an attempt to get at reality. As we shall point ™**T^£

sign-J^st as a scaffoTding can be made \f P™ closer
to-the form of the house and thus be brought to take on grao **W
a greater likeness to the house, 30 experimental science oan^
ever closer and closer to »?^\ & f™^?JS,el become the
likeness to nature, But oust as a. s^ + ._4.i on s0 aoienoe

house and mnst ever remain on extrins ic .^^f^ In fact ,
must ever remain an extrinsic constructor ^ nature the more

as we suggested in Chapter IV ^T^^f^SF^^M^^^"
extrMsJ £ J : t^comes O^ssuse^ofjh^^

E5cH5rcoHs~t55£qy r inc reases^ As we s, * the dialectical
XTTlheTe is a great deal of ^^%Xtix»ol movement of a
approach of science to nature _an a.™ - increasing sides to-

regularly inscribed polygon with ^tan^ y ^ ^ ^^ of the p0 _
J7arts_akrcle7) Just as the rnultipJ-ic. ^ ^ <rf

tyioT^akeTTt more like a °if cle ' c ^ le ( whic h has only@> "si-
ft polygon and hence. more unlike a circi ^^^esjiat^nce

de") so th e moveme j^of^gj^SgAJg^g^a

Pore objective a'H d^re_subjecbiv^.»

-257-

(T) . . A n ^*? r ° f ob J ection s may suggest themselves in re-

W gard to this identification of experimental science with dialactics.
In the first place, one may be tempted to ask: if experimental science
is dialectics, in what sense can it be considered as a part of na-
tural doctrine? The answer is: experimental science is natural doc-
trine principally 'b ecause of the limit (to wards which its d ialecti-
cal movement is orientated^)"!", e. natureV ~ln other words, it is"~na-
$ural doctrine not so much because of what it has achieved at any •
given stage of its development as because of what it has at all
^ times attempted to achieve. To get back to the example used above
~ a the circle is the limit of the polygon only in so far as the
Matter is in a state of movement through the successive multipli-
cation of its sides o If this movement should stop at any one given
polygon,, no matter how far advanced it may be in the series, the
circle can no longer be considered as the limit. Similarly, natu-
ral doctrine, in so far as it is built upon hypothesis, must ever
remain in a state of movement towards its limit which is nature,
that is to say, the absolute world condition. Ho_g iven stage of
the development o fexperi ros nt al sci en ce can be considered natural
foctrine"~in an ab ^blute^ jgnse. To^o consider it would be to iden-
tifTT~subje~ctive constr^cTwith objective nature - - which would
be comparable to identifying a polygon with a circle. Nevertheless,
just as a given polygon that is far advanced m the series which
fends towards the circle is already in some way a revelation of
I the nature of the circCe, so any given stage of «« °«f J»*^.
of experimental science is to some *f s ^? * r f e ^ - a of ob^ec
tive mture. And just as a polygon of a ^™ ?^f s " g^ows
to the circle than a polygon of ^^f a '<k^^^^
Uure_b jM erJ*^^ For

\lrn to examine these notions f fu ^ d ^ has just been said
the moment, it is interesting to compare what fcas ju
with the following passage from von Uexkull.

A nB n may have assimilated the -^f^ ^oy
ral science in the form of doct ™°^™/ of logic . but he
them in speculation, according ^^jature - - orj^any
still knows nothing whatsoever .^SEEiES—-^^^,
rate ^infinitely le^ jto^^rjf^ "

This s te t enB n V hich ^i^sight £*£■"
an externa operation, can be °; ^ p ^ Gr;ten tal science is a sub-
our foregoingWrks. In so *" aB ^oientist r«y be said to
jective hypothetical construction, the s° conal tion. Never-

know nothing about nature in ata. ^^J on is in soma m asure
thelass, because this subjectiveco »g™ t in tanediatejr qua

a reflection of nature von Uexkul » ^ there - s a sense in
lifying his initial absolute s^t,

-258-

wtaoh lu ib orue to say that experimental scientists know infini-
tely less about nature than gardeners and peasants, who Ire ttoueh
in an extremal obscure way, in contact with objective nSure! Tte
actual vegetables with which gardeners deal are'certainl^t cons-
tructed according to the hypotheses of biology. This would suppose
that biology had achieved a knowledge of tte essence of living things.
"Scientific vegetablea" are not edible.

A second objection to our identification of experi-
mental science with- dialectics raight be that in innumerable places
Aristotle and St, Thomas con demn the Platonists an d the Py thagoreans
for proceeding "logice sivTdialectice in naturallbus." (47 ) An
attentive examination of these texts, however, vail inraediately
reveal that they do not condemn the use of dialectics as such in
the study of reality. As a matter of fact, both of them have fre-
quent recourse to it, Yftiat they do condemn is the abuse of dialec-
tics, which cons ists i n granting priority to principles over espe -
gigncgjCwhen, as a roaster of fact, the former should ever remain

in complete dependence upon the confirmation of tSHLatterp Instead
of re jeoting _ principles in order to save appearances , t he Platonist s

made it a prac tic e of rejecting sensible appearances in order to

{ sa ve their preconceived p jrtngLpJLgg, This is evidenffrom the pas-
sage from the third book of the De Coelo quoted in the last Chapter,

.In other words, the condemnations of Aristotle and St. Thomas are
levelled against the logical errorC of oonfusi ng a formal consequence
vv^h^n_arguraent> which would mike dialectics self-sufficient and

I independent in the study of nature,

A final objection which might be brought to bear a-
gainst the identification of experimental science with the 'Aristo-
telian dialectics of the Topics is that the ^^^^Vf
the Stagirite gives of the latter (48) ^ j^£^a»?i>at
it is essentially a netted of p^^^^^T^cUos
consequently it presupposes aj^axagug. -^ ^ TC seen

essentially involves a kind of dialogue, since, as
its principles are always " toterrogatxones prob abil ^^ ^
bo granted that in writing the Topics Aristotle had p P ^

in mind the use of dialectics which ^^^^suppose such
But the dialogue of dialectics does not ne cessar ^ ^^ ^

a plurality. In dialectical F^ 30 ^.f^r^^^^ 1E ^ bveT>
what seems pro bable t o hto^ nJLggeK^nig-^j^^^ ^"JlTadversary,

even~^itt^ra-ilu^aTI^y^f ^ r3 ™*aZtZiS
VasuiKlyOhe other parto^ttie^trad^ctn^ .
/ " ' • , ' w nf experimental science

In this dialectical_ character of x^ ^ m ^

M. v/e find the basic reason why P^ sl °* ^Stitude within its own
~ V natioal physios. KoTTinding scientitic o

-259-

J0 Jm 3 j attempts to acquire for itself a substitute certitude
fr^eachingjjpj^ ^ om thiB gSSt-rfT lBg, Bertran d

Russell is correct in saying tj^pjiysicsjjsj^enatical, not
iBcaus^wjU^WjiO^^

toow^Jb-ttle," LM) What we have been saying diTthis Chapter

tDiobrings to light the reason why mathematics in the modern sense
of the tern is a natural prolongation of the dialectics of ex peri-
mental scien ce,^ Dialectics bestows upon physics the hypothetico-de-
d uctive method which is~so characteristic of "modern mathematics iy -
And in this connection it is extrensly interesting to compare what
we have said about the nature of dialectical reasoning from freely
chosen hypotheses with Bertrand Russell's famous definition of ma-
thenatios :

Pure mathematics consists entirely of assertions to
the effect that* if such and such a proposition is true of argr-
thin§ then such and such another proposition is true of that
thong o o , Thus mathematics may be defined as the subject in
which we never know what we are talking about, nor whether what
we are saying is true, (50)

This brings us to the -bask of analysing the proper .
nature of mathematics.

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CHAPTER SIX

THE MATURE OF MATHEHATTfiAL ABSTRACTION.

1, Mathematical Abstraction.

History has played with the terra "Mathematics" in a
way similar to that in which it has played with the term "science".
We have seen that the latter terra now has a meaning quite distinct
from,, and to a certain extent opposed to, the meaning it had for
the ancients: it no longer signifies certain knowledge of things
in their causes', but a purely dialectical type °f knowledge that
is lacking in certitude. In somewhat the same tray, the meaning of
the term'taathematics" has undergone a profound change, For the an-
cients it signified a strictly unified science specified by a de-
finite formal object, namely quantity. But in recent years mathe-
natics has been divorced from its essential relation to quantity
land given a range that extends indefinitely beyond its confines.

In former' days, it was_supposed (and philosophers are
still apt to suppose) tha t < £*nt£& was the fundamental notion
of ntittomtioB. But nowadays, quantity as bonished^together,
except from one little corner of Geometry, ^.^/^
aid more reigns supreme. The investigation of gf^niy^
of series , anT their rela tions is *ow a very large par t of m
^^^-^O^aTbe^Tfound that this ^ es £| at ^ ^
be conducted H^S^SSL^^Si^^^i ^>j7f series

most part, vi^^m^^^^^L Properties can be
are capable "^fo^lOT^g^^gg^^

de duced from i hr^H!i5^^2L^IE2^S-J^- -*-

A lgebra of ReJxiti yes^ (1)

' ' " , ~ „ a+rictlv unified science j

Mathematics is no long er a st rict jy^ ^^ . g ^

it no longer has a definite fo ^° n ^'± s not mathematics in
most of wto.t is now considered mattem *"^ ^^ In mB chapter
the original sense of the term; &-%~~tf^i3hB in the strict
ve shall try to analyse the nature of ™™ er whioh lt was under-
bid forraal sense of the term, xn the sense m

fl'uood by the ancient Thomlsts, (2), ^° .^ ^

„ • + + -°" e °5 th f objections brought against the relevance
of Peripateticisra for the question of science is that it necessa-
ry immizes the importance of mathematics because of the fact
that it considers quantity me rely as one out of ten predicaments.
L -if ^' AS a raatter of faot > bowever, Peripatetic¥"Eive~ always accord-
"' ed to quantity a unique position among all the categories, For of

all the nine accidents it is the one closest to substance . And it is
the only one of the accidents that can be a sub .i ect of a s pecial
science ,, For all science deals with a subject manifesting itself
through certain definable properties, and quantity is the only ac-
cident jjij ghich there is found both subject and properties. This
explains why quantity and the quantitative can constitute, in re-
lation to knowledge, a closed universe apart from everything else:

Sciendum autem est quod quantitas inter alia acciden-
tia propinquior est substantiae, Unde quidam quantitates esse
substantias putant, scilicet lineam et numeruia et superficiem
et corpus. Nam sola quantitas habet divisionem in partes pro-
prias post substantial^. Albedo enim non potest dividi, etjoer
consequens nee intelligitur individuari nisi per subjecting .
Et inde est quod in solo quantitatis geriere aliqua signif i c an -
tur ut sub.jecta, alia ut-passiones . (4) Coi«m- /•- K'**" ■ v > /•//-, ^fp^

But in order to get at the nature of this special science
it is necessary to point out that it is not quite accurate to call
mthemtios tte science of quantity. For the other *™ J^^*"^
sciences, metaphysics and philosophy of nature, al^ej^witt^pn ; :
tiby^nJome__wiy. Motaphysics dealswith it i|so^g^ti|_a

r^tu^de^iT^ith it in BoJ^r^J^l^o^re^^^J^

beingT^iiKJsveJ^ Consequently, ^ rt O g oonsider the

sic nature of ^themtics, it will be necessa^ ^ ^

particular way an which at deals wi™ possible the special

wilTbe _ n'ecessary to analyse as _ accurately as possi
nature of natheraatical abstraction.

A number of things were said about %£>£«»*>
abst^aoWon in Chapter II. Before pushing ahea
let us recapitulate the points already laid down. ^

Mathematical abstraction ^ ^^^f.-Sphysical
abstraction. It stands midway ^^S both. Yet from another
abstraction, and shares to some extent in ^ ^ thxrd

point of view, it is not midway between

-262-

degrees of abstraction, in the, c.o„ OQ *. , •

then. Rather it is out of line ofTfo ^"V" direct "i* v/ith
in this connection it i a intere s ?iL +o°f + S1 ^\ S ° to speak ° And
"metaphysics" is an historical accxlcn? it i ^P ^ tem

accident in the sense that it characWn'u! -T extre ^^^VPy
InaWe of the science it has been cho^n In T^ a f uratel y tte
point of vie. it ig highly sigSfican Cha S S™
comxng^dxrectly after physics ijiJh^de^^trackonTL
not called metaphysics. Nor isliiSiaihTilbi-eaTIirSa^Stios,
though it comes immediately after nathematics. And yet when^sics
begins to seek a substitute cause and reason to explain its facts,
10 is_not to metapnysics that it naturally turns, but to mathematics,
ihis is a paradox upon which we must endeavor to throw some light.

Mathematical abstraction prescinds from all sensible
ratter, though not from intelligible matter. By sensible matter
we understand matter with sensible qualities, and hence apprehensible
by the senses. It is important to distinguish between mathematical
quantity and the common sensibles. As we shall see there is a close
connection between the two, but they are not identified, precisely ^'.V^
because the common sensibles are sensible . A nmtheriaticalJLina,— - — v>*
a number, etc. are by definition not sensible. (5) — ^Tintelligible ___
matter we mean the substance (considere d as) the subject of quantity ,
|which_is _the ogder of the farts of tte^suGstance.j This abstraction
givis~~to~Satheniatics ' an dblecTlvFiich depends~upon sensible matter
for its being, (but__no t for its "bei ng_ kn own" , 3 that is, it is conceived
by the Hind, and defined~Tndependently r oT'all sensible matter, but in
order for it to exist outside the mind, it must be realized in sensi-
ble matter.

As we pointed out in Chapter II, this profound dichotomy
between subjective and objective existence is something peculiar to
mthenntical abstraction. It is found neither in phy sical abstraction,
in which the object is dependent upon sensible niatterboth for it a
existence in the mind and its existence outside the mind, nor in me
taphysical abstraction, in which the ob jec t »P^£ °f ^aS
natter both for its existence in the mind and ^.^£^ ^J^
the mind. We suggested that this dich otomy , g^^s n ow comfto
abstraction is extremely significant, and che time has now
explore that' significance.

■ we know of no better point of J^J*^ 2V

Ploration than a consideration of a text o actually

first sight might appear somewhat c °"^f cal abst raction. As we
contains the key to the nature of mathena oi question

noted in Chapter II, in the third article of the^ ^^ llf
of the De Trinitate, Aquinas seems to res ^ ^ mtlmtl(a i

nal absTr"Sction 7r to the type ° f abstract

-263-

aoiences. He points out in faot +h^+ +v,„

ion: the abstraction of a f orm fron JX *" W ° ""^ of at stract-
a universal from a partlcuC The fo^L^ the absfe action of
Ifctepar,- to mathematics, S thei +W ^ considers to be , ,-
ces, (6) We have already L£inet ff M COm T t0 a11 the scien "
sage must he interpreted; Bui'Sthis^unof^ 61 '^ ^ hW this paS "

-ac^on ^ W ^ PIDper mture of ^hen^iLl Ss-

. J n . simple apprehension the intellect is able to sepa-
rate certain things whxch in reality are not separated. It is in
this way that the mind gets at the things which form the object
of the mathematical sciences. Objects such as line and number oan
be separated by the mind from the sensible matter with vMch in
reality they are necessarily united. Now preoisely because this
union in reality is necessary the separation effected by the mind
in simple apprehension .cannot be _ transposed to the , s econd op eration
of the mind, the ju dgement For th e essence of the ^jud gement is
the_ ocypula j andTthis express"es_e xi3tence. reality_,TThat is why from
the conception of a line separated from sensible matter we cannot
pass on to the judgement: "the line exists without sensible matter,"
What about the judgement: "the line exists with sensible matter?"
Such a judgement can be made, of course, but then we are no longer
speaking about the separated line, the abstract line. There is,-'
therefore, a kind of indifference in this abstraction. On the one
hand, it does not say that the line, is with sensible matter. But
on the other hand, it does not say that it exists separated from
it.

This brings out the characteristic feature of irathe-
tical abstraction, and explains what is meant by saying that quan-
tity depends upon sensible matter in order to be, but «°* j^ °Kkj r
to be conceive! For on the one hand, in the case of /he sensible
qualities which enter intrinsically into the study of nature, there
is no possitaity of separation "secundum intellect^ sjnce_s|ssi
biHi^^ain^ ^ terl , a * S aTsubsW

it is the first subject of all tlle , Qel f . „H+v,rm+ mobility,
it, cannot be conceived as ffl^i^^^^^^fe)

On^hi-^th^rTiSdT^hile the objects with ^ch^ P^^ „ se _

are separated "secundum Intel 3e ctum , JJJ .^ntran spose the

ounflum esse", aidJbhaltejrtBLinJIS*?^

se paration taaSTET^^i^ ^^^^^^-i^Iir^^S^^
>^nt7 "Considerare BubstStSTs^quantitate,^ ^.^ goien _

Senus separationis quam abstractioms . .

-264-

tiao divinae, sive metaphysicae," (7)

m- +v, A11 i ^ 1S ^ 1PS US t0 see wlw St ' Thoms is justified
in calling the abstraction found in mathematics formal abstraction
in a very special sense. In it alone there is a form lifted out
of matter to which it is necessarily united in reality. And this

/ enables us to grasp the difference between the formal abstraction
characteristic of nathematics, and the "universalizing" abstract-

^ ion found in the other sciences. For it follows from what has just '
been said .that mathematical entities in one sense can and in ano-
ther sense cannot be realized in nature. They may be said to be
realized in nature in the sense that there are triangles, lines,
etc actually existing in the world of .reality. But mathematical
entities as such , that is, in their state of abstraction from sen-
sible matter, cannot exist in reality. This point. is important,
for not only does it reveal the special nature of mathematical abs-
traction, but it also enables us to understand the true nature of
mathematical physics. For as we have already pointed out, the ap-
plication of mathematics to physics consists in the application
of mathematical entities as such , that is, in their abstract state.
It is not merely a question of finding in nature quantitative de-
terminations as they exist in union with sensible matter.

But perhaps it is not sufficiently clear yet just how
mathematical abstraction differs from the abstraction found in the
other sciences. For all the sciences deal with abstractio ns, and
abstract things as such, that is, in their state of fraction
cannot be realized in nature, even though they may ^ ^l^e.1 by
the removal of this state. In what way, *^? n ' *° "^tSr^otenoes
titles differ fron the abstract things m * ^^^L other
deal? There is a vast difference ^^r^tllltlaT things,
sciences B For, although all sciences deal with ajstaao ^ng ^
only mathematics deals with abstract things S^-j-^ be

to say, the abstractions found in all the otn . ffll

Eredic^ted,^^^^ of nothing

^^h^T^^fs^BSrTSMr^ * e ^Tihev are defined in a way
existing' in reality, VE^S^-^^^^K-f^S^S^^
in wh ich the y cannQt_exist, that is, »b * between the abstract
rSttiFr5T5ther words, the only <^"°;™ lity is that of univer-
entities found in the other so « n *"^"£ tios there is much more
,s_ality and partio^ljri^.But in ^«ics ^ ,

^hSrtasTNsrsnjraTu&iversai ^themtic es do ^^

sistunt

ta^iSTiSSBd up wixn B— —

„a ™+heimtica non sue-

-265-

universaliter suupta non suli<n - =!+im+ fi,™

_„_ rations nff-i^, . s i stunt ( ho ° eni m esset ridiculura
pio rations afferre)* sed quod mathematica ut sic rarticulari-
•^sugjta, non subsistunt; seu, quod idem a^quoa mSS
tica ut sic,- non habent aliqoud individuum existed iTrSi
natura. Et propterea nequo sunt in universal!, neque in parti-
cular!: agjgrh oo bona esse non ^osmmt. Quod de aliis rebus
universaliter suraptis dici non potest. Et sic patet nullitas
consequents ad oppositurn factae: et quare singulariter dica-
tur de nathematicis quod non habent esse, (8) . '

This , then, is the essential difference between mathe-
matical abstraction and the other types of scientific abstraction.
In physical abstraction there is a kind of separation from natter
through simple apprehension. But the only kind of matter from which
separation is made is individual natte r. All the matter -pertaining
to the e ssence of the thing abstracted is retained . And this explains
two things, First it explains why the separation cannot be trans-
posed to the operation of judgement, for only individuals exist,
aid. things which have matter in their essence must have individual
matter to exist. Secondly, it explains why we can, nevertheless,
raake a judgement which predicates the abstract essence of actually
existing things, for the predicate of a predication is a universal
nature, and throughjhysicjO_abstractipn nothingjias been removed
from the nature except individuation.] )

In metaphysical abstraction there is a separation from
all mtter, and this separation can be transposed to the operation
of judgennnt, since there are beings existing without any m J*°£.
For thl sara'reason, we can predicate metaphysi ^entities xn their
very state of separation or abstraction of ^^jL^^tSc-
As Cajetan pointfout: "Metaphysicalia se cundum V*W™* individua
tionen sumpto subaistunt: quoniam habent in rerum ™^ ^ ™£
abstrahentia ab o.-r^ia mteria sensibili e^iMfeggig, u ^

do intelligentils.," 9) *^*"°£ t £ tta S£k»i22^
physical 'abstention VL^^S^^^^SSi^Sr^SSStraot
be transposed l> the qg eratagnrfjugS^.' t . be predicated

entities can V> p.^io^atelTErriaTIW they camOT
lin their verr 5t.yi,'i of separation.

D , mM tics there »^^£Z££%W?
of these two tvpe 3 of ^ st ^°^? n * ^ich depend upon sensible mat-
sics, mathematics deals with things which dep ^ exgJ&i &
ter for their existence outside the "^ - t dea i s with things
above), Like mataphysics and unlxk a pigs 103, concep tion and

which are independent of sensible matter 1 ^ me taphysics there
definition, Life the case °f .P^" 6 ^ of physics and unlike that
■Is separation from natter. Like to

-266-

of metaphysics this separating „„„„„.■. -u ^

Unlike both the case ofphJsicsXect^ t*™^ ***" ^gernent.
to do with natter which Stein, to T ^ separation w haa
tracted in so far ao hofe t, 1 V ?f S3S ? nce ° f thin S s ats "
the thi^s abstracted Xt^l^^*^ ™^ ±0S

^isunctrcS^t^SS SSJMg^

, . . . Ad vertendum est ex Cajetano quod quantitas potest du-
pliciter op.strahi, Uno modo secundum abstractionem generis vel
special abindividuis, reraanente. tota natura et quidditate quan-
titatis;, sicut omnes aliae naturae quando in universali conci-
. piuntur: et heac abstractio fit ab intelleotu universalizante
naturam; et hoc nodo quantitas in abstracto consideratur a me-
taphysico et sic non amittit ratione m perfeotionis neque boni .
Alio modo fit abstractio quantitatis denudando iliam a sensi-
bilitate, ^ejbfitper imaginationem: sicut imaginanur distantiam
quantitatisEPvacuo, lineas aut superficies in eo imaginantesj
et talis abstractio non est universalis a particulari, sed so-
lum quantitatis interminatae, seu imaginatae, a sensibili... (12)

We have already had the occasion to point out that
it does not pertain to mathematics to consider the nature of quan-
tity in_itself , nor its onlological properties, nor even the nature
and ontological properties of its two species: continuous and dis-
crete. All this bel on gs to rret aphysios, For quantity is a princi-
ple of being, one oFthe ten predicaments, and therefore conies un-
der the object of metaphysics whose object is the being thac is
distributed through the ten categories. It is evident, then, that
the mind is able to lay hold of quantity by another kind of abstract-
ion than that found in mathematics. And it is clear from the pas-
sage just cited from John of St. Thomas that this abstraction is
the kind that we have been opposing. to nnthemati ^ faction
since the beginning of this discussion, that is, the ^ersalizing
abstraction, jrtrfslLSSDaMsffi-^^

apart from the^a^dividuals in which i » J^^Soe,
traction lays hold of entity ^^^^SSSSSrT^i^^
a certain reality that exists ° nt ? log ^ a !^' B a prin ciple of being,
precisely in so far as it exist ^ ^f^l^T^gghJ^annot
and no t in so far as it ^ J^L^"-^!-^^

!!XSTiH5iility7Tra^ natter Co-

ration of quantity insoro3_way aD3 ^ r ^ & metaphysical considerat-
thorvri.Be it would be~a-?h^ical ^J.^^ consideration, expli-
ion). But it does not, like the mathematical

-267-

eitly serrate it from sensibility, "denudando illam a sensibili-
ze," and explicitly set it off in a world apart from the real
*orM. Ratter, while not taking account of its sensiSe dona-
tions, U considers it as it exists in reality a long with the othe r
accidents which constitute the strur.t.irg^;^?^^^ }Mh ~
mtical abstraction, on the other hand, considers quantity not in
so far as it is a principle of being, or a category of reality,
or a certain form or essence, but from the point of view of the
relat ions of order and measure that result when it is separated
froEt_all sensi bility and s et_ap a rjb_b y itself .

It rails t be kept in mind that physical abstraction
also lays hold of quantity in some way. For since quantity is the
first accident j i t is the matrix of all the sensible qualities ,
(w hich c o ns e que nbl'y cannot be conceived of except in relation to (
jX) All~the mobility in the cosmos is inextricably bound up with
quantitative determinations, and f rom this point of view quantity
enters into tba object of the study of nature. These quantitative
deterrainationsy incidentally, f°m the basis of the mathenatization
lof nature o But they are only the basis, for in mathematical physics
they are considered from the point of view of the mathematician _
and not that of the physicist. Quantity is also studied by the phi-
losopher of nature in a very particular way, in 'so far as in living
mobile beings there is found ajpMia2Jdrj4^£_S9Miiiy pertaimng
to the genus of quantity .

It is obvious that this consideration of quantity
is quite different from that of the mathematician,

Matteiiatica ex vi suae abstractions et conceptus, '

excludunt f estate statum ™**^°,£^i£Z
titatem secundum ilMn realitatem qua potest^dere ub s ,

sed secundum extensions ^^Srt Sneae et fl-
ans, ad demonstrations mathe ^txcas suf f ic^lx n ionl8

gurae in imagination *^^>**£Z£ Stest; non vero
proportionis vel °°n^ui' atis oonsiderari^po ^ geu

quantum ad id quod sensibilitatis est
in quantum ens naturalo est. K^/

+n be three distinct ways
There would seem, then, w ,,, t th g n ind»

in which quantS ™* * ^ Scttg gg^S sensible de-
First it may be considered wep}""^ otjeot of the science of -
terminations, and in this way it is the odj 1 a001 dent

Physics, Secondly, it W ^°^o5 5* the sensible accidents
in so far as it exists in reality ^™\ not explioiWy as de
- ~ abstracting from them m *™™fc£' aa separated from them,
■termined by thev.i, and yet noD ^1

-268-

In this way it is the obWt r.-p +■>,„

It ruy be considered as i^J"*^^*™*^*"' *»**,

a state in which it cannot have aoS Si^ lbl lty ' 3et off *»

sirs £^^ *£j£^~ -^s

of she most complete abstractions to which the human mind can at-
tain,, (14) The particular nature of the abstraction found in the
mathematical sciences has not been generally recognized. Professor

iLenzen, for example, writes: "The relational structure is a complex
universal which may be exemp lif ied in various instances , and hence

' the problem of the reality of mathematical objects is that of the

Lreality of universals,"' (15) We hope that enough has been said
to show that the problem of 'reality which results from the special
kind of formal abstraction found in the mathematical sciences is
something quite different from the problem connected with the "u-

ijiiversalizing" abstraction found in the other sciences.

This consideration of the abstract character of ma-
thematics brings us to an interesting paradox. In a sense it is
true to say that by the very fact that it is the most abstract of
all the sciences, it is also the most concrete. What we mean by
that is that in a sens e (the rayhbematical universal is th e same a s
the niathemtic^I ^ar^uTSrTJ lor^ hiraatical particulars abstraot
f?5m~si^sTble *o-v i^ vS^r^ STRghxaver^l^ioss . "Materm sen- _ _
s^HliTn^nTSditur in tatelleotu mathematicorum neque in uni- - #t»*. »
fersali, neque in particular!," (16) Nothing extrinsic ^ added r , -Jl

J to a mathematical particular to individuate it, A particular circle i *

U or b may be considered the ^ivejrsalj^le^

This truth has «n**^*%^ £ Svrfng
blem of mathematical physics as my be ^ed^rom everyt hing
passage from Ernst Cassirer. f^' no JJ~ 3 out effectively
contained in this passage We believe that it brings o
the point we are trying to makes

.,. • wr +h P lopic of the Wolffian school,
In his criticLsm of th ° l0 ^° give mrit of mthe-
/ Lambert pointed out that it was vob ^ determinations
matical 'general concepts; not to ^ f^ L jo_ i2 tain_them. .

of the sjp J Dcml_c i ^,jHLHL23^

Y/hen a "mathematician makes his torwu cia i cases, but

not only that he is ^J^J^^e universal formula. The
also to be able todeduce them trow

-269-

possibility of deduction is m i -pn,,^ ■ ..
Jastio concepts, since the™ „p ^ , the ° ase of the soho ~
ula, are for^bHeglectine the S^ *? ** traditio ^ f°mw
reproduction of the mg^^jg 1 ^ S^enc^e

e 5i™i5rTffirabitrtcUo¥ir^

hut onthe other hand, the detection of £ ^articTlarton
the universal so much the more difficult, for in ? tL Sclss
of abstraction he leaves behind all the particularities in such
a^3LJhatjia_camaJLrgoovgr them,Cm uoh less reckon the trSi^
formations of which they^are^pgOole^This single remark con-
tains, m fact, the gem of a distinction of great consequence.
The ideal of a scientific concept here appears in opposition
to the {schematic general presentation) which is expressed by ■
a raere vrord, The genuine concept does not disregard the pecu-
liarities and particularities which it holds under it, but seeks
to show the nec essit y of the occurence and connection , of just
these par ticujlatatiesT 1 What it g ives is a universal rule foythe
connection of the particul ars themselves . Thus we can proceed

from a general mathematical formula, for example, from the

formula of a curve of the second order, to the special geo-
metrical forms of the circle, the eTLipse, etc., by considering
a certain parameter which occurs in them and permitting it to
/vary through a continuous series of magnitudes. Here the more
universal concept shown itself also the more rich in content ;
whoever has it can deduce from it all the ' mathematical relations
which concern the special problems, while, on the other band,
he takes these problems not as isolated but as in continuous
connection with each other, thus in their deeper systematic
connections. The individual case is not excluded ^m conside-
ration, i^Js^^L^LlS^SSkM^^SS^L^^^
step in a- ^^Tpl ^ s^f^te ngirit is . e ^ent anev^that
STohSSoteristio f5SE5TonBr5bnoept as not the ™

aalily* of a (prese ntation ? 5S^«MJ»i?S^^^^|Si
oiplelof seri^ro7dern?/do not iso ^^JgiK 1 :
e^eVf^m-thTlSnlfoTd before us, ^"^^^^^b

b y an^^clJ i^illCTaJtod the further we pr establi-

mUSTSSSr^i connection -f^f^aeten.nmtion

,shed, so ^ch tho clearer does « f^, the intuition

of the particular stand forth. Thus, ior * olear

oT"o^tlclidian three-dtoensioml space onlyj^ ^ ^ ^
comprehension when, in modern ge >- l °™?> to tal (SHpSHcLstruc-

Vgher' forms of spacej) fffi^-igf^^f&feS^Tl^

ture of our av&GeJ£ft2£S£-ESV ■ ~~~

~ — ' — — — "" • A ed a strange universe

The mthemtical universe J ^n . ntelligiMlit y,

Cts abstract character gives it a high degr

-270-

the separated substances, this removal of mt.ter^oes_not^ontrlbute
wjjig pegf action of na tures. In fact, the sep^ratiolTfitoMtter"~~
prevents mthsmtioal entities from -bein g nntmg,, And yet, it is
to thelight of these entities that we shall try to understand- the
.natures) existing in the cosmos .

In order to add further precision to our notion of
mathematical abstraction, it seems worth while, before leaving this
question j to compare the way in. which mathematical entities are
abstracted from the world of sensible matter and the way in which
dialectical entities, such as the one discussed in the last Chapter;
the foaa of ange:,? considered independently of the sensible matter
to which it pertains, are abstracted. In both cases we have the
abstraction of a form from the r.iatter to which it belongs. Bui there
is a vast differenoo in the way this abstraction takes place. In
the case of the dialectical definition of ang er, we have the form
of a natural thing which is essentially inseparable from matter
both for- its being and for its 'being known". Hence when it is set
off by itself, it is in a purely logical state ; \it_is_a mere cons-
tructi on of the' mind. j Matheraatioal entities, on. the other hand,
arrby"their very~nature separable from sensible matter secundum
intellectu m, even though they are not separable secundumess£. _
Consequently, when they are considered as separated, they_ar^
th eir natural state ; |ihey are noTdlajgcji^Anger as a pure form
iT^^^^Tk^^^i^remty-Kr^^e form is an ens_m

turae

This brtogs us to the toportant question of the re-
lation between mathema tics and existence.

2, MxttonntioianaEd^tenge^

* «*» relation between mathematics ;.»!
,The question of the «£™"{V, problem ever since
&* existence has been an *=ute phi Jjgjg^ 1 the mtu re of
the tire of the ancient Greek. . The anaxy upon t
mathematical abstraction has ^^J^In fact, whatwe have
But the question demands f ^ratten throw the problem into sharper
seen so far in a sense on^y serv

-271-

focus, Fox- if mathematical entities mm „t „ • A
bub* wo not conclude that mttoSJLTSSa'SS enVr* ? ' -^
logical beings? John of St Thomas has PoneVn ~f , ratlo " is . " "
Cuv sus Theolog icus (18) to settle thl, * ° + ? raat P* ins ^ the
briefly his soUTbion: thlS ^ estl ^- I** ub consider

, . *■ • ^- a logical b ?ipg to understand: "ens habens esse
objective in ratione, cui nullu m esse correspondet in re". Conse-
quenoly, if mathematical entities were logical beings it would be
absoJAi^y_contraai ctory for thorn to exist i n reality, Ifow, from
what we have seen about the nature of mathematical abstraction
it should be evident that we cannot say in absolute fashion that
the real existence of mathematical entities always involves a con-
t:cadiction For there is 1 a sense in which it is true to say that
some mathematical entities may exist in reality, not indeed in
their state of separation from sensible matter. We say some mathe-
matical entities, because there are obviously a good many mathe-
matical entities, which are evidently mere logical beingsj and
whose real existence would necessarily involve a contradiction.
An example such as the square' root of minus one comes readily to
nind In fact, the whole point of John of St. Thomas' analysis
is to show that mathematics, by the very nature of the abstraction
it employs, remains indifferent to whether the entities it deals
with are real or logical beings.

And he illustrates this point by having recourse
to the example of predicamental relation. The essence of a relation
consists in the ordering of one thing to another. But a relation
may be of two kinds: it my be. either teal, that is, exiting in
reality, or it may be only logical, that ia, oreat ed by the mind.
A real relation is one of the nine accidental categories, and like
all of the other accidental categories it has a real ^"tence _
in the subject which it relates to something else A logical re
IStion does not have a real existence "1*^^ the proper
it is the mind which creates the ordering. Now since tn p p ^
essence of relation which distinguishes ^ gm £ ^^ or in
tegories consiots in the ordering of one want ^.^^ t0 ei _
Scholastic terminology, in the raBo_aOj lo „ ica i esistence.

ther real existence (the J£*3£J^ °*JflL; B f existence. In
The ratio ad is common to both of ™ e ?^$£ eren t to whether the
somewhat the same way mathematics i s . al ex i s tence. In

entities it deals with have real or oniy * and is a kind

this way it differs from all the °*J" ^^taph^ioB on the one
of medium between the science of ^™*\ h ieno a of nature and
hand, and logic on the other. For both th Logic deals neces-

raataphyaioa deal necessarily with real ^ Q . ther or both,

saril/with logicfelibeings, ^themtaoa^ ^ gcienc0 of

It is true that entia rationis enter into
nature and metaphysics,

-272-

liii-fc their existence in these stud-ina -i„ ~ ■.
is, the whole raison d'etre of ^ „W Actional, that
tionis is to embTe~tte^loL£r o? S ^ Cti ° n ° f thflse entla ra-
tolet to know reality- they Tnot con^ ^ t^e rnetaii^IHISn

sciences, and are not considered for ?heir^ v ° b f 0t f th93e
hmvpver the ent-W n+i™l theix own sake. In mathematics,

torevex , Uie entia rati onis are considered for their own sake. In
this respect, mathematics is similar to logic. It dlf?™ a SoA it
however, in that the entia rationis, it considers are based^n real
beings which also constitutes its object. In this sense Meyerson
is justified m saying; "...chez les mathematiciens, reel et idee
senblent en quolque sorte se confondre, on ne distingue pas tone-
diateirent s'll traite de l'un ou de 1 » autre ...0' est la, encore un
coup, la consequence directe de l'accord de 1« intellect et du con-
eret dans la mathematique , et c'est ce qui fait de cet element la
vraie et unique 'substance intemiediaire,' dans le sens de Platon."
(19)

As has already been suggested, this indifference on
the part of mathematics to real or logical existence is something
that arises out of the very nature of mathematical abstraction.
As John of St. Thomas explains, it is precisely because mathematics
considers quantity stripped of the definite determination and form-
ation that it 1ms in its state of union with sensibility that ma-
thematical entities can be simple concepts capable of being reali-
zed in sensible matter, or concepts that have been elaborated by
the mind into a state which cannot be realized in nature.

Mathematical ex vi auao abstractionis et oonceptus,
excludunt a quantitate statum sensibilera, nee considerant quan-
titatera secundum illam realitatem qui potest cadere sub sensu,
sed secundum extentionem imaginabilem praecise; quia, ut ctixi-
mus, ad demonstrations mathematicas sufficiunt lineae et f i-
gurae in imagimtione formatae, quantum ad id quod est arte n
sionis, propoVtionis vel oontinuita is conside ga potest^non
vero quantum ad id quod sensib il itat « ej*^ 1 «* et alio3
seu in quantum esn natural* est. Et ^^ ™ temlmta .
antiques considerabatur quantitas in ^™ra conside-

et ilia interminata dicitur W~™tSaC^um ad id
rat secundum quod praecise se ^£^ ™f Scit; teminata
quod de extensione potentiali eo iorma et for , m _

vero quantitas est ilia quae sub c ^.™f # . lte mthe-
tione concipitur, et sic red d«ur sensib ill ^.^ quod ^
matica considerat quantitatem quan tow .£ habet a m _

bet de extentione interminata, et secun ^^ & f

teria: non secundum terminationem °*^ u ^ ltitM mthematica
ratione cuius redditur sensibilis. W h ^ e0 modo

habet concepts positivum ^^f^inario, sivo sensibiliter
quo quantitas potest invenin, sive

-273-

in rations entia veri. Iinrln -n ^,j •

tls realis et vcri : ne^fe posS^ f , ^ ad ratiotlera e *-
adaequate, nequo positive excludlndo t tl ° 6t conside ^o
tatem ipsius quantitatis. Et in" hoc ^T 8 ™" 1 ^!' reali ~
jmginaria, quae est ens ration!^? w a ^ mntite te pure

habet ad quantitatera r a! n '™° ^.^gmrfor se

quantitas mather.atica non repu£Lter ~ ^ ^ At VSr °
tor: quia aeque bene potest faS^ !f *»****> aed indifferen-

realibus, vel imaginarSs- sicut^ ^ ?? monsteti <™s ^ eis
j a- •, l t> J - I "" J - J -s> sicut sx relatxo conaideretur qpran-

dun ratxonem ad praecise, nondum considerate ut enfSioni™
nee tamen ut determinate ens reale: sed indifferente/ad mud-
quia non considerate- adaequata ratio eius ex omni parte quae
requxrxtur ad realxtatem, ad quam etiam requiritur ratio in-
sed ex ea parte qua indifferens est ad realitatera, et solmi'
explxcat ratxonem ad. Sic quantitas considerate a mathematico
inadaequate, et sub ea retione praecise extensionis intermina-
tae: quae indifferenter se habet ad imaginariam et realem, et
sic non excludit rationem entis, sed permittit: neque repugnan-
ter se habet ad illud, sed | indifferenter r Unde nee ens ratio-
nis est determinate, nee ens reale determinate: sed indifferen-
ter et permissive se habet ad utruraque. Quod non solum contin- -
git in ratione entis in communi, quae abstrahit ab ente reali,
et rationis : sed etiam in relatione, quae abstrahit a reali,
et rationis, secundum inadaequatum conceptuia ad: et in quonti-
tate quae abstrahit ab imaginaria et sensibili, sub inadaequa-
to conceptu extensionis interminatae, (20)

•All. this helps us to understand more accurately the
meaning of the phrase to which we have already given some conside-
ration: "mathematica dependent a materia secundum esse". The pri-
mary meaning is that while it doesn't pertain to the essence of
a mathematical entity to be capable of realization, whenever it
is capable, the realization takes place always in natter. ^.f ere
is another important meaning which can also be attached to this
Phrase: in every nathenatioal entity, capable of realization or
not, there is always an essential relation to matter. If P£' B
mtter were impossible, mathematics would also be ^^J™*
prime matter is the principle of ^ & TrTtC^tlZ\Tol^Z
is the fundamental postulate of all mathematics , ™*V£ fe-

no possibility of mathematical science without an ^insi^ ^^
«*ce to prime matter. But the important point is ™ ^ ^

intrinsically dependent upon matter, ^^"S-!" for the capa-
Qlv/aya necessarily capable of realization in ma » foma iity,

^Hty of realization does not enter into «»£££, have this
^ is equally false to say that f ^^^mselves they are
capability, or that they do riot have it. in *
"different..

-274-

But this may seem to I'm n
0:0 at least in a sophism For in discu^I™^ 1 " a J . contrttdiotion >
tical abstraction wo stated that mattemScal t ^ ° f rnathem "
not capable of realization in nature ™a entities as such are

possibility of their realization "he ^tr"tioTh *° -^^
appoint; both statements are correct SS IT ^ 1S 0n3 ^
understood. And it is precisely because^ tJfl^i^.
bo ■*™* ?h *»t it gives rise L a^^S^oTSL*
tond, in the first place, it is obvious that abstract thtags^ro
not capable of realization in their abstract state. In tSs se^se
" 0t ^ven the concepts arrived at by mere universalizing abstraction
which lifts them out of individuation have such capability!^?
as to saw above, ;ra-=heratical entities are incapable of realization
:..n a deeper sense „han this. For not on]y does mathematical abstract-
ion lift them out of the. accidental determinations of individuation,
but it separates them from an element that pertains to their very
essence if this essence is to be real. Mathematical entities are
not capable of realization, therefore, in ,the sense that they cannot
exist in their state of separation from sensible natter. On the
other hand there is a sense in which they are capable of realizat-
ion,, for there are actually existing lines and circles and a plu-
rality of quantified things. These may be considered the realizat-
ion of mathematical lines, circles and number,. If. is true that the
realisation is not perfect, Mathematical entities cease to be truly
mathematical once they are realized,, The realization robs then of
the ideal purity and perfection they possessin their state of abs-
traction. The straight lines in nature are not perfectly straight,
nor arc natural circles perfectly circular. It would be a mistake
to identify the mathematical zero with the philosophical concept
of nothingness, or to confuse mathematical number with a plurality
of natural beings. And all this results from the nature of mathe-
matical abstraction which does not seize upon the ontologicaX es-
sence of the things it abstracts. On the other hand, the relation
between mathematical lines and circles and the ^V^J^Soal
existing in nature is not the same as that existiuig between logica 1
beings and their ftaixtlon in reality. We cannot say that logi^l
beings are realized in their objective foundation, as we can say
that mathematical lines and circles are realized in the lines an
and circles of nature.

All this makes it clear *ft mtham^Jeg i^
a radium between possible being, arrived as W orf ^ from

potion, and logical being, &b^*»*V£ ^LsiTorder
*e actual exercise of existence; it re ^ 1 "^ e very faot that it
to real existence. Mathematical being, oy stenoe , prescinds not
*o indifferent to either real or -WgioaJ. ox any ta „

°aly from the actual exercise of existence, but

n:

-275«

Ininsic order to existence; on the nthim h^a • *. ,

to3y exclude the possibility of act^xSe^ ^al^

lM t only prescinds from real existence; it positive^ exclude? it.

The mathematical world is indeed a strange world,
in it mind ana nature, the real order and the ideal order are in
soma sense fused. On the one hand, mathematical being is not a pure
nreation of the mind; on the other hand it is not a pure discovery
of the mindo For since mathematical abstraction never lays hold
of quantity in its oncological essence, a mathematical entity is
never a property of reality. On the one hand, mathematical entities
preocind not only from actual existence but from an intrinsic order
to real existence On the other hand, mathematical being has a ne-
cessary relation with the real, and the character of this relation
is unique, for it never retains the ontological essence of the thing
with which it is 'connected. Even the mathematical entities which
are capable of realization in nature have an ideal character about
them which they lose by this realization. Even those which are not
capable of realization in nature are. in one way or another elabo-
rations of something that is capable of realization. At the basis
of the whole mathematical structure is something found in reality:
quantity taken by itself with its proper forns and specifications
and relational structures. But right from the start the mind lays
hold of this quantity in such a way as to establish its own prio-
rity and its own autonomy. For as has been said repeatedly, it does
not grasp its ontological nature; to do that would mean a complete
submission of mind to ontological realiiy. Rather, i. transforms
quantity into a condition that is specially congenial to its own
nature: it establishes it in an abstract state and .deals with it
precisely as abstract. By so doing the Ff^^Z^ori-
freedom that is almost unlimited. Though dealing with W^^
ginally connected with sense matter, it no 10 n 6 There

ed with having its Processes term^ate - tteextern.^.^^^
remains an intrinsic connection with tne ™ s3 of

but as the mind exploits its freedom and pursues ^^ fo extrenB
intellectual elaboration this connect ion Oc ^ advan tage of

limits of tenuity. And as the ^lleot £ & ^ ^ mture

its liberty, it will tend more ana more to t> itatle g^rth
upon the mathematical world. There ^1 * J 3 * tia i it y of the con-
in spiritualization. The concreteness a * £ al3S tractnes S and
timmm will tend to be absorbed ^/f ^ aching cut ^jf
actuality of number. There will even J^ ^titude and pure
confines of quantity itself to ^anscenden te| ^vided

logical relations. And all this x ^g™ ^ what it is doing,
the intellect remains critically consciou ^ ^^ down to

And in this intellectual movement, them ^^ ^ ^ possi
dealing with real entities; i« roB

-276-

i> Uities of logical being. But in the ln<.+ „„„t • •_,
,,,, ,11 logical -the^tical *l££^^™%£°
vflW .cal bxengs, and that these real beings have by\ processor
^■ohemtxcal abstraction been lifted out of actual exp^ience with
■Sho real world. Thus the whole mathematical structure? is rooted
in real quant xty - - the same quantity which the philosopher grasps
ontologically, ^ e

All this is extremely important for the problem of
avthematical physics. As has already been suggested, mathematical
physics does not mean the discovery of the mathematical world in
the physical world. Nor does it imply the direct realization of
the mathematical world in the physical world. Rather, it is a quest-
ion of application. And by application we mean an intellectual in-
terpretation of the cosmos which always remains in some sense ex-
trinsic to the cosmos. This is true even when physics employs ma-
thematical entities which are real beings and which are consequently
capable of realization in the sense defined above, For, as we have
alveady pointed out, when these entities are employed in physics
they retain their mathematical character. In other words, they are
applied to the physical world in their abstract state. It is the
i^thematically perfect straight line' that the physicist has in mind
when he tells us that light is propagated in a straight line.

If the use of nathematical entities which are real
betegs is always on extrinsic application, that is a fortiori true
of the use of those entities which are merely logical beings. An d
it is extreraly significant to understand that by the very fact
that it is a question of .an extrinsic applica tion, i^s P s ^
for logical mathematical beings to be more f ™^ m A * h ^ =CT
prestation of the cosmos than real ™ the ^ oa ^S ly Vlcctrine
already pointed out, mathematical .physics is essentia /
of al/pl,. That is why a logic al ; W e <*£* be Sing upon
bettoFtfcm I- real being. And this P? 1 ™ Jf ** QQSa03 is Euclidian
the highly disputed question about *utKer solution of this quest-
or non-Euclidian, Wo do not wish to ««"** b ^i out as
ion here. But there are a few things that must^ j> ^ ^

to the meaning this problem must have. ^ iaian ge0 nietry assuch,
our cosmos is Euclidian cannot ^.^f realized in nature. Nor
that is, in its ideal geometrical Btattu . g ^le o{

does it necessarily Imply that »^J°° y and fruitful^ ss than,
"explaining" the cosmos with greater accuracy rinther natical entitle
oty other geo^try. It can ^."ri, system are real beings
which mteup the structure of the E^f £ ^ sans e explained^
art are capable of realization in nature appeal to the rela

Moreover, this question cannot f™%* goOT etries. For it is pes
tive explanatory powers of the a

-277-

, blo for a Euclidian universe to be more ra+i^ni *
tcr preted in terms of Riemannian geometry £*/?? ^V*"
in terms of Euclidian geometry. That ifwhv TL? 2 interpreted
,dduced by those who try to gove tit tuZll^ ' argUmntS
.o^clidianareinefficaoiL. The letti^Sr^oTc^Ltea
by the highly ambiguous meaning of "physical universe". BuTK
not wish to enter into the. problem at this point.

In connection with this problem and with the general
question of cho reJation between mathematics and existence, the
oft-quoted remark of Sir Jar.ies Jeans comes to minds the cosmos was
created by a pure mathematician. As we know, Jeans was led to this
conclusion because of the remarkable way in which modern physicists
have been able to fit the most abstruse constructions of higher
r.iatheraatics upon the material universe. But from what has just been
said it is clear that this successful and fruitful application does
not constitute a sufficient premise for such a conclusion. Moreover,
it 5.s worth while pointing out that there is a profound opposition
between the concepts of a pure mathematician and a creator of a
iraterial universe. The pure mathematician is indeed a creator, but
a creator in the abstract speculative order. And the world he cons-
tructs is, as vre have seen, not only cut off from concrete existence)
but even from any intrinsic order to concrete existence. He deals
with the abstract as abstract, and the whole movement of his science
is in the opposite direction from any embodiment in the matter and
motion which go to make up the substance of the material universe.
In another work Jeans states: "Kronecker is quoted as saying that
in arithmetics God made the integers and nan made the rest, in the
same spirit we may perhaps say that in physics God mad *^ maths
matics and man undo the rest." (21) Our analysis f^^^
of mathematical abstraction has led us to a some *^t Jofforait
elusion, and while it "/ouH not be completely true, it rraujn
be nuch closer- to the truth to say: in physics, man made .he m
natics and God made the rest,

v, V,.,* hpon said to make it clear
And now perhaps enough has been ^saio^ ^^.^ bet _

that mathematics and logic cannot be » notion of the mt ure

mem the two generally derives from a °°^" tity wi th such zeal
of logic. Nor are those who maintain wis ^ soieTce f

always anxious to explain what they mean oy b ^ it

logic is essentially a reflective B«e»^™ as n se? ond intent-
object is what is known in sch ?^ astl ° h S™he mind knows in the
ions". That is to say, it ^^.f te mind, Ifcthe-mtics £ not
other sciences, precisely as known by the ^ essentia lly

a reflectivo but a direct science. It joe properroalm of .

with second intentions. It has as .^oW ^ identi fied with logic
Wable "natures" , That is why it °™

-278-

This discussion of thp ™>i', + > -u
oyA existence would not be complete urPn + between mathematics
wore mode of the question of whether ilT* P 33 ^ *»"*«»
property of goodness. The ancient Thcw!+! -f belngs have the
tension to this question. In f act S i itlilT ° ons f deratle <*"
v,ith it that, they discussed the problem of K^t" T^ 1 ™ 1
r , a tios to existence; A^U^^T^^^ fthLfpr^ f~
ae ly because mathematics being prescinds not only from exigence '
but even from any intrinsic order to existence, it necessSily '
lacks the property of goodness. For the good is whatever can be
the object of an appetite, and appetite ha s a necessary connection
with the existential order. Or, to present the question in a slightly
different fashion: because the mathematical world prescinds from
all order to existence, it is an immobile world of pure essences - r
essences which in no sense are natures. Consequently, in this world'
there is no becoming, no seeking for ends, no finality. And without
finality there is no goodneas. For the good is formally defined
as: perfectivum alterius per medum finis.

In jtamobilibus non contingit aliquid esse per se bo-
nunio Unde in mathematicis nihil per hanc causam probatur, ne~
que est aliqua demons tratio, (22)

Ma.thema.tica non subsistunt separata secundum esse;
quia si subsisterent, easet in eis bonurn, scilicet ipsura esse
ipsorumj sunt autem mathematica separata secundum rationem
tontura, prout abstrahunt a motu et a materia; et sic abstra-
hunt a ratione finis, qui habet rational moventis. (23j

This doctrine roust be taken in the strictly formal
sense in which it was understood by the ancient Thoraists. It re-
fers only to mathematical being considered ^trmsical ^ ^ ^
is evident that extrinsically ^nality my enter into .nthematics,
am rath it goodness. Mathematical being can be an ^ £f ^
to an end. and thus in both ways involve ^^^/^there
Place, it-can be an end in the %£*£»£% £■"££. But
is truth in mathematics and truth is the goo ^^ ^ ^ mthe _
as John of St. Thomas points out, y*l * knowledge of evil

natical being intrinsically good, just as evil. things

things may be a good fab the mind without ^^ to the prac-
gooi. Mathematics may be good as a r«ans laa theuatics

tical order, as is evident from the l^g e ;^ in the purely

Plays in technology. It may also *° »**£* a a g0(A fo r the
speculative order. In this sense "atheuati instruraent to open
Physicist in so far as it becomes for hu . g the goodnes s of
«P the meaning of the universe, ^f^^nes its acceptance
a i-iathemtionl theory which primarily dotor

-279-

rejection by the physicist. For, as we shall « BO • m. ,

there is a sense in which it is trJ+n 1 e J m Chaptar

.ther true nor false; they are only^d ovt ?*$ ^V i6S are
view a scientist ie'ossentiallj a prTgmatLtf ' ^ ^

o:

n,

iiei

of vie\7

• , L ^ 3 rlng3 us t0 the question of whether or
noo there is truth xn mathematics. 'Since the world of mathematics
is a world of essences which constitute an object knowable by •
the mind, it is evident that there is truth in mathentios. But
since this world of essences is separated off by itself without
even an intrinsic order to existence, it is likewise evident that
this truth is of a very special sorte. (25) For the definition
of truth as the conformity of the raind with existing reality cannot
be characteristic Of a' world which is cut of f from existing reality,
and in which logical beings are accepted on equal terras with real :
beings The trtith characteristic of such! a world dannot .consist
essentially in a relation^ one of whose terms is found in existence,
but in a relation^ both of Whose terras are found within the realm
of essence^ or in other words, in intrinsic coherence. And that
explains why mathematics is the most deductive of all the sciences*
Free of all necessity of conforming to an objective order, it can '
follow out rigorously its own inner logics It does not, like phi-
losophy, have to keep in constant touoh with experience. It affords
the one chance that the mind has to triumph completely over mere ■
givenness. It is worthwhile noting here that the coherence notion
of truth is proper to the science of mathematics.. Every other science,
including logic, employs the conformity notion. From this point _
of view, mthama tics is even more detached from the real than logic,
although from another point of view, as Ave saw above, it ^is in
closer relation to it, It is- also worth while P^*^™*^* '
the word "real" is often substituted for he word -£
mathematician whatever is ^ &mt ^ G ^ y ^l^Iuon whether real
real. And this adds to the ambiguity * ^ gf^aU vMch
space is Euclidian or non-Euclidian. ine ** problem,

truth has in mathematics is of f^f+^^s a student of nature,
For a physicist by the very faco tha t to ^ ^ notion of

must adhere in so far as he is able to ci ^^ ^ brought

truth. What happens when these two »°^°" 3 later when we come
together in mathematical physics we snaio. theKatica i world

to discuss the relation between the physico .
and the absolute world condition. ^

T hou g h without -trTho^sr^iK.

beauty as well as truth. For as St. ™ *,„ ( 2 6) Aril thus
proprie pertinet ad rational causae iq
Aristotle v/rites:

~280»-

and daflrd^ri^nv^nJSS^'S'^ f^
in a special degree. And since these ^.f 63 d ™ trate
teness are obviously causes of Snv &*' ^ and defini ~
sciences nuet treaties ZlfoflZ.^T ^^T*
(i.e. the beautiful) as in some sense a cause, (fy) '

, „ + - ThesG J ellarks a ^e not gratuitous, for the beauty
of mathematics sometimes prevents the scientist from recognizing
the essentially functional role that mathematics plays in physics
fen that happens, the end of mathematical physics is made a means,
and the means an end, and the scientist becomes, as Professor Babin
has remarked, "un artiste egare ou frustre. "

This consideration of the nature of mathematical
abstraction and of the detachment from existence that is consequent
upon it helps us to understand the kind of causality that is found
in the mathematical vrorld, A world which is the result of the formal
abstraction in the strictest sense of the terra, that is, an abs-
traction which detaches pure forms from the material embodiment
in which they belong and sets them off by themselves, can be endovred
with formal causality alone » In other words, in abstracting from
matter, the mathematical world excludes material causality. Fur-
thermore, the abstraction -from matter involves abstraction from
mobility, since mobility follows upon matter. Hence the cathema-
tical world prescinds from both efficient and final causality,
which are, as it were, the two causal terms of mobility. Or, to
put the natter in a slightly different way,;" detaching i**J*
from existence the mathematical world detaches itsexf from coming
into existence, or becoming, and only f^mal causally can ex ist r
where there is no becoming, since the other three causes have an
analytical relation with coming into existence.

This point is of supreme i^or tance f or ^gj 8 **^
understanding of the nature of mathematical P^^ to taow ■
by the very fact that he is a P^-Cist, rnuot the ve ry fact that
the cosmos in terms of all four caus es, JW ^ interpret the
he is a mathematical physicist, am w» fQrml elemen t

cosmos in the light of mathematics f 110 " mt erial. he can see
in his study, whereas the physical is only -^ when these

things only in terms of formal causality.- jma ^ in chapter
two tendencies meet we shall consider m some

'in which efficient,

The paradox of <*^££'%?W* « \°?Zl
final and material causes are f f "^1 c^sality is *i the last
which positively excludes all duu

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y.wlysis reducible to the paradox of introducing <„*,
;hos e object is essentially mobile being tte SincS/ T"" 8 .
vWoh absolutely excludes all mobllityAe do noTiS^ & SClenc ? 9
, id er this problem here, but perhaps il wou id be weS S ll^ ■ +
to oliMnate a possible source of confusionAor I fight* ^^
,,,ed that there is mobility in the mathematical world/since the
Infinitesimal, vectorial and tensor calculus, for example deal '■
Mh the idea of variable quantities and the function concept,
j-ta ve can speak of an infinitesimal as a quantity -which approaches

zero as its limit. Moreover, the inherent constructibility of ma-

thematical entities seems to involve motion, for we can speak of
i surface being generated by a moving line.

There is indeed motion of a sort in the mathematical
.vorld, But it is merely dialectical and not real. It is a purely
imaginary and instrumental thing, and does not involve becoming
in the true sense of the word. Mathematical entities do not come
into being; and they are neither the principle nor the terminus
of becoming. We may have recourse to an imaginary movement in order
to generate the figures, but that is due to the imperfection of
our knowledge. The figures themselves do not originate that way.

Moreover, the exclusion of real motion from the ma-
thematical world does not eliminate the possibility of an appli-_
nation of mathematics to real motion. For, as we have already pointed
out, quantity is the primary accident and the matrix of all others-
&nd that is why all the determinations of mobile being are endowed
with a quantitative mode. This quantitative mode may be laid hold
of, and treated mathematically. But we shall come back to this
point later,,

It is clear from the «>^™>£&2£'
mathematics does not receive its aubgeot " or iginalty, from

■U is true that mathematical entlt ^ ur notion of a circle only
nenge experience. For example, we /°^ r °^ tib ie circular <*? e0 *
after having experienced a concrete per ceptib & pre „ n tific

nuch as a ball. But this sense experience ha ^ presupposition,
^notion. It is required by ^f*^ Itself, as i " r ^ ire
"°t as an intrinsic element in the n S ^S enc e, mathematical no
V Physics, Once derived from, sense expci

-282-

by virtue of mathematical abstraction ^

,so experience. They are stripped of °?L become . lnde Pendent

;^w*— ™~-*--°°^ red - and ? ^ d 5t^ e ^^LS xt

fclOi'S

o? aense

in \

not

ensu uAjjuiifiiuu. xney are stripned nf + v» . "^^"^nt

■hich they were discovered and invested ^Jf^^ial context
sensible character. That is why SSLtW.* -V' idealized >
have to terminate in sense experience judgements do

Recently a number of authors Vnw n»n^ • *.
, f this detachment of mathematics f^ll^t^LTXTl^-
pje, professor Hogben whose popular book, Mathematics for the mt
Uon, is written from the point of view of^EK5H5ST5EEiriSH5 m
sven co oho extent of being overt propaganda, says: "The statement
\J3 = CD does not mean < the line AB is exactly equal to the line
3D,' because no one knows how to make exactly equal lines with
any actual compass or rule. Its correct translation is ' measure
AB tp_get the length of CD as accurately as you need it./ [28j
lini as a refutation of the proposition that a straight line is
the shortest distance between too points he cites the example of
m experiment made on a shrimp whose directional movements are
controlled- by a certain organ connected with the nervous system.
If this organ is filled with steel fillings, the shrimp swimming
m a magnetic field will move in curves since the lines of force
in a magnetic field are curved. Consequently for the shrimp a straight
line is not the shortest distence between two points, (29) We
So not consider it necessary to give an explicit refutation of
this view of the nature of geometry. So much has already been said
about the essential abstraction of mathematics from sensibility
that it would be superfluous to labor the point any further. Nor
loos recourse to the etymology of the word geometry which signifies
the science of surveying afford any rational basis for the advo-
cates of "physical" geometry. In recent years the so-called con-
crete" methods of teaching geometry have become increasingly po-
pular, Whatever we may think of these methods as a Pf ^°Sicai
fevice to gradually prepare the mind to the effort of ^J^f
abstraction it is evident that one does not really enterinto the
realm of geometry until this abstraction has been achieved.

Einstein's views on the nature of SJ^aTa?
levant here. In his book Gecm^LS^BSS^ .* ^purely
fa-y into two distinct branches. The f irs t °f^^ fl P of . the
lorraal knowledge based on axioms that are " e enipty of

toiam mind and made up of schematic concepts w ^ . g ft m _

*«. content. The second is called practical geom yj. ^ branches
Wal science, and is in fact the most ancient o ■ ^ is real iy

<* Physics, Taken as it stands, this °P in *°" " first . branch of
a denial of the true nature of geometry. io ^ second

geometry seems to be nothing but di actios , ^ no place left

b ^nch is identified with physical science,

o:

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a si>

,cifically distinct and proper science of geometry.

Once again we do not feel it n»™
, refutation of these views. They have \lL TfT 7 to ™^ into
;. iHg into focu8 the point to Tl^ZsleTil thf ^^ te ^
;, at while on the one hand mat he^iTifinde ^^^
; ,p„.ence and hence not to be identified with physical scSce
-, a he other hand it is not independent of all reference S sense
in dialectics may be, . ■ u fasnse,

Though detached from external senses, mathematics
\ioa an essential connection with the internal sense of iniagination
it is in the intuitive imagination that all the judgments of ma- '
theiratics must terminate, either directly and immediately, or at
i.ooa'fc reductively. And this brings home to us once again the in-
toryrsdiary character of mathematics. Unlike physics and like netaphysic
it is independent of external sense experience. But unlike meta- I
jhysios and like physics it still retains a terminal connection
yi th sense life, Mathematics is at once both more free and less
free than metaphysics. It is more free in that unlike metaphysics ;
i.t not only does not have to terminate in sense experience, but
vin judgments do not have to correspond with anything that is given
in objective reality. It is less free in that it has to terminate
in the intuitive imagination* It is because of having abandoned
this intrinsic connection v/ith imaginative intuition that modern
atheiiBticians have arrived at the notion of mathematics as a _ science
that is empty of any objective content, as a science that is in
the last analysis identified with logic. It is evident that the
true view of the nature of mathematics holds a middle course bet-
ween the 'concrete 11 notion of mathematics which seeks to estaDiisn
vn intrinsic connection between it and ^'^SS'
"id the purely axiomatic notion which severs all °°™ e °£°™
the internal sense. Both of these extreme jxews will evi dentg
have repercussion upon our problem. By holding the £^ £ tg
0ii3 could be lead to believe, that mathematical P^ 8 " 8 ^ W
in discovering the mathematical world in the pnys ^

holding the second one would be forced to o?™ 1 ™* * objec tive
Won provides the empty forms to whi °JP^°J t | al rules of the
content, or that mathematics reveals the fs- 8 " universe.
gavr.3 which the scientist plays with the physical univ

Mathematics and the *^^^f£w"
'fi-eion to external sense experience. &** ly as a presuppo-

''^tion is dependent upon the external senses _ t can to

■*Won, Once it has received its material fiom context from
^■0 extent detach this material from the pe roj p ^ sical CO n

•^h it was drawn, that is to say from the

-284-

(Ui.oM which embodied it originally. n kp nn+ , ,..
-„-.,•, fcuiot and reconstruct this material int^^ "^ 1CS ' " can
;;irills . it can create new entities only r^T p tania ^ V&i ~
t5; , ,,-atorial to which they owe their origin Sd T ed ^
•^rottics must retain some connection- with the f™ ^ a ^? n wh y
lto!i though freed from the determinations^ B^Sf^S?. 18
u l0 not freed from all materiality and hence^t m swfsoT'
- :xf verrain bound up with a cognitive power rented to n^eriSitv
Chough prior to the whole sensible order by reason of its beine
fe primary accident, quantity is nevertheless known to us only
tlivo\igh sensible determinations, and hence even after it has been
letaebsd from sensible qualities there is still something of sense
-ai-oging to it. It is the imagination which, though a sense faculty
r,yl thus essentially distinct from the intellect, is nevertheless
in the existential order bound up so inextricably with the workings
3? the intellect, which makes it possible for mathematics to re- '■
tain its orientation towards sense, even though it is so far ad-
v.viwcd in the order of 'intelligibility. The object of mathematics
Is never purely intelligible.

But this connection of mathematics with the imagi-
rvtive intuition must be rightly understood. In the first place,
fe intuitive schemes which the imagination presents are not in
t.bssraelves the object of mathematics; they are only the sensible
illustration of that object. Moreover, not all branches of nathe-
fc-.Vdcs are equally dependent upon these intuitive schemes. As has
already been pointed out, arithmetic, because of its more abstract
ohavacter. is more remotely connected with the imagination than
bp™,p*™ V™ =™, n„ri n-P nV^ntasm will serve to represent number,

geomsti
pvovided
t-o.
patios

ter, is more remotely connected witn tne imagma uxun „^>
ty. For any kind of phantasm will serve to represent number,
■ovided there is plurality; but only a very definite kind of phan-
tom will serve to represent a circle of a *^'^"™u \
rtio B tatea fuller advantage of its Cerent liberty, and^ as it
Allows its natural tendency towards ^fherabstraction and^_
realisation, the connection with the ^^^°K £l
"i-ogly attenuated, lb would be ridiculous to ^^ erfect rec0 „s-
mthematical entities must be capable of direct an ^ ^^ ^
oruotion in the imaginative intuition, am * inmBdiate iy in the
<* the judgments of mathematics must ^Tl theraa tios to an in-
vagination. Such an assertion would Hm"
"nitesimal fraction of its actual range,

But it is possible * ^&^e%

>Ao orientation of mathematics towards^ ™ ^m^e

'^ essential relation which ^^^Lmtical atetraotio^
^ter, which enters intrinsically xnto m ^ . f ^ 'uS

*> Kvo explained that mathematics, wh ale V* ^thernatica;

"^er, clings to intelligible matter. Non

-285-

-..M.idcvai-i sine intollectu substantia quantitatt „,k^ *.
,, ; ; „„, abstrahi aatei, intelligible^. " o fV*
,, ;;n m g ible matter is understood the material sub tance as Xter-
,,,,d by quantxty in so for as quantity is the order of its mrts
;liy it is called intelligible natter is explained by St. Thomt*.
...nlxvtontia enim remotia accidentibus non remanet nisi intellectu
^rehensxbilis, oo quod sensibilos potentiae non pertingunt us-
ie o.d substantias comprehensionem. Et de his abstractis est ma~
M;w^tica " (31) Though this matter is rightly called intelli-
gible, it has an intrinsic connection with the imagination, pre-
finely because it is matter. For mathematical forms are not purely
intelligible as metaphysical forms are. They are like natural forma
in that they are in matter, "Sicut naturalia habent fornam in ma-
teria, ita et mathematical' (32) And just as the presence of
ramble matter in the object' of the study of nature makes it ne~
nonnovy for sense experience to enter into the understanding of
thin object, so the presence of intelligible matter in the object
3? imthematics makes it necessary for the imagination to play a
povt in mathematical intellection.

In his quae sunt per ' abstractionera, idest in nathe-
maticis quorum ratio abstrahit a materia sensibili, rectum
oe habet sicut sirauni, Haec enim matheraatica habent materiam,
sicut et naturalia. Rectum enirn mathematicum est, suum autem
naturale. Ratio enim recti est cum continuo, sicut ratio si-
mi cum naso. Oontinuura autem est materia intelligibUis, si-
cut simum materia sensibilis, Unde manifestum est, quod aliuct
eat in nnthematicis res et quod quid erat esse, ut rectum _et
vcoto esse; unde oportet quod alio cognoscat quod quid erat

esse horum, et alio ipsa, . . na+ „ nditur , quod intellec-

Unde sicut per naturalia °s*™J*™?' ^ ius a sensu
tus, qui cognoscit quidditatesnaturaliura, sit aliu ^

qui cognoscit ipsa naturalia singulars, ita e x ma
oatendltur quod intellectu s ^.^°£*^£«lit ijsa
rum, sit aliud ab imaginative virtute, quae a PP

rrathematica,, (33)

It is clear from this ^"^V^^
Bible natter plays the part of the matenal elemen
°al definitions, (34) ' . i na-

The principal role $fi^J$J^^™
thenatioa in connection with ^} l f^s^™ said about the na
Pointed out in Chapter II. W what has w provides the -

; '"ve of intelligible natter 'it is ev^cnt tn ^ ^ wh ole mathe
^.ogeneoua exteriority that is at the ^ waffl a mU^
'^tioal structure. Now homogeneous ex*

-286-

ntic

,:,; a of the same form « such a multrDli™+-i™ ^ .
■ s individuation And this individuation nHsTtaLXr^hf"
vv^Uve intuition. For since mthemtioal entities Se strta-
, a ,,c sensible qualities, the individuation cannot be effected
■ (lU0 1.itative determinations grasped by the sense* n« vZltl
.H the intellect of itself has L do'with ^S^^
•,,,,,, i-,-.i-oter, and hence if it alone functioned in mathematics we
.gild have no notion of homogeneous multiplicity. For things that
r,i outside eaoh other because of the form are formally different
Lonoo heterogeneous. Speaking of Plato's doctrine of the intense-'
vy position of mathematics, Aristotle says:-. "Further, besides
inible things and Forms he says there are. the objects of mathe-
,;i.c;i, which occupy an intermediate position, differing from sen-
sihle things in being eternal and unchangeable, from Forms in that
■■hey e ax ~e many alike, while fform itself is in each case uniqu e."
3!ff

There remains just one last point of which passing
ra'<Ao\\ must be made before we bring this discussion to a close,
in bis Coimientary on the Posterior Analytics St. Thomas explains
bint intelligible matter is ipsa continuitas . (36) Taken in its
itvicbest sense, then, it is essential only to geometry. Neverthe-
lean, even arithmetic must terminate in the imagination in some
.ay, in so far as number is caused by a division of the continuum.

4, Mathematics and the Human Mind.

There are a number of reasons why physic n W«*J.
holies out to mathematics for illumination, *s
Mraady been touched upon. But at * hl3 .P°^tInt Causes of this
ttrarfav attention to one of the most Bipa*™* exis ting between
i^'val gravitation: the profound congen iaxi ^ ^ of the

^thernatical science and the human ml " d ' ^"L noraen al development
Renaissance when mathematics commenced w y per f e ction, and

"hich has brought it to its present high P°g* * he fao t of this
r 'ten physics began to be increasingly q.«an . g ^ otei as

"onvuxtuvaliiy hSs been clearly recognized.Kep ^ taoW no thing

"Wing that ou, minds are so constructed that J ^ fo
f-feotly except quantities. "Just as the ey^ ^ a to .

^»vn, and the ear to hear sounds, 3 ° ^If quant ity. " ( 3V ) fand
"■"Wntanrl, not whatever you P lease >^ion between the mind and
fcWwa.' insistence on the close relation

-287-

AohemMcs ±3 too well known to need i, ^„
,„ fact of this congeniality £s Loom ^vS "S" ** ^
I.;, Ms not been so clearly recognised, It ilsSn^W^w^-n
lu eompar son. with moderndevelopments mto^SS^Jg™*
fc quantification of physics were only in an incipient IT.t I
, ;r ,e of Aristotle and St. Thomas, bot/of C pg 2 p h r ^
ynl.y grasped the fact of the intimate relationship beSn ?he
intellect, and mathematics,, but also gave a clear and adequate ex-
xbnation for it. (38)

As Aristotle points out, (39) difficulties which
liond in the way of the mind's perfect union with a scientific
object may coine either from the mind or from the object,, In the
-nse of Metaphysics, the difficulties come from the weakness of
I'.he human rdnd, For metaphysical objects because of their complete
reparation from all matter are of all scientific objects the most
Allowable in themselves. But in relation to the human mind they
r,;e the least knowable For their high degree of immateriality
■:eeps them from being within easy reach of an intellect which is
^saentially united with matter and which must derive all its know-
ledge from the material world through the medium of organic facul-
ties. In relation to metaphysical objects, as Aristotle goes on
to explain, the human mind is like the eye of the owl for which
the light of day is too bright to see well, and which can Bee with
ii-eater clarity in the obscurity of night. And this explains why
Cor Aristotle and St. Thorns metaphysical wisdom was southing
too divine to be possessed by ran except in a very ^equat? and
pvacarious fashion, something rather .loaned to man than actually
given to him outright.

in the case of physics, on the other handle dif-^ ^ ^

Cienlties come from the object. For 00 ^°JS'obsoure. It is
-tfcrnrd in the flux of mobility, are essentaaUOT triuKph over
tme that by remaining in generalities _ t he nin ^ . nevitable
this obscurity to some extent But as it j™^.^ aeriv ing from
progress towards concretion, the light a x physics is

generality gradually fades. Now inoa ^ n . e r far adV anced towards
a stage in the study of the cosmos chat is Jt is obs-

ooncvetion. That is why its object is &°fj. of imt ter and mouion,
™ve first of all because it is cosmic reaii ty ^ „

U is obscure, secondly, because ^ a^ts g ^n

rto reality in its concretion. In ^.Tfro^ a certain point
^tellect is caught in a kind of ^J^ to it. ^"J"
<* view, it is in a realm that is wost pr P ^ x things,

^ in h«n, its proper object is th ess ^ ^ no t 0"

»«* since it is an intellect it » f^fio con cretion. Andy
'* a general way but in their proper sp

-288-

■ following the instinct of its nature it ineiri+nvi v

,vs.;d i"-A rtoopo::- and deeper obscurity. lnevitab ly becomes to-

Now mathematical science occupies a m-^w^
Uion between those two extremes. On the one ha^d" sSc^S ?T
„,,^ s i'ror, mtter and motion, its object is more intelligible "
* S o linn vhat of the sclent of nature. On the other hand, since
i .~ not. completely iwnaterial, since it always retains an'essen-
:.ol ccnnection^with the imagination from which the human intel-
0;)t dorivou all its-concepts, it is more intelligible for us than
hot of metaphysics. "Sed mathematica sunt abstracta a materia,
i; i-omen non sunt exoedentia intellectum nostrum: et i£eo in eis
f.t requirenda certis3ima ratio " (40)

Another reason . for ' the oonnaturality of mathematics
i.th the human mind is given by Aristotle and Saint Thomas in the
ixbh book of the Ethic3o (41) The intellect finds the science
inch deals with sensible things difficult because it demands a
voat dsal of experience j it finds the study of metaphysics dif-
iou.lt because it transcends the imagination and is free of all
sference to sense. In 'between these two extremes stands raathema-
■fica, "quae nee exoerientia indigent,, neo imaginationem tranooen-
■nr.t," One of the signs of this connaturality is the comparatively
vsnuert oocurronoe of child prodigies in mathematical science - -
i phenomenon that is not found in the other speculative sciences,
42)

Shis profound attraction which hematics has to
fe intellect can. constitute a-danger. For it is easy for he mn
io try in one way or another to reduce all to °^ e ^™ ble
deal knowledge; and to reject ^^ *£**%£ %£>*»***.
io this reduction, Descartes, we know, foJJ. a P y .^.^ nlsi
in St„ Thorns remarks, "quidam non reoip^uno qu ^ ^^ ^ ^^
lioatur eis per modum mathematician,, \^> , denoy is gonetimes
?oes on to explain, that a similar monisD 10 ^^ ^

!W with regard of other typco of ^ e g^ of the connatteal
we acute in connection with mathenatios becaus ^ why Aris totle
faction of which we have ^Vf ^ mture must not be reduced
?A St. Thomas insist that the study of nana
to a kind of mathematics a ...... op ti-

Ostendit quod Ula ^^^f&f^f^
m 3 , non debet in omnibus ^[j^S I nmtb^cis, s d
idast diligens et certa ratio, ^f^ sunt | scicntaae, ^
debet recuiri in omnibus rebus, do q ^ ^ Ea

deb,t solum requiri in his, qua na ot veria tiom.

quae habent mteriam, subjecta sunu

-289-

: ;,loo non potest in ou omnibus omnlmoda asrhibifln v, ^ ■
vitur enir.i in ois non quid semper sit et 7 ^° ri * Quae "
q „i,l sit ut in pluribus " (44) ' ^^tate; sed

Prom all that has been said thus far it i s oleir
;-U this passage does not intend to exclude the possibilitTof
. x application of mathematics to the study of nature. It is merely
, 7 ing to point out that this .application is not an identification.

But we have not yet fully explained the connatural
ttvaction which mathematics exercises over the intellect. There
:; an innate tendency in the human mind to see one thing in another,,
hio is the root of all scientific endeavdr, whose purpose is to
oo tilings in their causes. And the source of this tendency we
,irv, r : every intellect is a reflection of the divine intellect which
ees all thing3 in their proper specification and in their ultimate
or.raretion in the light of the one divine essence. And not only
ocs every intellect seek to grasp one thing in another, it also
:go!cs to construct otherness out of sameness, It strives to become
i\e the divine intellect by constituting itself prior to things,
,y racing itself the creator of its own object. Because the human
n;-,Alect i3 hui.ian it will always in some measure be subjected
;o givenness; but because it is an intellect it will strive to
roiumph over this givenness by making itself the source of the
Mngs it knows, thus dominating its" object completely, Now t he
^limited constructibility of the mathematical world provides the
dullest freedom for this tendency of the mind, ^f^t it
intellect is able to construct its own object, f ^^g^ a
in able to construct a line, from a line a plane, from a plan
,olid, etc. And it is only after the °°»f ™g™ f ^ the mini
that the properties of the subject become >^ifest„^ ^ ^ ^
instructs the source of these properties, x ^^ them t(?

Tther sciences merely discover the Properties ^^ sK , enoes

lead it to a knowledge of a given subject, in a
tho subject is givenness there is obscurity.

Mathematical abstraction has £*^^£Z
■that the most knowable inje is tho most kn ow ^^ s

other two typos of formal abstraction, th^n^ pr inciples
in. the least krowable inj^e. Unlike ^ e ^% leB of nathanatioa
of the other speculative sciences, the 1^ ^ uni^salj^
«vo at the same time universal i\£^^tical world ^ « ins
r^aanfto. And that is why the f 1 " 16 "postulates. And this exp^.
--o>^7ew fundamental pr incipl g an ^ dom , as Courno ^ re _
rtV in some way mathematics is W? V^ property of visa
"ophiae gcrmana mathesis. For it is ^ x aou roe, ancl
voo:i::all thTnglTiH" the' light of an ori i

-290-

, : ,luctibility of mathematics enables tho mawi +~

„,ieal world_ as flowing out of tho 0x5^?^^ 2f

,,., wo expiated above, mathematical particulars fre'abstL
> ,ov.m souse identified with universal, this process of mi-
mical wisdom is able to roach ovon particulars. In a wav
ntios satisfies tho mind's innWnot for wisdom oven better
.-.■-; taphysics, for since m metaphysical abstraction the best
for us ir. the bust known in se, tho v/holo metaphysical world
3 t bo drawn out of tho original principles. That is why after
i.nd has -pursued its course from the original generalities
cough tho angelic universe to the divine being it must, in

i;o satisfy its quest for wisdom, complete its study by having
rso to a dialectical process by which the multiplicity of
s are derived from the divine source.

In our introductory Chapter we pointed out that Plato
lived the mathematical world as occupying a kind of interne-
■oosition, and we suggested that this was an extreraaly pro-
ana fruitful insight. There are, in fact, many ways in which
matical being is truly a median, Some of them have been touched
and other could easily be adduced, (45) But here we wish
.attention to one particular aspect of this -termediary
; c ter of mathematics, for it vdll serve to throw ligho upon
joint wo are trying to develop.

Mathematical being is a medium between g^J^
,uroly imperial being, and it par -^^^1
, In the first place, though i. is J^nc v , hich frees

, because of the nature of ^ th9, :^^^^le from it in the
■;om sensible matter, it ronnins inseparable t ^^

, of always being linked to it ^ an intrinsi ible ,

, As a matter of fact, if the '^^^n,. For it is
^thomxtical world would likewise be impos ^

in a world of composed essences, in whic that

incomplete because of the comaon matrix P ^mW
mtho matical world can ^S 1 ™ ^cnoity, and consequent^ of
; provides tho source of the bomege n ^f , ^jos. Th

univocal relations which are oBsontxg ^ , s Q s ^

^.^tical world is a world of ^^% ca a is » Aeneous
.olity, a Mr* of mtorial ^ g^ f rom the c £ g"
-'genoity. It is something quite c ^ ^Tti '«° rld

vality of the wo:,ld of Bopora^a* ^ ^ mathe^ ^
'X'.enoity an,l the common ^ rl *.:. v and tho pure dis* h0B0 _

,o ia /lack of the perfect uniWJ 1 thQ siU * W ™. ng ,-
-1 in tho separated substances. BJ laok of un^ a is
vity provides a substitute for tni jiiathom tical
one of the relations out of whion

Ld

the

-291-

or, trucked. On the other tend, the mathematical vrarlcl i s a vroria
fon.nlity ovon though this formality i 3 not pure. And that is
lY it transcends tho world of contingency and obscurity, and be-
,,,;<; : V world of rationality and necessity. This brings it close
^ the. spiritual world and transposition- from one to the other
,oor.os possible, it was indeed a profound intuition on the part
"■Plato to give to mathomtics an intermediary position between
-o "Sane" anA tho "Other". (46) By its very nature mathematics
in >ars to us as a principle of reconciliation betv/een areason and
\iorial nature* (47) And all this enables us to understand more
1 early why the mthovnatization of the cosmos ci'.nlead, and often
w ' j ec -|_ \ Q both lvatorialism and idealism, It is only by understanding
'-- true nature of v.iathemtical abstraction and the intermediary
boractor o£ the science that results from it that these two ex-
,-,;ov:cs can be avoided.

Now it is this intermediary character of mthemtios

tot «*** it the ideal ^^^%^^]^Z^ S
)u t mtter so^midu^nxelligx it J^ 1 ^^ retter seoundum
yf necessity and rationality; because it is vroh >.v_ -_-^
; 3S e it is applicable tocosmic "^^^ m tural obs~
u£trur»nt by which physics my be lifted out ot ^

ouvity and contingency into the ^ aim .oMM ^ ^

into a ata* that is in so Respect s s ^ lt is at the

toing a nediun between the ^^J™*^ the subjective, as
s.me time a nodium ^ween the ob^tive an ^ ^

™ saw in our discussion of * h %reLation ss M a aoientifio ™"
hi adds Measurably to its of ^*^^ k out its ovm rational
went For it leaves the mind free to wr g being

:S' and yet it provides the gssibxli^ rf ^ lS
applied to cosv.nc reality, inc.

cx'ci-erely relevant heres e du

p. * cue le Bothematique, se detach^ cu
reste du ^ "-^ jjg- - X; J-^

._,-,,„..„ „„*„■! 1'attrait que J- e ^ . ,„u ivers qu'i-L ° 01 .+. hu -

fruste irranediablo do 1 *™& n& vnt sur

exerco et cxercera sans doutc
uain, (48)

f m,^f

■2JL. A_S E R T A T I N

PRESENTED

TO THE -FACULTY OP PHILOSOPHY.
, OF LAVAL' UNIVERSITY

TO OBTAIN

BY ;'.'."

'■■'BERNARD I. MULLAHY, C.S.C

LICENTIATUS IN PHILOSOPHY

.. . GREGORIAN UNIVERSITY, 1 ROME

Thoiiiism and Mathematical Physics ,
: ' ' .)/■ TQMI. '■'■' 2 ■■'■■■..'■ .

_ _ _JHLY^1S46, _ _ . _ _

ro/xWi (TriVi^i (ft~3) vA.e.AAiwewti**' co^v WvCM.e. -\ke1e_ 7h1\e-vAwc&4 ovft^ »'vn

o

13

2, Synthesis}

a. The Principle of the Synthesis;

Vl) Science, Sensibility, and Hcrofionaity. ...... Chap , m

V 1 The ProMem.... ......... ........^..........^^^ 2g2

\/ 2 The Nature of Sense Cognition. 299

\/ Z Science and Sensibility,.. ...... .......I............ 312

v 4 Science and Homogeneity. ... 5 ,.,..., a ...... .

> o o o c e o

317

2) ^li22il£2i^^ chap, viii

</ 1 Science and Measurement. ....,...<,„.„„,;.„. 339

V 2 The Nature of Measurement. . .......................... 345

3 The Limitations f Measurements .,.........,.,....„, 3S3

( (JVvf )

vi

h. The Results of the Synthesis*
1) The Physico-mathematical world*

ft) The Mathematical Transformation of Nature..

1 The Transformation of Natural Science.. , . ,

2 The Transformation of Nature, ....,.'......,

h) k Shadow V/orld of Symbols, . , ,.,......

1 The Nature of Symbolism.

2 Symbolism and Mathematical physics. . . . ,

3 A V/orld of Shadows,.,

a « o o o a o e o <

0....0.0 Chap, ix

........ 394

............ 410

.■»......,. Chap, x

I......... 426

.......... 431

.......... 438

2) The Real Y/orld.

a) Relation between the Physico-mathomatical world

and the Absolute V/orld Condition, .. „• .>...,..,.. « Chap, XI

1 Isomorphism,

...coca .. ...... .eueo.ti

445

2 Logical Identity, ...„»..,.,

3 Movement towards Real Identity

oeoooco«<

b) Objective Subjectivity, .„..,,.,.„. . .. . . „ . .

1 Subjectivity and Objectivity,, ,,.,..«,.,.. .

2 Mathematical Physics and Kantianism^ . „ „ „ .

III. Conclusions

The Nature of Mathematical Physics. . ,. ....... .

1 The Essence of Mathematical Physics ,j

2 The Existence of Mathematical physics

. . . . i . . . 455

....... o 459

........ Chap, XII

• eaooieo 4e ( O

i • > * » c a

o Chap, XX II

O O O O 3 « (

,..* 490

ojooo»qpoi>»o 'ii/O

vii

SSCTION TWO

APPMDIX

Io NotOS. •o»oooo«i«..ooom....'»co«e<

EI» BiMiogrnphy......... ...... .............. (98)

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CIIAPTER SEVEN

SCIENCE, SENSIBILITY, AND HOMOGENEI TY.

1, The Problem,

This Chapter marks a turning point in our study.
In the last three Chapters we have been concerned with a deline-
ation of the salient characteristics of the two sciences whose
union constitutes the intermediary science of mathematical physics,
Y/hatever this delineation has accomplished, it has certainly brought
into clear relief the profound antithesis .which lies between these
two sciences: on the one hand, a science which sees everything
in terms of mobility and sensible matter, a science of contingency
and obscurity; on the other hand, a science which prescinds essen-
tially from mobility and sensible matter, a science of necessity
and rationality,, A more radical antithesis could hardly bo imagined
than the one which exists between those. two studies. And yet out
of this antithesis must come a synthesis if mathematical, physics
is to exist. It is to the nature of this synthesis that we .must
noir turn our attention. We shall devote three Chapters to an ana-
lysis of how this synthesis is effected. In the remaining Chapters
of our study we shall consider the results of , this synthesis,

Tho general problem which immediately confronts us,
then, is this: how does the mathematical world lay hold of the
vrarld of sensible phenomena and transform it into its own likeness
and image? Anyone at all acquainted with science knows that the
answer to this problem lies in one vrorfl.; | measurement,) But before
vre can come to an analysis of the process of measurement, a pre-
liminary question imposes itself: what is there in nature itself
which makes it amenable to this transformation through measurement

-295-

wl Tr °^ ° f '£ the ^ti°al symbolism? Measurement is the ins-
, -n ttio coaroos itself a basis for this nathoimtization.

. ,, _ nn Duhem has posed the question which confronts us hore
in oho i oilowing terms •

Pour qu'uno theorie physique so puis se presenter
sous la form6 d'un onchnlnomont' de calculs algebriques, il
iaut que toutes lea notions dont olle fait usage puissent e~
tro fxgure es par dos nombres ; nous aommes ainsi araenes a nous
poser cette question; A_o^eU^_oojidition un attribut physique
Jl gu^-i-i otr-e g ignj^fiej^a^jxn^yj:^, ? 10 nuni e_ r ig]jg.?" (l)

And to this question he gives the following general

answer

^Cette question posee, la premiere reponse qui se
presonte a l'csprit est la suivante:' Pour qu'un attribut que
nous rencontrons dans les corps puisse s'exprimsr par un sym-
bole numeriquo, il faut et il suffit, selon le langago d'Aris-
tor,e y que oet attribut appartienne a la categorte de la quan -
*iM e ' b non » la c ateg orie de la q ualite; il faut et il suf-
f itj pour parler un langage plus volontiersaccepte par le ge-
oraetro moderne ; que cet attribut soit une gra ndeur , (2)

This general answer is fairly obvious } and was al-
ready implicit in what we saw in the last Chapter about the nature
of mathematics and the link which binds it to 'reality, But it is
only a. general answer, and it stands in need of a good deal of
explication. And perhaps we can orientate ourselves towards a
more • definite solution by presenting the issue, in the following
terms: Since mathematical physics consists in the union of a sen-
sible world with a world which prescinds from sensibility;, the
suture v/hich knits the two together must be along the lines of
something v/hich is at once connected with sensibility and indepen-
dent of it j something v/hich while not sensible in the fullest sense
of the word, is nevertheless sensible in a secondary sense. Pre-
sented in this way ? the problem immediately calls to mind the Tho-
mistic doctrine of proper aensibles and common sensibles, of whic h
the latter ar e all reducible to quantit y, f even though in themselve s
they are not"~quantity .J by the very fact that they are sensible ,
17a believe that it is in this doctrine that the fundamental solut-
ion of our. problem is to be found „

And we know of no better way of bringing the quest-
ion into better focus than by having recourse to the well-known

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advcnture of Sir Arthur aldington's elephant:

Lot us then examine the kind of knowledge which is
handled ^ by exact science. If we search the examination papers
m physics and natural philosophy for the more intelligible
questions we nay come across one beginning something like this:
•■An elephant slides down a greasjr; hill-side,,.' The experienced
candidate knows that he need not pay much attention to this;
it is only put in to give an impression of realism. He reads
on; 'The mass of the elephant is two tons, ' Now we are getting
down to business; the elephant fades out of the problem and
a mass of two tons takes its place ,„ Let us pass on, 'The
slope of the hill is 60> , ' Now the hill-side fades out of the
problem and an angle of 6CP takes its place,.. Similarly for
the other data of the problem. The softly yielding turf on
which the elephant slid is replaced by a coefficient of frict-
ion^ which though perhaps not directly a. pointer reading is
of kindred nature o0 ,

'tie have for example an impression of bulkiness. To
this there is presumably some direct counterpart in the external
world, but that counterpart must be of a nature beyond our
apprehension, and science can make nothing of it Bulkiness
enters into exact science by yet another substitution; we re-
place it by a series of readings of a pair., of calipers. Simi-
larly the greyish-black appearance in our mental impression
is replaced in exact science by the readings of a photometer
for various wave-lengths of light. And so on until all the
characteristics of the elephant are exhausted and it has be-
come reduced to a schedule of measures » (3)

' This remarkable passage brings out with great exact-
nous the fact that it is through the instrumentality of various
[types of_j^asuremont)that the cosmos is mathomatioized. But it
also suggests what the basis of this mathematization is. For it
is evident from the concrete example hero given that when the ma-
thematician seeks to lay hold of the material universe all the
attributes of this universe which are known in Thomistic termino-
logy as proper sensible s and in modern terminology as secondary
qualities slip through his fingers. And no matter how many efforts
he makes to recapture them, they continue to elude his grasp „ With
their passing, the very natures of the things h e is~d oaling_ with
vanish. The characteristic qualities of the hill-side, the green-
ne3s~of the grass, the softness of the turf, etc, fade out of the
picture of the physicist - - and the hill-side fades with them.
And the same is true of the elephant itself.

Yet it is dear that the exact scientist lays hold

-295-

of so,. Mu hint; m the material universe, otherwise his sciexice could
■nVM n? ^. onll ? d Phyaips. It is likewise clear- that he lays
nold of something which though in a sense independent of sensibi-
lity is ao the same time essentially, connected with it. He does
not grasp the greyish-black colour of the elephant in its proper
mture ; yet the wave-lengths of light which register on his pho-
tometer are ^^sje^ially^onnecjte^w^ this greyish-black colour,
tu e \ ld ?" dy the ^^g^M^FT^la^Thold of can be approached
through the avenues of more than one sense For, a blind, scientist
can have a perfect knowledge of optics, (4) a deaf scientist can
be expertly proficient in acoustics, and if it were possible to
live and have sentiency without the faculty of touch there would
be nothing to preolii.de the possibility of the science of thermo-
dynamics, This common character of the object with which exact
science directly deals manifests its nature: it reveals the fact
that it is intimately bound up with homogeneity . And all of these
considerations lead us to this conclusion: mathematical physics
prescinds from proper sensibles; its object falls within the domain
of the common sensibles,,

The views of the modern scientists and philosophers
of science conf irm this conclusion, even though these views are
not expressed in Thomistie terminology. Max Planok, for example,
has this to say:

Now all physical experience is based upon our sense
perceptions, and accordingly the first and obvious system of
classification was in- accordance with our senses. Physics was
divided into' mechanics, acoustics, aptios, and heat. These
were treated as distinct subjects. In course of time, however,
it was seen .that there was' a close connection between the va-
rious subjects, and that it was much more easy to establish
exact physical laws if the senses are ignored and attention

is concentrated on the events outside the senses if, for

example, the sound waves emanating from a sounding body are
dealt wixh apart from the ear, and the rays of light emanating
from a glowing body apart from the eye. This leads to a different
classification of physics, certain parts of which are re-ar-
ranged.- while the organs of sense recede into the background.
According to this principle the heat ray's emanating from a
hot stove ceased to be the province of heat and were assigned
to optics ; . where they were dealt with as though entirely si-
raile.r to light waveso Admittedly such a re-arrangement, neglect-
ing as it does the perceptions of the senses, contains an ele-
ment of bias arid arbitrariness a (5)

But this concentration upon primary qualities to

-296-

tne w.cLusion of aacondary qualities is by no means pooulior to
Hocu-a-ii soiunoe, A definite movement in that direotion is discer-
nible almost froi.i the beginning o-£ the systematic study of the
cosmos. j.u is true, as Planck points out, that in the first stages
of xcs development natural scionce identified the sensible and
thc_ph ysical . This was inevitable, since, as we have seen, pure
na-cura. science ip o. study of reality in terms of sensible ratter.
Physics •'•ooic its o.-.-ijur. when man brgan to observe and analyse
porcov/oibio properties and to express the results in descriptions,
ittis or,.iD±ed hiis to introduce order into his cognitions by means
of cl^ifioayio.:;. Regular recurrences in his sensory experiences
<,e.„p\. hot bodiis Decora cold; a swinging object comes to rest, '
uco.:) uado it possible for him to arrive at general laws based
on qualitative univ'ormi oies , But the persistent attempt to perfect
this .rudimentary knowledge 5 to analyse these classifications and
unifovvvj^jes with greater exactness, and to render then more ra-
tional inevitably 3.ed to a dissolution of the relation of identity
between the sensible and the physical, and a gradual abandonment
of sensorial categories in the 'explanation of the physical world.
In some cases this abandonment became not only methodological .
bu t philosophical ,. Already in Democritus and Lucretius we have
an explicit, denial of the ontological existence of what were later
to be known as proper sensiblos 'or secondary qualities. It is only
by opinion or oonventicn that they can be said to exist, At the
tines of the Renaissance this doctrine of the ancient atomists was
revived by such :,\oy. as Vives, Sachez, and Carapanella, and this
revival;, together- with... bhe astounding success of the new mathema-
tical r.iit'.iod i:o. physics,, had a profound influence on the opiate-
molo£:'/--al views j.f tv.ibsoquent scientists. As we saw in Chapter I,
KopIej: ; while adir.it ting the objectivity of the qualitative deter-
minations of nature..- r,v.\iritained that they v/ere somehow less real
and fundamental ilian the . quantitative determinations. Galileo went
further than ftoplsr and made" the secondary qualities subjective.
Per him the quantitative determinations of nature were absolute,
objective and imputable,., and the object of true knowledge, whereas
•che qualitative determinations were relative, subjective, fluctuating
and the 3ource of mere opinion and illusion, Descartes' expulsion
of qualitative determinations from both the physical and the geo-
metrical world, nnd No'vton's subsequent discovery of measurable
correlate of ;jclour in terms of , differently refrangible rays (6)
p - x-ovit'.od both a theoretical and experimental foundation for this
position. And it remained for Hobbes (7) and Locke (8) to lend
tho weight of Iheir authority to make it the generally accepted
philosophical and scientific view. In mochanism tho divorce bet-
ween the sensible and the xJhysical was accepted as a fundamental
dogma,, And whore-v-ir mechanism was accepted as a philosophy, the
denial of the ontological existence of the secondary qualities

-297-

usually resulted,

, . . ^ Contemporary soienae has continued to maintain the
th^T-n^^V^ sonsiblc and ^e physical. Max Planck sees
the evolution of Physics as a progressive withdrawal from the world
oi sense •

But at the soma moment the structure of this physi-
cal world consistently moved farther and farther away from
the world of sense and lost its former anthropomorphic charac-
ter, still further,, physical sensations have been progressi-
vely eliminated, as for example in physical optics, in which
the human eye no longer plays any part at all, Thus the phy-
sical world has become progressive^ more and more abstract;
purely formal mathematical operations play a growing part while
qualitative differences tend, to" be explained more and more
by means of quantitative differences, ,.

'As the view of the physical world is perfected, it
simultaneously recedes from the .world of sense; and this pro-
cess is tantamount to an approach, .to the world of reality,
\")

The gap between .^the world of sense and the world
of physics has become so wide that authors dispute whether "qua-
litative physics" might not be considered a contradiction in terms,
or whether such qualitative propositions as "copper conducts elec-
tricity;" "the melting point of ice is lowered by pressure," can
be called physical laws,; (10)

Recent physics has introduced a new and significant
aspect into this progressive recession from the, world of sense.
In classical physaos, although the gap between the world of science
and the world of external sensibility has already grown wide, there
still remained a direct and immediate relation between the scien-
tific world and the imagination, The scientific constructions of
classical physics were susceptible of direct representation through
concrete images. That is why mechanism was essentially a physics
of models o Lord Kelvin's well-known. remark that he had to be able
to make a model of a thing before he could understand it is typical
of classical physics. But in recent years science seems to have
made a direct break not only with external sensibility, but even
with the imagination. This break was first effected by the intro-
duction of the theory of Relativity and the theory of Quanta, And
more recent developments have served to widen the gap immeasura-
bly. The theories of Schrodinger and Dirac, for example, seem to
be completely incapable of imaginative representation.

-298-

.,, , It is ira P°rtant to recognize the faot that this pro-
gressive Withdrawal from the world of sense has sprung from a f i -
nality^n trinsio t oj^pgri raental science itself . [Itwasnot braagM
^2gll^JjX_5J^trar3 ^ oxtrJ^sio~Tn"f luence ,J iTTpIrtioular, it did
1 ^L^2^ w _ouL2LiHBLj4g ali3tic bias . When Galileo made the secon-
dary qualities subjective, ho understood subjective in the sense
of cint ra-organic) and not in the sense of c psychic.^ They were fol-
ium the product of an interaction between an external object and
a sense organ. Even Descartes, who might perhaps be suspected of
a bias towards idealism, admitted the objective existence of a
reality which caused the secondary qualities, (11) It is true
that idealistic philosophers have seized upon this particular de-
velopment of science as grist for their mill. But science cannot '
be held responsible for the interpretations and generalizations
V of philosophers, .>

And yet the directions in which science develops
have great significance for philosophy. The particular development
we have just sketched presents several important problems which
we must try to solve if we are to understand the true nature of
mathematical physics.

This should be evident from all that was said in
Chapter II about the essential 'relation between physics and sen-
sible matier. In some way physics seems to depend upon the senses
for its very subject, , (12) and yet as it develops it draws far-
ther and farther away from the deliverances of the senses. What
than is the precise relation between physical science and sensi-
bility? Why has progress in science produced an ever widening gap
between the sensible and the physical? In withdrawing from the
world of sense; what is it that science is actually laying hold ^
of in the cosmos? What is the 'nature and validity of the knowledge
that results from this prescinding from the determinations of the

(cosmos that are presented by the senses? Is Planck correct in sta-
ting that this withdrawal from the world of sense is tantamount
to an approach to the world of reality? Has the progressive desen-
sibilization of physical science demonstrated that the objective
world is devoid of qualities or that qualities may in some way
be reduced to quantities? What is it that the intellect is attempt-
ing to achieve fundamentally, in pursuing this progressive desen-
sibilization? Docs this development in any way favor idealism?
These are 3ome of the questions that demand our attention,

At the beginning of this chapter wo suggested that
the key to our general problem might be found in the Thomistic
doctrine of proper and common sensib,les. But the recent develop-
ments in physics to whioh we alluded above might seem to challenge

-299-

^Jf e 7" P ? r ?? me aUth ° rs see in this bre * k with the ima-
Sferfbgs* d °^ natr S 1 ° n ° f th ° illuso ^ ^araoter of the common
^ ^t^-i^f "*, hGy SQ ° ln th ° P rwi ^ withdrawal from exter-
( nal sensibility a demonstration of the illusory character of the
proper sonsibles:

Or on constate sans peine que le discernement entre

10 sensible et le physique, si Men commence jadis, n'avait
Pas etepousse aussi loin qu'il aurait pu, et que sans doute

11 aurai-c du l'etre. De quel droit affirme-t-on la valeur im~
media.ement physique des qualites premieres et des autres don-
necs mather.iatiques percues? La force, et 1'inertie, sont des
notions issue?. ^ direotement de 1< experience sensible, Et l'i-
mage, car_e ; edt bien d'une representation imaginative qu'il
s'agit, l : ir.\age d'un corps a. trois dimensions, dans l.'espace
euclidien, d'un corps qui se deplace sans se defomer et qui
demeure impenetrable, depend indubitablement des. conditions
particul%vos de I 1 experience sensorielle de l'homme. Notions
anthropomorphiquss done, et qui ne sont pas moins liees a la
structure particujiere de notre s.onsibilite que ne l'etait

la couleur orangee ou le parfuia de la violette, II s'agit d'ail-
leurs de ce que los anciens appelaient des sensibles comrauns,
qui ne sont jamais percus qu'en liaison aves les sensibles
propres; si dene ces derniors sont transposes du fait de la
sensabion, il est normal que les sensibles communs subissent
le meme sort," (13)

Perhaps the best way of c.Qning to grips with these
problems is by considering the relation between science and sen-
sibility, But in order to understand this relation it will be ne-
cessary to recall a few fundamental notions about the nature of
sense cognition^

2i - r Jho Nature of Sense Cognition.

Sensation is in many respects an anomalous thing.
It represents' the first confused awakening of matter to conscious
life, It is at once an act of knowledge (which is defined in terras,
of immateriality) and an aot of a' material body. While on the one
hand transcending pure corporeality, it remains immersed in it.
By the fact that it is knowledge it involves a kind of immaterial
trans-subjective union between subject and object. But because

-300-

"n^S? a V°\° f a ^ tQrial ho ^> this union is bound up with
material subjective uninn mwii,^^ w „ „v,.._j_.n ..

a. mat

subjective union produced by a physical movement.

Pn „ , , . N ° w n a11 knowledge is by its very nature objective,
£°i :°, lQ1 1 °T , 1S *? ^ ec01 : 10 '"f^w th ^g in its very otherness,
Bu, not all knowledge is equally objective, for there is a direct
Proportion between the objectivity of -knowledge and its .perfection.
Only divine knowledge is completely objective, for it alone, is
perfect. This does not mean that knowledge vMch is imperfect is
subjective precisely in so far as it is knowledge. It merely means
that its objectivity is conditioned by a certain measure of sub-
jectivity.

Since sensation is the lowest form of knowledge,
it is necessarily the most subjective. It is immersed in matter,
and matter is by its very nature a subject and the farthest removed

| irom the state of object. It is to be borne in mind that an object
is^an object not in so far as it acts physically upon a knower,

\h\iu in so far as it specifies an act of knowing. As we have just
suggested, sensation is dependent upon matter not only from the
point- of view of its object as the intellect is, but even in its
own intrinsic nature. For the senses are. not purely psychic powers;

(t hey are p syc hosomatic. ) Sensation is an actus coni uncti, and mat-
ter enters~tnto it not merely as ajiecjssaiycond^ion/cbirb_as
a co-cause „) That is why it cannot~p^ssiis^ffi _ 61nerness .necessary
for pure objectivity for: "intus existens prohibet extraneum".
In the measure in which cognitive powers must conform to their
object in its entitative state, they cannot conform to it in its.
objectivity.

Professor De Koninck has brought out with great e-
xactness the profoundly, subjective character of uense cognition:

Mors que 1' intelligence est une faculte separee
qui attcint les choses sans lours conditions r.iaterielles in-
dividuantes, lo sens reste, a tous les niveaux, lie a ces con-
ditions de la matiere, Et cela est lo plus manifeste dans les
sens externes, Oeux-ci sont pour ainsi dire diffuses sur les
chos es dans leur concretion materielle, et, "par consequent
dans ce qu'elles ont d'obscurt en soi, sous ce rapport, ils
participant aux conditions mSmes de l'objet dans ce qu'il com-
porte d' irreductiblcment entitatif: la sensation est liee a

/ un organe corporel. On le voit le mieux dans le toucher, L|or-
gane de la temperature a lui-mome une temperature; il a lui-
uiSme durote~~et molTesse; il'''es't''Wtendu7* et* il"el't mesure par
le temps; il a sa masse a lui; il se repand sur l'objet eten-
du; il cede a l'objet dur , et il en epouse la figure; il s'im-

, prime dans l'objet qui l'enveloppej^etc. Bien que les premiers

-301-

nS^^n f, ^ S01ent .^o,^es dans leur explication de la con-
£ tt P I Un ° SMili * udQ entitacive qui serait requise
1m»T! 'T 1 ;™*' ilS ont n ™ oi »s enonce un prin-
sure oS 1 Verif ^-, dU "° nS ' Mais iX s 'y ^ if i° dnns ^ ™>~
sa»L ^M S ! elolgnt dG ln P"™ objectivity La connais-
InSion d "<™ J^aite 1™ quelle domando cette im-

M ^ 01 ■ ^ Cre danS la i;Eaure oi I;L to ^ e au pream-
ble une assimilation entitative dans laquolle le sens mSme
; aot p.-sBif. Le toucner ne pout sentir uno temperature sans
q -^. 1 i " l ;" !U1 « no Inline lui~m6rae cette temperature, Cetie pas-
sibilite, ou nous sqhmos, pour ainsi dire, assimiles par una
autre chose, est, comme telle, a 1' extreme oppose de la con-
naissance; Celle-ci ost, en effet, une operation vitale; mo-
tusab intrmseco L ' iimixion 'aux choses dans leurs conditions —
lm-oerielles l-oste purement instrumentale 3 (14)

, . , . , The subjectivity of sense cognition is so evident

tnax it; nas beconB proverbial; de_gu3tib us et de coloribus non
l r|i^£P^ndy3U The same subject may~receive differemTselSvBions
of the some object, as when, for example, a person touches a piece
of iuotal and a pitce of vrood in a cold room: Though both are of
the same temperature, the first yd.ll feel much cooler than the
second. The some subject may likewise receive the same sensation
from different objects, as when one's hands have a different tem-
perature and are brought into contact with bodies of different
temperature ,

Nov/ we can best get at the nature of this subjecti-
vx-'cy by having recourse to some fundamental principles laid down
by St, Thomas, "Nam sentire, quod etiam videtur esse operatio in
sentiente, _esjfc_^xtra L jMJiuramjinte^^ neque totaliter est

re.motum a genere actionum quae sunt ad extra," (15) Sensation
is at a ■ point in the universe where immanence first emerges from
tto_trans_itivj^cji^ty of jviatorial natures ," It" dFes''lioT'compTe'«
■Eely emerge from ft; lFTremains inextricably bound up, with _it.
For in every act of sensation (a p_hys^jL,_jiiaterie.l( interaotiorj) J*t)
takes place between the material object and the material organ,

?Jfi t .JiC.J^i^..yi^£iS .tiSE l l .°i ) S. e , s -,.ft..'l :> jE < 2^ u Pi , 9 w ' 10SG nature is determined
Doth by the character of "the stimuli which impinge upon the organ
(and these are dependent upon the nature of the medium) and the
character of the organ which receives them. It is this "mixture"
of external stimuli (already a "mixture" arising out of the inter-
aotion between the distant object and the innumerable, indefinable
elements v/hich go to make up the medium) and the complex structure
of the material organ which constitute the direct object of sen-
sation. What is immediately sensed is not an absolute , distant

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be£ l-^i-^ TV* ^/^^l infraction of which we' have

of ^oton^ to'-h ? t r iS ^° n ° f the Sensible ob ^ ot fr ™ tte state
Ss ^triL™ . ? ftoKigjigt apure actualization which leaves

Pi^icaXLy diffgrcnt from the sensible' object in potency sf Tho

-,™,« n+ •^ Pr ° b ^ b ( p hil°sophus) quod supposuerat; scilicet quod
anus ct idem sit actus sensibilis et sontientis, sed ratione
uit.crant, ex his quae sunt ostensa in tertio Physicorum, IM
enua ostensura est, quod ton motus quara actio vol passio sunt
m eo quod agitur, id est in raobili et patiente, Manifestura
est aatem, quod auditus patitur a sono; unde neoesse est, quod
turn sonus secundum actum, qui dicitur sonatio, quara auditus
secundum actum, qui dicitur auditio, sit in eo quod est se-
cundum potcn-ciara, scilicet in organo auditus Et hoc idoo,
quia actus activi et raotivi fit in patiente, et non in agente
e-c movente, Et ista est ratio, quare non est necessarium, quod
onme movens raoveatur,, In quocumque enira est motus, illud rac-
■/otur, Unde si motus et actio, quae est quidam motus esset
in movontej. sequeretur, quod movens raoveretur, Et sicut dic-
tum Gdt in tortio physicorum, quod actio et passio sunt unus
actus subiecto, sed different ratione, prout actio signatur
ut ab agente, passio autem ut in patiente, ita supra dixit,
quod idem est actus sensibilis et sontientis subjecto, sed
non ratione. Actus igitur aonativi vel soni est sonatio, au-
ditivi autem actus est auditio,

Dupliciter enira dicitur auditus et sonus ; scilicet
secundum actum et secundum potentiara, Et quod de auditu et
sono dictum est, eadem ratione se habet in aliis sensibus et
sensibilibus „ Sicut enira actio et passio est in patiente at
non in agente, ut subjooto, sedjwlumjrfc in principio_a quo,
ita tarn actus sensibilis quam actus^sensitivi", est in sonsi-
tivo ut in subjectoo (16)

Sensation, then, is the result of a physical, mate-
rial -action which takes place within the material organ, and which
produces there a material motion, and this involves a physical,

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mteiial passio on the part of the organ which, paradoxically,
is the source of both the objectivity and the subjectivity of .sen-
sation. It xs the source of objectivity because it is the reception
of an action coming from an external object; it is the source of"
subjectivity because it involves a physical chango ontherart
^_^^m^^ttt_^_BSDaa^a(h^r& reactiolTwhich contributes
- to ...t h °,°onsii_tut_ionj f the object^Ba"Ste^-se^ieaTrAs"st7Tho-
mas points out, "non enim oportet quod actio 'ag5ntis~ recipiatur
in patiente secundum modum agontis, sed secundum modum patientis
^i^^ntis.." (17) On a. number of occasionTbbT:h AHStotle
and St, Thomas state that sensation consists in a modification,
^IL^yS^^Vof tho_sense_organj[it is this, alte^bignlthiris
^i^^^^ii^ed.^'Sentire cons is tit in moveri'et pati, EsiTe-
nim sensus in actuYq^aedarn alteration quod autem alteratur, pati-
tur et movetur," (IS) — """""" ~~~— -"

Whitehead, then, is justified in remarking: "It is
an evident fact of experience that our apprehensions of the exter-
nal world depend absolutely on the occurences within the human
body .... Yfe have to admit that the body is the organism whose
states regulate our cognizance of the world, " (19) By naively
attributing absolute objectivity to our sense cogniticn we are,
as Sir Arthur Eddington has remarked, "continually making the mis-
take of the man who on receiving a ielegrain, thinks that the hand-
writing is that of the sender,". (20) And in the same context
he points out that to attribute the taste we experience in eating
an apple to the apple itself is something like saying that. the
pain we experience in a dental operation is in the de ntis t's drill.
It is necessary then to recognize the enonuouiTTLilFEance whicTTse^
parates is from the things that are the closest to us. The very
physical proximity of sensible things is a sign of their distance
in the order of knowledge.

It is important to note that this subjectivity of
sense cognition in no way gives aid and comfort to the idealists,
as some might be laid to think. For, as- we have already pointed
out, the very source of the. subjectivity is at the sane time the
guarantee of objectivity. That is why Aristotle, after pointing
out that sensations are really nothing but "modifications of the
perceiver" immediately adds: "but that the substrata which cause
the sensation should not exist even apart from sensation is impos-
sible, For sensation is surely not the sensation of itself, but
there is something beyond the sensation, which roust be prior, to
the sensation; for that which moves is prior in nature to that
which ia moved, - """ (21)

Moreover, to say that the qualities that are imrae-

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%t^l> l™ ,-'-^^- is not the s "'» ™ saying that

^Sunr^-f-^' t S * mtt0r ° f fnc *» thQ 7 «•" completely physi-
^Jl^^P9^nt_.of consciousness,} (22) They arc a pLt of
the^jraicnl world, even IttoGgTnihey do not exist in the place
in which ctay are localized by the mive view. And the reason why
,hey arc where they are is determined by the very structure of
2d Logic^ G ° Bertrmid Russe11 br ings out this point in Mysticis m

The view that sonso-data are mental is derived, no
doubt, m part from their physiological subjectivity, but in
part also from a failure to distinguish between sense-data
and 'sensations'. By a sensation I mean the fact consisting
in the subject's awareness of the sense-datum. Thus a sensa-
tion is a complex of which the subject is a constituent and
which therefore is mental. The sense-datum, on the other hand,
stands over against the subject as that external object of
which in sensation the subject is aware. It is true that the
sense-da turn is in many cases in the subject's body, but_the
sjAj J ec_t^s^odyJ : s_as distinct from the subject_as_ tables__and
SlBl^JirG, and is in^'acTlaere]^""a'p^ world.

So soon, therefore, as sense-data arc clearly distinguished

from sensations, ai^jig_jte:i£_s^jectivity is recognized to
^Pfei^^gical not_p_sycb^
way of regarding them as physical are removed, (23)

We have laid considerable emphasis upon the nature
of sensation -both because it is of great importance for the pro-
blem we are undertaking to solve, and also because the majority
of modern Scholastic philosophers have presented sensation as though
it possessed the sara purity of objectivity as intellectual cognit-
ion. It is extremely important to realize that sense and intellec-
tual knowledge differ ggjioricall^_and..n_ojtjitgrely_s pecif icall y.
Prom the point of view of objectivity there is a vast difference
between sense and intellectual knowledge, Kant brings out this
difference rather accurately when Jw jvrites: "Sensitive cogitata ,
esse rerum,rei3rae3entatione3, (uti apparent intellectualia autem,
(sicuti sunt.^J (24) The sense s*Tmve™~to"3o with phenomena, with
things' as" they appear and not as they are in themselves. Their
object is not an essence - - something absolute as it exists in se
in the external world, but something essentially relative to the
external sense-organ itself. It is true that' when the intellect
is brought to boar up^'sense-data there will be an instinctive at-
tempt to assimilate them to the condition of intellectual objects,
that is to lift the "uti apparent" to "sicuti sunt", arid' as we
shall point out presently, this is precisely what the intelleot
is trying to do in its mathomatization of the sensible world, but

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ohs fact rernuia that in thonsolves the sense-data are purely phe-
nomenal. To lose sight of this and to project into the external
vrorl</oho sense-data as sensed by us is tantamount to identifying
•chc sensible in act with, the ,. sensibleTn^otenqyTl^e^p^nted
ou-o above, because of the material'- naiure of ^the sense-organ, there
is _a difference between the two, not only from the metaphysical
porno of view, but even from the physical and material point of
view, ^camot^oyJust_how_grQat this_difference is, I To do that
^wouldJ5o_necGs^a£v__f or us to~know actualTQhXJe n H iDl e~In~p^
1 ig ? ^.' ^ich is a contradiction^ Only the separated substance" know
j actuary the .sen^iliojir^ontia, and, we may add, they know
the ^nsib^l^^nactu in the only way in which they can be known:
as sensed by material sib jects, as existing within the organs of
Lbemga endowed with sense life. But even though we cannot say just
how much a difference there is between the sensible in act and
the sensible in potency we know that there is a difference. Things
do not exist exactly as they are sensed by us. And we cannot in-
sist too much upon the fact thatjre _neyor_scnse the sensible in
E2*enoy;< Qh?-t is) ttTC_separate_d .^ ,

f Perhaps wo can sura up this point succintly in the" following terms,
i On the one hand only the sensible in potency exists (i.e. outside

the sense organ); on the other hand, only the sensible in act is
:. known by us. Consequently there is a real gap between^the^sensible
9^-Jhe_physical (i,e. the extra-organi c world jT^And the°~with'dra-
^lJ*LjL2.i2£ c iL froiajeho sensible world is a__clear rec^TrtTon'of"
this_gap_a ) "" "'"""' "" ' "" ~ ~— -—-- — ,--

Paradoxical as it may seem, the attribution to sen-
| sa-,;ion of the pure objectivity proper to intellectual knowledge

comes closer to idealism than the clear ' recbgni'tibn "of "the^'subjec-
Itivity that is characteristic of all sense operations. For in the
last analysis this attribution consists in projecting into the
external world something that is the product of the sentient sub-
ject. In other words, idealists identify the sensible in potency
wj^h^ne^s^wiW^^^in^a^y'Hifio^G'who"" attribute pure objectivity
to the' sense s""ideni;ify the sensible in act with the sensible in
potency. Ultimately, the two positions coincide,, Aristotle and
St. Thomas point out the consequences of this fatal identification:

,Si orano apparens est verum, nee aliquid est verum
nisi ox hoc ipso quod est apparens sensui, sequetur quod ni-
hil est nisi inquantura sensibilo est in aotu, Sed si solum
sic aliquid est, scilicet inquantura est sensibile, sequetur
quod nihil sit si non erunt sensus, Et per consequens si non
erunt aniirvata vel aniraalia. Hoc autera est impossibile. Nam
hoo potest ease voiAun quod sensibilia inquantura sensibilia
non sunt j idest si accipiatur prout sunt sensibilia in aotu,

^

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<^L n ™ ..?""* .3^0 sonsibuai Sunt enim sensibilia in actu se-

actyfest quaedam passio sentiontis, quae non potest esse si
sentientia non sunt, Sed quod ipsa sensibilia quae faciunt
banc passionem in sensu non sint, hoc est impossible, (25)

■a *.■*■ If tho sensible in act and the sensible in potency
are identified, either tho objective world depends for its exis-
tence on sensation, orjvors ;thing in the objective world is actually and cons

l y sensed ,(^or nothing is sensed „> This ^ lasT"cSnse quericlT follows ™— - •■

because in oHerTor an object to be sensed there must be a phy-
sical mutation produced in the organ, and this mutation necessa-
rily involves a transition from a potential to an actual state
of sensibility. It is only by clearly distinguishing between the'
sensible m potency and the sensible in act that we can escape
idealism and angelism.

And now a few notions relative to the object of sen-
sation must be touched upon before. we can consider the relation
between science and sensibility. Aristotle and St. Thomas distin-
guish between objects that are sensible per acciden s and those
that are sensible per se . Objects are said to be sensible per ac -
cidens when > although they themselves arc incapable of being sensed,
they are connected with something that is the actual object of
sensation. Thus, for example, substance cannot be .actually sensed;
nevertheless in so far as it is tho substratum of the accidents
that are sensed, it is said to be sensible per accidens . Objects
that are sensible per se are those which are actually sensed in
themselves. They are divided in two types: proper sensibles and
common sensibles. It is this latter distinction that interests
us particularly.

The proper sensibles are those which constitute the
specific object of each individual external sense, and are conse-
quently the^2Mlj^^e^pjropertyj3f_piily__one sense, as, for example,
color for the eye, sound for the ear, e tcT~TKe~common sensibles

.are those which are the common_ property of more than one sense.
There are five principal common sensibles*: figure, motion, rest,

! number and magnitude; and to these are addedT"tHreT*ot"he?sT'"^Kie
which is connected with motion and rest; position which is connect-
ed with external figure; and place, which is connected with magni-
tude,"}

These common sensibles comprise all of the predi-
caments except two. Action and passion are included under motion
and rest; quantity comes in under number and magnitude'; quality
under figure; habitus is taken in by figure; situs has already

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are dlrectlv tta- -T f T ^^ sonsiW -^ and ubi and cjuando
no? wSS r ° dUC ^ 1 ? t0 P lace and t^». The only tie-predicants
not included arc jMAatonoo, which, as we saw is only a sensible
per accidens, and relation, .which cannot bo sonsed because it in-
X9iYos some Oiuig ._that .is(03rope^_tg.the,..intollect:| an~^rdSdnTl)7
S™ *!?«& Jo another,) Henc.^In_sg far as expert kaTicTen^ii ■
««gluponJjw_ggmijonjBgMi bIeB' it will bo"If!5aTOblel>f "gTttaining
lhg_Jubjtancos_o_f things or true predicarnenteOglati7^7Tn7i" y P >.
quantity provides^ substitute for* both substancelSr"predicamental
relation. Because of~t"he unique, position it occupies as the first
: accident and consequently the one closest to substance there is
\ £_2£2i sutetanjiiality^abput^it which, as we saw in the last Chap-
tor, explains why it alone of 'all the accidents is capable of being
the object of a^special. science. Because "in solo quantitatis ge-
nere, aliqua significantur ut subjecta, alia ut passionos" quan-
1 ?.3r.'SL° an constitute a world apart. And in this world mathematical
V^rdor •■substitutes f or real_ predicaincntal relation, — —

Now perhaps the most important aspect of these com-
mon sensible s as far as vre are concerned is that they are all re-
^^J^_i°_auMLtityo (26) Number and magnitude are species of
quantity; figure is a quality which is proper to quantity, since
it consists in the termination of magnitude- j motion (rest) and
' bii ' le a ££JE°A ( ?!Li 3 i! Jiuantity, "ex eo quod dividuntur secundum quan-
titatem ad divisibneiiialicuius quantitatis"; (27) and position V
and place, by being connected with figure and magnitude are redu-
cible to quantity. The fundamental reason for this reductibility * "
to quantity is that quantity by being the first accident isjbhe Vv -:
!il9^I^^jL°^jfall l L.-OJhe,. : Eg C^ n ^ h ence contribuj^_to^thom^_gjjanti-
feiiZSJ2ol5-"' BlIj^.o.I^ojlJI&iEiiL^^^ G >

. foundati on of the _ common sensibility_on the part of the senses,,

(The very homogeneIty"Th~whTch"air'"6f the coSiibn son"sib"les"are~'rooted
makes them common to several senses and prevents them from being
proper to any one sense.

In connection with the proper sensibles a distinction
must be made the importance of which will be apparent later. Among
the external senses there is a hierarchy in which sight occupies
the highest place and touch the lowest. Of all the external senses
sight is thn most perfect because it is the imst JLramterial and
the most objective. It is the sense which onables us to know the
[greatest number and the greatest variety of objects. Of all the
( senses it is the most d etac hed from its object, (28) Touch, on
j the other hand is the most material and the most subjective of
i all the sense faculties. It is the least detached; it has the weak-

est capacity for apprehending things in their distinctions. And
I yet it has a quality which makes it excel all the other external

-308-

sent .dans SS^— -^?° 8 chosGa telles Relies
W condition!!^ ^uTr^loTlV ^ ^ ^^

ses d"L Sur U o COnt ^- re ^ daVantagG a °^ s "u °hoc des c ho _
SLS Qans J.eur concretion er>aiqqp T1 «= + ^i„™» i. .

. des anciens .grossior- et S^or^i/ X^ Sele™

lui donne des avantages au point do vue de la sobre sertit,iL

En tant qu'elle implique 'subir' la c^nnaLsLce oLIrSnta!

le est essentiellement imparfaite, mis olio 1'LS^hS

sance, et pnnwpo do toute certitude; < Veritas principiorum
■ TSZT*" ^ SG n ° ta ' ^5°^5 SSSPer est reSlL

Ju 1 ?T Sen3US de ? endet °' Jean de s, Thomas,

\Curso Thool T, i. Po 392b.) C'est sous ce rapport qu'il

S d l0 T ? lus Pleinement a la premiere exigence de 1-intel-
' + & * °?° P a ^ ^ une aff "inite a 1' intelligence, qui se
'ferfin cSt^T ^^l' '^ Se ° Undura ta°WmltuTdif-
''S wn h ^ in !.° 0gniti ° nis ab alii ? ^imalibus. Unde
; quia homo habet optinm, tactum sequitur quod sit prudentissi-
\rarn onmum aliorum animalium. Et in genere hominura ex sensu
tactus accipimus, quod aliqui ingeniosi sunt, vel non inreni^
osi et non secundum aliquem alium sensuiiu Qui enira habent du~
ram camera, et per consequens habent malun tactum, sunt ine P -
tx. jec^duEyrentora: qui vero sunt molles carne, et por~5Sw5-
quens bom tactus, sunt bone apti mente. (in II de Antoa.
leot, 19 nos„ 482 -485) (29) — ~ ' "~ •'

11 is olear > the "> that though from different points
of view we vr say that both sight and touch are at once the most
objective ana -cho most subjective sense faculties, the objectivity
of couch has a very special significance for experimental science.
In spite of its lack of distinction, it provides us with the great-
est certicude, and .in this jLt is. like something that is found in
^^i. n tollQcMl._prde.r: (the most confused knowledge has '"the >roat-
^est certitude for us»»

-309-

Now ln s ° far aa the sense of touch is the sense
ot homogeneity, the sense which comes closest to the quantitative
i aspects of material objects,' the sense that coires closest- to pure
, corporeity .and pure exteriority, it is the sense that is the most
•closely ^allied to mathematical physics,, Modern science wants to
j reduce its sense experience with the universe -to the minimum that
i.^J^MtJJLJ. he songo of touch,, and that means not merely tcTthT
Cgeneric>sense of touch which includes perception of temperature,
| etc. but to pure_ taction, cthat is to soy to pure contact of point
Vijojpoint,) ™ ' — ~

This brings us to the consideration of a final dis-
tinction that has a bearing upon our problem, The distinction

between external and internal experience. External experience con-
sists in the experience of the external senses of which we have
been speaking. Internal experience consists in the experience had
of one's own proper reality through the operations of the inter-
nal senses and the mind. Now all too often it seems to bo taken
for granted that the study of nature depends only upon external
experience. This is far from being the case, especially when it
is a question of the study of living nature. As a matter of fact
it is true to say that in a certain sense the study of psychology
is based principally upon internal experience. We cone to know
what life is originally and primarily through our own proper ex-
perience' of living-, St„ Thonas brings out this point in his Com-
mentary on the De Anima of Aristotle: "Hoc enim quilibet experi-
tur in seipso, quod scilicet habeat animam, et quod ahima vivifi-
cet," (30) This internal experience is so" important that if ■ one were tc
abstract completely from his own personal experience of living,

I he could not speak of life existing in anything, And it is impor-
tant to insist upon the" fact that this internal experience is not
the flimsy and untrustworthy thing that many modern scientists
attempt to make of it. On the contrary it enjoys the greatest cer-
Vtitude. In the t ext just_j^ted_J^_ _T^omasJbases the_ eminent cer-
titude which psychology poss esses precisely upon" the 'fact that
lif e is known through inte rnal experie'nlaeT'Er'comparrsl^lYith
the certitude which we have df~15ur' own life, our knowledge of the
existence of life in other things ? which depends upon external
sensation, has only a greater or less degree of probability. It
is precisely because psychology is based upon the experience we
have of our own soul that ithe) basic Aristotelian treatise on liv-
ing jiature is called D e Aniim ,. In it the soul is considered in
quadam abstractione - - not in the sense that it is studied in
complete abstraction from the sensible matter with whioh it is
united, for then it could not ; f orra a part of natural doctrine,
[but in the "sense that it is considered to some "degree "'in and" by
1 itself , And this dopendonco upon intornai experionce introduces

-310-

wo sLlt S Ph ? ^ r 0rd °. rim ° f thG mtural troatiaes about which
wo spoke in Chapter IV. since tho basic methodological principle

™+, ° gl ? ^ t V hat is bost k" ™ t0 us, the study of living
nature must start with the soul as it is experienced by us, to
quaton abstraction^ and then pass on to things that are mor~in~
S* J? b ° Un f t0 mtter » That ^ why De Sonsu et Sensato comes
alter the De Anima. In the introduction to his Commentary on De
|ensu__ot sensato St, Thorns explains this ordering" (31) Vege-
tative life which is not attainable by direct internal experience
is the most hidden form of life: "vita in plantis est oculta."

But it would be a mistake to believe that internal
experience enters only into the treatises on living nature. It
is also used in the Physics,, For example, in book three when Aris-
totle is looking for an illustration of motion, he has recourse
to the example of a man building a house. 'One might be tempted
to wonder why he deliberately chose the example of the becoming
of an artefactum and not of a natural generation. But the illus-
tration like all the illustrations "of "Tirlstotle, is not without
its profound significance. For in the example of the building of
a house we have a case of motion in which both external and inter-
nal experience enter. As a matter of fact, the striding of an a-
gent for an end, which is so essential to the true concept of mo-
tion, is most clearly apprehended by us in our own internal expe-
rience. When this internal experience is completely set aside,
it is all too easy to lose sight of the fact that motion involves
the coming into being of a new actuality which is the end of an
agent, and to look upon it as a pure degra dation. As a matter of
fact many modern scientists have corns to loolFupon motion merely
in terms of the second, lav/- of thermodynamics which states that -
the world is continually in a state of degradation, that is to
say, continually losing actuality, and consequently destined ul-
timately to arrive at a state of thermodynamic equilibrium in which
all of cosmic reality will be in a state of utter chaotic diffusion
and formless homogeneity. In connection with this question of en-
tropy which constitutes time's arrow for tho scientists, it is
interesting to note that in his commentary on Aristotle's treatise
on time in the fourth book of the P hysics St, Thomas teaches that
if we abstract from the agent of motion and from its intention,
time is a degrading factor: "mutatio est ad peiora ex natura sua,"
(33) Mutation and time must be joined with the idea of an agent
acting for a certain end in order to have the generation of a new
actuality.

All this may appear to be an irrelevant digression,
but as a matter of fact it is very a propos. For it serves to bring

-311-

c^nJS- ?n thS startin 8 P° lnt °f mttematical physics is
diametrically opposed to that of philosophy of nature, I&ithemati-
; cal physios socks to take its start from a minimum of experience.
It excludes internal experience, andTXl-elu^es^aer^rFxperrence
to its very lowest form: pure corporeal contact. And out of this
^Miu^j;_expe_r^enc^r^seeks to construct tho wh^E~0H3?rorior
ihilosophyox nature on the othe^"HBa7^a^lTs-3t¥-p5iHt^bf^depar-
ture a.mximum^o^orionoo. It employs not only the whole range
,ot external experience, "buFlxlso internal experience. And in con-
nection with its dependence upon internal experience, it must be
pointed out that this method of investigating problems is neither
anthropomorphism noysub jectivism. ' On the contrary it enjoys a high
degree of objectivity. For one's own internal states and experien-
ces are_^^j Q ^Qtivo_as_any thing in the universe,

Thi s contrast between the points of departure of
mathematical physics and the philosophy of nature brings into re-
lief a. striking paradox. While from the point of view we have had
m mind in this discussion philosophy of nature depends upon a
maximum c.f experience and mathematical physics upon a minimum of
experience, from the point of view from which we considered the
problem of experience in Chapter IV the situation is < complete^
reversed: a minimum serves as a starting point for philosophy,
while a maximum is required for mathematical physics and all the
branches of experimental science. We may say, then, that because
of a significant effort on the part of the intellect to shake itself
1»S3 f rora its dependence upon the senses, mathematical physics
tends towards a minimum of experience,, This tendency is seen first
in the vast use of hypothesis by which the mind seeks to antici-
pate reality. It is carried forward by a reduction of sense ex-
perience to its lowest form: pure taction. But it is a tendency
that can seek its end only by binding the intellect down to a ma-
ximum of experience.

3ut in order to become aware of all that is involved
in this question it is not sufficient to consider the difference
between the starting points of mathematical physics and philoso-
phy of nature; we must also consider tho terminal points at which
they aim. Precisely because philosophy of nature begins with a
maximum of experience it has as its ultimate goal and. as its im-
portant object the noblest being existing in nature, the being
which in some sense transcends nature, and yet is a_par_t__pf it.
The _being_ which possossesythe highest degree of" he terogenoous in-
te rior ity'in3h~9"'univer¥eAthe"' spiritual soul of man, (34)' On
the other hand, precisely because niatheimtical pliyaics begins with
a minimum of experience, its ultimate goal must be to reduce the
whole cosmos to. pure hqi.iogenoous exteriority, to a state of pure

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st^ZlJ&Y^^ As we «holl have oooa-

actollv^rrLrn^^^" 10 -, 1 ^ ^ if ^tatical physics could
it S ^ I^ • G , SOaI towarda whioh " is constantly striving,
it would succeed xn reducing the cosmos to a state of pure empti-

ness

oonnpn+nri «rt+v, 1 ^ obvious that this question is closely
^hilosoSnll Urgent forms of measured employed in the
we all,X S" ^ le ? ces T and in ^ experimsntal sciences, to which
w. alluded in Chapter I and which we shall consider in greater
detail in Chapter DC. The method of mathematical physics has its
^.^antagca and its rich ^turns, but then, as has often hap-
pened, the knowledge that it provides is proposed as the only va-
lid _ knowledge of nature, then we are asked to accept an epistemo-
logical monstrosity, an exaltation of the superficial, a radical
form of nihilism.

3q Science and Sensibility,

We are now in a position to consider the problem
of science and sensibility. From what Y/as said above it is clear
that it is especially in relation to the proper sensibles that
the ever widening gap between science and the sensible world has
occoredo We must now try to see what has created this gap. Perhaps
enough has already been said to show that it is not an artificial
and arbitrary creation, nor a fortuitous occurence, but something
that has come inevitably from the very nature of experimental science
and .. the_ na Jbure_of _scns ibility ,

The first cause of the withdrawal of science from
the sensible world is obviously the subjectivity of sense cognition.
Natural science is orientated completely towards the absolute world
condition, and its whole inner finality urges it to draw ever closer
to this goal. The inherent subjectivity of the ministrations of
the senses is a direct obstacle to this tendency. For the delive-
rances of the senses present an anthropomorphic world, a world
that has been refashioned, to some extent at least, according to
the structure of man's sense organs. They consequently present a rela
tive world, a world of appearances. If science is to be true to
its inner urge to strive for the absolute world condition, it must
find a way ^.^aanthropOTio^lia£e_thoso_deJ.iverjince3} it must,
as we have suggested, strive "to transform the~~"utf "apparent" of

-513-

Kaiit to siouti sunt". And it does this by means of a double subs-
titution: one on the part of the subject and one on the part of
-che ob oc-c On the part of the subject, it puts in the place of
2£ta^..2£s^rur lS nts °f perception inorganic (artificial ins truinents
ot measurement especially designed for the purpose' in' accordance'"
with scientific theories. On the part of the object there is a
corresponding substitution of quantitative for qualitative deter-
mina-cions. The scientific world thaFliTbuiit 'by £feHni"^f these
artificial inorganic instruments of measurement will inevitably
draw farther and 'farther from the sensible world that is built
up by the organic instruments of perception. (35)

. Ii; is "to bo noted that the subjectivity of the sense
is an individual subjectivity. The corresponding sense of ten dif-
ferent subjects will not necessarily represent the soke object in
the same way. Ten different men, for example, may get ton different
perceptions of the temperature of the same body of water. Now this
is contrary to one of the ideals of science, which has core to
be known in recent years as intersubjectivibility. And science
. has found that by [the d ouble ^b s'titatibri" yiientioned above almost
I perfect_intersub gectivibility can be acHTeved, Norman Campbell
has shown that the "only" exact judgments with 'regard to perception
that are universally accepted are those that are based on quanti-
tative determinations, and particularly those which have to do
with the three categories of space, time and number. (36)

Another important reason for the withdrawal of science
from the world of sense is that from the point of view from which
experimental science approaches the cosmos, ttie„proper_sensible s
9^Jj-Eaii9SS?r. s » A* 14 tll£vt for two_reasons.'ln the first place7~
their proper sensible s ?anQO^_be|;'defineJ> It is impossible to de-
fine heatj it is impossible to define a c olour or 'T3o u P^«" They"
are utterly ^incapable of analysis.; They" p ossess nei inherent com-
raunicability ~[It is impossible to explain to a man born blind what
red~1uidrb"lue are".^, Arid the reason for this is that the proper sen^
sTbles are "the primary and immediate data of sense cognitions Hence
there are no prior notions in terms of jvhich they may be defined;
thore'are no more l^ndamental 'elements into which thliy ~moy 'Fe" ana-
lysed,

itow it is different for the mind to rest satisfied
with this state of affairs. It has an instinctive desire to define,
('to express 'to itself*) the quod quid est of things,, That is wiry there
have always been attempts to_ liberate the proper sensible s from
the incommunicability that is native to them. The mediaval,* Scho-
lastics made attempts of this kind, For example, they defined ..
white as disgregativum visus. But it is evident that such attempts

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can never yield strict definitions.

There are no ^f 1 ^'. th - e P ro P e *- sensibles are indemonstrable,

they r,nv be do^ n Pr ^ C i? leS in the sc »sible order from which

princiXs o? t° <*"/* * h ° *<** ttoe, they themselves are not

lEver it L T S lT ati °^ N ° tMns Can be deduced f rom them, ■

be plrceiveV ^ + y - th ^ Ugh tbem that the cora sensibles can

is known In °hr ? ViV*? ^ ey r ' ny ±n a ^ be COI - 1 P ared to *ha*
whiS T- ■ « intellectual order as the supreme dignitates,

in LrZ-1, T^ f .° r GVery d ° mons tration > but ^MchlS^ot
fficSo^? 3 ^- E ? ln ^le3. of _any demonstration. Indefinably
S Snttro + ?™ y ^ 3 ' lnd ^ onstraa ^' incapable of being a source
ol demonstration, the proper sensibles are merely given. Is it
any wonder that science instinctively draws away frolTthem?

i -i -,-, • \ The SGDond source of their irrationality is very

Sv SC o V ^ d Iu th thG first: by tbG ve ^ f *°* that they are pro-
ber sensibles, they are irreducibly heterogeneous; they are is£~
lated one from die other; they are noTunTfiecTby a logical pat-
tern. As we shall attempt to explain presently, not all types of
heterogeneity are essentially and completely irrational. Neverthe-
less, in the measure in which heterogeneity is incapable of being
reduced to some kind of unification it always presents an eleront
oi irrationality to the mind. Mayerson has laid considerable em^"
phasis upon the isolation of the proper sensibles j

. II suffit en effet, de reflechir a. la nature de la
qualite pour se rendre compte a. quel point elle se prete dif-
ficilement^aux tentatives consistant a rolier, montalement,
le divers a l'identique, qui constituent l'essentiel de toute
explication du reel. Car toute qualite nous apparait couae
--■quelque chose de complet en soi; non seulement le fait de son,
existence ne postule rien en dehors d'elle-meme, mais elle
est quelque chose d'intensif et ne parait done point suscep-
tible do so combiner, de s'ajoutor a quelque autre chose. (37)

Material qualities lend themselves admirably to des-
^ipA^Y 6 knowledge, but they seem refractory to explanatory know-
ledge. They appear to be closer to sentiency, whereas quantity
seems closer to rationality. Once again from this point of view,
the proper sensibles are merely given, and this givenness is in
direct opposition to the necessity that science seeks. Not being
able to find this necessity in the realm of the proper sensibles,
it will look for it elsewhere.

Another reason for the withdrawal of science from
the sensible world arises from the extreoily restricted nature

-315-

of the senses. The crudity .of our sonso organs allow us to percei-
ve only an infinitesimal!/ snail part of the cosmic occurences.
By the substitution of inorganic instructs of icb asuremont for
the organic instruments of perception the scope of science is in-
creased immeasurably.

In general, then, wo may say that we experience the
outer world through small samples of it coming into contact
with our sense-organs,.. Yet not all samples of the outer world
affect our sense prgans. Our ear-drums are affected by ten
octaves, at most, out of the endless range of sounds which
occur in nature; by far the greater number of air- vibrations
make no effect on thorn. Our eyes are even more selective; speak-
ing in terms of the undulatory theory of light, these are sen-
sitive to only about one octave out of the almost infinite
number which occur in nature* „,

Science has of course provided us with methods of extending
our senses both in respect of quality and quantity. We can
only see one octave of light, but it is easy to imagine light-
vibrations some thirty octaves deeper than any our eyes can
see. While philosophy is reflecting how different the world
would appear to beings with eyes which could see these vibrat-
ions, science sets to work to devise such eyes they are

our ordinary wireless- sets. We also have means for studying
vibrations far above any our eyes can sec. Actually a range
of vibrations extending over about 63 octaves can be detected

and has been explored 63. times the range of the unaided

eye. And oven this limit is not one of the resources of science,
but of what nature provides for us to see. In the same way,
the spectroscope makes good the deficiency of our eyes for
analyzing a beam of light into its constituent colours, and
further enables us to measure the wave-length of each colour of light
to a high degree of accuracy,

Science has extended the range and amplified the
powers of our other senses in similar ways, in quality' as well
as in quantity. We cannot touch the sun to feel how hot it
is, but oui- thermocouples estimate its temperature for us with
great accuracy. We cannot ta3te or smell the sun, but our

spectrocopes do both for us or at any rate give us a bot-

- ter acquaintance with the substance of the sun than any amount
of smelling or tasting could do. V/e are entirety wanting in
an electric sense, but our galvanometers and electroscopes
make good the deficiency, (38)

As Hermann Weyl has pointed out, this' crudity of
our senses leads us to identify things which are physically dis-
tinct and thus runs counter to one of the most basic principles

-3:1.6-

of scionce:

For the question forces itself upon us- why is nhv-

nr %Z + t tensions, what urges it to put oscillations
or the ether or something similar in their place? After all,
l^tin^JNT ^ rce P tio ns wo know nothing about the oscil-
-W^ i ^ h6r; What We are Siven are precisely only

£ZL l° U ^'J hG ^ We cncounte r ^em in our perception.
Answer: To light rays which cause the same impression to the
eye are in general distinct in all their remining physical
and chemical effects. If, for example, one illuminates one
anr. the same coloured surface with two lights which visually
appear as^the same white, the illuminated surface usually looks
quite different in both cases* Red and green-blue together
'give white light, equally light brown together with violet.
•But the first light produces a dark hue on the photographic
plate, the second a very light one If one sends two lights
which visually appear as the same white through one and the
same prism, the intensity distribution in the spectrum arising
behind the prism is different in both cases. Therefore physics
cannot declare two lights which are perceptually alike to be
really alike, or else it would bo involved in a conflict(with
iTO^ominat^ng_p_r^cip_lVi) equal causes under equal circunstan -
22s_2I°d^_£3^__effects. Perceptual equality therefore ap-
pears to physics only as a sojnewhat: accidental equality of
the reactions which physicallyjlistinct agencies produce in
the retina. The ac^cTental^CSuality of the reaction rests upon
the particular nature of this receptive apparatus. (39)

In connection with this point" it is not superfluous
to add that the deliverances of the sense are extremely fluctuating
and unstable. As Heyerson has remarked: "le retour de sensations
veritablement identiques est excessivement rare." (40) That is
why science must look for a source of permanence which is so es-
sential to its nature.

Moreover, the qualitative determinations of nature
permit of only general and loose propositions. In order to achieve
accuracy, and in order to make its propositions capable of unara-

^iSyP™. °.?.E f i ri 'i la ' fc: !:° n ° 3 L ref '} 1 ' fc ? L ' t . : i-.? n > science mus tTiavo"l ; ec6tffse
to quaHtltaTiivF'ae tormina .ti6ns7'Pbr example, the stateircnt: "fire
causes water to boil", is not true unless a number of precise de-
terminations be added with regard to temperature, pressure, res-
pective masses of the water and fire, surface of radiation of the
fire, etc, A certain arrangement of those conditions could actual-
ly keep' water from boiling,,

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<,„„ -, It seems necessary to add one final observation be-

ZZ 15 °T - 3 (luestion - The whole rnaterial univorso is a mix-
umo of quail ca.ive and quantitative determinations. As we go up
the scale of perfection in cosmic reality, the qualitative deter-
minations assume an ascending importance, for they manifest the
^r^aj^g triumph jrfjTpm^^^

^2^£^25,^M2ip!^4^Si°ncesi) BuTl!HTnorgfnic-Wttef it
^i_jj}QH£ggative aspect that iF'in tag"asbendencyT"Ana that
can perhapsDe-Koauced as a further reason^vhFghys'ics as it pro-
gresses becomes more and more immersed in the quantitative,

, And now > having considered the relation that exists

between science and sensibility, we must try to see the way in
which the mind triumphs over the limitations of the senses.

4, Science and Homogeneity,]

In order to understand the part that homogeneity
plays in science -it is necessary to begin by making an important '
distinction between two types of heterogeneity. There is first
of all a kind of heterogeneity which is found on the part of the
object of knowledge^nd which we shall call"natural" , This is the
heterogeneity that exists between man and brute, between the num-
bers two and three, between the different angelic species, between
the logically distinct rationes formales of the divine essence.
This type of heterogeneity obviously springs from a difference
of form (in the broad sense in which it signifies a. ratio forraa-
lis). It.-.-is---oonseque_nWy„a„hetexQgejieXty„Jhat is essentially ra-
tional, ( PL-has its source in i^eUagibili'tyT? And the more perfect
an intellect _57~tjJOJjc^^
proper and irreductib^ i _hetoFogpneT : £y7''* ""*" """ ---•-->-■-■•• •■- ■■■■->■ -■■

There is another type of heterogeneity that may be
termed "noetic" because it is found not on the part of the object
of knowledge but on the part of the intelligence itself „ It con-
sists in the multiplicity of media or concepts, or intelligible
species which the intellect needs to employ in order to know rea-
lity. The more imperfect an intellects, the greater is this mul-
! tiplicity. This heterogeneity the refore is essentially irrational.
It is a reflection of the original "''potentiality of '"'the "''"'intellect.
It is clear that perfect knowledge will consist in knowing natu-
ral heterogeneity in all the fullness and richness of its proper

-318-

»i fi ° flistinctness by moans of absolute noetic homogeneity,

of knowledge J^L?" 1 ? *" A± * im ]ai0vfled &° that this perfection
SaweSr^ iT'i^ UniqUS intelli S^^ species which is
W.tiv^v ,^ ■ fv ^ n th ° inclividual natures in existence, ex-
is no ™S-iS-i n ^ lr Ultimat * s ^ cif i° concretion. Here there
Peneitv ?« f ? ?f ^ 5 ? nf ^ bet ™ e * heterogeneity and homo-
geneity. In fact, xt is only because God sees things in the one

Xolutetef " Hin f lf *S at hG 1S able t0 ™ them in their
nnn?S £ ^"'S 6 ™ 1 ^- But as we descend the scale of beings,
'^o.^torogeneite^^uallj: increases. The higher separated
substances can know a large number 'of individual natures in their
specific distinctness through a snail number of intelligible spe-
cies, m che lower separated substances a great multiplicity of
media are required. And the limit of this process' is found in the

. human intelligence which because it partakes of the diffusion of
mat-cer with which it is united, can know things in their distinct-
ness only trough a multiplicity of intelligible species equal

kte uhe multiplicity of ontological species,

In the intellect of man there is a profound conflict
be Ween homogeneity and heterogeneity. On the one hand, he is in-
capable of sharing in noetic homogeneity. He can, indeed, attempt
to * riun, P h ° Ver this linlite tion by having recourse to the dynamic
gS| h ? d ..glL^iE-ts , and this method is not without fruitfulne'ss".""'"'
But it always remains only an attempt, since dialectical limits
cannot be attained. On the other hand, natural heterogeneity,
though something basically rational, will always. present to him
^ n _H ra .t iona -l aspect in the measure in which it remains in its
pure isolated g'ivenness, in the measure in which it cannot be re-
d ";5S c l.to^ome kind of .. unification^' to soias type of homogeneity^
It must be remembered that even though" the" soured of natural he-
terogeneity is fundamentally something rational, in so far as it
is found in the material universe it also involves an irrational ,
element in the sense that a plurality of really distinct forms
is possible only because they are imperfect and limited.

The problem of the human intellect then, is to see.
the heterogeneity of nature in term of some type of homogeneity.
Here we are touching upon -a conflict in the intellectual order
of which there is something strangely analogous in the sensible
order. We refer to the distinction pointed out above between the
faculties of sight and touch. As we saw above the first is a fa-
culty of heterogeneity in that, bettor than any other sense, it
is capable of grasping things in the richness of their specific
distinctness. The second is a faculty of homogeneity in that it
has the least capacity for grasping distinctions and in that it

-319-

sc,ms to come into closest contact with the quantitative determi-
nations of nature. It is also the most important sense faculty
from the point of view of certitude, and this carries out the a-
nalogy still further, since, as we have seen, it is only by remain-
ing.^n the, homogeneity pf generality that the SAis' : cM.eJboar^
m^-Jil-^^MMS^±q, S e^i^^^:^^ionto'^ cosmos. I The
i^a^OTOrda^vhich mn will ever strive . will"' be"a union" of this
5Mtin.otnoB8 .and this ^certitude VJ In the sensible" order this"' is
.possible, since sight and touch can be brought into a combined
j operation on the same object: "unless I see in his tends the print
IS , + • ?? •' and £ ut my fin ger into his side, I will not believe."
| But m the- intellectual order separate faculties of distinctness
| and certitude, or heterogeneity and homogeneity do not exist. Tfen-
1 ce the mind will have to discover soma other means of striving
v. towards its ideal. Let us see how it goes about it.

There are two important ways by which man tries to
triumph over the heterogeneity of reality through homogeneity.
The first is by retreating into, generality and consequently into
J2gi°£LP°*®B*^.Wy« CEt is_ iri thi swny that philosophy of natur e
studies the cosmos." ) By reducing the~speciflc _ heterogeneity of the
universe to the logical homogeneity of generalities, it is able
to procure for itself a number of .important advantages. It is able
to get at the fundamental, common structure of the physical world,
and to know it with certitude. It is able to view the cosmos in
terms of_unity<a nd in terms .of what is most knowable for it.") But
the price it has to pay for these advantages is great. For all
the concrete richness of the universe remains untouched. At the
limit of this process of homogehization the universe would, be re-
duced to the emptiest, most vague and most potential concept

that of being, abstracted by mere total abstraction.

It is in order to get at the richness of nature
that the mind. starts its march towards concretion. But by advanc-
ing in this direction it soon gets involved in an "intellectual
crisis. For it gains from the point of view of heterogeneity which
means a loss from the point of view of homogeneity, and hence an
increase in irrationality. And this increasing irrationality for-

ces the mind to seek for some kind of homogeneity through whioh
\to triumph over it. But it will have to be a homogeneity that is
quite different from the one from which it is emerging, i.e. one
that will not lead it back into generalities, but will carry it
forward into concretion . It will have to be a homogeneity that
is not logical but ontological. It will have to be something which
will afford at the same time both a unity to provide for what is
lost by drawing away from generalities, and a distinctness to e-
nable the mind to press forward towards concreteness. It will have

-320-

nature i "£S5 nA^ 11 n ° kB " P ° Ssible for the *** to sec
uS for whJ T i /^V* I" 8 * kn ° Wable for " ( and ^us make
same ^ L L^! J" ?T- nS ™7 fr0, ' A g^ralitics) and at the
up for the U? ^ ^ 1S m0st knOT ^ le ilLse (and thus make
f Liz - W r. T? 3 - ? f - g^J£-i ^gg_ric knowTidg e^ , And the mind
oft,2r 1 ■ Xt ^^^ini^Fin a general structure

OoiEo ealit) , in a oaMflonjaatrfxJnwhich the heterogeneous

of'tlt u^LerL^^t g gnifiCanCS ° f ^ - tte - tiz ^ i0n

LlnM«r- •> N™ fiance gets at this homogeneous matrix by dis-

Ig^nj^^objoc^fro-a the realm of the prope r sensibles t fjtot
'SOh^con^pnsibies. And these conwon sensibles sexWiSTpE-
pose e^celKTtly by the very fact that, while, the.y are not m Z -
% ~ J ln tte mselves, (th ey are all reducible to quantity! ) Since They
are sensibles, and h^c~e not quantit y specifically, t % science
Vvh.ch studies them is able to remain within the realm of. physics.
On the o-cher hand, since they are all reducible to quantity, the
nund is able to find the homogeneity it is seeking for, and phy-
sics becomes mathematical physics, Since quantity is the primary
accident and the one closest to substance, all the specific de-
terminations of cosmic 'reality are rooted to it, and hence they
all assume ,a_gu &ntitative mode „ Because of the principle "quidquid
recipitur ad nodum recipientis recipityr", quantity necessarily
modifies the qualities that are received into it, ((427)- (a.6.

In order to understand the nature of these quanti-
tative modes it must bo noted that in the structure of physical
reality, the qualitative and the quantitative determinations are
notjr elatcd to each other after the manner of two contiguous lay -
ers. Rather, there is an intimate, dynamic union between them.
And this explains why the qualitative determinations can be "trans-
lated" into quantitative equivalents, why the colours and sounds
and heat of the universe can become functions of the space, time,
mass and other derivative relationships that exist between various
parts of nature. By getting at these quantitative modes, science
is able to construct a physics that can be informed and rationa-
lized by mathematics.

But at this point it must be noted that it is pos-
sible to study qualitative perfections in a quantitative way with-
out having recourse to a physical quantitative mode. Intelligence,

jfor example, is studied in experimental psychology in terms of
quantitative measurements based on an association between certain

[psychological reactions and a scale of numbers. Mathematical phy-
sios is primarily concerned not with an extrinsio and artificial

-321-

correlation of this kind, but with on_lntrin sio correlation which
springs from the very structure of physical reality. This intrin-
sic correlation is not a discovery of modern science ; it was clear-
ly recognized by the ancients, and was the basis of thoir mathe-
matical physics. (43)

. BnJ ° in order to understand this point accurately
it is necessary to introduce a distinction here, which will not
only help us to clarify the present issue, but will also be use-
ful 1 for us in the next Chapter when we come to discuss the relat-
ion between science and measurement. We have in mind the distinct-
ion be tweenpredicamental and transcendental quantity, St, Thomas
explains this distinction with great preciseness in the following
passage :

Duplex est quantitas, Una scilicet, quae dicitur
quantitas molis, vel jquantitas dimens iva. ) quae in solis rebus
oorporalibus est. Unde in divinis personis locum non habet,
Sed alia est quantitas virtutis, quae attenditur secundum per-
fectionem alicuius naturae, vel formae. Quae quidem quantitas
deslgnatur, secundum quod dicitur aliquid magis, vel minus
oa.lid.uni, inquantura est perfectius vel minus perfection in ta-
li caliditate, Huiusmodi autem quantitas virtualis attenditur
primo quidem in radice, idest in ipsa perfectione formae, vel
naturae; et sic dicitur magnitudo specialis, sicut dicitur
raagnus calor propter suam intensionem , et perfectionem . Et
ideo dicit August 6 de Trinit. Cap 18, quod in his quae non
mole' magna sunt, hoc est maius esse, quod est melius esse.
Nam melius dicitur, quod perfectius est. Secundo autem atten-
ditur quantitas virtualis in effectibus formae. Primus autem
effectus formae est esse; nam omnis res habet esse secundum
suam forraam. Secundus autem effectus' est operatio: nam homo
agens agit per suam- forman. Attenditur igitur quantitas vir-
tualis et secundum esse, et secundum operationem.
Secundum esse quiden, inquantura ea quae sunt perfectioris na-
turae, sunt maioris durationis. Secundum operationem ver ,
inquantura ea, quae sunt perfectioris naturae, sunt magis po-
tentia ad agendum, (44)

The more or less of transcendental or virtual quan -
tity is baaed on heterogeneit y 1 while that of predicamental
or formal quantity is based on homogeneity J And it is interesting
and helpful to view the latter as the dialectical lyLmit towards
which the former tends as the hierarchy of immaterial things des-
cends towards the realm of corporeality. The difference of forms
gradually diminishes and at the Admit the definition of each part
is the same as the definition of the whole. The diversity is no
longer formal; it is purely material. In all material things both

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2S°L wSS*^^^^ ^P***' Th9 hetor ogeneity of the

St, Thom. -|a^qe5t W3_Di a putatae do Virtufribus' in Coiamu ni.
and foms arS^f. lat ^ SFc ' S tte ^S ni ^e which qualitl— '
that i? attrlbS J°' ?° SSGSS *2^» there is another magnitude
Sll +£ l * , them ££*LS2£tt22S. It is this quantity per
accxdens that is of special signifi^n^c for mathematical phytics:

., ,. 9 innibua lualitatibus et formis est communis ratio
w+ X f^ 1S ^ dicta est ' scilicet perfectio earura in sug-
jeoto. Aliquae tamen qualitates, praeter istam magnitudinem
sou quonuitacem-guae competit eis^r^e> habent alien magni-
tudinem yel quantitatem quae competirelS per accidensj et
hoc dupliciter, Uno modo ratione subject!; sicut albedo di oi-
|H£_qggt|_per aooidena,<^uia^ubjeotum eius est quantum;) un-
o e augmen.ato,subieoto, a^j55tati5na^oir^or'7 ^7i5^^q^'
secundura hoc augmentum, non dicitur aliquid magis alburn, sed
maior albedo, sicut et dicitur maius aliquid album „ . . Alio
modo q umtitas et augmentum attribuitiu/alicui qualitatiCpSg)
kaccidensp ex parte obiecti in quod agit; et heac dicitur quan-
titas vir cutis; quae magis dicitur propter quantitatem obieoti
vel contmentiam; sicut dicitur magnae virtutis qui magnum
pondus potest ferro, vel qualitercunqua potest magnum rem fa-
cere, sive magmtudine dimensiva, sive raagnitudine perfectio-
nis, vel secundum quantitatem discretam; sicut dicitur aliquis
magnae virtutis qui potest multa facere . , , Sed consideran-
dum est, quod eiusdem rationis est quod aliqua qualitas in
aliquid magnum possit, et quod ipsa sit magna, sicut ex supra
dictis patot; undo etiam magnitude perfectionis potest dici
magnitudo virtutis, (45)

It is clear from this passage that in so far as forms
and qualities. are found < in corporeal beings they may become quan-
titative per accido ns in relation to predicamental quantity. And
from the last line of St. Thomas just oited it is evident that
there can be a direct relation between the transcendental per se
quantity of these forms and qualities and the predicamentally quan-
titative modes which make them quantitative per aooidens. This
ma kes it possible for science to deal with the transcendental quan -
tit y of the specific perfections of reality in terms of predica^ ~
mental quantity .

By fixing its attention upon the quantitative modes
of the specific determinations of the cosmos, physics obtains for
itself innumerable advantages. For, in the first place, nothing

-522

» ^th I™ t SG ^ e aS 1 uantit ^' As Spaier v has remarked,
En nn not q ^ n ^ t . e .^ 1 represente la realite la plus solide . „ ,'
S^ntit"!' (y ettllSraC hnbituel est avant tout un realise do la

By adopting the quantitative method the mind enjoys
an experience that is in some way similar to that of being able
to reach out and touch and handle an object of sense,- Whether or
thlnri ^ 7?" * *hw there is the advantage of being able to grasp
things in their distinctness in the way that would bo similar to
the perfection of the sense of sight is a question which we shall
consider a little later. Moreover, nothing is capable of being ■
so abstract and ideal as quantity. And this gives almost unlimited
scope for -che mind's desire for perfect rationality,

■ . Thi ^ reveals the profound significance of the homo-

fgeniz ation of the cosmos . Because man is composed of bothliiatteF
and spirit, there are two fundamental tendencies in him: To draw

^everything from matter, and to draw everything from mind, The per-
sistent recurre nce of the extremes of materialism and idealism
in the histo ry of philosophy have been a constant manife statio n
of_this. Now the quantitative homogenization of the cosmos makes
it possible for. man to realize both of these tendencies simulta-
neously. The mathematization of nature means something far deeper
than an attempt to escape from the anthropomorphism involved in
the subjectivity of sensibility. It is really an attempt on the
part of the intellect to shake itself loose from the senses. This
is_i n a way (a g riatur al)movenTfi nt, \ since intellect in its perfection
is indepe nde nt of s"ense .J To "construct the universe out of a minimum
of_ex p_erience i sthej iexVlshing to positing the universe . To a
certaTn extent the mind is successful in this attempt. But by an
ironical paradox this success involves a falling back upon some-
thing similar to the very lowest form of sense life • pure tact-
ion. It is a conception of the universe in terms of the homogeneous

^ exteriority o f pure materiality .

All this _ explains why the goal towa rds which science
is ever striving is [to^reconsliruct ; the universe out 6T sameness.]
"The aim of the analysis employed in phvsios>_writes Eddington,
"is to reso lve the -universe into stru ctural (units ) which are pre -
cisely li k e oiie ■ another ." (47) TheTanalysis of matter has gone

far in this direction; it has succeeded in rosolving cosmic rea^
lity imto protons which are all alike and electrons which are all
alikej) And when nature seems to present an irreducible dualism
in the heterogeneiy existing between protons and electrons, the
theory of relativity will attempt to disolve this heterogeneity
by suggesting that "they are actually ^similar units; of structure,

-324-

Sw° $ iff °f no ° ^ i3es in -*holr relation to tho general distri-
bution of matter which forms the universe," (48)

I . . 3 Th ° Gnd towards which physical science is aiming
is to reconstruct the whole universe, i.e. to conceive the universe
in terns of structural knowledge determined with~elSctness by ma-
thematical formulae. Knowledge of this kind prescinds completely
irohi che nature of the units which constitute the structure. In
'^ X Z P r^ Ce ?, r ° subst ituted manipulatable mathematical symbols,
which while they serve as admirable instruments for knowledge of
structure, at the saiiB tine blot out all that lies beneath the
structure. Mathematics is e speciall y competent to express patterns,
but incapable to reveal the (pr55|gmtures of entities and o perat-
ions, ihrough group-structure mathematics is able to lay hold of '
realities which in themselves are not directly susceptible of ma-
thematical conceptions.

All this explains the increasingly important place
of mathematics in physics, for it is only in mathematical form
that purely structural knowledge can be adequately expressed. In
particular it explains the central role played by the Theory of
Groups o

This structural Icnowledge is at once extremely ob-
jective and extremely subjective. It is objective in the sense
that by prescinding from the proper determinations of things, the
knowledge of which involves so many subjective elenents, it is
able to constitute a type of knowledge that is exactly communica-
ble to all minds. It is at the same time sub jective in the sense
tha -t thg_ essential plasticity of the sameness (out of which ) the
structure is formed gives unlimited scope to the constructivity
V pf the mind . In fact, this whole process must be looked upon as
the mind's imposition of its engrained forms upon- reality. This
is a point that has been stressed by Eddingtons

Granting that the elementary units found in our a-
nalysis of the universe are precisely alike intrinsically,
the question remains whether this is because we have to do
with an objective universe built of such units, or whether
it is because, our form of thought is such as to recognize on -
ly systems of analysis which shall yield (gaFE^ precisely like
one another . Our previous discussion lias com-.iitted us to the
latter as the true explanation, We have claimed to be able
to determine by a . priori reasoning the properties of the ele-
mentary particles recognized in physics - - properties confirm-
ed by observation. Accordingly we account for this a prio ri
knowledge as purely subjective, revealing only the inipress

-325-

/-

and doducibr? th L 0U S^ e <&**» knowledge of the universe

on tte l™™" \k tl ^*7r EE ^ Ifi 5?Hi-s of our frame of thought
thercfo^ °S^Se forced into the frame . . , I want to show
r^tZtTt , h ° °°T ept ° f idenWc ^ structural units cx-

wMrh hn Very ^IMn±aryja.nd instinctive habit of thought,
Wlotw U " c ? n « clou ?ly directed the course of scientific de-
vS?^f I* Briefljr ' " is the ^bit of thought which regards

^t^te OTg-produot oF atm^ir ^TSS^Sr S^Brie^ ■ We keep
v? n ?f ^ g 0Ur SysteVa ° f ^lysia-uT itil it is such as to
yield the sameness which we insist on, rejecting earlier at-
tempts (earlier physical theories) as insufficiently profound.
The sadness of the ultimate entities of the physical univer-
H . ls a f o^s ceabl^consequence of forcing our knowledge into
this form ofTttiought "TT-T-T-Sonclude therefore that our en-
grained form of thought is such that we shall not rest satis-
fied until we are able to represent all physical phenomena
as an interplay of a vast number of structural units intrin-
sically alike.:) All the ■ divers 1 tyhf 'thp pW^n™^ wilT~thcn~
be SQen to_oor respond_ to different forms of relatedness of
ythese units, or, as we .should, say, different configurations,
49;

1

The foregoing analysis makes it- clear that it is
precisely through the source of homogeneity that the common matrix
of quantity offers to the mind that it is possible for science
to rationalize the cosmos. Much has been written on this point
by modern philosophers of science. Professor Whitehead, for exam-
plehas this to say in Proce ss and Real ity;:

It^is by reason of this disclosure of ultimate sys-
tem that an intellectual comprehension of the universe is pos-
sible. There is a systematic framework permeating all relevant
fact, By reference to this framework the variant, various,
vagrant , evanescent details of the abundant world can have
their mutual relations exhibited by their correlation to the
common terms of a universal 3,ystem . Sounds differ qualitati-
vely among themselves, sounds differ qualitatively from colours,
colours differ qualitatively from the rhjrOhmi^hrobs of emot-
ion and of pain; and yet all alike are periodic and have their
s patial relations and their wave-leng ths. The discovery of
the true relevance of the mathematical relations disclosed
in presentational immediacy was the first step in the intel-
lectual conquest of nature, (50)

But perhaps the author who deserves particular at-

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tention in relation to this question is Emile Meyerson for we
of hirvn\ lnG - herG UP ° n th ° Central *«*» which r^s?to ugh all

Snd c^oHndor T^ ' "FT™* 1 ^ a laWd to show *£* tto
To ITS, S -f a?* reality except by 1 reducing its diversit y)

Sole^To i T~ITfT S ^ , -^ that tte ^noity in which it comes
lUy ?Unfortu^W S ^ S ld ° al iS ,' btevk °f U gaifferentLated smtia-
SiS hS ^ y - f^ 1S USUa1 ^ a fairfTThick penumbra sur-
rounding his analysis because he fails to make a number of important

cSnfulTt^ di + stinQt f na - Lite ^amenides and Anatgoras, Z
f^SH^— 5 -— ^ ° ntol °S ical ^oblems of the one STd
3hf^P' ? not seelTto recognize the dif ference between

Sn ^" 1 °^. ta0 7 able f °r u- ^d what is more knowable in se, bet-

Ihev hnl o i°? al -'^^ ich thingS W for us and ^e rationality
difk™+ ^f 03 ^ 1 ^' Proni thi? arises a confusion between the.
, £? w? • lldS ° f dlversitv a *d ^e different kinds of unity by
which the mind seeks to triumph over the diversity, With regard
to diversity, he fails to make the all important distinction
o? -r n + - + m ^ al and n ° etic he *erogeneity. And in his treatment
of identity there is no attempt to distinguish clearly between
the horaogenization arising from the reduction of singularity to
universality, from the coordination of laws in theories, fronTthe
relations of causality,- from the quantification of reality, and
from the methods of limits. It is especially important to 'keep
\this laso type of unification distinct from all the others,

, " ■ , BuJc in spite of these limitations, his fundamental

tenets are quite correct. The following passage is a good expres-
sion of his central theme :

Ce a. quoi la science tend de la nianiere la plus im-
mediate, c'est a etablir jmrap^pjrtj.ogijgue entre les pheno-
menes » aJLes_dijduire les una des autres, Mais cette tendance
n'est au fond, qu'une consequence, une expression particuliere
d u postulat de la rationalite du reel - o'est, en quelque sorte,
■ deIa~meme~moniie de rationalite. II n'est done point eton-
nant qu'en I'accumulant nous finissions par reconstituer,
au moins partiellement, le capital primitif, e'est-a-dire qu'a.
force de deduire les phenomenes les uhs des autres, la scien-
ce finisse par faire crouler les murailles qui en divisaient
le domaine en parcelles distinctes, privees de communication
les unes avec les autres,

Cette operation, cela est de toute evidence, ne peut
s'accornplir qu'en renoncant a. ce qui est qualitatif, au pro-
fit de la quantity En effet, tout ce qui est affecte d'un
indice qualitatif devient, par la. meme, specif ique, isole , . .

Mais ce qui apparalt certain, o'est que l'eolosion
de la notion de quantite dans 1' ensemble des conceptions du

-327*

eSLf en etant favorisee par des constatations
G^IT SUr 1SS ? henome nes,.. est cepindalrtlurW
s°tion Zi oo^ + C t "T 1 Ae 1,ex * licati ™> de la rational!-
l^oute^n^re^CsiT ^ ^^^ fonflamn tal de n °^ P™*

tbp ™h, q - If th<3 id ? al 0f scie nce couH be adequately achieved,
would t ™i7 rae W °f d be r6duced to an ^«nse tautology and '
n^lWr ITJ and I^ cora P le te^. "La raison, en cherchant
a expliquer, a rendre rationelle la realite exterieure, la fait

Sn^^f ^ leme , nt d T le tout ^-tinot de Impact St
cur t.J L Acc0rdln 8 t0 Meyerson, this collapse will not oc-
ZTil°T ^V^T Tillkgvgr_rg min propped up , so to apeak,
bX~£ rational _elgmgnts ^SdhTSTelientiaLly refectory to the

I mind s process of homogenization. As we have already suggested,
Meyerson fails to make it clear that from a more fundamental point
° f vie * these props oxe rati onal element s jC ln the sense that they
aepg^r^mt ural . heterojineity pit is because of them tbaTTur
attempts at^ rationalization are kept from issuing into the utter
irrationality of a purely homogeneous and amorphous universe whioh
wou^d_corregpon d to the original "irrationality" of the human inTiT ^
leot m its st ate of tabula rasa - It is a striking and highly si-
gnificant paradox that if our attempts at rationalization could

.succeed the universe would be rendered completely irrational.

Better than any one stateramt of Meyerson himself,
the following passage of Prince Louis de Broglie sums up the es-
sence of this doctrine:

Selon lui (Meyerson) dans la recherche scientifique
comrae dans la vie quotidienne, notre raison ne croit avoir
vraiment compris que si elle est parvenue a degager dans la
realite mouvante du monde physique des identites . et de s per-
manences , Ainsi s'explique en particulier la structure commu-
ne des theories physiques qui tentent de grouper des catego-
ries de phenomenes par un reseau d'egalite s, d' equations , cher-
I chant toujours, autant que faire se peut, §. elirniner'la diver -
I site et J Le ohang e ment reel et a. montrer que le consequent e-
Vfcait en quelque sorte contenu dans l 1 antecedent. La realisa-
tion complete de 1' ideal poursuivi par la raison apparait a-
lors comme chimerique, puisqu'elle cons istait a resorber . tou-
te la diversite qualitative et toutes les variations progres-
sives de l'univers physique en une identite et une permanen-
ce absolues, Mais 3i cette realisation oomplete est impossi-
ble, la nature du monde physique se prete neanmoins a un suc-
c&s partiel de nos tentatives de rationalisation, II existe ,
e n effet , dans le monde physique non seulement des objets qui

-328-

rn^S a P^u Pres aemblablea a eux-momes dans le temps,
we nS °1 eg ? ries A'objeta assez semblables entre eux pour
concept ™n 10 n S los / dentifi ^ ?n les reunissant dans un
i2Z " ° e !° nt CSS ' fibres ' de la halite, comme dit
v^e I ' qUS n ° tre rais ° n sai3±t dans 1' experience de la

SontSw^ T Ur C ° nstit ^ ^ec elles notre represent^
men- JS, P ^""^ exterieur; ce sont ces fibres egale-
"l n \ ^ f S P 1 ^ subti^s,revelees a notre connaissance
?' 2< me ^ hodes Krffmeos de la recherche experimental , dont
Zl?^' du , savant s'enrpare pour chercter a extraire de la
oSllP ^ Ce ™ ntC ' ^ P artd ' Antique et de permanent
qu elle renferma. Aussi, grace a 1' existence de ces fibres
bien que 1< ideal de la science sort en -toute rigueur irreali-
sable, quelque science est possible: C'est de la grande mer-
Zv~ B ? ° e ™ e situation se trouve resumee par une phrase de
;.;,. .eaua. Valery,, phrase sans doute inspiree par la lecture me-
me des ouvrages de M. Meyerson: L 'esprit hunain est absurde
\par ce qu'il -recherche; il est grand' -par ce qu'il trouve. Mais
cemme en definitive l'univers ne peut pas se reduire a une
vas-ce tautologie, nous devons forcement nous heurter ca et
la dans notre description scientifique de la nature a des e-
d-enients 'irrationnela' q ui rp qjstent a no stentatives d' iden-
tification, 1' effort jamais lasse de la~raison hum aine s ' ach ar-
nant a circonscrire ces elements 'et a en reduire le domaine,
(53)

I* is clear, then, how the mind through the homoge-
nization of the cosmos succeeds in triumphing over the irrationa-
lity that arises out of the pure givenness of the deliverances
of the senses. Unlike the isolated perceptions of sense experien-
ce, the quantities with which mathematical physics deals lend them-
selves to the mind's desire for deduotion: they can be both the
conclusions and the .principles of deduction. And to the highl y
( integrative value of quantity jwhich m akes them derivable from each
other is added the advantage of the wide scope of rational possi-
bilities which arises from the, extension of the -quantitativ e sys-
tem to_ include zero values'; "liegative values, in finite v aluesTjTco

But what is the price which the mind must pay for
this triumph? Prom what has been said about the movement of scien -
ce towards tautolog y., one might be led to suspect that the price
is^atter high, and. to wonder what has actually been gained by
t: abandoning the logical homogeneity of generality in which the spe-
cific distinctions of things are swallowed up. It might seem that
the homogenization of experimental science is contrary to the ve-
ry nature of that science, which seeks, to get at things in their
specific natures and consequently in their heterogeneity. To put

-329-

ion TtiZoln^^T 11 ^ ^ n0t thS ^^tative homogenisat-

M,.

difference belS^F^'tP^^^ There is a " essential

cantetenunciS' 1 ^ ° f f ^^ ^— - ^Vsec" n"
by locating i^ nblY • ^ V ^ ±!Xl ° P ° r ns we gained above,
Scna^wL a 3 °t ln the rea:bn of coraraon aensiblas, nathenia'

tEfl t,™ f °^, t0 USe the ex Pression of Meinong, the '-quan-
tified surrogates" of the specific demonstrations of mture. And

,but ^n m themaclcs * s not °**-y a science of great general^,
but also a science of great exactness, mathematical physics can

sSci!ic a dr°T ? f ri8OT0US Physi ° ai — ment, eta t ^
,2m££liX IS "? I^ h fa ? greater C ° nCrete P^^ision than
^^~hff- f ^ ^ alitati ^ aspects of nature have

mutation. aS VG m ? d f s ar V heir variations involve quantitative
mutations. And we pointed out above that there can be a direct

t™ a - n et T! n the ^scendental quantity that is intrinsic
to qualifies and forms, and the predicamental quantity that is ■
measured by physical processes. That is why the homogeneity of
mathematical physics is not a complete renunciation of the hetero-
geneity of nature. Prom one point of view it is a means of knowing
it betcer, and' in this sense there is a distant resemblance here
of the perfection of co-nition found in the separated substances
m which it is precisely through the homogeneity that the hetero-

/geneiuy is known. And even though in its superstructures mathema-
tical physics moves towards undifferentiated spatiality and tau-
l? Pi Xt alwe ^ 3 st arts from, and must inevitably lead back to,

Uhe heterogeneity of nature. This makes it essentially different -W
science based on logical homogeneity , •

1 T bus the mind is able to enjoy an experience remo-
tely analogous to the combination of sight and touch in sense ex-
perience. It is able to gat at nature with something that resem-
bles the certitude that is derived from touch, and with southing
that resembles the distinctness that comes from sight. But it is
extremely important to recognize that in both cases it is a quest-
ion of a mere substitute. Mathematical method affords a kind of
exaotnoss and certitude in dealing with mture, but from all that
was said above about the essentially dialeotiaal character of ex-
perimental science it should bo clear that it cannot provide true
objective certitude. The same must be said of distinctness, 'For,
with whatever extreme precision we get to know the quantified sur-
rogates of the qualities and forms in nature, it is always with

-330-

Innfl 5n™ • ^^ t ™ are dealin S and nevQr ""* the • qualities
NTnS^ ° Vm Pr ° per ' s P ecific ^ture. Exact knowledge
I ?f not th f , s ^ lg-4^_g£gojjj : g_taiowj £ dgg Moreover,^" surrogates
1^7+ an*xvalent; at the same time that it unites us with the
v object for which it substitutes, it separates us from it,

-,..,. , , To ^tempt to geii at the proper nature of the qua-
litative through purely quantitative methods is to accept one of
the fundamental principles of Hegelian and Marxist dialectics:
every quantity if sufficiently increased turns into a quality.

That many have actually been led to identify the
qualitative with the quantitative is well known. Spaier, for exam-
ple, holds that our physical experiments succeed in measuring qua-
lity directly, (55) For him quantity is not something that exists
objectively in the physical structure of reality, but a conceptual
construction which_ _re suits from our process of me as urement . (56)
But ordinarily this identification has been approached from the
opposite direction by a sacrifice of quality to quantity. The e-
vident dependence of the sense qualities upon the organic struc-
ture of the sense faculties, and the immense success of quantitative methods
in .science have led some to deny an objective status to all qua-
lities and to conceive of the cosmos as a purely quantitative struc-
ture. Such a p osition in completely gratuitous . We have already
shown that even though conditioned by the instruments of percept-
ion, the sensible qualities are not psychical, but physical and
hence existing objectively in nature. And the fact that they do
not exist in the distant object in exactly the same way as they
are perceived, is no argument that the object is deprived of all
qualitative determinations, (57) Moreover, the success of quan-
titative methods cannot be adduced as a demonstration of the non-
existence of qualities without transforming a methodology into
an ontology, (58)

■ As a matter of fact, the existence of an infinite-
ly homogeneous reality is hardly conceivable. And' even if it we-
rd a possibility, it could nevor be a source of knowledge. - (59)
It could not even be measured, For, as Professor Thompson has re-
marked, "quantity, per se, in other words, pure undetermined quan-
tity, is as unmeasurable as quality . It is measurable only when
bounded, stamped, or permeated with quality. The quantitative pic-
ture of Nature, in spite of its satisfying accuracy is not self-
supporting: it is executed in a framework of qualities, with which
the savant must maintain contact." (60) It is worth while point-
ing out, moreover, that the numbers out of which the structure
of mathematical physics is erected, are conorete measure-number s.

-331-

Pm even thoSh S^ ^T Domethin S ^o than pure quantity,
t rhtil2. y ° ¥1 * ece ssarily have a direct and iwnedia-
the ontolo^^l ^qualitatively different sensations or with
lita?ivSy S differo^ Xtle3 ° f ^^^ J^ey are the results of qua^

in the scien+4 S J!!* 8 el ^ bl f f us to see what is actually involved
Sting tte ^pM?- geni S atl0n ° f the COsmos ' The Carriers iso-

rTllvJtLTJ lt° properties of -nature are broken down; The P u-
forrSTnto \ L °l ■ pr0pe ? ties are ^stered; nature is trans 1
SLlnW? 8 astern; reality is rationalized; the most
profound aspect of the cosmos: the order of the. whole, is in a

eTwi^telf V to nlind : At . thG """^ ^"fac^is^ain-

S5y$§*!*u1S^ a^hll^nViuTlfira

without its price. For the determinate properties of things in
££% s P°°^io essences, the very inner natures of things have

stflLT, o? * ? P ^ tUr ?- ! hS hillslde With its 8»^« and its
* «^ L? ' the ele P han t ^ its own proper essence - - all

of the things in Nature which seem to be of the greatest signifi-
c , anoe _ for, th e other science s of reality, for all the arts , and
fglLaun^iif e itsel f, have slipped through~the fingers "of "the"
physicist and have left in their wake only a series of pointer .
\ readings, (61)

This raises the question of the relative rationali-
ty of the qualitative and the quantitative determinations of rea-
lity. It has often been stated, that the latter are more rational
than the former. That there is a sense in which this is true is
evident from all we have been saying,' But perhaps one might be,
tempted to question this superior rationality on the score that
quantity is said to follow upon matter which is the source of ir-
rationality, whereas quality is said to follow upon the form, John
of St, Thomas gives us the answer in the following terms:

Non est intelligendum, quod quantitas sequatur ad
m ater jam nudam sine forma , cum constet sequi ad gradum corpo-
reitatis qui praebetur a forma. Sed intelligitur sequi mate-
riam, vel quia solum invenitur in rebus materialibus, quali-
tas autem sequitur actum, etiam si immatorialis sit, et sic
proprium est qualitatis qualificare sicut et.formae; turn etiam
. quia quantitas se habet in genere accidentium, sicut materia
I in genere substantiae, quia non est activa, sed medium recep-
Vtivum aliorum accidentium et inter reliqua primum, (62)

Quantity has the greatest advantage of being the
accident closest to substance. Material substanoe is a substance

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tasTthev arSnn°^r r \r b :e aUSe ^ 3atter arS qUAlitiSB, but

Dc -oausc the y_a re sensible . Mathematical beings are more -Dei-foot -

S^^^S^? 1 ^^^ ° f ViGW ° f -<^S certitude,
t^j— ^51— ^ggag£3^g_Jhe ^ource of precision . Moreover,
thexr very^tlHe^iaes them ntore manipulatable by us, Final-
seen^ ^ ^^ ^ C °™ mtrix ^ich, as we^ave just
nlTAr +h necessary for the rationalization of the cosmos. For
all of these reasons quantity has a source of rationality which

t^^T^T?- Pr ??! r tiGS ° f reali '^ d0 not Possess, And it is a
type of rationality that is particularly amenable to 'the methods
01 physical science,

. . , 0n the other hand, the specific properties of rea-

lity are far more rational from another point of view. They reveal
the proper natures of things. Consequent^, ■• it is in physical scien -
ce that their rationality is particularly relevant. As we' explained
in Chapter I, the rationalities proper to physics and to philoso-
phy are related to each other in inverse proportion. In the last
analysis, it all comes down to a, difference' in \thg_t ype of raeaau -
V remenc ) profier to_each_science„ In the following Chapter we shall
return to this point.

And now, having seen the way-, in which the mind triumphs
over one of the sources of ; irrationality connected with sensible
perceptions - - their isolation and pure givenness, we must turn
our attention to the other element of irrationality about which
we spoke earlier in this Chapter - - the indef inability of proper
sensibles. By the same process which we have been describing scien-
ce succeeds in nastering this second irrational element, it suc-
cee ds in definin g, the indefinable ., Through, its quantitativelnethods,
physics is able to define heat and colour in terms of movement
\of molecules, light waves, etc, A non-scientific person with the
faculty of sight cannot define what he means by redness, but a
blind physicist can. And the advantages of this definability are
so obvious that they do not need to be mentioned.

But once again we must remain erratically aware of
what is actually involved in this defining of the indefinable.
Prom what we have said about the impossibility of attaining the
qualitative in its proper, specific nature by means of the quan-
titative it is obvious that the scientific definition of heat,

k

-333-

(SrTL^world^f SS^y^^^^ thSSe ^perties,
I movement of molecule^' ISYh^ T ex P re ^ io ^ as "heat is a
a (correlation betweenSthf L t I actually mean is: thesis

T;oj.j.(why^there is such a correlation,

for qualitS wS e H tiSt ^° eS "■?* Seek a derivative measure
orler to find w^ ^ lnoa ? nble of dire °t measurement in
of on obiecf li f ^lities really are. The measure

wLf the obiect L ^ araenta \° r deriVed ' does not ^ess
oeofT™l n T' ^presses how the object, as an instan-

senL\7tZ^tT ,t r i ±S , rel ^ ed to another object cho-
Iter. (63) character or for a emulated charac-

vious inst™?^ 10 ^? 5 li 5J e ?.° f DeW3 ^ ^ spite of their ob-

TO ^,^Sn?ii£i ' ins out rather Gccurately the point

to *w + h ! ^ esol ?' tion of objects and nature as a whole in-
^ l *? ! ? exclusively in terms 'of quantities which may
be handled m calculation, such as saying that red is such
a number of changes while green is another, seems stFange and
puzzkng only when we fail to appreciate what it signifies,
+o +^ , I • 1S ? declara tion that this is the effective way
to think tmngs; the effective mode in which to frame ideas
of chom, to formulate their meanings. The procedure does not
vary in principle* from that by which it is stated t hat a TT
a r ticle, is wort h_ag_inan y___aollars and c -j n ts . The "l.n-H-.^ a f..4-o-
nent does not say that the article is literally or in its ul-
tima ue 'reality' so many dollars and cents; it says that for
purp ose of exchange , that is the way to think of it, to judge
it. It has many other meanings and these others are usually-
more important inherently. But with respect to trad e, it is
what it is worth, what it will 'sell for, and the price valu'e
put upon it expresses the relation it bears to other things
Lin exchange... The formulation of ideas of experienced objects
in terms of measured quantities , ( fas theoo are est ablished by
an intentional art or techniqTieTV does not Rny ttTTlM. ■;„-
the way^ they must be thought, the only valid way of thinking
them. It states that for the purpose of generalized, indefi-
nitely ex tensive translation from one idea to another, this
is the way to think them . . .

There is something both ridiculous and disconcerting
m the way in which men have let themselves be imposed upon,
so as to infer that scientific ways of thinking of objects
give the inner reality of things, and that they put a mark
of spuriousness upon all other ways of thinking of them, and

-334-

se Stlfie "f ylng 0tem - Xt 1S ^-ous taoauae the.

mdebv If • oonce P^ ons » like other instruments, are hand-
ttet of +h? , ? UrSUlt ° f reali ^i°» of a certain interest - -
that of the i maximum convertibility of every object of thought
V mto any and every other, (64) "i°ugm;

succeed in arJ.Vf, f^ar then *hat mathematical physics does not
succeed in actually defining the specific properties of nature

SLrTto^^^^^ Mt e™th

be made \t ? 0rrelKir °? a further-important qualification must
SonVl +K rf* since scientific definitions are necessarily opera-
nbSl definitions of physics do, not give us an absolute,

objective, quantitative element that, is in correlation with the
specific prope^tiesjthey necessarily involve the whole operatio-
nal Pjgcedu reCb y whiSH) this quantitative element has cone to be
icnown by us. This obviously removes them still further from a di-
rect rendition of the jup^uid^est of the sensible properties.
Ana in_ chis connection it is necessary to point out that though
the pointer readings which issue from our processes of measurement
are not abstract but concrete numbers, they are not concrete in
the sense that they directly correspond to certain sensations,
but only m the sense that they are produced by concrete proces-
ses of measurement into which a multiplicity of concrete determi-
. nations have entered,

This brings us to-another significant question. One
of the important reasons given above for the adoption of quanti-
tative methods in physics was the attempt to overcome the subjec-

+hvnn?v 1 a ? d =,^ t + ? ? p ? ,ilorph J sra of sensibility. We pointed out how
onroukh a substitution of inorganic instrum ents ofi
mea sure; rent for organic m struf.Lnts of v)o rceirEiofrsnien- '
ce has been aBIe^fco triumpn over the su'Blectivily of sense cognit-
ion. But just how complete is this triumph? Do our measuring ins-
truments provide us with a perfectly objective rendition of rea-
lity? Until fairly recently, it was not uncommon for scientists
to think so, (65) Yet a greater eseror couH hardly be imagined.
In the next Chapter when we come to analyse the process of measu-
rement we shall try to show just how much subjectivity this "process
involves, and for the moment it will suffice to merely mention
the more important sources of this subjectivity. In the first pla-
ce, there is the mental operation involved in the conception and
method of application of the measuring instrument; all instruments
are constructed and applied in accordance with certain scientific
theories, \\ and honce participate in the subjectivity of these theo-
ries,)! In the second pla'ce, there is a physical operation involved
in the actual process of measurement: the instruments of measure-
ment enter intrinsically into the process of measurement in such
awaythat the results are not independent of them.

-335-

«°ipiont 8 !XS^tiSS , T tB are - not raerely passivQ

they play an aotivTS ' fu he rayS inT P in S in 8 ^pon them:
a causal influenc^on l£ ^ iX^ ° f raQas ^ing and exert
consideration fo^Lr+^M eSUl *' The P^ioal system under
cess of rileas " r gf^^lxt7 subject to law only_ifthepro-

'the measuring inl^^T^f^^^^ 3 inseparable from "*"*

it; and similarly T"^° r ^ 0rgan ° f Senae tbat Perceives
,ple from the" invent f "^ ° annot be separated in princi-
™ e lnve stigators who pursue it. (67)- fl^

objects we succeefon^^ t0 - Set ^ fr ° m a ndaetura of 3ens °s c^
and object, y ln arrivln S at a mixture of instruments

crued to scienS^rorthe^ 1 ; 111 ?-^ 1 M thS advanta S^ that have ao -
of measurement for nl • s ? bs * ltuti °n of inorganic instruments -
tant to realiL ?Lt our 1 " lnstruments ° f perception, it is impor-
n*mt, and that f r™ tM T *" alS ° inst ^ents of measure-

ference beWn tte £o! P ° f ™ ttor ° ±S n ° SSSSS^Sl *!*-

... There iT^t 1 ™ + * ? ^ ° f Crude P^oal measurement .
seres S the 1 eSS<3ntla ^ distinction between scientific mea-
quSntanL^tnt^ 63 .^ thS Senses ' ^ either case ~our ac-
S chants \t Z* term ?- world corae = to.ua through mate-
hS labo^S ' observer's body can be regarded as part of

^t ffr U T nt ' ^ S ° far as We *™«> " o^eya the
scTentifi;,t therefore group' together perceptions and the

welnclude ^T^' •" SPeaklng ° f n P^tioulor observer-

we include all his measuring appliances.. (68) -i*.,-^

nf . , , T he grater objectivity that ot.es to us bv means

rrL-,~r i rr — t^^ — -■•■-■ ..>, ^ m- uum-m ujac-cer. xr it were T5c

sible to know the physioloii^airiTate of. theTTnger with great

T^ZZ COUM F mennS . 0f " arr±Ve a K the de S- e °f*»*e-
In ^nor^ f gr f^ Pf ao "f oa as that achKVed by a thermometer.
In general it must be kept in mind that in our perception of the
common sensxbles, even without the aid of impersonal instruments,
we already have a (compar ison^

, , In connection with what was said above about the
advantages of the homogenization of the universe deriving from
the greatly extended range which meaauromnnt adds to our limited

-336-

( powers of DBrfpnt-in^
is true that there ^ a r ? s ^ at ^n ™^ bo made. For ^-n- O
which can be rZ ? »uoh_in nature which cannot he sensed but

deal Vaioh Jt„ ' ; XS llke wi a o true that t.hore is a great

'M.oh oan bo sensed and cannot be measure.

sensibility would S , 1 o^ ySiS °, f the rel * tion between science and
tempt v^renotZ n lt 0mP tS v if ' bef ° re °™ cl ^™S, some at-
remains linked to tv, detorraine how ^sely the scientific world
seems to have H roJ^ sense ., world. From one point of view the bond

r-thennticS pf™s is bSf ,*"""»■ . A ! tos ^^ be - -old,
onlv kind nf ;1„ m.-,^ ?? upon a minimum of experience. The
tlst tooal™K bllliy , ttat is diro ° ^ squired for the scien-
iS oblect^r, ? ^ W ° rk " that Which is necessary' to recogni-
fLS Snl on n r 116 "^ Md t0 P^ive the coincidence of a

dommvi g TTTCTTT^rrT -^ """"ig^rarxapxe .Line. All th at this .

vMed it aff ordf n rS £• this + 4ualitative differentiation is, pro-
ions In other 1% X ° Xent means for raakin S necessary distinct-
^1,1,?? ^ 8 1 ', S ° 19Me has corae as close as- possible to
S^p^SpjLn' (ST 6XPerient?e " ~ ^^tit^iv^on-

[„*.,+ , .. ^ ^ is important to keep in mind that in spite

of its tenuity the bond between the scientific world and the sen-
\ ses remains essential t

What I mean is this: we rig up some delicate physi-
cal experiment with galvanometers, micrometers, etc., special-
ly designed to eliminate the fallibility of human perceptions:
but m the end we must trust t o our perce ptions to tell us
Vag_re|ult of the experjji^nt.[Evgn _ifthT apparatus iTlelf -
- g ggording we employ our senses to read the records J (70)

The desensibilized processes of physics are not self-
supporcmg. Independent of the whole background which they have
in ohe sensible world they are i meaningless,. Moreover, it must not
be forgotten that by the very fact that mathematical physics is
physics, it must realize the red uctio ad sensu m mentioned inChap-
V J}j^Y^ i s characteristic o f_eye ry. science of nature} It ■
must< both) toTre~Tfe origin^ in the sense woTId and termi nate' iiTit,
Planck explains this very clearly in The Univ erse in the Light
of M odern Physics: ~ ; : : " —

In my opinion the teaching of mechanics will still
have to begin v/ith Newtonian. force, just as optics begins with
the sensation of colour, and therrrodynamias with the sensation

-337-

sis is subs-

sos

/tS t f ! t aeBpitB the fact that a raore P^eise tes
S f cancf of °li £f iU " mUSt n0t be f - g otten ttt the"sT

is ao^Sp^s tfcoi^rto the r opcaltions uitiniate -

is indfPd r*™ 1 •!'■ reJ - aolon t0 *^e human senses. This
SvsioS r^?°i r £ t10 ° f thS P eculiar ^thods employed in
applicable to S"- 1 ™ WiHh t0 f ° m C ° nce P ts and Xpotheae.
S? S2i! „? ? hys ^ cs I. we ""s* be 8in ^ having recourse to

obSln Sir t ' arS tbS ° nly S0Ur ° e ° f our ideas ° ^* to
obtain physical laws we must abstract exhaustively from the '

x'elevant^r^f ' °? PBnovo the ^inltiox* set" up all ir-
relevant elements and all imagery which do not stand in a lo-

vefoJ^tT^ ^ th , thQ measurements obtained. Once we ha-
bv ^t£S f yslcal laws > and cached definite conclusions
rLt ^ Z °? \ P fT S ? e ?' the results _whichwo_ have obtained
must be transl ated_b ack into the lant^age-pTTKi world of our
||MgplLJho£^35Tg _of any use t o us , In a manner this
method is circular; buTTt is essential, for the simplicity-
and universality of the laws of Physics are revealed' only af-
ter all anthropomorphic additions have beun eliminated. " (71)

. As physics progresses it inevitably becomes more

abstract and more highly symbolic. But to even its most abstract
symbolism there always remains attached a dictionary which links
up the symbols with concrete entities. And these concrete entities
ultimately lead back to the world of sense. Thus modern physics
presents the paradox of an ever increasing detachment from the
sense- world, and at the soma time an essential attachment to it.
And this paradox is comprehensible only in terms of another para-
d ° x - modern physics is at the same time physics and not physics :
that is to say, it is a hybrid science ^an intermediary science"^
It is formally distinct from pure natural, science, but at the sa-
me time it is a valid study of nature. Because it is formally ma-
thematical it must in its development draw ever farther and far-
ther away from the world of sense; but because it is terminativeta
physical it must inevitably lead back to it. >

This brings us to the final point that must be tou-
ched upon before we leave the general question which has formed
the subject of this -Chapter. In setting up the problem which has
been occupying us we mentioned that some authors see in the recent
developments of physics an abandonment of the common sensibles
similar to the former abandonment of the proper sensible si and com-
plementary to it. Vfc do not believe that this is the correct in-
terpretation of the newer scientific constructions. It is true
that they are not susceptible of direct imaginative representation.
But this does not moan that science has removed its objoot from

-338-

from fi ^alm hS of X senaiW -^ as earlier it had removed it
thi^s! p£ So Ut P "° Per r naibles » " P™bably ircana several
hove to do with So S? ^ S ° ? &r aS thQSa recent oonstruotions
beginnLp to d? ,™ "^o? 08 ™- world > " means that science is
S not L o^K £ ^° ? henomena ° n ^b microoosmic level
n7on the mfS ° f ? lrG , Ct ^P^ntation in terms of phenome-
re et Lmdlre : ' ^ ^ 0&1±G poirit8 this ° ut * n 3fe*±&-

fl « 1, rl . + -? 1US n 10US descen dons dans 1b a structures infines
^L Li • 6 P n ° US nous tt PP8K»vons que lea concepts for-
ges^par notre esprit au cours de 1' experience quotidienne,
nL,T Partxculierement ceux d'espace et de temps, deviennent
urpuissantsa nous permettre de decrire les mondes nouveaux
ou nous penetrans. On dirait que le contour de nos concepts
aoit, si 1 on peut s'exprimer ainsi, s'estomper progressiveraent

pour leur permettre de s'appliquer encore un peu aux reali-
tes des echelles subatomiques, (72)

* ju „ •, But in S eneral the most fundamental significance
of these developments seems to be that science, by using as its
instruments mathematical entities, which, as we saw in the last
Chapter, oan stretch their connection with the imagination to the
extremes of tenuity, lias so intellectualized its subject aa to
place it outside of any immediate relation to the sensible, The-
rg_i 3 no reason wh y it should not do this , provided all of its -
intellectual constructions can be made to lead back ultimately
<to_verification in the sensible woriep In this way that can be
said to "explain" the sensible world. But this does not mean that
these constructions give us a direct and immediate revelation of
things as they exist in the real world or that they prove the com-
mon sensibles to be illusory.

And now, having seen the basis for mathematization
that exists i n nature, we must see how science, by laying hold
of this basis |through the instrumentality of measurement J succeeds
in transforming natu re into a nevt world of symbolism . This Chap-
ter has attempted to show that in mathematical physics the mind's
ambition is to transform the universe intoQijjurel y rational s ys-
tem) in which multiplicit y and difference twill be constructed out
of unity and sameness J It is in measurement that th e mind finds
a road towards its goal, Porjoien^urement oonaint.q in the nrpnn t.Rfl

application to reality of ^the same uriiliy ) a unity which the

jnit id has determined )

-339-

CIIAPTER EIGHT

AN__ Am_LYSIS OF MEA RTTTOfTCNT.

J^g— §£J^B° 9. and Measur ement.

B , lo . , v , Thi f, C^Pter is in a sense the pivotal point of our
whole study For the central idea in mathematical physics is that

SticlrSF^ T lvinS * Uni ° n ° f the ^ si -l <** "atto-
ma.ical worlds, and it is precisely_ti2rough _measurement that the -
sot wo worlds are brought into oont aotrThis~was already recogniz-
ed by John of St. Thonns, for in speaking of the mathematical po-
sies of his time, he writes: "Astrologus non agit de coelo at Sla-
V et *5» ucguM_entia mobilia,Csed _ut mensurabiles sunt eorum mo-
ws^ {i) The reason why measurement is able to achate this
union between a science that is essentiall y_ex perinBntal and one
than prescinds from experiment is that, while remaining a physi-
cal instrument of experiment, it is not an instrument which mere-
ly reveals physical phenomena; it both reveals them and_tr ansforms
chem into numeri cal values . "Ce qui distingue notre science," wri-
te , s Bergson, "ce n'est pas qu'elle experimente, mais qu'elle n' ex-
periments etplus generalement ne travaille qu'en vue de mesurer."
(2) It is significant that the names of practically all of our"
modern experimental apparatus end in "meter " whereas formerly they
v ended in "scope ". " "

In other words, there is something both physical
and mathematical about measurement. It is, as it were, a transform-
i ng machin e into which [physical determinations ) enter and from which
I numbers ) emerge. And oven though the concrete msasure-numbers which
issue from our pointer-readings are not in_th emsolves a mathemati-
zation of the physical in the full sense of the word, they are
the jinotatetion of this ma therm tization . They are the stuff out

-340-

already have something ^^ V^ With thrpiySiSnTtETy

And just as the whole nnth^^S?"? *° th > "^matical world,
ses qutofthe phvSo^iS t interpretation of nature ari-

of measurement! For no Sthfn,?^ , ^ phl£aoal thTOU S h Processes
ui- vcrnied m concrete pointer-readings,

science i s direlSlV? 1 ^" 3 W ^ tte Who]i3 P^ess of physical

t^( g ,_ m}= £Sei? r PO feSfi^f rm-

^^SenT ^"S £? - ~ SL^e^tTte-

extent in 4it *?>. 22^^^ ^SS^lSffSS outsf

its Physiol 2- nBa r BUBrt " ^-^ To^efi a hedyby
X-S J Pr ° pertles lnenns simply to enumerate the optional
?o ^Hv, measurement to which this body can be subjfeted^ anl
to list the series of numbers which the instruments used in the-

Jhv5£wT Tf P ' P01 : eXaniple ' What meanins for a mathematical
Physicist- can hydrogen have, with its various properties: color-
less, of a certain density, liquifying at a certain temperature,
L,'nV^ noraeaning except the following: a body will

be called hydrogen @)when subjected to the instruments which de-
iine fluidity, viscosity, compressibility, temperature, refraction,
etc,, ijLjjroduces a collection of pointer-readings which square
with the numbers cited in the definition of hydrogen,,.

Among modern philosophers of science no one has la-
bored with greater zeal to make this point generally understood
than Sir Arthur Eddington. (6) In connection with the ad ventu-
re of the elephant which we discussed, in the last Chapter, he
writes; '

The whole subject mat ter of cxaot science consists
of pointer-readings and similarjgai cationg ) We cannot enter
here into the definition of what are to be classed as similar
indications. The observation of approximate coincidence of
the pointei with a scale-division can generally be extended
to include the observation of any kind of coinc idence - - or,
as it is usually expre ssed in the language of the general re-
lativity theory, an (Intersection) of world-line s. The essential
point is that, although we seem to have definite- conceptions
of objects, in the external world, those conceptions do not
enter into exact science and are not in any way confirmed by

-341-

m;t B be°SpSd hf enCe ,°? n b W^2Jl2^ the problem they

physical measurement .

tor-readingTeX^n^ 1 ob ^ ot * tat although only the poin-
nonsense of the ^r„hf ^ ^otuol calculation it would mice
else. The probleHo *° 1<3 ^e out all reference to anything
ing background T- Z ^^ll lnvolve « ^ohb kind of connect-
intmachS S,% JT , thG P° inter -reading of the weigh-

of™ o'eStt Stn ^ th !, hill! And yet frora the P°^
the hSl can onlv hf, ^ , thlnS tha * roall y did des ^
Ct+ «£ if ? y be desc nbed as a bundle of pointer reading

by pointed SadinT^I 1^ ^ ^ als ° ha " *™ r?Kf '
+1^ ? \ dlngs » and the sliding down is no Ionizer an i^-

mel urelT ^ wo V ^T^ "^ ° f ^^^
Sul elephant calls up a certain association

- as uS LTr 1 ^' ^ " ±S ° lear tbat raental Sessions
such cannou be the subject handled in the physical problem .

of words such ™f U ^7 0f \ thS ^ B±a± ^ comprises a Liber
current *?~ ^ ? gth ' a ?? le ' velocity, force, potential,
3 ! ' °i WhlCh We ° al1 "P^sical quantities". 'It is

SoraSrffiS - S eSSen ^ al tfet these ^ould be defined ac~

siPn^fSll ^? m ' and n0t aooo ^IST?o-the-Sel5p5;ical
nlf if T£ , We ™~ y haV ° ant i°iP^ted for them. In the

old fcext-books mass was defined as 'quantity of natter'; ■
but when it came to an actual determination of mass, an expe-
rimental method was prescribed which had no bearing on this

nf« n I * 0n \l h t bClief that the ^^ity determine! by the
accepted method of measurement represented the quantity of
EH ln * he 0bject was mor ely a Pious opinion. At the pre-
sent day there is no sense in. which the quantity of natter
in a pound of lead can be said to be equal to the quantity
{xn a pound. of sugar. Einstein's theory makes a clean sweep
of these pious opinions, and insists that each physical quan-
tity should be defined as the result of certain operations
of measurement and calculation. You may if you like think of
mass as something of inscrutable nature to which the pointer
reading has a kind of relevance. But in physics at least the-
re is nothing much to be gained by this mystification, becau-
se it is the pointer reading itself which is handled in exact
science; and if you embed it in soma thing of a more transcen-
dental nature, you have only the extra trouble of dicFine it
out again ...

Whenever v/e state the properties of a body in terms
of physical quantities we are imparting knowledge as torfche
response of various metrical indicators to its proseno eTand
nothing more , , „ .

-342-

troatPd 1» S r ? cos,1XDlon *l»t our knowledge of the objects
ond othor ^^ S1 f co " sists solel y of readings of pointers
sical t^nwl 1 a -° rS transf °™ our view of the status of phy-
tS STi 8 + *\£ fundama ntal way. Until recently it was
S£ ih5 fi f'^ ed that we had knowledge of a much more inti-
mate kind of the entities of the external world. (7)

llv ™ m j„. + P ° r ^ s Q word °f explanation should be immediate-
r l^?f n x. S P assa S° le st confusion arise. When we say thai

mX.Th^ • I l ySXCS dCals ° nly ^ th P° inter readings, we do not
mean t^,t it begins and ends in numbers alone. If this were the

Koxae. it would be mathematics and not mathematical physics. The
numbers it deals with are | measure numbers ^ la other words, the
experience which gives rise to these numbers has something more
than a pre-scientific- function as in mathematics. The physical
process of measuring the quantitative determinations of nature
if a V" tegral P^t of mathematical physics. Consequently, even,
though ohe numbers dealt with do not . represent things in the ob-
jective cosmos, as we shall see, they are always tied up with ob-
jective determinations of the physical universe out of which they
have issued through measuremen t. In this sense there is a physi-
cal background in which they are embedded. Yet the mathematical
physicist cannot get at this background in any other way than by
measurement, and that is why as long as he remains true to the
nature of his science this background will always elude him. Of

I course it is possible for him to go out beyond the limitations
of his science and embed the measure-numbers in a background .*
of his own choosing, but as Eddington remarks, in so far as mathe-
matical physics is concerned, there is nothing to 'be gained by

1 doing so, .. ■

We shall return later to discuss the nature of know-
ledge which grasps reality only through measurement, For the pre-
sent we merely wish to emphasize the fact that this is the only
type of knowledge that is had in mathematical physics. Of course,
in actual practice, scientists never, restrict . themselves comple-
tely to measure-numbers, (8) As Poincare has remarked, they can-
not be denied the liberty of using metaphoros any more than poets
can, (9) Bu t in the last anal ysis their grasp of the cosmos is
restricted to |metrio knowledge.^ 'It is beoause this is not always
recognized that much of the confusion about the (meaning, of modern
science has arisen. This is particularly true of many of the abor-
tive criticisms of the Theory of Relativity. Einstein's great me-
rit is to have clearly recognized the complete dependence of ma-
thematical physios upon measturariant . and to have seen the impli-
cations and limitations of this dependence..

-343-

of measurenent S \ ^ emt ^ Cal ^ s±oa ^ essentially a science
not generallv roo T- b r° rain S generally recognized. But what is
sentiallv n^L gn T d ±S thnt GVer * 8cience <* reality is es~
SSSX %n^°- f n meas ™ G ^.. This, statement is, of course,
stood ?n the «i ° bV10US ^ the te ™ "measurement" cannot be under-
i™ +„ V WQy 1U Which we bavQ been employing it in relat-

the SiK: ^i* 4 ?^^

^^^^r^¥^^^^^S^^^rShSi±n order to
understand accurateT^Tfchi part that m^u^ielrt plays in physics

tLuLrlvXe Lh 50 ^^ , t0 SSe h ° W thG other scianoesfS par-
ticularly che philosophical sciences, are related to measurement.

«f^v+ ~ + u Tak f n in its general sense, measurement implies an
?«?£ ?" , ^ f ° f thS intelle ct to see a certain complexity
v?,lod C lg 1 f a P rinci P le of simplicity. This principle is pro-
vided by a standard, and the attempt of the mind to reduce com-
plexity to simplicity will be more or less successful in proport-
lonto the degree of simplicity possessed by the standcoMrThis -
explains why i n physics there is a continual search for a miiTiiua
measure. But it is not only in physics that there is an attempt
to see the complexity of reality in terms of the simplicity of
a standard. This is found, in the philosophical sciences as well,
although the nature of the standard and hence the nature of the
measurement is something quite different from what is found in
physics,,

St, Thomas defines measure as "that by which the
quantity of a thing is made known. " (10) But as we saw in the
last Chapter., there are two kinds of quantity: predicamental and
transcendencali The former consists in homogeneous exteriority
and the latter in interiority, that is, in perfection of being.
Now whereas in physics it is predicamental quantity that is made
I known through measurement, in the philosophical sciences it is
V transcendental quantity. In both 'metaphysics and philosophy of
nature it is the principal subject of the science which provides
the ultimate principle of simplicity in relation to which every
other subject in the science is measured. For, as John of St. Tho-
mas remarks: "mensura importat perfectionem, cum semper accipia-
tu^fpro uensura id quod perfectissimum est in unoquoque genere ;
nee requiritur quod sit notificativum rei mensura tae, ut fundans •
imperfeotam cognitionera; sed per modum alicuius magis simplicis
et porf ecti < quo res mensura ta _, magis ad unitatem et uniformitatem
reduoitur . " (TT)

I In every order in which a relation of more or less

is possible there is measure, and the"maxime tale" is always the

[measure of everything that is found in the order, (12) In meta-

-344-

standard

It is by^aSf^^^^^^HL^He is the cause of being,
quantity of all LTa'X,? fl ^ Act ' that thgjganscendental
siS^ection rovealS ^^i-r 88 ^ '^^ed^ndlTOTS^n-
subject is »n and H ? ■ ^Ph^lo|ojDhy_of J]a ture the principal
tal quantity of aS rltn^^^T^^^ tte transLnde n-
PJ^ta^^^TrfshT 3 ^^

^thlTE^omS^
f-irtho a + ram j f 6 n th0 cosm °s and the one thnt is the

one Sat il Zf/™ ^ Standard of '-ssureraent and hence the
S£ point of view oT^'V-? ^ P rocesses of measured, from
Pie beins in Thn Philosopher of nature he is the most aim.

degree of interio^r"^- 0136 ^ be ° aUSe he Passes the highest
sur™?. ^ te ^ 10 f lt y- " is extremely significant that the mea-

te directions^ho C o ^ ?° ? hiloso ^ of ™ture lead in opposi-
Plicitv of ™L- \ determines things in relation to the sm-
fhe s£plLitv of hC T g ~ S exteriority, the other in terms of
ml .pwT \ of " lterlori ty. B^rphysicsJ.n teriority is irratio -
SerirtLl ^ ^ ^^^ntaT^cie^cT^h ^h deals wit lTmTmr -
2?S«i Psychology- - isjhejnostj ^rational of all the ex-

?^22^tythap i s_ 2 S^^

go^s^^Pg^otionjNo.ono, perhaps, has hand led this question
with greater skill than Professor De Koninck:

_ Toute science s'ef force de reduire le conrolexe au
plus simple et do l'expliquer en fonction de lui, Mais il faut
a enxendre sur la signification du terme 'simple'. La nature
de la simplicite a laquelle on doit tout ramener difference-
ra profondement les savoirs. Or il est facile de montrer que
ce que nous appelons L simple en s c ience experimentale, est tout
oppose a ce que nous disons l s imple en philosophic. , En scien-
ce experimentale une pierre est infiniment plus simple qu'u-
ne cellule; le va-et-vient d'un piston est beaucoup plus sim-
ple que le bond d'une panthere qui se jette sur sa proie; de
tous les etres qu'etudie la science experimentale, l'homme
est incontestablemcnt le plus complexe. Or en philosophie o'es t
tout le contraire qui est vraio . L'animal est plus simple que
la plante, et de tous les etres qu'etudie la philosophie de
la nature, cjest 1 ' homme,_qu i. est le plus simple : de m§me qu'en
metaphysique la rnesure et la cause do tout etre est la simpli-
city absolue do 1'acte pur. En physique on mesurc par la mi-
nima mesura le temps par lo temps atomiq ue par_exgmplg ;

en philosophie la rnesure est toujours riohe et comprehensive

le temps est rnesure par 1' evitornite. et tous les deux par

-345-

lest invor» proS?i **"%' ^ S ^ lici ^ expert ntale
Le philosophe dira cue lT \ & ^ /i.nplioite ontologique.
I'inf&deu?, la mrfait ° f T?^ e *? liqUe 1<3 ^P^ieur par
dire par avance !,", T ^ "«*"*?«. Ainsi nous pouvons
rimentale dTl?^ *? 1& mesUre °* ulw explication expe-

dans la perspeoSrdo 30 P ° S ? ibl °> elle consistora a 1'eldier
Pie que lui non ^ °° ^ 6st ewimentalement plus sim-
1'ellLntaire n ? POUP ^ dentifier entre eux le complex et
tout natSefLr^ ^ f r±V<3r ^ de 1,autre ° IjJgt done

dl^es^cS n. i, ^^^P 1 ^^^^ !^ toute la hierar cW
tion tnn^ mtur ? lles s'eriger dans le sens d'une organisa-
S 1 Z^?.^ 33 ^ 6t P1US °°^^e. Le philosopte qui
sence nine do ^ '^ & me th °° ri ° ^olutioniste nL Pes-
de^ait nior , rae ^°de scientifxque. S'il etait logique il
dcvrait.nier aussx la valeur d'une mesure de longueur. (13)

relation to th!^! b ^? gS "! back to what we saw in Chapter I in
relation co the possible extent of the mthematization of nature,

+v„ , n ^ ^ in ordGr to understand the peculiar nature of
in tSS nr g l ^ iS based c ™plete^ on a JLur'emen? of things

^r f°La h s=x s exteriority we mist ^ to anai ^° the

. 2. Nature of Measurement.

«•>, 4. v u- u ^ aaure » according to Aristotle and St. Thomas, is
that by which the quantity of a thing is made known, (14) . This
definition immediately gives rise to a difficulty. For quantity
may be known independently of any measure. In fact, homogeneous
exteriority is an immediate datum of cognition, and consequently
does no-c depend upon any medium such as a measure. Moreover, we
have already pointed out that quantity is known and studied both
by the philosopher of nature and the metaphysician, and in neither
case does the knowledge of it involve measurement . This difficul-
ty did not escape Aristotle and St. Thomas. For after laying down
the fundamental definition just cited, they proceed to qualify
its meaning by adding the phrase: inquantum quantitas . That is
to say, measure is that by which the quantity of a thing is made

tit y independently of „*,£„£» in «^^* 5£^ ^

ling: "Addit auteV, ?{£T th f ow , s , 1:L g ht upon tte question by writ-
LX VT 1 " nutQm (Philosophus) ' inquantum quantitas' ut hoc refe-

ti^ntitS^^ Uant f atiS ' ^ P^ieLes et alia accidS
there^re So fun^n ™f? c °g"oscuntur,» (15) In other words,
ce!in so far a??f 1Gntal aa l°f ta to ^ntiiy. In the first pla-
ce, m so far as it is one of the nine accidents it is n certain
essence and consequently can be too™ in the sal way th4 all

^Sll"? 117 r kn0Wn ^" S ° far - « °^-s the parts
rioritv 1^2" SUb3t ^ nce h * contributing to its homogeneous exte-
ft° It Y U \- • f" 1 "?^ and ^rasdiate datum of oognition^En so
Studied L "v, lnV S-r d ^hL Jgobility of the cosmos . it can be
studied by ote philosopheF^f-HRture^n so far as it is one of
th^princn|les_of_being it can bo studied by the me taphysicIHH;
ae thnf? {!! oasoT-it is-a question of,»quidditative» knowled-
?°9 m ^' k ^ rlQd S° tbat answers the question: What is quanti-
ty?_Now while this question "what" can be asked ofldl the cate-
gories of reality, t here is a special question that can be asked
2"£°fj!uantijy - - ^owmuch^C^an^; And it is knowledge'
which onaweriThia quiitiolTthSF^l^ealed by measure. Since, •
then, the question "how much" (Quantum) is proper to q uantity '-
lone, Aristotle., and St. Thomas ore justified in saying IhaTSa-
suro.is „hac by which the quantity of a thing is mode known: and
they are speaking with strict .formality when they add the phrase-
xn. i3^^iL_quant_itas i

It is extremely important to insist upon the preci-
se nature of the knowledge of quantity that is given to us through
measurement. It_is_not " q uidditative" knowle dge: it does not in
any way answer the question: w hat is qua ntity. It rarely tells

I us h ow much quantity there is .TThis, knowledge is mediate and de-
rivative, ^ since it cornea to us through the medium of meTisureJBut
a measure is a very special End of [cognitive niejH umTTUnlltce'a
sign, it does not substitute for the thing known, nor does it

lin _any way manifest its(nature^ > And the practical conclusion to
bedrawn from those consid.oraHpns is that in so far as science
is based upon measurement, not only does it riot toll us the "What-
ness" of all the determinations of reality -wlvVh fall outside the
category of quantity, but it does not even tell us the "wha tness"
of the quantitative determinations that are dealt with . This point
Is frequently lost sight of, " ~ ~

But in order to understand more clearly the nature
of this peculiar type of knowledge we must try to see just(how)

-347-

Stity of a f T^ me * s ™ nt - A measure ^ifests, the
reduction oA ! not Cin_asy_H» whatsoever, but th^ugTTthe
Sterraimtionto ^f n tyP<3 ° f com P le *«y to simplicity, of in-
E^sSSf^ 2 -^ 2 ^^. 11 ' of variability to uniformity^ -
^S^f 5 ^^^ 11 ^ to , jnteUifHHHy , When the
another t^Z ^"T^k^e^ 555 ^^ of

that the tl ^l^S^JH^a^it would rem ain indetermined^ we say
sure isV^-^ ^T^^^^^^^iS^-STm^^the mea-
lows tL? ?W ^^l™ of the thi "g measured. From this it fol-
ciBloS r,Sf + - r A^ essential elements in measurement: a prin-
§^o£_|erfectxon)and uniformity and simplicity, which isTfe^
measure, and a^agaagaa Cpf reduction") of the conplex and variable -<M« ■> Vf-^
:-2-H^P£ig£^?„. (16) This second element obviously involves
some kincTof (union, between the measure and the thing measured.
In order to understand the nature of measurement it will be neces-
sary to analyae each of these two el ements .

With regard to the first it is clear that in order
I or a thing to be a measure it must be one and indivisible, for
in no other way can it be simple and determined. That is why in
the- tenth book of the Metaphysics St. Thomas begins his explanat-
ion of measurement by saying;- "cum ratio unius sit indivisibile
esse; id autem quod est aliquo modo indivisibile in quolibet ge-
neris it mensura; maxime dicetur in hoc quod est esse prim am nsn-
su ram cuiuslibet generis ." (17) But itlist be pointed out that
the "one is not as such a measure. That is to say, indivisibility
of itself does not necessarily constitute a measure; it must be
indivisibility (in_a_ce rtain given order .j The transcendental One
is_no t a measure because it is not in a d efinite genus . Moreover,
it does not possess strict unity. (18)

Aristotle and St. Thomas make it clear why indivi-
sibility is one of the essential qualities of a measure:

Assignat autem rationem quare mensuram oportet es-
se aliquid indivisibile; quia scilicet hoc est certa mensura,
a qua non potest aliquid auferri vel addi , Et lideo, unum est
mensura certissima; quia unum quod est principium numeri, est omi-
ro indivisibile, nullamque additionem aut substractionem suse
cipions-manet unum, (19)

, A measure is a certification of the thing measured.
But it can be a certification of somethin g else only to the extent
in vrtiich it is fixed in certain ty only by being fixed jn indivi -
sibility,

A thing can be a measure, then, only to the extent

-348-

in which it is indivisible. But as St. Thorns goes on to explain:
"sed'quaedam ^J^ 11 ^*^ °^ ±hUa invenitur indivisibile;

"indivisibilia secu^ SUnt OEmin0 ^ivisibilia, sed

"instituentiuVa ISSS^

(J^aivisibiiis_est prgjoortione,) sed^no^i jatura ., (20) »

lE2ssibilein genere suo" ??ft ? ° e *> ^\^o^^SLmsi^t

IS p™£e 2 "twT^ ii4i^nly the one which is
hr^-^^E^-OlLSante r that has absolul e-IHaTB^i h-i Hk, ^^

fS W?1^, secundl^uod est prf5535Eon numeri." (22) And just
ouantitW 2^ ° n o ^^^^rement .are derived from^ iimL^f
S all'o, r L ^ thS ^^ 0t ' P^icamental quantity it-

in d-!^ T , e , + :PrilT10 °f* e * ait luod ratio mensurae primo invenitur
in discreta quantitate, quae est nurnerus dicens, quod id quo
primo, cognoscitur quantitas ' est ipsum unum' , idest unitas"

■ Snt^ „ Prin °^ Um nUra ° ri ' ^rUSETijrSiris speciebus quan-
^itatisnon^os^ipsum unum ^sed aliqu id cui aocidit unumj) si-
cuo dxcim Sl unarajTanun,) aut l unammagnTtudinem, , ttST^allitur .
quod ipsuni unum, quod est prj5TTHel5u^7TIT>inoipium nume-
r * secunduiM quo d est nurnerus . . . Hinc scilicet ex numero
et uno quod est principium numeri,, dicitur mensura in aliis
quantitatibus, id scilicet quo primo cognoscitur unumquodque
eorura, et id quod est raensura cuiuslibet generis quantitatis,
dicitur | unum in illo genere,) (24)

. r~ -, Por "s 'the "one" which is the p rinciple of number

istthe_model)for every measure. It is (that_b^wH^ quantity is
£irst made known to us: "id quo primp cognoscitur. quantitas,"

In the measurement of other kinds of predicamental
quantity only q uasi-indivisibility is possible, It is impossible,
for example, to have a length which will be a universal measure
for all lengths as the one which is the principle of number is
\the universal measure for all numbers.

Hoc modo derivatur ratio mensurae a numero ad alias
quantitates, quod siout unum quod est mensura numeri est in-
divisibilo, ita in omnibus aliis generibus quantitatis aliquod

-349-

^o^SS^uw" 8 ''^ 6t ^^ Sicut in mensu-
pedali- irlcT^„-, tUntm i h ° m " ea -g - uaai ^ivisibile 'mensura

ra ali quid iSi^-f 1S5 J f MqUe ' enim oritur pro mensu-
a axiquid indivis^bile, quod est aliquod simplex. (25)

mitation of thtl^- ^f 1 . ^divisibility is nothing but an i-
mutation, for it cannot bo b y itself an absolute measure „

hoc ,,m,n ^ ^nsurae aliorum generum quantitatis^i mitantu r,
£o Zntr? ea t lndivisibile > accipiens aliquid ^S^t"
alLn ? d ™ S<3CU f^ qUOd P° ssibile e ^. Quia si acciperetur
aliquid magnum utpote stadium in longitudinibus,. et talentum

nL! + !' lateret » si aliquod. modicum substraheretur vel
adderetur; et semper in majori mensura hoc magis lateret quam
m minon, ^

Et ideo omnes accipiunt hoc pro mensura tarn in hu-
■ midis, ut est oleum et vinum, quam in siccis, ut est granum
et hordeum quam in ponderibus et dir.iensionibus, quae signifi-
cantur per grave et magnitudinem; quod primo invenitur tale,
ut ab eo non possit aliquid auferri sensibile vel addi quod
lateat. Et tunc putant se cognoscere quantitatem rei certitu -
dimliter, quando cognoscunt ' per huiusraodi mensuram miniraan.

This attempt on the part of the measur ement of ma -
gnitude to imitate the . measurement of multitude must be considered
in the light of what was said in Chapter II about the difference
between arithmetic and geometry; We pointed out that the higher
abstraction and superior intelligibility of arithmetic was based
upon the superior rationalit y of number in comparison with magni-
tude. Number is in fact more immaterial, more determined , more
actual than continuous quant ity. The continuum is something~essen-
tially obscure, indetermined and potential because of its intrin-
sic divisibility into infinity. As ; a result, the; measurement of
discrete quantity is something clear and absolute, while that of
continuous quantity is always something obscure and relative „ In
the latter there is always a background of irrationality. (27)

But since measurement is always a rationalization
in the sense that it manifests the quantity of the thing measured,
the mind can never rest satisfied with this background of irratio-
nality. That is why there will inevitably be a constant attenipt
(_to _ assimilate as nuch'as possible t he m easurement of co ntinuous
quantit y to that of discrete q uantity. J "Omnis mensuratio quae est
in quantitatibus continuis aliquo nodo derivatur a numero. Et ideo

-550-

^^ZSy%) eCnn ^ ^anti^tem continue etiam attrd-

subjective and^bdo^ve^lftho'Sf fT " U1 b<3 &t ° n ° e b ° th
unit of measure is not nlve^ nS f " at P^ oe * since a definite
for multitude one r^A f ctively for magnitude as it is

by fdX. constructed by the mind, established

divisibili^^/ 61,0 n ° n sunt oranino indivisibilia, sed in-
inltituentLT?^ \ SSnS r j 3222 §m quod voluit auctoritas
mstituentium tale.aliaujd prTSniul^ . t™\\ U , L ;

le uno^ J f-avitate ponderum accipitur ut unron indivisibi-
^ronLtT^ ' ld6St a ^ odd ^ nnnimum mondus; quod ta-

bile Z nil ^ ^ ° ranin °- 1 qU±a <l uodlibet Pondus est divisi-
on? 'tsof rgr^L?^^^

^■ii -u „ T , Ms P ° int is of considerable importance for the

sics is P Saf o?^ 1 ^ 1 r lenCe ° P ° r thS ^--c measurer^Sin phy=
sics is that of magnitude. Though science employs a great variety
of measurements, theyare reduoible in the las t analysis to^he

out of which the whole structure of mathematical physics is erect-
!.» ^-Bgj ^sed on a oq ethingahaoluti 3 , something perfectly objec-
1^° T^tF^ as , such in tjglreT TEuTupon a construction of t he
mnd^Both the intellect and the wIlThdv¥~to~o' nter into the p Fo-
cess of measurementto determine a standard and establ ish a uni-
|y_4ga|-T g as_notLexisg> Magnitude is lifted ^^^^T^-Tpj^i-
ligibi,.icy tho.t is not native to it , ) And all this obviously il^ol-

ves _a separation of some sort from the real worldTT Ihat is not

by nature one and indivisible is considered by the mind as if it
were. Once again, from this point of view, mathematical physics
'is a science of als_ob, '

However, this construction is not purely subjective
and arbitrary, In order to assimilate the measurement of magnitu-
de to that of multitude it is not sufficient to declare by fiat
something indivisible that is by nature divisible; it is necessa-
ry that what is declared indivisible approach as closely as pos-
sible to that which is objectively indivisible. In other words,
tjje _less extensio n the standard chosen possesses, the more perfectly
will it be able to serve as a measure. That is why science is al-
ways searching for the smallest possible measure - - the minima
nensura. And this is true of ancient as well as of modern scien-

ce:

-351-

ra illius SnS !• T™ n 1 " un °W°^ 8«iere, est mensu-
uncia et in »!' • S1CU * ln nelod ^ tonus, et in ponderibus
2s notus eSt a ? TT'i S '" ^Jj^tujn_est autera^uod ratol -^

poSs* ^ TMo S P^SijWsum ™ e * I**te brevitatis tem-
(31) %tSL '" attendltur secundum minimam magnitudinem,

vens as the ^tSj^^l^ tte speed of the, movement of the hea^
lies L St S Was + Wd >P°n an hypothesis of ancient phy-
sitioin auod nf S P01 ? tS ^ " P ° nit (Aristoteles) hanc supjo-
s^ionen quod motus coeli sit mensura .'omnium motuum » (32TTS-
^ffij%S^^ the speed ^ j he ~ _

Ihfg^p^p^^r the stahdS^h^n-be-thT^peid^f
iSh" o; ?h, ^^If" 3 ' or the speed of the propagation of
S£ U, + £ n the . w ^e-length of a red .spectral line emitted by cad-
'g™' ^£l2Si°al, structure of t hg_j ^_asurement of oontinu oug_QuAr,-
S^^KT^P^; ^ isalT ^^4uestion,of a staI3a^rwhieh
" ^lilZl t e -^g-£ai4hgaBhnpJ^rjafax^ and which represents
~n attempt to cor* as closely as possible to the minim mensura .

i * • n 1 *, ls clear > then, that there is something profound-

ly paradoxical about the measurement of continuous quantity. On
the one hand, it is necessary for the scientist to search for the
minima mens ura, | a_nd the dialectical tend ency towards certitude)
about which we spoke in Chapter, ,VTecome^J : n_ : bhis_field the search
ior_ an absolute^ small measure . OrTthe other hand, "this infini- -
tesimally small measure does not exist. "Sed in lineis non est
invenire minimum secundum magnitudinem, ut sit scilicet aliqua
Imea minima; quia semper est dividere quamcumque- lineam, Et si-
militer dicendum est de tempore," (33) An infinitesimally small
measure would involve a contradiction, since it would consist in
(a^QntjjM^UEJJhQ ^t extension^ It is then a__p urely dialectica l
limit_ that can 1 be approached indefinitely: it~is not a limit "gi-
ven in nature that can Ultimately be arrived at. And this imr>o=i-
sibxlity of arrival is not due to any lack of precision^n P
our part; it is due to the very nature of continu6{il n

quantity. We must then be satisfied to accept the minimum measu-
re that is possible for us to have - - "accipere aliquid minimum
pro mensura | secundum quod 'possibi le est,," )

■'How is it, possible for the mind , in spite of this
paradox to succeed in some way , to assimilate the measure of magni-
tude to that of multitude? In order to answer this question it
is necessary to recall that it is possible to know that two or
more classes have the -same number, without knowing what that num-
ber is. Thus, for example, if all the tickets to a certain thea-^

-352-

Uny w4 the number of Z e t2 JE *" **^ W±thoUt knowin 8 **
it is possible to know tLt two T,° S "T 1 ^-. In the **'* way
without knowing what «J« £ o^ases have different numbers,
lar is found S Snitude B^W "*? Now something very simi-

™ «,« structure of mathematical phy" sics is based

reoucxDie in_the_l ast analysis ftoTtoT owledge of ratios. -.When f oV

ZT ™i indicated by the measure number 0.000065628, this does

existinr L^7. ^ 1U n te BgBg^ it merely tells us the ratio
existing between the length of a wave, of H 2 light to that of a

ner Z wholl of J 8 ? b ™^ *» "*"W -tenflnrd. In like mn-
reT,t?nn J ^ P^ics ia built up out of ratios determined in
relation to arbitrary standards,, (35)

It is, clear, then, how it is possible for the mea- •
i^f^Vl WLtudo to imitate that of multitude. Just as I can
know that two classes have ,„the same number, so I can know that
two rods have the same length. The two oases- remain similar until
1 attempt to get at the meaning of the "same" .In the case of mul-
titude this meaning can be determined absolutely since it is based
on cardinal n umber, (and_congejuentlj) it is possible~ to esc ape from
rare knowledge of proportion. In the ^STaT^ffa^Ae , the ^an=~
ing of the "same" cannot be determined absolutely: it is impos si-
ble to e j cape from knowledg e of proportions . (36) ~~

From a 11 that, has, been said thus for about the na-
ture of measurement of magnitude it follows, that from the point
of view of the physicist, the standard of . length has no length.
Sir Arthur Eddington has brought 1 , out this point very forcefully
in the Prologue to Space , Time,, : and Gravitation . (37) But lest x
confusion arise it is necessary to make several distinctions,, The
term "length" is in fact extremely ambiguous and is susceptible
of a groat variety of meanings. It may be taken to mean: l) di-
mension as such (and this is its most proper meaning); 2) a line,
that is to say, a finite leng th; (38) 3) the measured magnitu-
de of a finite length; 4) a geometrical line; 5) the, measured ma-
gnitude of a finite line; 6) a sensible line taken as a dimension;

-353-

tudc of a sensible line Now ^ f^^ 5 8) the ^^red magni-
indicated under numb^ s £ '^ *•?"> taken in tho se ^e S

dard of length i s a length t?4 ^ Xt i S ° bvious that the stan-
dard: "opo £ te : y^ su ^X; pL^ n W r e n0t ** COuld not be a sta «-
plTysicisT^peTDS-HrKnlS^^ But when a

by number eight that he l\V ia .^ifE^T®* sense indicated

magnitude thft is expXsShle w ' ^ " ±S a 1 Uestion of a
the question "what iX T i? measure-number which answers
is tie to say tSt the "t^ S ^ °^ ^ line? In this sen ^ "
far as it is a standi ?t l ^ °£ ^" gth ^ no len S th - *« °°
in no other way! ?he s ^ if?* 6 d f £* d 2niZj^de^igmtion and
derstood in tM wjv th^ lh f^° Ma suremelrt^f-EiBr„ Un-

taining that £ aTobieS no ?7 ° f Re l ativi ^ is correct in min-
it wouM be outsSe of tL Jn/r ""V** Vel °° ±ty ° f U ^
liKht is hfcnnTiv, * ,' the speed of the Propagation of

light xs oaken as the fundamental, standard of the ^asulement of

standard in ^J'*? * 3 ^ 2 *' ° f covrse > to define a certain designated

standard tt^nT nete ' la - But odiously in.this definition the

™ s "° lon S er ^e meter but the centiliter, and yre are
faced with the question: how long is a centime te^Ttere are lust
• two ways by which one might attempt to answer this questic^ one
is by saying that it is theWlredth part of a ^erfoS ?his

se'tTf^iirSll" ^tT f r ° le; the ° ther ia ^ haSng^cour-
infSitur^ ^ T I standard > ^d this involves a procesi ad .
V^lnUumj) by the time we have corns to the Anstrom as the standard
^-are-still as far from the answer to our question as we we"
m the beginning, ±c . .

_ The yifinity of the vmcbus circle and the indefini-
te, process is a sign of what is at the .bottoSTof 'this whole^u^i-
tion: thjynexhau^tiM^^ And m £ st

of the difficulties that arise in c^ec~tion with~this problem
have their origin precisely in this that we attempt to confer u-
Sgn_the__continuum a dcgree^f_i n Aelligibilit y that beTo TSToTay-
to discrete quantity. It is extremely important to keep iHlSnd
that the measures of continuous quantity are essentially inadequa-
te and imitative. They do not do away with the inherent ^intel-
ligibility of the continuum, for they, cannot change its nature
Measurement consists in\ the juxtaposition of an unknown with a
scgiej It is usually taken for granted that this scale is something
definitely known by itself. As a matter of fact, it is not. And

-354*

^l^?Z^^^^^ t ^^L^^B is merely the
^^^r^Jir^~^~^^^^' But perhaps this whole
ion between! intrinsic and v ? ° n When We take u *> the distinct-
is ^rth^it^S^i-^^^iSiaS^ ^ the runtime it
gnificoaoe ^or the wlX J£ V ^ ls odiously of extreme si- '
ticulorly foAheVheor^ ^S^gf"™**™* ^ Sl ° S and ^

first element o?^ iSibilit { ±S then the P rij ^7 Quality of the .'
Section ™d i *™ ent mentioned above; the principle of

uniform tv In nra^l * led t0 lt: the rae ^ure must possess
xity b S,pit?tv • ,° r + meaSUreraent to be able to reduce o™>Ple-
bilit: to S Slf^:™^ to ^terr,ination, and varia-
and i' ,vS % i 1S not ' suffic i°nt that measure be one
othe^vinn ii ^ n ls fl als ° necessary that it be uniform. In no
the W r T ? T dS ob J ecti ^ certification in respect to

dSd S is conS;i?°M e<1Uently " ±S neC8SS ^ t0 cho ° se - ^~
aard that is controllable, precise, uniform and invariable,

, . ., . Per fectio raensurae consistit in uniform! tate et sir*,
pj-icicate, qua aliquid de se est notificativum alicuius quan-
txcaoxsj hoc enira exigitur ad rationem raensurae ex parte suae '
periec-cionxs, eo quod perfectissimum in aliquo genere est men-
sura ceterorura. (39). .

■ And obviously the uniformity required is uniformi-
ty wi-ch respect to the particular genus in which the measurement
takes place.

Sola uniformitas seu regularitas, sumpta in abstrac-
ter e.vt communis ad omnem mensuram. . . Ergo oportet quod deter-
minetur ratio talis raensurae essentialiter et intrinsece, per
hoc quod sib uniform ta g talis vel talis quantitatis , vel ge-
neris , . . Ergo"pertiriet _ ad ipsam essentialerorationem mensurae
non_ solum habere uniformitatera, sed uniformitatem talis vel
talis c o:cditionis seu generis : . ratione cuius sit apta et ha-
bilis menritira ad mensurandum talia mensurata, '(40)

7he perfection of a measure of length, for example,
requires that 'it be uniform in the genus of length , in other words,
that its length be objectively constant. Here we are touching u-
pon one of the most important problems of measurement' in so far

as it affects mathematical physics - - the problem of the rigid

(rod, Vie shall have a great deal to say about this question -later
on, and at this point it will be sufficient to merely touch upon
the fundamental issue. In every measure of continuous quantity

-355-

an ossontir^rSoftSr ° f Vff ™^ «* invariability
the point of view of ind^H ^-? a f llel3 its perfection from
nuous quantity is an SeL» ? * *' l° r 6Very ™* avre of conti "
bile (and consLu'eVtlvf »S + ? iepC ° f raatter which is an
i^W^raS^g^ ^ a °° ntlnual state _gfjWTTt-
every moment u^der -oifXT^Tn and ™ table S^iT^is at
cessarily produce ch-^l^-^ 16 ^ s± °^ influences which ne-
he eliminated T^t^t\ v ll™+ ^".P^iool influences cannot
rial .+^^^ n °7. X ° ^y^^hgutchang ing the nature of the rrat P .-

eSSent. But in'nrd^ + f S P^^^^^Tbe controlled to~loS-

h^e~aTT^h ^+t i n la ]!!^_2fj3aturerrE^irould he necessaryTo
,dent tLt^e^S^ 3 ^^^^ 311103 - 0nce ^ in " ^i-
liSt Sat oa/hn y T f ° m standa ^ is only a dialectical

^ir^gS^g^Eoaohed indefinitely^ot a natural li^

mSt ateilSL?? ?*' b6ing ^^tedTOn^S- again the J ni

K P !aw. h "MeV Vhat SV J""" hL to U sa?S%h^fLidiW
seTin rlbu, ^?Ti -f °* eSSe P e ™™s, ^ntum est possible;

have hP-r, ,=n In ? onnection ^th the first essential elemant we
Sly there if of ^ V *•' * rinci P le ° f Perfection and sintpli-
S ST ml PO±nt that ^ be touched u P° n - We have

saia that a measure is that by which the quantity of a thin? is

Sw 01 "?' ^ ^-T T ^'WB in which one thing may manifest
another. In the first place, a less perfect object may serve to
raamfest a more perfect object. It is in this way that creatures
manifest their Creator, and this is in keeping with the limited
nature of our human knowledge which in the order of generation
progresses from the less perfect to the more perfect. But it is
also possible for a more perfect .object to manifest a less perfect
oboect, and it is obviously in this way that a measure manifests
the thing measured, since, in relation to the latter the former
is always a principle of perfection .

Licet mensura de se ordinetur ad notificandam quantita-
ten formalera vel virtualem rei mensuratae, non tamen est de
ratione raannurae quod notificet nobis quantitatem rei mensu-
ratae modo imperfecta, seu juxta modum quo procedit nostra
cognitio de imperfecta ad perfectum; sed requiritur quod ex
ips/ratione mensurandi notificet nobis mensuratum modo perfec-
ts seu procedendo a ■perfecti ori.-ad minus perfectum sen tiinns
nobis notif icatum ... Hoc enim modo mensura notificat, sci-

-356-

' S,™ ff' m0dma P erfect i°nis et stoplicioris, quia perfectis-
"erfeSnis^of nSre ? Bt ~* "teroJuS unfeTer mo~
, Corfeoto S ^if V ? ™ Senamtionia sive processus de
I luporleoto ad perfectum), debet mensura notificare. (43) - Jrf. , fr-P-i

which science f™* ^f^ that in ' ^ type of measurement with
- - there T s no nil T^ conoe ™ & ~ ~ the, measurement of length
feet TiZallx ? b ^tively perfect standard, no absolutely per-
has bee^d Y^f 1 * °f l 1 ra P lici ^> <* is evident from all that
search of TnnS^ f • + T ^ t iS Why ScienC ° ^ ever rer '^ in in
perfect And?h« + Pe G u t ^ andard ln ° rd ° r to ^^test the less
lli°l' A ? d * hat ls wh y xts measurements will always remain to-
periect ana obscure.

„- JL And now > living analyzed the first essential element

of measurement we must consider the seconds the union between the
measure and the thing measured. In order for this union to be pos-

^•f?-, 1 ? " obvl0UBl y necessary that there be sons kind of com-
patibilxty between the two. And this prerequisite condition is
expressed in the fundamental Thomistic principle : ' "mensuram opor-
tet esse honogeneam mensurato. "

Mensura semper debet esse cognatum, scilicet eius-
dern^ naturae vel mensurae cum mensurato: sicut mensura magni-
tudinis debet esse magnitudo: et non sufficiet quod conveniat
injj atura communi , sicut omnes magnitudine s conveniunt : sed
I oportet esse convenientiam mensurae ad mensuratum in natura
/speciali secundum unumauodqne T sic quod longitudinis sit lon-
gitudo mensura, latitudinis latitude-, vox vocis, et gravitas
\gravitatis, et unitatum unitas. (44) -VjfM».

But this immediately gives rise to several' difficul-
ties. In the first place, number is measured by the "one", which
is not a number. Consequently, in this case the measure and the
measured do not seem to be in the same genus. St. Thomas answers
this difficulty in the paragraph which follows the one just cited:
"Undo nihil aliud est dicere unitatem esse mensuram numeri, quara
unitatem esse mensuram unitatum." (45) In other words, even though
the "one" is not a number, it belongs to the same genus in the
sense of bein g_the jrinciple of number . Though not in itself dis-
crete quantity it pertains to the order of discrete quantity in
so far as it is its principle . A more serious difficulty arises
from the fact that God is said to be the measure of all beings,
and eternity is said to be the measure of time; yet in either ca-
se does it seem to apply the principle: "mensuram oportet esse
homogeneam mensurato." St. Thomas suggests, the solution for this
difficulty in the Sumraa: "Mensura proxima est homogenea mensurato,

-557-

non autem mensura reran tn " ( ao\ t j.-u

ve measure in fht =+-? + { ' In othcr words > in ° rder to ha-
that the me'sur'e an £? f^ 6 ° f ^ WOrd " is not necessary
the strict ^Z of i th , lng measured *° in the same genus in
diately proxir^L 1 W °^' ThlS is ™^^ only of the ir,™-
that the? be In ?hr' ^ f^ ° ther measm ' e " is sufficient
case of tine ™a S ^! 8<3 ? eral cate S or y a s for example in the
ion or even in tho ty ^ ioh bel ° ng to the cate S°^ of d ^ a *-

oTLTZiT^ts s ^riTt ? rd ^ r of b ? ing / s in the case

the -DrincinlP vSm£ ■ ' - ln the realra of magnitude that

IT-h t requires the measure and the thing measured

a leL?n Sene r S i^Boai^orf ectly realised . For the^asurTof
inhK,^ a P01 ^ bUt an ° ther length ' That is wh y st ° Th or<as

„ h ^Tv^^ ° n ^ fifth b00k ° f the Meta physics in speak-
ing of the difference between number and mainitude uses the phra-
se: magnitudo sive mensura" . (48) Magnitude is, in fact, a nea-,
sure, whereas number is not,

a 4-u j-u • But this basic compatibility between the measure
and the thing measured is only a prerequisite condition for the
ituxtillment of the second essential element in measurement. In
order for the indetermination of the thing measured to be effec-
tively reduced to determination some kind of union between the
Vtvro is necessary. Now there are two ways in which a measure can
be united with the object measured. In the first place, it can
be united to it gxtrinsically by means of some kind: of ap plicat ion,,
This application need not be physical; it may consist in" a^ure^ - '
ly intellectual juxtaposition ^ comparison,, as when, for example
the transcendental quantity of creatures is measured by the Supre-
me Beihg i n physical science the application is in one way or
J another physical; but it does not have to be direct or immedia -
Ite, otherwise it would be impossible to measure objects in motion
\and objects at a distance. Yet it must be pointed out in passing
that physical measurement acquires certitude and. objectivity to
the extent in which the application becomes more direct and imme -
diate Now whenever a measure and an object measured are united^
by means of an application, the measurement is extrinsic „

But there is another and more intimate way in which
a measure can be united with a measured object: by identification .)
and when this type of union is realized the measurement is known
as_ i ntrinsic .

This brings us to the distinction between extrinsic
and intrinsic measure which is of considerable importance for an
understanding of the nature of measurement,, St. Thomas touches
upon this distinction in several places, (49) but perhaps
the clearest end fullest explanation of it is/ found in John of

VDe. Vwr. & ILSeX

-358-

St, Thoraas:

cam, ExtriSeca eat J*T lne "™ intrinsecam et extrin-
per applicationen J + q ^"T at ali 1 uid extra se ? e * "so
sicut furttio ctn J contl ? entiara iUius dicitur mensurare,
qjiSeS^r C ??, 1:L raensurat "notus inferiors tam^

libra poSTs Undo ?T ^ et Ulna KBnsurat pa ™> et
sui nensuratl j^T^ 5 raensura torninat relationem reale m
3uratae--Trl; IntHHi^a mensura est illSquae inest ^eTW
SiS^. ,t^T4. nanaUrat EHLapplicationem, sed per iofor-
S ™J e . habe * P^^ctiSKiTie^HeT-Iicet nofrilEtiS*
S^^L^^^^SES^SS"" iua mensura tum da-
MtlSiil^Uf It* V ln un0( l uo< l ue geSre perfectissimum

^™ n SSSiS. et (Sf erQrum ' sui quidem intrinsec a > ali ™

^ ffDV , . J* is obvious that this distinction rests upon a

theolTot ^ kind of union, existing between the measure and

extent in wM^* V^V^ aS a meaSUre is raore perfect *° «»
Si-w + ^ S flrst essential eleHBnt, that of simplicity
rt JrfS ^' " T 6 P! rfectl y realized, so likewiseit is mo-
uni™ 2th ^ proportlon *° the de S^e^ of intimacy found in the
re^ tn measured object. This has already been noted with

regard to union by application in physical measurement: the cer-
tj^4lina_pb£ctivi^ of the measurement depends upon how dirict
and immediate the application . is . But obviously union by identi-
fication is more perfect than any kind of union by application,
no matcer how direct and immediate it may be. That is why, speak-
ing absolutely and objectively, intrinsic measure is more perfect
than extrinsic measure. Thus John of Bt. Thomas writes:

Quanto perfectior est mensura, tanto perfectius co-
niungitur suo mensurato, illudque magis ad se trahit quantum
possibile est, Et ita cum aeternitas sit mensura .perfectissi-
rcia, aumme coniungitur suo proprio mensurato: ita quod habet
identitatem cum illo, (51)

The difference, then, between extrinsic and intrin-
sic measure comes down to this, that, whereas the former measures
and manifests a certain object per applicationem . the latter mea-
sures and manifests per informationem . In the first case there
(is a real distinction between the measure and the thing measured;
(in the second case the distinction is only logical. That is the
meaning of the principle "omnis mensura in suo genere seipsa men-
sura tur," In Thomistic terminology, an extrinsic measure measures
its object ut quod , that is to say, per contactum rei ad rem . In-
trinsic measure, on the other hand, measures its object ut quo,

-359-

that is to say, it is the very form of the thing measured;

with Bxtrix^^^^^^^.^^J^^e coterminus

of intrinsic measuT And vet ^f.^^f 1 * . *° fom a clear notion

[simplicity and SormSv o? V . ^^ that the P erfe °tion,

1 only by mnife.twT l • thlng can "an**** another thing

^^S^^l^g^a^w ^this sense, intrinsic

^ Thomas writes! foundation of extriniTc measure. John of St.

ra intrin<,Sr d ° f n f ura est intrinseca, idem quod est msnsu-
intrK^T n S^txamJ^rma: alioquin non esset mensura
intrmseca, id est, per inforraationera mensurans; cum tamen
ZT^t ?° ne ^ aliqUas raensur as intrinsecas, quia id quod

est mensura m aliquo subjecto esse debet, et non mensuratur
per aliquid extnnsecum, alioquin de illo inquiremus per quid
mensuratur: et_s3 £J re^r^roces J 5u^ J L^^ V el deve-
nienemus ad -(J^x^-SSfScaSSr&^^^^^^ieati^t

I fSTga.etmensur^respectu vero aliorum extra se sit mensura
S2"S5; Kec tamen sub~e53em formalize es t forma et mensuf a:
sed est forma ut constituit formal! ter; est autera mensura ut
respicit quantitatem aliquan virtualem vel formolera, unifor-

^raatate affectam, et sic mensuratom„ (52)

... In other words, by the very fact that a thing exists

it has a certain perfection and simplicity, independen tly of any
( comparison with ) another object . [ J3onsequentlyJ itio^ie"sse~s a mea-
sure intrinsic to i tself. And since it is the Torm of a thing which
makes it both be a nd be 'known , this intrinsic 1 measure is the form
which gives perfection, simplicity and uniformity to the thing
it informs and by so doing manifests it. It is only because this
perfection and uniformity is possessed independently af any com-
parison that there can be a basis for the comparison necessary
for extrinsic measure.

Esse mensuram homogeneam mensurato potest intelli-
gi vel ut quo vel ut quod, et respectu subiecti recipientis
est homogenea ut quo, scilicet id, quo tale subjectum reddi-
tur homogeneum et uniforme alteri extrinseco, respectu cuius
est homogeneum ut quod, si mensurat illud per applicationem
et contactum rei ad rem. (53)

But the relation between intrinsic and extrinsic
measure must be rightly understood. It is extremely important to
keep in mind that the extrinsic measure does not reveal t he intrin-
sic measure , as some might be tempted to think.

-360~

With regard to the nature of i„sin -i

tions suggest themseW ArlT does \7T*V ^^^tant ques-
of the thing in the sense of ™ t manifest the quantity

secondly, if it IvJtteZ absolut^^L qU ^ ti0n " h ™ ch ">
but we shall consider +hfn!\ Se 1 uesclons are connected,
it is difficult at firS^ Par f e ^° With regard t0 the fi ^t,
nifests the '."ho W rich" of 8 t^ Ce \*? ^ intrinsi ° *««ure ™'
we wish to f-^nd n,^ L u quantity measured, since whenever

vifably IVeto fall blTon TT^ ^ ±3 ™ a thi ^ v ^ne~
YfTT^hd^nTdN^ 1 -^^^ 0n the otheFlSnd/

of qZltfty^TZl^^ the *****

finiSm is valid," f ? S P ° 3SeS .? es is mde lcnown > ^ if this de-
^^^—^^^^pp^J ^ntrinsic naeaiui eTPgrha^--

S PaLVgTof^o?^^ ^ ^°-

™Wv= ™- Aliudest °onsiderare raensurara et raensuratum, ex

Ix parte 2 i iUd . 6X ^ 6t 6X Parte °oS"OBcentis„
S™^^ f "J. cognitae, server raensura est perfectior
SHS^V no ^^°ativa illius, atque explicativa confusio-
corno^LI,- Perfectl0nis et simplicitatis. At vero ex modo

nen ^St • T*^ f M n ° Stra > pr ° pter suam ^P^rfectio-
nom, attmgit simplicitatem et uniformitatem rei mensuTcU-tis
supra mensuratum: hoc tamen non tollit rationem mensurae ex
parte ipsius rei cognitae, licet per accidens ob defectum co-
gnoscentis non possit uti ilia mensura ad cognoscendum per
lXlam, tamquam per medium, rem mensuratam, (54)"'.

. x , Intrinsic measure does make the quantity of a thine
known m the sense of manifesting the "how much" , and therefore
realizes the defJgiitiontfjngaBure . But this manifestation is" de-
pendent upon two factors JOTn the first place, it is dependent u-
pon the nature of the subject to which the manifestation is being
made. It is possible that an intrinsic measure may manifest the
quantity of a thing in a clear and adequate way to a superior in-
tellect but only in a vague and general way to an inferior intel-
lect. In this case the inferior intellect will have recourse to
extrinsic measure. This is true of the intrinsic measure of -prg-
dicame ntal magnitude . The intrinsic measure of an isol^-b^ w+^x^
object manifests adequately the quantity of that object to the

(divine intellect. But to the human intellect this manifestation
is only vague and obscure. Before comparingone extended object
with another we know the quantity of the first in a very loose
and inadequate way. If we did not there would be no basis for com-
parison. To answer the question ; | how much quantity is ther e in
an extended object , we can point to the object and say:) that much.
But the intrinsic measure does not give (us) any accurate and deter-
mined knowledge of the quantity. It does not give us the precision

-561-

absolute" S,^ quantity of this object is southing fixed and
This is trie S f^ 3 ^ in a definite and absolute fashion,
j! S i *™L • thG transcendental quantity of imperial things.
But the extension of material objects is not fixed and absolute.
1^\^ !1 P 01 " ted . Q «^ all material objects are entia mobj u
lia and are constantly m a state of flux. The extension of .eve-
ry material object is always in a state of becoming since it is
forever undergoing the changes being produced in it by the innu-
■ merable physical influences to which it is subject. That is why
I even to the divin@j-.iind the intrinsic measure found in every neH
terial object cannot manifest the quantity of that object as so-
mething fixedMiddefinite. flf it did, becom ing would be idelrti-
i ^fied with being!; " '■

And this brings us to the answer to the second quest-
ion: is the intrinsic raeasure^of material objects sons thing abso-
lute? The answer is yes and no^ ^Itris^absolute in the sense of
not possessing the relativity that is proper to extrinsic measure
and that derives from the comparison of one object with another .
It ( dsfnaE) absolute in the sense of manifesting a quantity tha t.'
is fixed a nd definite . The partisans of absolute d-ir^ng-irma iW •
the cosmos consistently overlook this second point, (55) To their
argument: orone ens est aliquid , must be appended the qualificat-
ion: in quantum est ens .~fo~the extent in which a thing is becom-
ing it is not a being and hence is not absolute. And from this
point of view it is likewise true to say that the standard of length
lhas no fixed length. Through a progressive refinement of scienti-
fic processes, physics is constantly drawing closer to the .abso-
lute world condition. But in so far as' the process of measurement
is concerned, it is important to keep in mind that though this
absolute world condition is absolute in the sense of not being
relative to our ways of knowing, it is not absolute in t he sense
o f being fixed and immobile . We are not drawing close to a static
cosmos o

We said above that extrinsic measure differs from
intringig measure in that whereas in the latter \the relation bet -
ween/ the measure and the ob j ect measured is only logical , in the
former it is something real. Since all scientific measurement has
to do with extrinsic measure it might be well before finishing
the discussion of this point we try to determine as exactly as
possible tho nature of the relation that arises out of physical

-362-

measurement .

I relation- trnn^^^ 1 ? 3 trad±t±0 ^-^y distinguish two types of

constitute a s^l + ^ P redlcM ^l. The former doel not

Uveral ter-H t? T y ° f beinE and hence is realized in

tMnTabSute in^f if ^ V ' her<3Ver m entit ^ thou S h SGHe -
ccssarfoSp^.f" ! ^', haS in its ^.intrinsic nature a ne-
anr^ote^v f i tOWard soraethin S e ^e. The relation of act
lation o7 + h S a ^ ay \ a r lati ° n ° f this kind - Predicamental re-,
ded ?n'thP £ ? + 3r handj is a special accident that is superad-

f.? + h f-, abS0 ^! entit y which " elates to something else.
As Aristotle and St. Thorns point out, (56) there are three spe- - f h Eft*
+ -l S °^P" dlC0 f» ntal relation:, l) those based on number and qui-
tity; 2; those based on action and passion; 3) those based on mea-
sure, St. Thomas clarifies the meaning of the third species by
explaining (57) that measure here means something distinct from
the measure of number and magnitude, otherwise there would be no
difference be Ween the first and the third species. It has to do •
with the 'measurement of being and truth." In this sense our know-
ledge of things is measured by the things known, that is to say,
the truth of our speculative science is determined by' objective
reality,

These distinctions throw light upon the nature of
our physical measurements. In the first place, there is a trans-
cendental relation between the standards and the measuring instru-
ments used and the reality that is measured, for neither standards
nor measuring instruments have any intrinsic meaning except in
relation to an object to be measured. In the second place, there
is a real predicamental relation of the first species between our '
units of measurement and the quantity measured. Finally there is
a predicamental relation of the third type between the knowledge
that we gather from our measurements and the object measured. But — /
here it is nec essa ry to in troduce a distinction . The knowledge
i that comes to us from physical measurement in science is at once
J both sp ecula tive and practical ; from one point of view it reveals
I to us objective reality, while from another it reveals an article .
\ which we have jman uf acture d.) Hence there would seem to be a double
predicamental relation ofthe third type involved. Prom one point
of view objective reality ' is the measure of our knowledge;
from another point of view our mind is the measure of the object.
But of the two the first relation is the most fundamental, for
the second has only a functional character in relation to it . That
is to say, the only reason why we become the measure of the object
is to make it possible for the object to become the measure of
ouy . knowledg e in a more perfect and adequate way . It is true that
we choose the standard by which the quantity of reality is revealed,

-563-

j-buic pnysioists tend to overlook this point. -f-

3. The Limitations of Measurement ,

sifvinr » flrttHWv, 1 ^ s^oolmen had measured instead of clas-

ff a V ^' u^ ! Whitehead, "how much they would have learnt." - S^w i>\y*M*.,r*

)vn ™ S r hl ^°^ 1 cal reasons indicated in Chapter I it is doubt-

Iv leaded "-if It Z rf the mfidiaval schoolmen would have actual-

sureS t£ S^ d r° teA theraselvesto scie «^ hased on mea-
surement, Buo there can be no doubt about how much has been lear-

V n m u m tm<3S throu G h *e systematic processes of measure-
nent ' giejna gnificant structure of modem physics i s an eloquen t
Broo f of the nmazl ng~fruitfdlnea3 of metrical method. y B t. +.h P ZZ
pistemologist must not allow himself to' become unduly, impressed
by this tovrenng structure. He must strive to remain completely
detached, and examine its foundations with as much objectivity
as possible. His task is to assess its value, not from the point
of view of practical success but from the point of view of
\pure knowledge.

This is the task we must now undertake. Having on-
ce recognized the amazing success and fruitfulness of the proces-
ses _ of measurement it is necessary to try to analyze their limi-
tations. Many of these limitations have been more or less impli-
cit in what we have been saying about the nature of measurement,
but it is important to try to make them- as explicit as possible .
It is only in this way that we can come to see the true nature
of the value of the knowledge that is found in mathematical phy=
sios, since , as we have seen* all this knowledge is in the last
a nalysis (derived from measurement.)

In the first place, metric knowledge is able to co-
ne to grips only with the quantitative determinations of nature.
As we explained in Chapter VII, it is utterly , blind to all the
determinant properties of things in their specific essences, to
the very inner natures of things, to all that seems to be of the
highest significance for philosophy, for art, and for human life
itself. The proper realm of metric knowledge is the homogeneous
exteriority found in nature, and from the point of view of pure

-364-

W* 'h 1

S°of eC the honored SXt f f lv Poverty-stricken area, both becau-
se ol the homogeneity and because of the exteriority.

make the ou+H^ r ^ P +u* h<3 - f0ll ? wins considerations may serve to
the fir^t S,™ t * hl ! ^^ton* Imitation more clear-cut; In
ture to l P nt^'- , mSt bS n ° ted tholj I! easure m ent can reveal m -
l^r^h^^r^^- 1 ! 3 aif^?5H5ii7 ^his- i s in itself
^" A^Jff^i -^ riiT ' diati °n» *ut it is only half of the
handl: fS v^* 1S thG ^^ ^^tation that measurement can

pLsessereSio^tv the /^ ct ^7^?oi5FfieTd of measurement is one that
possesses exteriority andjience differences, and at the same ti-
me _ homogeneity and_hence sameness. But perhaps we can make this
point still clearer by rendering it more concrete and precise,

• + a-** . There are two types of variety in nature. Some ob-
jects differm kind, as e.g. green differs from large and hot
from hard Other objects (or states of objects) though of the an.- -A

me kind, differ by the fact that they possess their common cbarac- V^' .

ter m various degrees. In face of the first type of difference w^ v e "\

measurement is wholly incompetent for _the sim p le reason that it fc^*" ^ °"
is__a q u estion of difference withoutl^SnigsT ' (59V-TTe^nremCTTh Vsi***
can come to grips with these differences only in an indirect way
by introdu cing sameness > through an artificial construction., That
is to say, if changes in the one object are functions of changes
in the other,, or if certain occurences in the one determine in
some way corresponding occurences in the other, then a correlat-
ion can be established between them. But it need hardly be remark-
ed how limited is the type of knowledge that results from such
correlations.

Measurement has far greater competence in relation
to objects or states of objects which differ by degree. But even

here an important distinction must be made the distinction

between what has become known as "intensive quantities" and "ex-
tensive quantities". Examples of the former are density, hardness,
temperature. The most important examples of the latter are length,
tirne and mass, but there are many other examples of less importan-
ce, such as volume, electric resistance, momentum, etc. The mea-
surements of both of these types of "quantities" have this in com -
mon that the ir e differenc es) can be determined by a serial arrange-
ment which will be both asymmetric and transitiv e. This is possi-
ble because thero is /a sameness uniting the dif f erences .) But they
are distinguished' from each other by the fact that in the case
of intensive quantities the serial arrangement is not additive ,
f whereas in the case of extensive quantities it is.') It makes sen-
se to say that eighty feet of length are twice as large as forty

-565-

arises from the fact thot tv, v, • ?f ty de S ree s- This distinction
an nttempt to !LS?J?^ inaa that ° U »*»sure™t consists in

£;H s -r- ----- - -

since, as we explained above, the measurement of magnitude <m
^never escape the limitations of ratios. ^gaxmas can

_.-„„,„, „ trough processes of correlation similar to those
can to !Z 7 ■"+ £ 6 '°' the raeasu ^nt of intensive "quantities"
fihVLT extent be assimilated^ that of extensive "quantities".
This is done when the serial order of an intensive "quantity" is
\ found to corres p ond^? the serial order of an extensive "quantity".
The most common examples of this are the correlation established
between degrees of heat and degrees of length of a mercury column,
between the degree of color of a- light and the degree of its re-
traction, between the degree of intensity of a sound and the length
ot a wave. Measurement obtained in this way is called derivati ve,
whereas direct measurement of additive "quantities" is" called fun -
damental. Now the indirect, artificial and arbitrary character
of derivative measurement is so evident that it is hardly neces-
sary to call attention to it. And obviously the ' knowledge that
results from this measurement is extremely limited.

But even in the field most proper to it metric know-
ledge cannot get at the quantitative determinations of the cosmos
in the sense of being able to tell us . what these determination s
are. Precisely because it is "quantitative" knowledge it is not

("quiditative" knowledge. It cannot answer the question "what",
it can on ly answer the question "how much "? This is a profound
limitation which must not be lost sight of. It makes little dif-
ference to what extremes of refinement we succeed in pushing our
measurements , ( in the end the nature of the thing bein g measured
is just as inscrutable as it was in the beginning^] (60)

But the metric knowledge .that is found in physics

-366-

;ell

t^lT^tVl^Tr Ch " f thG *™*«^ detect

based nZZo S^^g^lF^'^^^^^SS^P
jIsT omethiii^ nh ^^^a^-g^gf* agjgunting, J^lauHjEel

is alwayVHueS^ And it

sic measure ?m "f^^HHn^ic-lHiSiU re, never of intrin-

of the object ^SoJTi+^V- f VGI " tellS US anythin S absolute
only tells S S? 7 i ^^ lnde P ende »tly of the standard. It
object under c^l °^ ect . stan ^ ^ comparison with another
taowledPe in tT ^^ circu ^^nces . I n_ other words metric

Soperties of tho'T 6 f* t0 transfo *^e ratios into absolute
it S ?hn? fh»^ aS f S T aSUrea - W»i«n f for example, we hear
look utJf^i Slty ° f 80lA is 19 ' 32 ' " is ^sy enough to

S belong +n nea iT e ~ nUraber aS de «iE^tinS something absolute
Ite.STf + - f ^ 32-SS' AS a mttcr of fact > it merely indi-
Vof a voW S°v, f ^ ^ T^* ° f any P^ ce of S°ld and that
out +M^ ■ I -^ r u° f 6qUal Si2e ' Sir Arthur Eddington has brought
out this point with his usual clarity: ■""

. So- in any statement of physics we always have two
ocgects m mind, the object we are primarily interested in
and the object we are comparing it with. To simplify things
we generally keep as far as possible to the same comparison
object. Thus when we speak of size the comparison object is
generally the standard meter or the yard. Since we habitual-
ly use the same standard we tend to forget about it and scar -
cely notice that a se cond object is involved . We talk about
the properties of an electron when we really mean the proper-
ties of an electron (and) a yardstick - - properties which re-
fer to experience in which the yardstick was concerned just
as much as the electron. If we remember the second object at
all we forget that it is a physical object; for us it is not
a yard-stick, b ut just a yard . (61)

From what has been said thus far it should be fair-
ly^ clear that strictly speaking metric knowledge does not reveal
things to us. As Professor De Koninck has remarked, "les entites
fondanen tales de la physique ne symbolisent*- que des coupuros me- p , ( ^ t "^
triques dans les choses dont elles ne representent qu'un -aspect,,, ^ Zfyt^'^
II est absurdo de considerer un atome comme une chose." (62)-"'^ ^ u
One of the most common errors in science is to reify provisional
metrical segmentations and to attribute to them the status of on-
tological entities. In this connection the following lines of Cas-
sirer are extremely pertinent:

It seems almost. the unavoidable fate of the scien-

-567-

formed at once into a thw and establishes, should be trans -
tl^lrirth^nf^ 12 ^^ 2 ^ 2222 ^^ Ever doea it believe that
tude a^eassureJ onlf ^ °? J*" physical conce P ts of leni-
ties to correspond to Sf "r ^ rmitS Certain absolute r " ali -
correspond to them. Each creative .opoch of -nhvcnV*

to S talItr o ? n bei° rnlUl f GS ^ ct — teri S tic P 2 s Sef ^r the
ge! of ta°ip e thLf d n ?. Uml ^ CSSS ' but each stan ^ ^ dan-
Krtftti? Preliminary and relative measures, these
2 lefSitU intellectual instruments of measurement,
torv of the L^T*?™ ,° f th<3 onto^SioaUy real. The his^
of the JZL f P ^ ° f ' mtter ' 0f the atora > of the concepts
Pies of thi^ T?-, e + ne ^. °«^ the typical proof and exam-
SnlSl' ti f\ te *; lalism - ~ ^d there is materialism
not only of tetter but also of force, of energy, of the ether,
e-Lc., - - goes back from the standpoint of epistemology, to
this one motive. The ultimate constants of physical calculat-
ion are not only taken as real, but they are ultimately rais-
ed to the rank of that which alone is real. (63) _ <^h^ & $>**<" "f

She fact that metric knowledge in science gives us
nothing more than the ratios between two objects brings to light
further limitations that are intrinsic to it. If nature itself
determined the standards, the resultant ratios would have a fixed
and objective meaning. But as Bergson has remarked, nature does
not measure. And since the standards of measurement are not given
in nature they must be established by convention. The intellect
and will of man must enter in the process of measurement to deter-
mine the norm in relation to which the ratio must be established.
Man becomes the legislator for nature. As Professor Beneze has
remarked, "dire que le choix de l 1 unite est arbitraire, c'est di-
re que la volonte de 1'operateur va introduire dans la connaissan-
ce un element sur lequel la sensibilite n'a plus aucune prise.
lEt cela ne signifie pas que le nombre qui va apparaitre ne soit \i^.^>° r
I pas lie au sensible, mais il ne lui est lie que justement parce ^ c if-
\que la volonte de 1'operateur en a decide ainsi," (64) - Ov^X'

All this evidently introduces an element of subjec-
tivity and to a certain extent of arbitrariness into our metric
knowledge. As a matter of fiact, most of our systems of measure-
ments derive originally from extremely arbitrary sources. In the
English system of weights, for example, the weight of an average
grain from the center of a head of wheat was originally selected
as the standard, and the pound was consequently defined as the
weight of seven thousand of these grains. The bloek of metal pre-
served in the United States Bureau of Standards now provides a
much more uniform standard, but the basic relativity and arbitra-

-368-

Sroflhe^rSn? of?™ £" ** ^ otan **« The same is
own whimsical way: S * h ' aS Eddin e.ton has shown in his

year 1120 ^i^Kj iS *? ^ trusted > King Henry I, about the
vifof Scotland riH?^ by stretchin G out his arm. King Ba-
the inch shonS t'l } ] m0re deraoc ^tically ordained ttat
men "an nS * ^ meaSUre ° f the thunbs °f th ^ ee
nnn-' the Sh T a "*" ° f ffleasu ^le stature, and an 'lytel]
meter les, S" beir «sured at the root of the nail. The.
ceodesis^ P th ?f qUely smbod ies the mistakes of the early
is to deterrS 3 r ^^ ° f a11 ° Ur Careful measurement
Lnrthnfl^i for example, how many hydrogen atoms to the

That L^ ™? S "^ S am ° r to the thmbs ° f ttoe Scotchmen,
ture (65? """^ "* ^^ ^^ ^ ^ riVsteries of Na ~

+ , „ , .. Tt f s true that science does not rest content with

the pure arbitrariness ' of the standards just mentioned. It has
been possible to discover certain constants in the cosmos, such
as Planck's constant, the velocity of light, the mass of a proton,
^ tc " qna t hese to some _e _xtent enable the scientist to meas ure na-
turejrajhJier__own - _g£uge j _so to "speak . But even these constants - '
are determined in relation to the originally selected standards.
And no matter to what extent science may go in its attempt to pu-
rify its processes of arbitrariness, in the last analysis the es-
sential relativity intrinsic to the measurraent of magnitude will
remain untouched.

This essential relativity imposes an infinite limi-
tation upon the metric knowledge that physics affords us. For no
matter what extremes of refinement the progressive perfection of
our processes of measurement may reach, the resultant measure-num-
bers are always an infinite distance from any absolute meaning.
Sufficient attention is not always been paid to this infinite
limitation. The impression is often given that an absolute measu-
re actually exiswLn nature, though profoundly hidden and extre-
mely difficult to gat at. This is, of course, an illusion,

II pense volontiers que le nombre exact est la, ca-
che dans le sensible, et il l'y poursuit comr.ie on poursuit
un gibier difficile a. attraper. Metaphore trompeuse: l'impos-
aibilite do l'atteindre ne tient ipas au fait que la mesure
exacte serait profondement cachee, mais au fait que le nombre
est le resultat do cette tentative du Jugement d'iraposer a.
la raatiere l 1 influence d'un element, l'unite pure, qui lui
est originairement etrangere, ( 66) - Cvi'tii" ^ 0> bmwn. — <f., flev^-u-

-369-

this ciuc s tion/a s tome C \uthor, 0a fl r n V ^ " *? ille Sitinnte to dismiss\
sure-nunbers are on^approximS^ ZT * stattn f. ttat °- »»*- *■*
a relat ion to a dcfini? i V ~ r a PP roxl ^tion imp lies 1-
nus exists J LiUl££y3il; ^^ /

science must seelct'remi^t UP ^ BOm W for this limitation

4.11 , . xt » But here we are broupht ut) short befn-
re another restriction. For even though theoretical^ thi' inde-

^, f' ehnlt ^ llmts to *he accuracy of our raeasuremants in

of ne^tlnl^'^ ™ f^^ h ° W highly refined our instruments
n+„ ST ? become, they are in ihe last analysis made up of
atoms themselves, and as Planck has remrked, "the accuracy of
^~ lns instrument is limited by its own sensitiveness .<•
(67) Moreover it is impossible for us to receive any message from

S Xton g ^ aCSr refinenent than tnat brought to u/by a comple!
te Photon. This is a very serious confinement, and at present at
least there seems to be no way of evading it. As Sir James Jeans
has said, "we have clumsy tools at best, and these can only make
a blurred picture. It is like the picture a child might mate by
sticking indivisible wafers of colour on to a canvas," (68)

In relation to this question of the limitation of
the accuracy of measurement in atomic physics, the much discussed
problem of mdeterrainism readily comas to mind. So much has been
written about this problem in recent years that it hardly seens
necessary to go into detail in explaining its nature. It is well '
known that classical mechanics was rigourously deterministic. Its
whole structure was built upon the assumption that every given
state of universe was completely predetermined in its antecedent
state, in such a way that if all the elements entering into this
antecedent state had been known, it could have been mathematical-
ly deduced from it. And this applied not only to the universe as ■
a whole but to every individual particle contained in it. The fu-
ture state of each particle was already precontainod in its pre-

1 son * state, Past, present, and future was* perfectly convertible .
It is true that the existence of statistical laws was recognized
but this existence was attributed merely to subjective ignorance

^and not to any objective indetermination in nature. That is why
thermodynamics was for a long time considered to be the least
scientific of all the branches of physics, and it was taken for
granted that as science progressed the role played by statistical

-370-

laws would inevitably decrease,

has taken place! StatistLS iS " ^ ^ tte ° PP ° site that
sics, and classical phvsicS 2S T rel f supreme in atomic phy-
completely dissimte^ i £ ' • dremi of de ^minisn has been
Press ii7the rcf?nori* + ? SreSS ln science > in general, and pro-
vided us with w, ° f r T SUrenent iri P^icular, has not pro-
0n the contra^ it L^T to l*f iot fu *^ states'of particles,
un tno oonwaiy, ic has demonstrated with increasinp claritv ™,r-

S^alTr^Si'oT mk ^ G ? UCH P«^««». » £5 noTbeLT
ne Sth ?he 3+- ^ Posies that it is impossible to determi-
Sn5 Tt X f° Sli 1 1 ? n *" S velocity of a particle at the same
hvn^l^l 1 T ^ deteraine wii; h great accuracy its position

Xri*l^^.^^ a -V > * OCii * r > ° r ±ts velocity by prescinding
fr 01 .i ios position, buj_it_i s_ impossible to do b^ j^ TW^,!

fe*' " 17 *?*' ^t there is a constant profo^ttoTin'^Ftaow-
ledge of these two facts; that is to say, in the precise measure
i* which our knowledge of the position increases in accuracy, our
know t edge of -one velocity decreases, and vice versa. And this pro-
portion is equal to Planck's constant, h, the quantum of action,

All this has become known as Pleisenberg's principle
of indeterminacy, and a great deal has been written about how this
principle should be interpreted. It would take us too far afield
to attempt to analyze its philosophical significance here, but
m so far as bur present purpose is concerned^ it is necessary
to point out that there are two fundamental issues involved in
this question, and both of then reveal an intrinsic limitation
of the process of measurement.

In the first place, the velocity and position of a

(particle cannot be simultaneously measured with a high degree of
accuracy simply because such a thing is. a contradiction in terms .
A particle in motion is not in place; it is passing from one pla-
ce to another. And the ■ higher the velocity,- the less is it connect-
ed with any one definite place, At any given instant one can speak
of its position only by prescinding from its velocity. It is true
that by being satisfied with rough and inexact measurements we
can determine both the position and velocity at the' sane tine,
especially if the velocity is low. But as soon as we try to deter-
mine both of them with a high degree of accuracy, we shall find
that they are necessarily mutually exclusive , for a thing is mov-
ing to the extent in which it is not in any one position, and it
is in a definite position to the extent in which it is not moving,,
It is not surprising, then, that science finds it impossible to
measure both the position and the velocity simaltaneously with
any great degree of accuracy,, And all this shows how the process

-371-

of measurement, by the vpto f n „i. - .. ,

us inevitab^ into' - SsfSo^^Se-^r^ --

I quate solution^ Ihe"^^^^^^ M u th± * *« on ade-
a good deal more involved tn+h! 1IKle *?muiac!yv There i s in fact
sue° is, of courserwholher ? he Ldot ^ ^^ Principal; **"
discovered in its processes is t, ^^ f ich 3 ^nce has

Vterminacy actually existin, ?« « + Vel ^ lon of an objective inde-

fersn^riSS^^ But we feel

tific indetermi™ *« V n said: m the nealureTn which scien-
it is in ™^T f -f velatlon of ontological indeterminacy
it is in perfect conformity with Thomisra - - all the writinps of
c^ntgajora ry Scholastics to the contrary notwi th JJ^w^SS 5 ^

gnjiOhi-.o,^ oi Aristotle and at> Thomas ^ts a-S^ ™f

Passed by the large measure of. contingency and true objective
indeterminism that they attribute to the material universe. It
is something tha t is a pivotal, point in the whole Thomistic sys-
and ' p^V^^^diate^gr^ ^lary oftthe doctrine of matte r
andjbm, To deny objective indeterralnisn to the material univer-
se and to affirm at the same time that one. of the co-principles
which constitutes the very essence of the things of the universe
is a principle of pure inde termination - - prime matter, is a con-
tradiction m terms.

An adequate discussion of this question cannot be
given here . That has already been accomplished with admirable skill
by Professor De Koninck (69) We have introduced the problem
only because it reveals another important source of limitation
of the measuring process. For, as we pointed out at the beginning
of this Chapter, there is something at once froth physical and ma-
thematical about the process of measurement. The mathematical cha-
racter is revealed in its attempt to arrive at exact determinat-
ion. If measurement were, being carried on in a mathematical world
from v/hich all contingency is excluded^ the refinement of its e-
xactitude could go on ad infinitum , but as a matter of fact, scien-
tific measurement is oarried on in a cosmos th at is fill ed with-
chance , and that consequently is refractory to~~the exact determi-
nation v/hich measurement seeks to realize.

This discussion of the progressive refinement in
the exactitude of measurement raises a question which cannot be
overlooked. We have said that the definitions which result from
measurement can never be anything more that operational ; physical
properties are defined in terms of the concrete processes by which

-372-

they are determined. And at first sipht +M« *. ■ -,

in an insolvable problem Fnr^«n! ? • , ^ t0 lnvolve u ^

limitations the whole process of t^" ^ SenSeS fr ° m Whose
I V er USo " process of measurement is intended to deli-

.1, nhv ^ ml It *! true > as we pointed out in Chapter VII, that

sense^But thi^n^^f l0n inV ° 1VeS m ultimate dependence upon
L? S ' M *£" < ioQa not «"« a going back 'to the limitations of
the senses which physical science encounters at its point of de-
parture And we can escape this without getting involved in a vi-

inTsMral ^U ^ ^ *■ qUeSti ° n ° f a oi ^ but ° f an "»Se£-
ing spiral. Inthe beginning, science, in making use of ordinary

t?^L t a arrlVSS a * an eleraen W Physical theory. The substi-
tution of measuring instruments .makes it possible to correct the
primary theory; the new theory helps to reveal the deficiencies
°1 tte . inatrurjen 1* employed and -makes it possible to perfect thera:
through the use of more perfect instruments science is able to
arrive at. a more perfect theory, and so on ad infinitum .

There are two thingsbhat must be noted about this a_ ~ .

process. In the first plo.ee, it never arrives at perfect exacti- Qh XK)
tude. And this is an important point to keep in mind. For it means •— ■*

that from this point of view mathematical physics does not have
an absolutely certain point of departure. Its primary data, the
measure-numbers, are not truly certain,, And the fundamental rea-
son why they are not certain is that they aim at a kind of certain-
ty that cannot be attained in the realm' in which it is being sought.
5^om_tMsjDoJjitj3f_v^ew .the primar y_data _pf the parts of the stu -
p3^ ^nia^ure~^h;itare~not matheraaticized have greater certitude,
Th^_Ti^ tru e above all of the philosophy~bf nature. But lesTThis
limitation appear greater than it actually is, attention must __

be paid to two points. First of all, even though the measure-num- fjM
bars are not certain, they are certainly an approximation, and ^-^

science is of ten able to determine with great exactitude the li-
mits within which this approximation certainly falls. Secondly, (S~\
because of its highly theoretical character, mathematical physics —
is not so essentially interested in the certainty of its points
of departure as a purely inductive science must be,. In a sense
it is true to say that it is more interested in its point of ar-
rival. It is satisfied with any point of departure which will, pro-
vide a sufficient basis for a theoretical structure which will

-373-

eventually "save the phenomena",

have been dlaoSjiSIf tS^ha^r "^ " the *— "°
more implicated it become Tin theo^T ™f Y refine * " S ets > tte
deeply ^ in ^roSiS^S^roTS^SSti^ doel
net e^aCanlT "T ^T^ — Ptioi. But the ex4-

I ,_ „ ,,i tt „„ nf T„, a vcriuable maze of postulates and assumptions .
as a matter of xact, does not our method of deciding that one pro-

teihiining that it is more m accordance with our theories and with
'the laws which we have .assumed to be true? ■

i j. i. ^ This 1orin Ss.us back to what we saw in Chapter IV ■
about how the subjective logos is injected into nature through
^^^BS^ol-SS^^n^tion. Everything that was saiOn" that
connection applieV^THSFtiSular force to the processes of mea-
surement. For measurement is an operation which we perform upon
nature, and this operation has a double aspect. In the first pla-
ce, it involves a mental procedure which gives the operation a
meaning only by placing it in a teohly complicated pattern of in-
terwoven assumptions, In the second place, it involves the actual
physical procedure of measurement. Both of these aspects implica-
te measurement in a manifold of complex limitations. But for the
moment we are interested only in the mental procedure by which
hypothetical elements enter into the operation.

Measurement has been considered by some as a pure-
ly empirical procedure, dependent only upon perception and its ,}
means, and completely free of hypothetical assumptions. (70) O^
Nothing could be more. false. Not even the 'simplest measuring ope-
ration has a purely empirical and immediately certain starting
point. There is always a multiplicity of conceptual presupposit-
ions lurking in the background, which, though subtly implicit,
determine, nevertheless, the whole meaning of the procedure. If
all the implicit assumptions upon which the ordinary process of
measuring temperature by means of a column of mercury could be
disengaged and laid bare the results would probably be; startling.
Hot/ much more is not the elaborate and complicated scientific pro-
cesses of measurement dependent upon hypothesis. Innumerable theo-
retical assumptions go into the whole conceptual setting up of
the experiment, into the construction of the instruments of mea-
surement employed, into the precise way in which they are used, :
and, in fact, into every operation that goes to make up the expe-
rimental procedure ,1,^(71) And every attempt to verify these as-
sumptions only leads into a more complicated network of presuppo-

-?374r

sitionss

derable stress upoj thL poiH™ r6r ' ^ ^ ^ C ° nsl " .

on p.v+^-J^ atly '. ev ? n the simplest, measurement must rest
WtheS," T* 10 - 1 Pf su PP°^tions, on certain 'principles-,
„5° S n ' , '™ s '> Which " dces not takc ^ora the '

of h ^ ^^fhich it brings to this World as, postula- "
tes of thought In this sense, the real iV of the phy sicist

existing properties, butof^DiteacTintellectual symbols,
which, serve to express certain relations of magnitude and mea-
sure, certain functional coordinations and dependencies of
I 'phenomena, , . ,

In tnis sense,, each measurement contains a purely
ideal element; it is not so much with the sensuous instruments '
01 measurement that we measure natural processes as wi th our-
own thought s. The instruments of measurement are,~alTit v/ere,
only line visible embodiments of these thoughts, - for each of
them involves its own theor y and offers cor rect and useful
results only in so far as t his theprylXassurnecTljo be valid .
It is "not clocks and physical measuring-rods but principles
and postulates that are the real instruments of measurement,
For in the multiplicity and mutability of natural phenomena,
the thought possesses a relatively fixed standpoint only by
taking it. In the choiee of this standpoint, however, it is
not absolutely determined by the phenomena, but the choice
remains its own deed for which ultimately it alone is respon-
sible, (72)

But not tififtl^ dp inMii&rablti lifttLtatlohS result from
the mefttal operations" Which construct "the processes" of measurement,
they also result from the physical operations involved in the ac-
tual concrete processes. This is an extremely important point and
too much attention cannot be paid to it. It immediately reminds
us of all that was said in Chapter IV about the operational character of the de
^initions/of experimental science. But a few special considerations
must be introduced here which apply in a particular way to the
process of measurement.

In the first place, it is important to keep in mind
the proper reason why definitions of magnitudes are necessarily

-375-

ZTa^L'lln^Ze:^ ° f ™*?^ °°» never give us .ore

e^eHo^e^^

u-joii the \v»v -in wh^i, +t, *" lu ^- e ''leaning of the results depends'

ne^n whlc^ £ I e%i5ed ta and1li S thi° S - n °f ^ ^ ^
arbitrary P ip r ,Pn+. u " r P-<-oyea, and all this involves innumerable

taowleS whiSh tU -~ ™ haVC 4 .^ lrc ^ suggested. That is why the
Ss^tiallv relate' Ure, '^ nt ° f ^nitude gives us is alwaya
essentially relative, even when it is a question of the determi-
nation of the proper length of an object. By proper lenrthiHhv-
in C whLVf e r st °° d 1 the le »Gth which resuitf f?rr r ae::SLnt P y

I^K"™ t S .^ nd T arc J 1S applied to m oh * ect that is at rest in
relation to it. Later on we shall see that a second kind of rela-

Uivity enters m when. measurement is rade of an object in motion.

Because number is sorting absolute, counting is
an absolute operation. No matter how many different ways , of c ount-
ing a_ certain given plurality may be devised, their Results ?SsT
coincide exactly if they are to be true. Aa a natter of fact, count-
ing is not <gssentially?an experimental proces^TToF it does " not
ne_cessarilyJjiTOlye_a _manipuTation of bocHgsT ~It is true that~bhv-
sical manipulation may be used as an aid, but in itself countin g
is_ a purely mental operatio n. Magnitude, on thTother hand, is '
not. something absolute, nor can the operation by which it is de-
termined be considered absolute. It is possible for a number of
individuals to measure the same extension by means of different
operations and all arrive at different results. And it is possi-
ble to consider all of these results as equally true,; To conoei-
ve the results of a certain measurement of magnitude as the reve-
lation of some thing absolute in nature tojy/ hich all other opera-
tions must^co nfom) is to miscons true the whole nature of roagni tu- .
de. | That_is_wh y such measurement can never have an y meaning ■
independently_o F^the concrete operations involved7~J ~~

And all this means several things.' In the first pla-
ce, it means that if we wish to g et at the exact significance of
a_dcfinition of a length we must be able to specify com pletely
^and with perfect precis ion ja ll of the operations which~~have enter -
ed into its determine tiond Becauseof "fche extreme complexity of
even the simplest kind of measurement this seems to be an impos-
sible task, not only because of the innumerable elements involved,
but also because the operations interfere with each other, and
there is no way of fixing upon the exact nature of the different
^interferences"^) But even if one could specify the operations com- .
plete~Iy and with perfect precision, the results would be very mea-
ger. For in the last analysis this specification would consist
in merely pointing out certain processes and certain material ins-
truments. One does not reveal very much about the nature of man

-376-

by merely pointing out an individual r,

man.

I means that wherihe P operSioL C ^ Ct<3r +l ° f the defin "i°ns of length
I definitions changes As S ■ 2 S °> the significance of the

W-iple the opo^tion lyr^l^^T h ™ * oi f ed ™*> "A
niquelfTpccified Tf ™T -length is measured should be u- \

wAav! rt\ £££ ™ *£? - e «* <* operations," U.fl.

parate name to correspond -o I, ?, ^ 1Ctly there should be a se-/

(73) The primr? nS ^ T different set of operations ."^ ^{^ w^

found in thTKrSSnlS T a ™ ent ^ in ^ sics is ^* fe* »* ^
or JuxtapoaitSrSTSS^JSSl^ ** ^ a F liCati0n ^ ^ 3#

■srs.ri^^ ^ ------ s&£ t ::c ^

^ operations^TSt^^ ?

Sat °hl C df USi °? arlSG " ^ be wel1 P-haps to p"nt "t
sSelv uln th^n ^ ean ^ thS « sul *«'°f the'nsasuLment depend
aTTcf ier^ ™ 7 e °l the °P eratio ^ employed, for othe rwise

Ve shall have-T^imilar relSriETomkT in connection with the~ie-
cond kind of relativity mentioned a moment ago: the result of ?he
measurement of a body in motion do not depenl solely upon the fra-
me of reference m relation to which it is measuTedT for otherwi-
se every body measured/In relation to the same frame would have
^the same length.

This relativity of measurement is often lost sight
of. One type of operation is constantly being substituted for a-
nother on the presumption that they are equivalent and interchan-
geable. An operation proper to one field is projected into another
field where determinant factors are different, , and it is tacitly
assumed that the operation preserves its original meaning . How""
is it possible to have any assurance that operations which give
similar results under certain circumstances, will necessarily gi-
ve similar results under any , other circumstances?

Perhaps a few concrete illustrations will serve to
bring out more closely this important limitation of the measuring
process, (74) In the first place, a very simple example is found
in the difference between the fundamental and derivative measure -
ments. All too often tl-eso two types of measurement are consider-
ed to be practically equivalent; yet there is a Vast difference
in the operations by which they are determined, A more important
case is that of the measurement of a body in motion. Such a pro-
cess involves operations that o.re quite different from those in-
volved in the measurement of a body at rest, and the
higher the velocities of the motion, the more complicated do these

-377-

operations become » As n T-«ai,i+ ±.\.

goes a profound change .' We -wi? * raeanin S °f the process under-

se later on because of its Sl,^ 8 T to s ^ ^out this ca-

01 its capital importance in modern physics.

ed beyond its Siti^ ^, ? 10 ^* ° f len Sth is extend-
tremely large objects Herf ?h £ ^ ±n the raea surement of ex-
Ployed^n measured ; tha ^^X t^*^*** are ex-
perience, and which consist n ho , ra ^' 6 ° f ordinar y e *~
of the standard rod to ?he obleS * UCcess f e di ^ct application
opticalfopeTaTKn^arerl^TTTrT^ Su- n ° longer be ^P^yed, and
ie-^xtenV^^T^^ is alread fQund > —

of sol^r and stolStlT?* 8 ',^ " ±S Particularly true

i„ tne complexity of the operations aJncreaaerTnlDroW^^rTo^ ''
(the remoteness of the distance measured. As Bridgrln has ^S,

At greater 'and greater distances not only does ex-
perimental accuracy becoms less, but, the very nature of the
operations by which length is to be determined becomes inde-
Imite so that the distances of the most remote stellar obiects
as estimated by different observers or by different methods
may be very divergent , „ ,

We thus see that in the extension from terrestrial
to great stellar distances the concept of length has changed
completely m character,, To say that a certain star is 10 S light

years distant is actuall y and conceptuall y an entirely dif-
ferent kind of thing from saying that a certain goal post is
100 meters distant., (75)

Something similar to this occurs when measurement
is extended in the direction of the infinitely snail. The opera-
tioms involved change; they become more indirect and more highly
complicated. Consequently, the results of microscopic measurements
have a different meaning than those of molar physics.. In this con-
nection it is interesting to note that though in the determinat-
ion of the number of molecules in a certain pd&e of matter we
are forced to use indirect and complicated methods, and though
different methods may give results that are systematically diffe-
rent, there can be no doubt but that the number of molecules is
something absolutely determined in nature; consequently the results
do not depend for their meaning upon the operations employed. In
' so far as these methods are theoretically good and accurate they
must all arrive at the same absolute result. But it does, not seem
to cake any sense to say that in the determination of length, mass,

-378-

dently of the operations "Si^^L" ^}^

suit do not occur? V?^f weaning,, the changes which re~
s o L t 1 fortuitous and uncontrollable way. That

£ fasS^tLyTre'se'lec?^ Wo^ ^^ * * ***** ari ' lte -
in the reilm i/ w m , ! !u !u y lesiGn ln such a W that ^th-
in the realm in which both the original and the new operations

the SrSJ o 16 f e*^ b ° + th , SiVe thG san " i«H resultfSin
V *J + w v + ^ x l ,erimental ^ror. Yet there is never any assuran-
^i* 61 ? th ° f w °P^ations are applied outside tnis^ reata
Sll™^™^^ 003 °" inV ° lved ' ^ *W -^ence

I* ^ possible for several divergent definitions
ot length to be employed in circumstances in which direct measu-
rement is impossible, such as,, for example, in intense electric
and magnetic fields. This is quite legitimate, provided that, as

T\ l- • teM t0Ward 2ero > the y a11 converge towards the accept-
ed definition. It is impossible to say that one of these dif init-
10ns is right and the others are wrong, for they will all be con-
firmed by observation, since the very observation will depend u-
pon the theory that is originally accepted. But as Eddington has

! pointed out, it must be kept in mind that the distances thus mea-
sured will be pseudo-diitfbanoosy since they lack the most funda-
mental characteristic of the metrological conception of length,
namely the correspondence between similarity of length and simi-

■ larity of physical structure," (76)

The second. thing that must be noted in regard to
this operational character of the measurement of magnitude is that
the operations in question are concrete, physical, material. ope-
rations. No matter how completely mathematicized or how highly
theoretical physics may become, the definitions of the quantities
involved in it are never independent of singular , concrete, mate-
rial operations , nor do they ever have any meaning except in re-
lation to them. The definition of length of a Relativity physicist
is the same as that of an ordinary metrologist.

If, instead of length being defined observational-
ly, its definition were left to the pure mathematician, all
the other physical , quantities would be infected with the vi-
rus of pure mathematics ...

In all orthodox physical theory, the metrological
practice - - or more strictly the principle which it attempts

-379-

I it t77ec^tat^i UPP l ie \t he thQOTeti ^l definition. Thus

bothare TefL^.\ £ thG e ^ e rim3nter checks the theorist,
\ Dotn a±e lef erring to the same thing.

what the ™?™v ±ng l y ' by 1Sngth in rel ^tivity theory we mean

Tn acco^^r S 8 i -f ana ' ^ what tte » ure Seometer means.
In accepting relativity principles, the physicist puts aside

ta-Vv^r° Ur f™/***?***", clismisses^heir go-between me-
taphysics, and enters into honourable marriage with metrolo-

+ u n ■ . * ^° n the point of view of logical structure of scien-
ce the limitations which all this implies are simply enormous.
N ° definitions ln Physics are detached an d universal; they are
all tied down to particular material operatioHiTrhiy have no si-
gnificance independently of the concrete instruments of measure-
ment employed.

All too often measuring instruments are looked upon
almost as if they were immaterial cognitive faculties which regis-
ter events in a purely trans-subjective fashion. But a moment's
reflexion will show how far this is from the. truth. In the proces-
. ses of measurement the instruments employed do not remain purely
(passive; they enter into the experiment in an active wa y. For ob-
/ viously a physical instrument can reveal an event to us only if
I there is a physical causal connection between the instrument and
V the event.) And this causal connection inevitably involves an in-
terference of the instrument in the event.

The seriousness of this interference depends upon
several factors. In the first place, it is clear that the inter-
ference will ordinarily be greater in proportion to the greater
imperfection of the instrument employed. And in this connection
it is necessary to recall that perfect instruments exist only in
the mind of the scientists; they do not exist in reality. Conse-
quently, there is always something defective about every measure-
ment made. Moreover, measuring instruments never remain the same;
they are constantly in a state of flux. The very fact that instru-
ments wear out is a sig n that they are at all times subject to
minute derangements. But even if measuring instruments were per-
fect there would still be considerable "interference in the event
that is measured, For purely material things cannot register ob-
jective events in a purely trans-subjective fa shion.

Another important factor upon which the seriousness
o£ the disturbance depends is the degree of refinement demanded
by the experiment in question. In molar physics the interference
is relatively light, though even here it cannot be overlooked.

-380-

But in the microscopic vtorlt] +Vm i„i„,p

gnitude as the quontitieTre-xaS^ uei ? orenoQ is of the .son* ra-
tions of measurement in thi™^' consequently the linita-
of intimacy in thTl thls . realm are simply enormous. The degree

^rsrie^tirmisrSs 10 ^^^ r e ~-»s *»&»-

the seriousness of the disSa^ce r^T° h & ° in detenntoi «S
copic phenomena the causal^is ^^^rl^^T^

S\™por5:n'to e tl iS •' ^"^ ^^de/Thir^agniLde 3 decrla-
tmment InTevent L J -. lncrease of ca ^al distance hetvveen ins-
trument and event, but xt can never be reduced to zero, since as
Planck has remarked "if the can=n1 fl-i=-'- „^ ■; «'"i i>™ as
«„ Holv „„ , . vi causal distance xs assumed to be in-
finitely great, x e. if we completely sever the object from the

M^J*? T & f >:Z l6arn n ° thing at ^11 about'.the real event.
(78) Nor must the fact be overlooked that when experiments depend
upon a multiplicity of pointer-readings, there is necessarily mu-
tual ^ interference between them, — '

Perhaps one might be tempted to think that this li-
mitation of measurement is not so. serious as it appears at first
sight, since it is possible for scientists to take account of the
xnterferences in question and to make compensations for them in
their computations. It must be admitted that 'certain possibilities
of this kind lie open. But they are extremely meager in compari-
son with the problem in question -.-if for no other reason than
that every attempt to account for a disturbance involved in a mea-
surement demands another measurement for its verification, and
\this obviously starts us out on an infinite series. (79) i-.V}>^

In our discussion of this limitation of measurement
arising from the causal influence of the instrument upon the quan-
tity measured we have been using the term "interference" and "dis-
turbance" because they are the expressions which have become cur-
rent in the modern scientific litterature which has treated this
problem, but perhaps they do not bring out the most profound as-
pect of the question as accurately as could be desired. For they
tend to give the impression that the causal influences of the ins-
trument is a purely accidental and extrinsic thing, or,.. in. other
vrords, that the measure-number emerging from a process of measu-
ment is essentially a revelation of the object measured, but this
revelation has been accidentally and extrinsically modified by
the instrument used* To conceive the problem in this ligh t"" is to
miss the main issue. For measure -numbers are jessentialTy] the (pro-
duct) of both the object measured and the censtrument. employed . And
here we have in mind something more than the point brought out
above about measure-numbers being mere ratios resulting from a
comparision of an object with a standard of measurement. We have
in mind here something that has to do with physical causation* ■

-381-

gyausaiity of both th £ajaidLltt^^350EZ5ii^H^^. |<

bv a simple diST thi i P oint can be clarified to some extent
pon the Suits or" m T' ^ ^ fluence that an instant has u-
re causal an* in f ™ easu ™t _ are of two kinds. Some of them a-
labor to corrlt , f ? eX * rinsic > ^ these the scientist may

iSluencea wM^ !-\' t0 aCC ° Unt for * But there are °^er
naSre of tS S f SSSSStial, since they result from the very
?~^,f the instrument and from the very purpose it was designed

atteS to'.V • h f e ^JT* bS ™n<*»»ioal for a scientist to
attempt to eliminate, (80)

Professor De Koninck has brought out with great e-
xactness the fundamental issue involved in this question: '

Ehtre ces nombres-mesures reperes sur l'echelle .
graduee d'un instrument et le sujet m ateriel , ilya la fabri-
2&Jl° n ^Hi_°IL£ejeeu^aj£e_abs^

subje ctivisme . Ne confondons pasTa^onnee prescientifiqTS
avecle nombre-mesure qui n'est pas une traduction immediate
et adequate de cette donnee. Ce n'est pas l'objet sur le pla-
teau de la balance qui sera le point de depart propre de l'e-
laboration scientifique, mais tel nonibre, sur l'echelle graduee
auquel' s'arrete l 1 aiguille. Une fois definie la propriete,
je ne puis l'attribuer telle quelle a. l'obje'3, comme si la
balance n'etait qu'une espece de rideau et que dans la pens ee
on epiait 'derriere' la balance pour surprendre l'ob jet tout
nu (Et c'est Men ce qu'on croyait faire avant la critique
einsteinienne des mesures d'espace et de temps, oubliant que
les circonstances memes de mensuration font partie d'une de-
finition et que la difference de circonstance change qualita-
tivement cette definition. Dire que des definitions de longueur
qualitativement differentes doivent avoir la m§me valeur quan-
titative c'est tomber dans ce relativisme dont Einstein nous
a liber es» (8l)-.^«w» FWt\i/^

One of the reasons why this point has often been
lost sight of, at least to some extent, results from the innate
and inevitable tendency of' science to idealize the entities with
which it deals. As we pointed out in Chapter IV, the physicist
tends to substitute in his mind an ideal geometrical model for
the physical apparatus with which he is working. He tends to de-
materialize his instrumen ts, in such a' way that a concrete meter-
rod, for example^is transformed into an immaterial metQr. Speak-
ing of this question Sir Arthur Eddington writes:

-382-

se a Freat^^^-'V^ && rathet * han ' yard-stick becau-
se possible TtTT^? substitute - for the yard-st ick a-
ic possible But we do not generally think of a yard as a ee-

systemr^e do not f* 1 ^™^ «* Physical ob^t^ £
systems, we do not think of it as an object a* all. I grant

£;^?^ S1 fl 0hiB f* W be an equivalent substitute
for^a yard-stick, but I donot grant that a de-materialized
yard is an equivalent^ substitute for a yard-stick. When the
quantum physicist employs a standard of length' in his theory,
he does not treat it as an object; if he did, he would accord- '
ing to the principles of his theory have to assign a wave funct-
ion to it, as he does to the other objects concerned in the
phenomena. In my view he is wrong. Either he, is using the stan-
dard length as a substitute for the second body concerned in the observ-
ed relation of size, in which case he ought to attribute to
it a wave function, s^_bhatjjg_g an. bring it into his equat e
ions in t he same way that, the second body would have been brought
in; or he is treating size.; as -though it were~not an observa- "
Die relation between one /.physical ^object and another, and the
lengths referred to in his : formulae ; are not the lengths which
we try to observe. We have." to recognize then that what are
called the properties of.k'h electron are the (dbmMnec!) proper-
ties or relations of an electron ani. some othfeph^pl
cal system which _ constitut es a com barison ob.jectj For an elec-
tron by itself has no properties. ,H" it were absolutely .^lo-
ne, there would be nothing whatever 5 to be said '.about it

not eve n that it was an e lectron. And wo muslfTnot be misled

IbyfEcTTact that in current quantum theory the comparison is
replaced by an abstraction, e.g. a.meter, which does not en-
ter into the equations in the way that an observable compari-
son object would do; for that is a point oh which current quan-
tum theory is clearly at fault. (82)

These , considerations will serve to bring to light
the position occupied by the instrument in the process of measu-
rement. In some sense it is an ambiguous position, for the instru-
ment belongs at the same time to the object who is measuring, and
to the object measured. For on the one hand, it is a kind of pro-
longation of the cognitive powers of the subject ; it refines the-
se powers" and enables them to arrive at more exact and more sen-
sitive discriminations. On the other hand, it is one with the ob -
ject ^both becaus e it is o ne term of the c o mparison which ever y
measurement implies , and because of the physical causality it exer-
ycices in the measuring process.

In connection with this limitation of measurement
arising out of the part played by the instrument, another closely

-383-

f^rrinf to ^r^^™ fT ^ t0UChed **»• We - ™-
concrete measuring processTheS^ 63 th&t enter into eve ^
have a very definite effll't llon^^T *** legi ° n > aM the ^
It is true that it is possiblffn^ J^" 3 ,^ the "^urement.
them to a certain extent tI I scientists to cope with

is an attempt to achieve \l±aT,7 H^ 3 ° f measu rement there
as arise from electw/L lde al state m which such influences

coLiimit. In order to be able to accoW for aTXThe~5o3Sc-T^:

£ve ?o S bf^etff ^ " Part . ±n the raeaSUring P™ M "» one luld '
would de^nJ ^ a °? uai , nted wit * these influences, and that

would demand an exhaustive -knowledge of Nature. And perhaps it

£ rL S Tf U T t0 add that this Evolves much more than a per-
fect knowledge of all the laws of nature. For chance plays such
an important part in the co s ros that many of the influences that
actually bear upon concrete experiments are pure chance events
which have no determined cause, and which are therefore outside
the pale of aUTawTTt seems safe to conclude, then, that our
actual knowledge of the influences entering into our experiments
will ever remain infinitesimally small. And in this sense there
is a great deal of wisdom in Planck's remark that "measurement
gives no immediate results which have a meaning of their own."
(84)

What is it that we actually measure in our concre-
te processes? Perhaps it is not an exageratioh to say that even,
in such a trivial measurement as the weighing of a pound of meat,
we are not merely measuring the vreigh^ of $he meat we are ac-
tually measuring the whole cosmos .OPor the object measuredand
the instrument employed never constitute an isolated system. Nor
can an isolated system ever be achieved through successive appro-
ximation in the control of known cosmic influences. A perfectly
closed system, other thaii the entire cosmos, is a pure idealizat-
ion, It exists nowhere but in the mind of the scientist. The fol-'
lowing lines of Louis de Broglie have considerable relevance
here:

Le concept d' unite physique n'est done vraiment clair et
bien defini que si l'on envisage une unite completereent inde-
pendante du reste du monde, mais, comme une pareille indepen-

-38d,~

dence est evidemment irrealisablp ]o . ,

pris dans toute sa St& w ^'V ^ Pulque
alisation, comnB un cnfnn^ * S ° n tour comre une ld<5 "

a la realite. n en est d J l," 8 ^*^ rigoureusement
systeme. Le svstemo a T-> d ' aille ^s, du concept de

isme o^S "^^ f ffffi striate est Sn orga-
le conceptl^^x- a ■■- — e y. san s relations avec l'exterieur;
i£ ^^ 2 2OJL2SLdonc_ap P ; icable^^ ( £ 5)

a question whic^LISe's^ 3 ^^ 10113 leaa US inevi ^bly to

an? discussion of ^^gni^icance of Lo° St ^^ T blems in
ion of , the ri F id scale m^^ V *""? aaur ? ment " ~ the quest-

\raenu for any standard of measur ement in ™*-,-,. <™I require

as standards, nor are easily^stlfm tals^r ^1^^

br^oufST And n the fUnd£mKn fal -son f or thifSleen
brought out m our analysis of the nature of measurement.

„ But to what extent is self-congruence possible?

ur, to put the question more pointedly, does the concept of self-
congruence even have any meaning? If it is irapoaalBIeTo arrive
at any definite determination of rigidity, and if the very notion

i^-;" COn8rUenCe is ^thout meaning, then to say the least the
validity and significance of the whole measuring process will be
extremely questionable. And at first sight it might seem that we
must be laid to this conclusion. For if the statement which we
made a moment ago, that a length must be measured with a rigid
scale, is to have any meaning for us, we must be able to define
v/hat we mean by a rigid scale, And the definition which natural- '■
ly suggests itself to us is: a rigid scale is one that preserves
the same length. But this immediately involves us in a vicious
SSTi^Sj for we have defined length in te rms of a rigid s cale, 'and
a rigid scale in terms of~le ngth. (86) And as long as wcTcling -
to these two definitions we shall be confronted by an impasse,,
For, obviously, if length is a quantity obtained by means of mea-
surement with a rigid scale, it will be necessary to have recour-
se to another rigid scale to decide whether or not the length of
the first scale changes, and this sets us on an infinite series, - —
The only possible way of surmounting this impasse is to revise
one of the two definitions. And a moment's reflection will show
that the definition of length cannot, be the one revised, since
length can have no definite meaning except in terms of the self-
congruence of a standard, lie must then attempt a solution of the
problem by seeking for a determination of rigidit y independently
gf_thg notion of Jength . At first sight this may seem an impossi-
bility, for it is difficult to see how one can decide whether an
extension has increased, or decreased, or remained the same, except

-585-

5™if irSSffi. 8 * lf "—* * «****, a vici-

ithe way is supSeTbf^^ 1S ? ^ ° Ut ° f this i^asse.'Arxl
the standoff le^thVaa 'T&St ^^ ^ this + *»*«*»

and even nonsensical; tel^t £ SSi^ rgffig^a

mediately that, far from leading us out of our impasse thi* m -

len^7t\ZtrT? a11 thS m0re ' P ° r ^TtandSd £ ■
length has no length, what sense is there in speaking of self-con-
gruence or rigidity? No matter how much an elastic m^ter tape mea- ' "
^J*l ^ftched everything that is measured with i TrllT
always be a meter in length. As a result the whole process of mea-
surement loses its significance. ^ *>s, 01 mea

A moment's reflection will show that this objection
arises from a confusion over the meaning of the term 'length".
As we have already pointed out,, this term is susceptible of a mul-
titude of meanings. But since we are dealing with physical scien-
ce, we _ have been using it, and shall continue to use it, in the
sense m which it is . employed in physics: the measured magnitude
of a sensible line. No standard has length in this sense. That
is why we cannot employ measurement to determine rigiaity, for
then the standard would be a measured magnitude. But obviously
every standard has length in the sense that it is an object with
a definite extension. And it is possible independently of any pro-
cess of measurement and merely by having recourse to identity and
non- identity (87) to determine the constancy and inconstancy
\of this extension, Cs^ai'cKt. \ ~£w>U<m\cww Cri'hv^c ><n m«i )'&])« of k fa-f*

A number of bars of different material may be taken
and their identical_ extension aeterminea by noting the. coinciden-
ce of extremities. These bars may then be subjected to a variety
of influences such as pressure, temperature, atmospheric condit-
ions, etc., ana by comparison their coefficient of exgansion or
vpon traction observea. The bar which comes (closest to (identitjfo with
thg_ orig inal extension is chosen as the standard . A special room
is prepared in which conditions considered to be ideal are kept
as constant as possible, and every effort is -made to exclude dis-
turbing influences. The chosen bar is then placed in this room,
and at last a rigid scale has been achieved. This is in substan-
ce, the way in which the international legal standard of length
was arrived at - - the Metre des Archives, which is a bar of pla-
tinum preserved in Paris at a temperature of melting ice and un-
der atmospheric pressure.

-386-

pear extrenclv^^T^A^ det ? rmini ng self -congruence may ap-

Sble for ^ S S 1V ^ ° nCC a Standard tes been chosen > " ^ 4os-
xll IT lt 1 t ° Change - ^ standard . The question might also mean:
does the scale remain absolutely rigiOs far as scie W ^=on^

C °™T\ T ■ \t* imp0ssible t° answer such a question in the af-
firmative, in the sense that the whole structure of science is
\ based upon the assumption that the scale is rigid.

_ Perhaps the word "assumption" will be immediately

seized upon and the question pressed home: "But is it really ri-
gid?" T he answer to this question depends upon what is'^iSnt by
jreally , If it means that there is existing somewhere in the cos-
mos an ultimate and absolutely immobile ideal standard in relat-
ion to vrtiich the constancy or inconstancy of the chosen standard
may be objectively determined, it is extremely doubtfu l just how
muchjsens e a question JL ike that can have . It certainly'has no sen-
se fronT'Ene point of view of physical science. (88) We do not
see how it can even have sense from the point of view of philoso-
phy* Bu t_i£ the question means : does the scale possess absolute '.
objective immobility, then a definite answer can be given . And
the_ answer is : certainly not ^ for the very, notion of an absclute -
jy_i ramobile mat erial obj ect is a contradiction in termsT^ )

And this brings us to the central point towards which
most of this discussion has been directed: the whole significan-
ce of the measuring process depends upon the rigidity of the sca-
le that is employed as a standard, and it is impossible to arri-
ve at an absolutely rigid scale. The rigidity that is spoken of
in science is one that is determined by fiat; it is a convention.
And this obviously introduces a profound limitation into the pro-
cess of meaning,, But it is impossible to have a clear notion of
the nature of this limitation except by pointing out that, while
it is meaningless to ask whether this convention is true or fal-
se, it is extremely important to determine to what extent it is
ar bitrary . It is obvious that like every convention, the determi-
nation of the rigid rod is in some measure arbitrary. But it is
likewise obvious from what has been said that it is far from being
purely arbitrary. In other words, it, is something that is at on-
ce both subjective and objective . And though it wiH always remain
impossible to determine the relative degrees of subjectivity and
objectivity, it is important to note that purely objective rig i-
dity is a dialectical limit to which science may draw constantly

-387-

closer and closer, by mean* nf ■?+ ■

proximation through an asoenaf™ - SU f ? eth ° d of successive ap-
ed above. When we sfoted S ^ P ^ Similar to the onG ^escrib-
it cannot change, ^ n t 1 ° / ^ scale has been chosen,
reject a chosen standard in favor o/aTV^ SC±en ° e can neve ^
In fact, it is of th^Stu^e 0?^^^? ^ ^^ m ° Te perfect »
in search for a nore rarfW* * P 7 s J cal science to be constantly

Paris standard v^lleventuallvr^'i". 18 Pr ° bable that the
such as, for oxoWle l£^£Xin SUpplanted ^ another standard,
se latice stated ^ the ^/V^^ a Calcite W*™-* who-
withpure numberT It i„ 1^ anta S e °£ associating the standard

d,5aln^i5^; a " r 1 L^SrtL prob ^ 1 ! that science v ' m s ra -

only important point to keef in Snd 8 f "^ ** itB standard ' The ■
cussion is concerned, ^tSt^^er^t degr^ ^rifidSv'

r"Lble™in of^ 6 ^ -f W b<3 ±n the ^Lf anlnt L-
mimble margin of subjectivity deriving from the free i ntervent-
lon of the human intellect and will. mtervent-

■ j ' , This dis cussion of the rigidity of the measuring

trtcZn^irsI ^V? ff thS qUSStl0n °^ the ^tSrSd On _ .
SS"™' £ lrs * P ostula ted to account for, the absence of any in-
dication of ae th©r drag in the Michelson-Morley experiment and
later confirmed by the electromagnetic researches of Larmor and
Lorentz. According to the postulate of Fitzgerald, a material rod
moving at high speed contracts in the direction of the line of
motion. The consequences of this postulate for the problem of Mea-
surement are immediately apparent. What determined meaning can
measurement have if the standard scale expands and contracts ac-
cording to the velocity at which it i'S moving and according to
I the direction in which it is turned - - especially if (as is the
I case) it is impossible to know in any absolute way the velocity
\^of the scale. In ordinary circumstances this contraction is negli-
gible; for example, the diameter of the earth contracts two and
a half inches, or one part in two hundred million, in the veloci-
ty of nineteen miles a second of its movement around the sun. But
at the speed of one hundred and sixty one thousand miles a second
the contraction would be one half. And is there any way of know-
ing whether in relation to some point of reference in the cosmos,
the whole solar system is not moving in a manner that approaches
/this velocity? What is worse, is there any way of knowing whether
the whole frame of reference in relation to which we make our mea-
l surements is not moving in relation to other frames of reference
in different directions and at different velocities, which perhaps
^.do not remain constant?

It becomes immediately evident that all of our de-
terminations of length (and of time also, as we shall see presently)

-388-

rence be Veen Classical Ind £e1 ^° • ^ T? the P rofound diffe-

■ not that ClassicalXsicJ failed In 7 P ^ Si °!; But the P ° int ' is
cities and diffWr-n? * railed to realize that different velo-

thc ^roSs oTmeLurST °Z JfTT *"? *" ^^ UP ° n
each observer con d ^i n fa °*' Xt P rovi ^d formulae by which
eacn ODservcr could apply "corrections" to reduce his "fictitious"

natfer lief in ?*"" NeTrt ° nian length '' The whole crux of the
Cd "^e" Tn oth^ lnS , 0f thG W ° rdS "erections", "fictitious" ,

and unique . In other words, Newtonian physics realized that mea-
Uurcments made by different observers will give different results.

^^gg^gl# 7 ^v3^gll^iitign -^jTjjj it ion th at was Na ture « s
o^ositio^JAndTrWth^^

tulates: 1) that spatial relations determined by the measurement
of length could be reduced to an absolute meaning; 2) that tempo-
ral relations had an absolute and indep endent cha racter. Einstein
was astute enough to see that both of these postuIivEes~were per-
fectly gratuitious, and he proposed to do/wlth them. But in order
to understand the significance of his doctrine for the, question
of measurement, it - is necessary to return for a moment to the Fitz-
gerald contraction and try to fix upon its exact meaning.

At first sight, this contraction might seem to be
in the same category with the changes in the standard scale, dis-
cussed in connection with the problem of rigidity, but as a mat-
ter of fact, it constitutes an entirely different problem. Indeed,
it is true to say that, paradoxical as it may seem, the Fitzgerald
contraction has nothing to do with rigidity. The meaning of this
statement will be fully explained in a few moments, and for the
I present it is sufficient to point out that the contraction is de-
termined completely by the velocity of the motion and not by the
specific nature of the rod in question . All rods moving at the
same velocity undergo exactly "The same contraction, no matter what
degree of rigidity they may possess in relation to such influen-
ces a3 temperature , stress, etc. The contractions of a rod of pla-
tinum and a rod of rubber moving at the same speed are identical.
Hence this contraction must not be looked upon as an imperfection
of the rod . It must not be considered a deficiency in relation
to an absolute rod. Such a rod does not exist, nor can it exist.

In order to come to understand how the problem of
the Fitzgerald contraction differs' radically from the problem of
rigidity, it is important to note that the length of an object
measured is in a sense' completely independent of the difference
between its temperature and that of the measuring rod. A cold scale

-389-

te pro -

^e^nrifriSS^fp^^ ""V» «*»»»* *>* body and
measured is not indeplndenfof ?h^ • ?f ^ lensth of <* ob *= ct
and that of the stXf a / ^ence between its motion
pletely dependent upon it. " ' 1S ' ln a aense > com -

mediate oontooW^^hS notiof ^ ^ * ^ ou S ht -to im-

be intensely si^lified if it were always possible^ arrive 11

re^L°f r »it n fs not the °^ ec ^ + — ed.^ut, as Eddingtol has

t^gf the^b ra ^-TLV* pS oTa" S??" ^
pie," (891 particles, for exam-

_ Perhaps at first sight the difference between the
determination of the proper length of an object and the determi-
nation of the length of an object in motion in relation to the
scale may not seem to constitute any serious problem, since it
appears to be a fairly easy matter to reduce the one to the other.
Let us suppose, for example, that a straight rod is moving with
uniform velocity with respect to a certain frame of reference.
It is possible to mark on the frame the simultaneous positions
of the extremities of the rod, and then measure the distance bet-
ween the two positions marked on the frame. Will the results cor-
respond to the proper length of the object in motion? One might
be tempted to answer in the affirmative, since the two positions
were marked simultaneously.. But then he will.be obliged to tell
us what he means by simultaneity. And therein lies the whole crux
of the matter.

As we have already suggested, Classical physics at-
tributed to the notion of simultaneity an absolute meaning. But
Einstein pointed out that this attribution was based on an impli-
cit assumption which was utterly incapable of being verified ex-
perimentally, since this verification would presuppose that signals
announcing distant events could come to all observers instantaneously,
that is, with an infinite velocity. Concepts have no meaning in
physics unless they can be defined operationally, and Einstein
made it very clear that every attempt to define simultaneity ope-
rationally inevitable results in making it something relative to
the frame of reference in which the operation was carried on. in
other words, the only kind of definition of simultaneity that has
any moaning is such that if two events verify it in one system
they will not verify it in another system that is in motion with

-390-

reapect to the first, The measurement of tim<= +h«, i,
sentially relative to a Piver, ZrJ+Z ? ^ • ' ' becomos es ~
of the lenrth of \ hnflv f lven + ?y stera - And since the determination

the velocity v it will have the length 1< = 1 /i _ 5L ,
where c is the velocity of light V 2

That is, length of a body has in each system a different value,
depending on the velocity v of the body with respect to the sys- .
tern in question." This difference in value is equal to the Fitz-
gerald contraction. And since the determination of the other quan-
tities which enter into physics is bound up with the reckoning
of length, it follows that mass, periods of vibration, electric
and magnetic fields, etc, become relative to a certain frame of
space, • ,

Because of the way in which simultaneity is involved
in our determination of length, it is clear that not only space,
but time as well is implied, in all our measurements. In other words,
to quote Eddington, "the fundamental measurement is not the inter-
val between two points of space, but between two points of space
associated with instants of time." (90) Events in nature are
exterior to each other in four different ways, of -which three are
spatial and one temporal, and the order of these events constitu-
tes one indissoluble four-dimensional space-time order. It is the
purpose of the laws of physics to express this order in the form
of numerical relations, and this can be done without ambiguity
only by having recourse to a system of reference of four coordi-
nates. That is why non-Euclidian geometry has become the instru-
ment of Relativity physics.

Science has been laid to reconstruct the world in
this four-dimensional order, not by any arbitrary choioe, but by
the very nature of extrinsic measurement upon which its whole me-
thod is founded. Because the bodies 'which constitute the cosmos
are in motion with respect to each other length can be measured
only in relation to time, and .time only in relation to length,
Consequently, observers with different motions will have diffe-
rent reckonings of space and time, and each observer by merely
changing his motion will make a different division of the four-
dimensional order into space and time. In other words, each ob<~
server, according to the different operational definition he gi-
ves of simultaneity, will cut up the space-time continuum into

-592-

(Hifx'erence equal to the Fitzgerald contraction? Taken in this
sense the question will receive an affirmative answer from the
scientists. And this -seems to be the only sense in which the
question can have any significance for them. For in physics the
phrase "actually takes place" can Only refer to what actually
takes place in measuring instruments.

But perhaps one might be tempted to push the quest-
ion further and ask: But does, the' velocity make the length of a
rod contract in the same way that a change in temperature does?
This question is still ambiguouB, since it attempts to establish
a comparison between "lengths" which have entirely different mea-
nings » But perhaps the issue can be clarified by putting the pro-
blem in these terms: does the motion of a body decrease its in-
trinsic measure? And then the answer is: first, the Fitzgerald
contraction certainly does not imply such a change, since it has
nothing at all to do with intrinsic measure; there is no way of
knowing what actually happens to the intrinsic measure of a bo-
dy in motion, for in order to determine the dimensions of such
a body we are foroed to have recourse to extrinsic measurement
made in relation to a particular franc of reference.

In this discussion of the limitations of measure-
ment it has been' necessary to restrict ourselves to rather ge-
neral and superficial considerations. A more refined analysis
of particular processes of measurement, such as those which ha-
ve to do with time, for example, would throw fuller and more de-
finite light upon the extremely limited character of knowledge
which measurement affords us. But perhaps enough has been said
to show how highly artificial and subjective this knowledge is.
There is indeed great wisdon in Bergson's remark that nature
does not measure. It is man that measures. And he cannot measu-
re Ydthout projecting his own logos into nature. At every step
in the measuring process there is a projection of the human in-
tellect and will. And the more perfect' this process becomes,
the greater becomes the part played by the subjective elements.
In a very true sense, measure-numbers are not found m nature.
They are imposed upon nature by man.

But lest all this seem to give too much aid and
comfort to idealism it is worth while pointing out, as we bring
this question to a close, that measurement is after all a real
Physical operation which comes to grips with the real ^ld.
And the relations which arise out of it are ^^^fforeover
ions, in spite of the large margin °f f^^^^S^k
the subjective element is purely functional; it exists only to

-394-

CHAPTER NINE

THE MATHEMATICAL TRANSFORMATION OP NATURE

±„ The Transformation of Natural Science,

"The rnathematician" ., Goethe once remarked, "is like
a Frenchman: if you speak to hira v he translates it into his own
language j and at once it becomes something altogether different,,"
In this chapter we must endeavor to see at least in a summary and
schematic way, how the mathematician who is called in to assist
the physicist. in the study of nature translates the world of the
physicist into his own language and makes of it something altoge-
ther different And we shall consider this transformation from
two points of viewo First we shall ""^e the way in which the intro-
duction of mathematics into physics affects the very structure
of physical science itself; and secondly, we shall attempt to bring
out the change that this produces in the reflection of nature that
is found in physical science.

In the last Chapter we considered the preliminary
step in the mathematical transformation of physical science. In
order for science to be mathematicizcdj, all of ' its processes of
experiment must be transformed into processes of measurement; all
of the phenomena .with which it deals must be translated intp point-
er-readings a This preliminary step provides the scientist with
a collection of measure-numbers, by which are determined various
Properties of bodies such as mass, volume, temperature, pressure,
viscosity, valence, molecular weight, various optical, electrical
and magnetic properties, etc. But just as physics is not a collect-
ion of phenomena, so mathematical physics is not a collection ol
reasure-numbers. In order for science- to emerge, the unifying pro-
cess described in Chapter IV must undertake, by using measure-num--
Wa as materials, to construct out of them an integrated and cooi
dioated system. And the first step in this process is the est--
■blishment of law.

-395-

Since the only materials of construction available
jvs numbers, laws in mathematical physics can be nothing but the
Expression of relations between numbers, Since a law mast be uni-
versal 5 that is to say formulate a constant relation,, a physico-
pathonntical lav; will express a relation between variable magni-
tudes;) and consequently will not be algebraic and not arithmeti-
cal (in t ne restricted sense of the term "arithmetic") The uni-
formity of association which constitutes the essence of experimen-
tal law finds its best expression the language of numbers because
it is at once both exact and universal, This expression usually
lakes the form of differential equation;:.;,

"A physical law", writes Planck? "is any preposition
snunciating a fixed and absolutely valid connection between mea-
surable physical quantities - - a connection which permits us to
calculate one of these quantities if the others have been disco-
vered by measurement o" (l) In other words, a physical law is
a constant relation between variable quantities; it takes the form
of on algebraic equation which expresses a functional relationship
indicating the precise value of any one of the measures that cor-
responds to any given value of the .ither' measures, Once the con-
crete measure=numbers are absorbed into mathematical equations
they become susceptible of all the pliancy of mathematical : mani-
pulation,, The mathematician is free tc have recourse to all of
the resources at his disposal: powers, roots, divisors, dividends,
sines, cosines, vectors, etc. There is nothing to prevent him from
squarring the symbol for time., for example y These manipulations;,
obviously, do not affect the concrete properties from which ^ the
original measure-numbers have arisen, but they may lead to ; the
discovery of new properties <,

It is extremely important to grasp the true nature
of this functional relationship of physico-mathematical law„ As ^
is evident from our analysis of the nature of raxthematioal *strac
ion, mathematics prescinds from all causality excep-o a type ol
forrhal causality that is found in formal relationships- For exam-
ple, the geometric "law" B = S/H: the base of a rectangle is equal
to the surface divided by the heigit does not mean that a Surface
ram actually be divided by a length.- And if B varies it _ is, not
because (in the sense of true causality) S varies., or vice, versa.
The law merely states that if the base is changed, ho na .re o,
a rectangle is such that the" S urface of * ^^Sle^ cSsSity
80 a proportional change,, The "if" makes all efficient «£?*£&
oxt-oinsio to the law, and the phrase "the nature of a ^c^gle
i« such that" shows that the law deals with formal causa lity, sun
«* U is the form of a thing which determines its ™^e° J^Led
frequently, in the measure in which physicax laws are expressed

-396-

iu ri -Abhcm=:.-cical equations they are stripped of all true causali-
ty. Genuine causal statements' are irreversible, that is to say
they always involve ontological symmetry and usually tenporal as-
gy,x!3tiy. The effect depends upon the oauae for its being and not
vice versa. Formulae of covariation and purely functional state-
Eients, on the other hand, are essentially symmetrical,, Any one
of the variables may be arbitrarily considered as independent or
dependent

When a mathematical physicist states that the move-
ment of the pianets is in accordance with the following law; the
foroe of attraction between bodies is directly proportional' to
the product of their masses and inversely proportional to the squa-
re of the distance between them, he is not expressing the cause
of planetary movement^ He cannot treat force as a true cause sin-
ce for him it is reduced to a measure-number which is a product
of the multiplication of the numbers derived from the measurement
of mass and acceleration. He is merely expressing a formal inter-
relatedness emerging from a comparison of the mass., distance, and
acceleration of planets „ Force and movement, then, are not related
an cause and effect, . They are simply two data which are mutually
dependent in somewhat the same way as the diameter and circumfe-
rence of a circle, (3) Poincare has insisted upon this point
in La Science et l'Hypothese ;

Qu'est-ce que la masse? C'est, repond Newton, le pro-
duit du volume par la densite II vaudrait mieux dire, repond
Thompson, que la densite est le quotient do la masse pa5p le
volume, Qu'est-ce que la force? C ! est, repond Lagrange, ■ une
qause qui produit le mouvemenb d'un corps ou tend a. le repro-
duire, C'est, dira Kirchoff, le produit de la masse par I 'ac-
celeration, Mais, alors , pOurquoi ne pas dire que la masse est
le quotient de la force par 1' acceleration?

Ces difficultes sont inextricable s B Quand on dit
que la force est la cause d'un mouvement. on fait do la me-
taphysique, et cette definition., si on devait s'en contenter,
serait absolument sterile , Pour qu'une definition puisse ser-
vir a quelque chose,- il faut qu'elle nous apprenne a'mesurer'
la force, cela suffit d'ailleurSj il n'est nullement necessai-
i"e qu'elle nous apprenne ce que c'est que ia force 'en s;oi',
ni si elle est la cr.use ou l'effet du mouvement, (4)

Ohm' s law merely signifies that the numbers obtain-
^ d by the measurement of the intensity of an electric current,
fch e electromotive force, and the resistances are so related that
^ey always verify the equation: I = E/fc, whatever be the numeri-
"^ values of the symbols in the individual cases. The law of

-397-

Ifavio-fcto is likewise stripped of causality when 'it is translated
into a mathematical equation. It does not mean that the pressure
is the cause of the increased volume; in so far as the mathemati-
cal pliysicist is concerned, Both the pressure and the volume may
be considered either as the independent ("cause") or the' dependent
("effect") variable. The law merely states that when all other
iseasuves are equal, if the measure of temperature increases, the-
re is a definite corresponding increase in the measure of the vo-
IwiiSo Or to put it in other words which Yri.il bring out the assi-
milation of a physical law to a geometrical law, and show what
type of causality is in question; the law states that if a cause
should, increase the temperature of a gas, _bhg_ nature of "the; gas
is such that there' will be a proportional increase in its volu-
me

The same is true of all the laws of mathematical
physics: they do not declare that A is the cause of B; they. mere-
ly state that one set of events B is a function of another set
of events A, If the mathematical formulation of the law expressed
causality, the causality would have to be reversible. Perhaps one
might be tempted to think that the intervention of a time measu-
re into a lav/ might introduce causality since this measure will
indicate which of the variables is the antecedent and which is
the consequent. But a moment's reflection vri.ll show that this is
not true,, This intervention of a time measure merely expresses
the fact that the other measures vary in relation to the time mea-
sure,, An expression of antecedence does not involve causality*

It is clear., then, that the mathematical formulat-
ion of physical laws empties them of all true efficient causali-
ty, And the same must be said of formal causality. Just how pro-
found this change is beconcs evident when one stops to consider
that all law essentially involves finality. By its very nature
law i.ieans an inclination, an ordination to an end,, We shall return
to this question later on.

Prom all that was said in the. last Chapter on the
nature of measurement it follows that, in spite of the exact ma-
thematical formulation by which they are expressed, and in a cer-
tain sense precisely because of it, the laws of physics do not
Have exact and absolute validity. (5) In fact any mathematical
Session of nhysical constancy is only one of an infinite num-
x ? of slightly different expressions which might possibly be em-
ployed to formulate the same phenomenon, All physical laws are
^.entially provisional,, And they are provisional for two reasons
;-n Particulars first because they are merely approximative „ and
,n *is sense neither true nor false; secondly because tfiey are

-390-

sohemtic They are approximative because the measures whose re-
lations they express are never made with absolute exactitude. That
is v;hy they must ever remain open to successive corrections, for
progress in the refinement of Measurement will continually intro-
duce slight changes in the numerical coefficients, and there is
n o limit to this process of refinement.

Laws are schematic because they include only a small
fraction of the possible measures that could have been made; that
is to say, they express a relation between certain chosen proper-
ties, independently of all tha other properties which may be con-
nected with the ones choseru Consequently; as sciences progresses
its laws must be constantly modified in such a way as to. take in-
to consideration attributes previously omitted,, Physical proper-
ties are defined by the description of their process of measure-
ment, and as we noted in the last Chapter, all of the circumstan-
ces entering into this process can never be enumerated,, Progress
in experimentation reveals an increasing multiplicity of circums-
tances which have a definite influence upon the results of .'the
treasuring processes, but which were neglected in the original for-
imlation of the general law. That is why all laws, must remain for-,
sver open to a progressive modification by which these newly dis-
covered influences are integrated into its structure. This modi-
fication does not change the form of the law or its numerical coef-
ficients, as does the modification occasioned by its approximative
character. The newly discovered circumstances can be introduced
only by the introduction of new measures and consequently 'new
properties. Thus progress in experimentation with gases revealed
the fact that in order to determine with precision the relation
between pressure and volume attention must be paid to the mutual
rttraction of the molecules and their proper volume. A determinat-
ion of these additional circumstances results in the transformat-
ion of the l«i.w of Mariotte into that of van der Waals,

Thus 'as science progresses its laws become increas-
ingly complicated by the integration of newly discovered influen-
ses, This complication results in investing general laws wxth greater
precision and accuracy. But as we saw in the last Chapter, even
ths .atoplest measuring process involves the whole universe. That
is why a perfectly exact law would require an exhaustxve descript-
ion of the entire cosmos.

But while this process of complication is taking •
Place there' is a concomitant process of. simplification going on,
■'hich consiats in the reduction of the ever incc ^f an S ™^ P ;£~
°it,y of measures to a few fundamental measures. This is done in
'*"> ways. In the first place it is discovered thau a number of

•^39 9. v

^■(Tov-Tivt instruments give the same results. Since physical pro-
pvtl"^ arco defined by their proooases of measurement it remains
v^ov.vSicalfy true that tv/o different processes define two diffe-
vr.'.sl; wcopsrties. Nevertheless it sometimes becomes evident that
;; ; o vaults of two or more different processes coincide, as for
8 yov,7.tfo when heat is measured by the expansion of a metal spring
oiid ivy 'She expansion of a column of mercury. But even more import-
ant tliM this is the simplification, resulting from the discovery
that the results of certain processes of measurement coincide with
i.altoatical combinations of other processes. Laws repeal constant
relations between the measures' of different properties. These cons-
i-ont relations make it evident that certain measures can be replaced
by a combination of other measures 4 In this way it is possible
to reduce a vast multiplicity of measures to a few fundamental
r^aouvea. In fact science has been able to push this process of
Bjrfolification to the extent of reducing all physical measures
to combination of the fundamental ■ measures of length, mass and
Mr/o, in such a way that the former ma,/ be considered as functions
of the latter. It thus becomes possible to define the> multiplied
t,y of physical properties in terras of combinations of a few irre-
ducible properties. This does not mean that bodies have no other
propsrties hut three that are measured b/ a rule, a balance and
a olock, It merely means that when the variety of physical proper-
ties are measured by different measuring processes the- results
"re numerically the same 'as certain mathematical combinations of'
the msasure numbers provided by a rule, a balance and a clock,
(6) By this simplification 'the scientist is able to synthesize.-.
iu.3 knowledge into a small number of propositions into which- on-
k' a few basic measures enter, and from the relations existing
befe'oon the fund-wnental measures- it becomes possible to deduce
'As ailtipiioity of rslations existing between '-the particular inea-
3urc-s,

All this shows' how t'tr.Wi process of simplification^
\nns the way for the scientist 'to take the -next step in the uni-
fication of his knowledge - - to ascend from laws to theories,
Cut before massing on to ah -analysis of the nature of physical
ilisBiy it is necessary to remark that because of the approximative
M'l schematic character which we have, been discussing, physical
Ware always a simplification of the mind and in this sense a.
product of the mind, And their provisional nature cannoo be lost
w-fihl of without undermining their objective significance, Oasc-
l »8 Physical reality in mathematical form has the advantage _of
providing it with great openness, that ia to say, of opening u
<P to the ^limited reaches of mathematical peculation which
^foixis such abundant sources of oplamtion. ISut at t ha ,ame time,
l * tea the disadvantage of imposing upon reality a frame which

"40.0...

because of its exact determination is too closed. And in this con-
nection it is worth while recalling the well known rovfxvk of Eins-
tein that in so far as the theses of mathematics are certain they
ao not refer to physical reality, and in so far as they are ! nade
to refer to physical reality they are. not certain, (7)

But perhaps one might be tempted to object to the
statement that all of the laws of physics are provisional on the
grounds that there are certain fundamental laws known as princi-
ples' which are not subjected to the successive change about, which
v?e have been speaking and which consequently seem to have an ab-
solute and not a provisional character. The conservation laws,
the law of inertia,, etc, are all laws of this kind. The answer
to this objection is that the absolute character of these princi-
ples is a pure gift "of the mind. The principles, of experimental
science are laws which have been merely suggested by nature'^ but
which the mind has arbitrarily erected into fixed and absolute
prineipleso The reason why progressive experimentation does not
modify them i3 simple: the mind has accepted them as conventional
definitions of "the very objects to which they apply. Consequently
it is impossible for these objects not to be in accord with, them.
And now, having examined the nature of physical laws we must take
up the problem of physical theories »

For reasons explained earlier in this study, the
iaind cannot rest satisfied with an a pos teri ori possession of phy-
sical laws. It will never feel that It has assimilated them per-
fectly until it is able to possess them in an a prio ri fashion,,
Just as the formulation of laws makes it possible for the mind
to arrive at the results which a certain measuring process would
give without actually affecting the process^ so the scientist ^ ins-
tinctively seeks for. a point of departure which will enable him
to ayrive at a certain law in a way that does nob depend upon ex-
perience. In other words, having arrived at physical laws by in-
duction, the scientist is led to attempt to arrive at them by de-_
auction; having posited their existence, he must attempt to explain
themj having arrived at universal functional relationships, he _
p"st;try to show that these relationships are necessary. This is
aone by making the lam appear as logically necessary conclusions.
(8) Since the laws themselves are numerical relations, the point
<* departure from which they are to be deduced musy be general
'"merical relations. These .general numerical relations <^itu-
<* v;hat is known as a mathematical theory, A ^^ ^s teen de
fined by Duhem in the following terns: "un syster.ede propositions
^.themtiques, deduites d'un petit nombre de P rincl P^> <^ on '
Pouv but de representor aussi simplest, aussi °^*^"*»J* „ ,*
»»ssi exactement que possible un onsemble de lois expennEntales. ^

-401« •

Not only does a physical theory synthesize the laws
W hich experience has suggested, but it tends to fill in the gaps
vf hioh observation has left open by substituting what Cassirer has
called "a continuous connection of intellectual consequences,"
(10) In this way science becomes a coordinated system. And this
system is perfected by a continual simplification and reduction
of the principles which form its point of departure and a conti-
nual increase of the experimental propositions which constitute
its terminus o As Whyt© has remarked, "the highest possible aim
for science is the formulation of a self-consistent closed, chain
of concepts and principles permitting deductive argument in one
direction at every point of the chain," (ll) The dialectical '
limit of this movement would be a science in which the whole uni-
verse could be deduced from one mathematical formula, (12)

On more than one occasion in this study we have in-
sisted upon the fact that the fundamental reason why physical scien-
ce reaches out to mathematics is to discover an explanation which
it finds itself unable to provide for physical phenomena, in other
vrords, to discover a reason or prop ter quid for its experimental
propositions But perhaps what has been said thus far in the pre-
sent Chapter about the mathematical transformation of physical
science may give rise to doubts as to, whether this goal is actual-
ly achieved. As a matter of fact, a number of authors explicitly
ftcny that a physico-mathematical theory is an explanation, Duhem,
for example, writes: "Une theorie physique n'est pas une explica-
tion," (13) We' believe that the difficulty here arises from the
ambiguity of the word 'explanation, " As a matter of fact, it is
a term that is susceptible of a variety of meanings., In its. most
fundamental sense it means to give the proper reason for a thing
by presenting one' or several 6>f the four causes by which reality
is constituted. This is the type of explanation that is employed
in the philosophical sciences.

There is another sense in which the term explanat-
ion is used and which has long been associated with experimental
science. It consists in presenting a, model whets e structure and
functions reproduce the structure and functions of the phenomena
to be explained, We understand the term "model "here in the sense
of a mechanical construct or at least of a pictorial image, and
not In the sense in which it is now sometimes used and which in-
cludes mathematical "patterns" such as "tensors and matrices, ma-
nifolds and their curvature, differential f ^ n ^ *^J!;^~ t
^nt 8o " (14) It is well known that mechanical ^\°™g™ oc
<* out of. pulleys, wires, rubber tubes, etc were the £vorite
"om of explanation employed by the classical .physicists, parti
^lavly the™ nf t.hn Tiiolish School, such as Lord Kelvin, Oliver

t-

~402r.

Ledge P Faraday, Maxwell, etc. Wo have already quoted Kelvin's well-
Imo.v.i remark that for him to understand reality meant to be able
to construct a mechanical model of it and apart from such a model
no explanation of reality could. have any meaning for hira„

But even when less emphasis was put -upon concrete
isclwnioal models and more upon abstract mathematical conceptuali-
zation, there was, until recently, always' lurking in the background
of mathematical theories physical models of some kind. For example
in the background of the mathematical kinetic theory of gases there
has always been a fairly definite physical model constructed of
roleou3.es which are so idealized and so simplified that they are
susceptible of accurate mathematical treatment,, even though spec-
trum analysis has given abundant evidence 'of a considerable, gap
between the idealized and simplified molecules and the actual mo-
lecules. These idealized and simplified physical models have served
as a kind of bridge between actual physical reality and mathema-
tical theory o Because of their physical character they have been
considered to be in contact with" reality,* at the same time their
simplified and idealized state makes them directly amenable to ma-
thematical manipulation. Recent physics has discovered however
that it can get along without this bridge, that independently of
any physical model it can set up a correspondence between the re-
sults of its mathematical constructions and the physical system,,
This has been particularly true of the quantum mechanics of; Dirac,
(15) Speaking of this significant change Professor Bridgraan wri-
tes:

What we now have is in effect mathematical models
'rather than physical models This emancipation I feel to be
a very important step forward toward greater theoretical po-
vev, because there is an enormously greater wealth of possi-
bility among the structures of mathematics than in the physi-
cal models which we can visualize and which have a simple e-
nough mathematical theory. It cannot be denied, however, that
a mathematical model cannot be visualized in the same sense
that a physical model can be. Although we may recognize' with
our intellect that the mathematical model is just as good as
the physical model if it only enables us to answer any quest-
ion that we may propose about the behaviour of the physical
system, nevertheless we have an uncomfortable feeling that
we have lost something. (16)

„ ■ Professor Bridgmn is correct in + 1 ^ in ^J^ b *£,_

cl »-s recent change in physical theory represents significant pro
2«* , As a matter of fact, the identification of scientific ex
P- 1 option with the construction of mechanical models such as is

-403-

famd in the writings of Lord Kelvin, and the classical physicist'*
insiR'tonce upon physical models as the criterion o*" the value of
•jaeories, make the intellect, the slave of the iKaginat-i o" . More-
over, they destroy the true notion of science, since they seem
to pake the sensible as such the formal object of science. In a
word; they amount to a confusion of the material and the formal
objeot of science.

It is true that this tendency to explain reality
in terms of physical and mechanical models reveals a trait that
is native to the mind in the sense that it is natural to nan to
want to reduce the unfamiliar to the level of the common and fa-
r.iiliax'o But to tie science down to this tyge of explanation can v
only result in creating insurmountable obstacles in the path of
progress,, For reality is infinitely richer than any fixed frames
that derive from ordinary experience Moreover it is presuming
a frvcafc deal to expect to find in familiar molar experience coun-
terparts of microscopic reality,. Scientists are coming to realize
this nore clearly every day, specially in the field of wave mecha-
nics, and the work of Dirac, Sc'nrodinger, etc. has put particular
enphasis upon this point, But the most important aspect of this
question is ifefc true progress in science, as we saw in Chapter
IV, does not consist in transforming things into what is most know-
able for us, but in approaching closer and closer to what is most
knowable in ge, though least knowable for us, In other words, it
does not consist in imposing our measure upon reality, but in al-
louiiig reality to impose its measure upon us. And if it becomes
necessary to have recourse to art, the only reason is, as we have
seen, to open up reality more and more' as an object s

But in this question it is not necessary to be a
purist,, The remark made by Dirac to Schrodinger: "Beware of form-
ing models or pictures at all", must not be taken too litterally,
0.7) Even though physics has recently taken a definite step in
assorting its emancipation from physical models, it is doubtful
''''at this emancipation will ever be complete, or oven that such
* complete emancipation would be desirable. Imaginative construct-
ion Inescapably accompanies intellectual activity. Moreover, _ this
"aginative construction may often prove useful for the physicist,
^ Professor Bridgman has pointed out:

I think that the ordinary physicist will want to
keep his physical models as long as he can... Unless one has
supreme power as a mathematician, one may well find it i-fefu.L
to have at his command methods of reasoning by analogy that
wU.1 give him an insi S ht into tho nature of the solution of
"Fooia.1. problems B and one nay cheer from the sidelines any

■^IVA-

. x ; ; lrj,pt co invent combinations of the elements of the mathe-
,;.u-'0. analysis which nay bo handled somewhat like tho clc-
, ;i n;s of ordinary experience, and of which we nay hope ulti-
iV.oly to acquire a more intuitive- con. and. I suspect that
.Giro's attempt to find a dualistio aspect of nature is an at-
: : -.!.-.:)t of thin sort„ (in)

Even mathematical conceptualization is necessarily
\ up with the imagination, as we saw in Chapter VI. The imagi-
i.v.,- construction which accompanies thia conceptualization, whi-
rr.! tho one hand less free than that found in metaphysical know-
;;.;, is freer and less determined than that found in physical
^.'.:V;c„ .m mathor.ntical theory it is of little importance what
v^'ive of the imaginative construction is> provided that it
. .:; useful and that it remains in continuity with the measure
out of which the theory is evolved.

; <. •>

Consequently tho physicist i3 free to employ any
.nail no.lels that may prove useful to him, provided he remains
finally conscious of their true significance. He is free to
r.-sivo of li.'dit in terms of "waves" or "corpuscules" or both,
v idol ho do03 not allow himself to slip into the delusion of
■I'nnr; that the ontological nature of light is actually like
■".■-. of water or like tiny pellets, The most important function
'; these models can play is to provide suggestive sources of
i-M.-atical manipulations and an imaginative support which will
'<;\:a mind in coordinating experimentally observed relations o
eatitfulnoss of Bohr's theory of the structure of the atom
■oo consist so much in the piano t-like circulation of electrons
'-I a nucleus as in the fact that this structure provided a circula-
''". for mthemtical speculation,, Dy considering seven electrons ting
rio atom and eight electrons in another, one is enabled in so-
"■'"i-y to seize upon tho difference between nitrogen and oxigen,

In La S cience ot JUHypo these Poincare has brought
■•!•'; tmo function played by models in physical theory and show-
'■> at they are essentially transitory while the mathematical
'^ions -,/iiich they suggest constitute tho essential and pcrma- •
: '' part of physical theory:

.... ceo equations cxpriment des rapports et, si
^:i equations rcstent vraios, o'eut que ces rapports conser-
vont lour realite. Ellos nous apprennent, apres ooims ayant,
1'i'il y a tol rapport ontro quelque chose et quelque autre _
^oso; seulemz-nt/ce quelque chose nous l'appelions autrefois
'■■-.uvoiVmt, nous l'appelons mintomirt courant elcctrique. Mais

40(3-

; apyellations n'olaiont que dos ii.ngea subatituees aux ob-
;- ; ; r-'cb quo la nature noun cachora otcrncllemcnt. Lea rap-
,-;•:; vcritaolea entro con objots reola aont la aoule reali-

qr.o ivvut; piisaion.-s aUoindre, ot la aoule condition, e'est
i ,i.l y ait l'.:n i.-.enoa rapports ontro cos objota qu'entre lea
-,■;,.:; (jio mil:; aor.t -.on forces do r.wttro a leur place. Si cos
,iportr, ncu.i aont connu.'i qu' im.iorto ai noua jiu.;eons commode

v.j.volaccr line image par uiyj autre, (20)

And ha goes on to explain tlvat tho scientist may
i.-.>lolr, that arc mutually contradictory:

II pent ye faire qu'cllea cxpriment l'uno et l'au-
.; dos rapiort.'i vrniS et qu'il n'y ait do contradiction que
(?r, lea imgos dont nous avons habillo la realite. Les hypo-
)oca do eo genro n'ont done qu'iui sens mctaphorique. Le sa-
nk nc doit pas plu3 sc los intcrdire que lc poete ne s'in-
k-Vx'c lea uo taphores; mais il doit savoir ce qu'elles valent.

Wo believe that this view of tho meaning of scien-
.•oK;lr; io correct and that it fits in perfectly with the
l;ic doctrine of tiie nature of mathcivatical physics. For in
fico which is formally mathematical and tor mina tive physical,
plana tory constructions vri.ll be essentially matheiititipal.
1. ivt b ( > nooonsary that in these constructions there be phy-
rc-enbodij/ionts of nature. All that is required is that the
\tical com tructions be in the end verifiable in physical

."!-ilt

But to return to our original question: is a mathe-
1. theory an c-xnlanation? Professor Bridgman, after noting
:c emancipation from physical models gives U3 an uncor.if or- _
feeling that v/c have lost something, goes on to say: "I think
-; discover on analysis that it is the explanation which wo
5 hrvo lost". It is certain that a mathematical theory is

explanation in the sense of a reduction to familiar expe-
, nor does it provide an explanation of the type that phi-
f affords. That is to say, the purpose of physical theory

to give us tho real foundation of the laws, but a logica l
Won. For th^ISTaro mental constructs, and it mat bo
i mind that mathematical physics is dialectics. Nevertheless,
'- that a physical theory my be called an explanation m

aonsu of the torn.

Ou'cst-ce done qu'cxpliquer? C est tout uniquonent
ive rontrcr un fait dano une forme. Le fait est expliquo

-406 ;

lorsqu'il apparalt identiqUe a l'un des phenomenes qu'engen-
dre un de cos sorites indefinis que nous abpelons theorlo ou
form, (22)

Physical theor? provides an explanation of reality
.in the sense of making it deducible and thus rational. It is an
explanation in the line of formal causality, even though it is
not a question of the proper ontological formal cause that is found

in nature. It is a mere substitute formal causality and never

mors than provisional. Nevertheless by means of it mathematical
physics truly achieves the aim of subalternation,'

And it must be pointed out that the emancipation
from physical, models of which we have been speaking dees not in
any sense disolve the intimate union between mathematics and phy-
sics that subalternation implies » To the question: what is; the
theory of Maxwell, Plertz is supposed to have replied: "The theory
of Maxwell is Maxwell's system of equations „" (23) And Poinca-
re writes: "Une loi pour nous .. en un mot, e'est une equation
differentielle " (24) There is obviously a sense in which the-
se expressions are correct. And yet it would be false to suppose
that Maxwell's Theory or any other theory in physics consisted
merely of mathematical equations and nothing more. In so far as
science remains materially physical, . there must be a link binding
these mathematical equations with physical reality - - even when
the bridge constituted by a. physical model has been removed,, This
link is provided by what is known as a text or a dictionary,, This
text reveals the physical significance of the mathematical^ equat-
ions and shows how these equations are to be used in order for
that significance to be maintained „ For example, the formula v
s= { gt 2 + v t, has no physical significance unless it be .accom-
panied by a dictionary which explains that it is the formula for
falling bodies and that the symbols s, g.„ t, v refer to distance,
gravitational attraction, time and original velocity, or, to be
more accurate, to sets of concrete measuring operations whose re-
sultant measure-numbers re-present the properties of distance, at-
traction, etc... To say that this is the equation for falling ho-
lies means that the numbers obtained by the concrete processes
of measurement determined by the text satisfy the equation.when
they are substituted into it. The text determines not only: the
nature of the raeasurementsdnvolved, but also the precise connect-
ions between the various symbols used .in the equation. If for e-
mniple the tine and the distance must be obtained by simultaneous
measurements, this must be specified by the text. It is clear that
in the dictionary we shall find the way in which the multiplicity
<* individual measures are reduced to the fundamental measures
and how particular measures, such as that of temperature, for

W

-407...

ffiwn^, become absorbed by the' theory and lose themselves, so
to speak, in combinations of the basic measures' of length time
oud Basso

_ It is easy to lose sight of the importance of the
dictionary m physical theory „ And yet its function is essential,
for :U maintains the intimate union between the mathematics' and
the physics. It is precisely by means' of the text that the mathe-
matical physicist is able to keep in mind that 'what to is dealing
with' directly is a physical elements and that the mathematical
element enters into his object only by way of connotation. To quote
Bvidgman once again:

It appears, therefore, that a complete mathematical
formulation requires equations plus text, and the text may
perform a variety of functions The necessity for a text is
almost always overlooked, but I think it must be recognized
to be essential, and a study of what it must contain is as
necessary for an adequate conception of the nature of the ma-
thematical theory as is the study of the equations themselves.
One of the functions of the text, we have seen? is to tell
us how to set up the correspondence between the numbers! given
by the equation and the numbers obtained by manipulations of
the physical system, The text cannot tell us what it is' that
the correspondence is to be set up with without going outside
the system of the mathematical theory and assuming an intui-
tive knowledge of the language of ordinary experience. In clas-
sical mechanics, the geometrical variables in the equations
of motion are the coordinates of massive particules, bu^ un~
less we know intuitively what a massive particle is, we! simply
cannot make connection with equation and theory. Not only is
the theory powerless to describe, .slther in text or equations,
what the elements are to which correspondences are to be made,
but all the more it is powerless to explain why the elements
have' the properties that they do 3 (25)

The, truly great physicist never allows the symbolism
tf mathematics to 'make him lose intimate contact with physical
reality. Of Einstein Langevin could write:

Pour lui jamais le voile du symbole ne masque la ,
realite. Nombreux sont les esprits pour lesquels le signe ca-
che souvent la chose signified; Einstein so meut a 1 aise dans
le monde des symboles, mais jamais oerao-oi-ne lui dissimulent
1'aspect physique des choses c (26)

But this union between mathematical construction

--40 8.-

OT 1 physical reality must be correctly understood. In the simple
sxamplc cited above of the formula for falling bodies there is
lone to one correspondence between the mathematical symbols and
operationally defined physical properties „ Must we expect this
some correspondence to be found in all mathematical formulations
an d throughout the whole of physical theory? Such an expectation
,vouM misconstrue the proper function of mathematics" in. physics
and would impose sterilizing restrictions upon the theoretical
power of mathematical construction,, (27)

There is no reason why each symbol in the mathema-
tical equations, nor even each steiD in the structure of mathema-
tical theory, should have a definite counterpart in the physical
system. Nor is it necessary that all the operations performed by
the mathematician in his interpretation of nature should have a
physical meaning, or that all of the quantities manipulated be
accessible by experience. It is true that all physico-mathemati-
cal theories must originate in measure-numbers produced by physi-
cal processes and must ultimately terminate in formulas which ha-
ve direct physical relevance and which correspond to concrete mea-
sure-numbers , But between this, point of departure and this termi-
nus the theoretical physicist is free to create any auxiliary ma-
theraatical quantities wnich will help him to carry forward his
task, even those whose realization ■'hi vn-bxee would involve a con-
tradiction. Nor is there any contradiction in maintaining that
fictitious entities can make a positive contribution to the expla-
nation of reality. It is well known how the" fictitious constructs
of the Theory of Relativity both provided an explanation for phe-
nomena previously .inexplicable , such as the anomaly of Mercury,
and led to the new discovery of the deviation of luminous rays
in the neighborhood of the sun. And if pure logical entities am
fictitious constructs can be efficaciously used to^solve practi-
cal problems, as in -the rather well-know case of Stemmetzjs
use of the mathematical surd, V^T , to solve the Problem' of
getting electrical locomotives over the Continental -Divide they
can a fortiori serve as efficacious explanatory devices to solve
theoretical problems o (28)

Modern physi'cs has exercised wide freedom in this
regard. It has felt free to push the theoretical power and the
creativity of mathematics to the limit, .Prided only *^J» ^ ■
end there result formulae that oan be given a physi cal ^™S*_
V'eyl has claimed for physics the right to make us e °? ^ery P°? }
Bible resource no matter how strange the results any appeal. ^
In this connection Eddington writes:

j • • „ „ + f-ir^t called in as a servant,
The pure mathematician, at tirst caxx

-409*

presently likes to assert himself as master; the connexus of
mathematical propositions becomes for hira the main subject,
and he does not ask permission from Nature when he wishes to
vary or generalize the original premises „ Thus he can arrive
at a geometry unhampered by any restriction from actual space
measures J a potential theory unhampered by any questions as
to how gravitational and electrical potentials really behave;
a hydrodynamics of perfect fluids doing things which it would'
he contrary to the nature of any material fluid to do. (30)

We see in this exercise of freedom a confirmation
of the Aristotelian and Thomistio interpretation of mathematical
physics as opposed to that of the aru.ri.ant and neo-Pythagoreansp
As Bridgnrm lias remarked, the feeling that all the steps in the
structure of mathematical theory must have their counterpart in
physical reality derives from the pythagorean belief that the ma"
ihsimiical interpretation of nature means a discovery of mathema-
tics in nature^ (31) which is in the last analysis a mathemati-
cal construction. In the doctrine of Aristotle and St. Thomas the
mthematical world is extrinsic to the physical world (in the sense
already explained) ■■■and consequently the use of mathematics in the
study of the physical world is not a discovery; it is an applicat-
ion. As a result the . thaorist in making this extrinsic applicat-
ion is granted all of the freedom that is native to. the world of
mathematics. It took the genius of Einstein to fully realize that
geometrical conceptions, raus t be manipulated with the utmost free-
dom in order to provide an explanation of physical phenomena,,

The following lines of Cassirer are relevant here:

For it is precisely the complex mathematical concepts,
such as possess no possibility of direct sensuous realisation
that are continually used in the construction of mechanics
and in physics. Conceptions, which are completely alien- to
intuition in their origin and logical properties, and. trans-
cend it in principle., lead to fr uitful appl ications wxthnn
intuition. This relation finds its most pregnant expression
in the analysis of the infinite, yet is not limited to the
latter, (32)

This brinps us to the mooted question of the geome-
trical structure of, "real" space. It is a question that has-been
.soldered obscure by the ambiguity of the terms employed. As a mat-
ter of fact, the wokL "real" can have more than one meaning. For
the physicist; if ha so desiren, is entitled to °™*^J^
*» "^al" when the geometry to which it °?^P°^ P™^ the
Gveateat theoretical power in explaining (m the sense determined

-410--

above) the concrete measure-numbers derived by actual experimen-
tation with the physical world, and has the greatest success in
synthesizing in an exact, simple, coherent and complete fashion
all of the experimental laws. The only meaning that reality can
have for the mathematical physicist is the numbers that are the
results of concrete measuring processes, That is why the geometry
■that best "explains" those results is for him 'the geometry of
reality o , . .

When the question is understood in this sense, it
is clear that no particular type of space, and no particular sys-
tem of geometry is priviledged. Any geometry which at a given sta-
ge in the development of physics provides the greatest explanato-
ry power for all of the discoveries that have been made up to that
point may bo considered to be the geometry of real space. And just
as soon as any other system of geometry provides greater explana-
tory power or is better able to meet the problems arising from
newly discovered phenomena, it i.iust supplant its predecessor and
become the geometry of 'real' space. In this sense it is perfectly
legitimate to say that "real" space is spherical or elliptical,
or that the geometry of nature is not Euclidian but Reimanian.

But for the philosopher, the geometry of "real" spa-
ce is not merely the geometry which best "saves" a collection of
measure-numbers. It is the geometry which is realized (though not
of course iiyfche abstract state, proper to the mathematical world;
in the quantity of the objective world condition. There is a vast
difference between this objective realization and an explanatory
saving of a collection measure-numbers, and that is something that
more than one modern scientist and philosopher of science have
overlooked. Theoretical continuity between a geometrical system
and a collection of measure-numbers dees not constitute on expe-
rimental proof of the objective character of that system. In tact,
it seems necessary to iUsist that as long as it remains true to
its proper method mathematical physics can ne ^ er ? r °Y^ ™ r jT"
prove tnat the absolute world condition is either Euclid ^n or
non-Euclidian, Nor does the theoretical continuity oust planed
prove as some contemporary scientists have claimed, that the dis
tinction between geometry and physics has been wiped <^» £ ^
a ray that the fomer must be considered an experimental science,

Tf the ability of a mathematical system to provide
-erical valued v^iefcoiiide with those derived from physical
measurement were a sufficient proof of th o °^°g ictitious
of the space proper to that sy stem, * ^ ^ '^^ meaaure nurabe rs
constructs which could be put into cent ^y ^ scientists
would have to have objective existence. Some mouern

•411 •*

seem 'to have recognized this fact and have consequently felt the
necessity of attempting to establish the possibility of some con-
nection between non-Euclidian space and sensory perception. The
results of these attempts have only served to show their utter
futility o (33) Sir James Jeans, for example, while admitting
the obvious difficulty encountered in trying to imagine "spheri-
cal space" believes that this difficulty derives merely from its
unf ami liar ity. He holds that' our intuitive belief that space is
Euclidian is similar to the "common sense" belief that the earth
is flat, and compares the difficulty of imagining non-Euclidian
space with the difficulty that a child has in imagining people
existing on the other side of the earth without falling off. (34)
A mopent's reflection will show that there is no parity between
these two cases „ It is -possible for the imagination to cope with
the sphericity of the earth, but it is utterly impossible for it
to cope with the concepts of non-Euclidian space .

Consequently, when "real space" is understood in
the philosophical sense of the term it becomes necessary to say
that' the geometry proper to it can be nothing but Euclidian. The
modern non-Euclidian geometries are purely dialectical structures
and they cannot be applied to real quantity without a contradict-
ioijfinly the entities of Euclidian geometry are capable of construct-
ion in the imaginative intuition, and this capability is necessa-
ry for realization in the objective world, since this realization
raeans to exist with sensible existence. The entities of non-Eu-
clidian geometry require Euclidian geometry as a foundation of
their conceptual existence; consequently, their objective existence
would involve a contradiction since it would deprive them of chis
foundation, For this reason the non-Euclidian constructions; which
have proved so fruitful in modern physical theory cannot be con-
sidered to have actual physical counterparts in nature; they must
be looked upon not as something which directly reveal the quanti-
tative nature of the objective world., but as pure geometrical svm-
bols of this objective world, And the same must be said, mutatis
mtandis, of the mathematical constructions of Quantum physics:
thSTari pU ro mathematical symbols, which, without possessing any
direct physical counterparts in the objective world, provide the
best theoretical schema to explain and synthesize the results of
our measuring processes «

Tt imi-t be pointed out however that there is a sen-
se in which- the'^ttotical "constructions which institute modern
Physical theory have objective significance, though withoub direct
Physical counterpart, they do, nevertheless, ^j*™^^
fashion in seizing upon the Btruotuw of th^objoota^ ^ *
providing an intelligible scheme of relations nip vniou

•dis-

continuous connection between the members of the manifold which
constitutes nature, they succeed in reflecting the interrelated-^
neas of cosmic reality and the harmonious order that prevails in
it„ Were this not so, the value of modern science would be extre-
mely dubious. For it would have gained very little for having con-
demned decadent Scholasticism for transforming facts into mere
names, if in the end it resulted in nothing more 'than the trans-
formation of facts into symbols,, As a matter of fact, however,
the sacrifice which raathematization has imposed upon it of renounc-
ing the inner natures of things is repaid by the reflection of
the all-inclusive structure of nature.

All that the 'thing' of the popular view of the world
loses in properties it gains in relations; for it no longer
remains isolated and dependent on itself alone, but is connect-
ed inseparably by logical threads with the totality of expe-
rience. Each particular concept is, as it were, one of these
threads, on which we string -real experience and connect : them
with future possible experiences, (35)

By reducing nature's manifold to a rational unity
through relatedness in a number system P mathematical physics pro-
vides a quasi solution for the problem of the one and the many,,
By creating an order of pure homogeneous relatedness it affords
a quasi sapiential view of the universe which enables the mind
ttf derive the manifold from the one,, even though the one be a pu-
re substitute and the manifold be reached only in its purely ma-
terial and nuinericajfriversity and not in its proper specific na-
ture ? This explains why it is so easy for the mind to mistake ma-
thematical physics for true wisdom,,

All this is a great achievement, But it is paid for
with- a great price. Perhaps nowhere does the adage Traddutore - -
traditore obtain with greater force than in the mathematical "trans-
latton r= o'f physical science. And that is .whr *. we must now try to
see by considering the transformation that • • this translation
produces in the reflection of nature that is found m physical
science.

■-41 3~

2, The Transformation of Nature.

^ It would be virtually an endless task to attempt
to bring out in full detail the profound metamorphosis that the
natheraatization of physics produces in the scientific view of na-
ture, and we must limit ourselves to touching briefly upon a few
of the most characteristic and significant points. And a moment's
reflection will suggest that the pivotal point of this whole trans-
formation is found in the concept of motion, which, as St, Thomas
says, (36) is, so to speak, the very "life" of the world. In Chap-
ter II we went to considerable lengths to _ show that physical scien-
ce is essentially a study of mobility,, For nature is necessarily
defined in terms of motion (37) and that is why 'Aristotle could
say that he who is ignorant of motion is ignorant of nature, (38)
On the other hand, we explained in Chapter VI that mathematics
essentially excludes motion. The mathematical world is a World
of immobility. It is true that mathematicians speak of a kind of
notion, but as we pointed out, this motion is only a dialectical,
imaginary, intraraental thing, which does not involve true becoming.
As we mentioned in Chapter I, this opposition between the mobili-
ty of nature and the immobility of mathematics was traditionally
one of the stumbling blocks for those who T/.ished to raathematicize
the cosmos, and provided one of the bases for Aristotle's criti-
cisms of the Platonistsand the Pythagoreans. We must now try to
see in what sense mathematics may be applied to motion and what
is the effect of this apx)lication

Aristotle and St. Thomas explain that motion may
be considered under two different aspects. In the first place,
it may be considered in its proper and specific essence, and in
this ; sense it signifies a coming into being. Considered in this
way, it is something profoundly obscure, since it lacks the deter-
mination and actuality of being. Consequently it can be correctly
defined only in a way which will bring out this profound obscurity,
Aristotle has given us the essential definition of motion in the
third book of the Physics: the act of a thing in potency in so
far 4s it is in potency. This coming into being is realized both
in the substantial and in the accidental order, and in ^he jitter
case (which is the strictest meaning of the Thomistic term 'motus )
it is found in the three categories of quantity (growth in living
beings), quality and place (local motion). All substantial change
involves accidental changes, and motion in the predicaments of
V'antity and quality always involve local motion of some «£*.
(39) This, in a sense all motion nay be reduced to local motion.

-414-

■phis kind of motion is the most superficial and the one which re-
alizes the least the concept of becoming. It involves essential-
ly an extrinsic denomination. To say, therefore, that all the mo-
tion in the universe may be reduced to local motion, is to say
that it may be reduced to a system of extrinsic relations.

The second aspect under which motion may be consi-
dered is brought out by St. Thomas in' his Co mmentary on the Fifth
Boo k of the Metaphysic s. (40) In analyzing the notion of quan-
tity, he tells us that there are various kinds of quantitative
modes. Some things are quantitative per se , such as "line", others
are quantitative per acc idens . Among those which are quantitative
■per accidens some arc such by the fact that they are accidents
inhering in a quantified subject; others however are quantitative
by the fact that they are divisible according to quantity. In this
category are found motion and time. St, Thomas writes:

Alio modo dicuntur aliqua quanta per accidens non
ratione subjecti, in quo sunt, sed eo quod dividuntur secun-
dum quantitatem ad divisionem alicuius quantitatis; sicut mo-
tus et tempus, quae dicuntur quaedam quanta et continua, prop-
terea quod ea, quorum, sunt, sunt divisibilia et ipsa dividun-
tur ad divisionem eorum, Tempus eniih est divlsibile et conti-
nuum propter motum; motus autera propter magnitudinem; non qui-
dem propter magnitudinem eius quod movetur, sed propter raagni-
tudinem eius in quo aliquid movetur Et eo enim quod ilia ma-
gnitude-' est quanta, et motus est quantus. Et propter hoc quod
motus est quantus, sequitur tempus esse quantum. Unde haec
lion solum per accidens quantitates dici possunt , sed magis
per posterius, inquantum quantitatis divisionem al> aliquo pri-
ori sortiuntur. (4l)

In his C ommentary on jbhe De _ Tririitate he shows how
this quantitative asp'iot makes "it possible for mathematics to en-
ter into the study of motion;

Ad quintum dicendum, quod motus secundum mturam _
suomnon pertinet ad genus quantitatis, sed particxpat alxquid
de natura quantitatis aliunde, secundum quod divisio ™txis
sumitur ex divisione spatii vel ex divisione mobxlxs: et ideo
considerare motum pertinet non ad mathematxeum, sed tamen prm-
cipia mathematica ad motum applicari possunt: et xdeo secun-
dum hoc quod principia quantitatis ad motum applxcantur, na
turalis consiclerare debet de divisibne + et oontinui, et mo^us,
ut patet in VI Physicorum. E^iLBgienfaxgj ^^ inter mathe
maticam et naturalem trj ^^nr^J^S^i^S^^ S1CUt in
. icientiis de sphoara raota, ct in astrologxa. \M)

-415-

These distinctions mate it clear how it becomes pos-
sible for mathematics to be applied to the motion in the univer-
se. By reducing all motion to local motion or movement in space
by considering this local motion not as a coming into being : but'
as a pure extrinsic, relation, and by considering this extrinsic
relation purely in terms of its quantitative aspect, nathema'tics
is able in some way to seise upon motion. But in doing so it trans-
forms it into the' only sense' in which it can have meaning for a

mathematician the simple variation of the relations of a point-

with coordinated axes. And thus in mathematical physics mevement
becomes nothing, more than a variation of spatial relationship bet-
ween two or more bodies which remain intrinsically unchanged,
Lenzen, for example defines it as a "change of position in space
with time", (43) A continual series of spatial points areiuni-
ted with a continuous series of temporal points and the four di-
mensional curve which results becomes the model of motion'*

It should be immediately evident that such a notion
of notion empties it of its proper physical essence. It is ho lon-
ger a true change,, but a mere displacement of a point, no longer
a process but a relation, no longer a becoming, but a state which
has a certain determined value that can be measured, (44)

Things do not c ome into exist ence at a certain pla-
ce and at a certain instant of time - - they simply exist at a
certain point in a continuum.

That physico-mathematical motion is emptied of all
becoming is clearly brought out by Sir Arthur Eddington:

Events do not happen, they are just there, and we
come across then, "The formality of taking place" is merely
the indication that the observer has on his voyage of explo-
ration passed into the absolute future of the event in quest-
ion; and it has no important significance. (45)

It is clear, then, that there is no true becoming
in the .physico-mathematical world, and consequently no rest, (46)
A. good analogy of the difference between the physico-mathematical
world and the real world may be found in the difference between
a piece of 'music played by a symphony orchestra and a record made
of the piece. There is something on the static record to corres-
pond to all the movements and nuances of the piece, but the move-
ments and nuances themselves have been lost. They have all been
spatialized.

Because mathemataticized motion is not a coming into

-416-

being but a pure relation,- it is perfectly reciprocal. That is
why Descartes who had identified real notion with raathematicized
notion could say that it is perfectly indifferent whether we say
that we are moving towards a goal or that the goal is moving to-
wards us, since in both, cases the variaiion of the relations of
distance remains exactly the same. (47)

It is easy to see what has happened in this mathe-
natization of motion,, Nothing is so irrational, so refractory to
the intellect as potentiality. That is why the mind in its .attempt
to rationalize the universe as completely as jiossibleis inevita-
bly led to the', attempt to wipe out potentiality and to reduce e- ■
verything to the plane of actuality., From this point of view Berg-
son is correct in maintaining that experimental science deals on-
ly with the "tout fait", (48) But in wiping out potentiality
it destroys all true mobility, It thus succeeds in explaining na-
ture only at the expense of destroying it. It reduces . motion to
something that is perfectly clear and intelligible, but in ;so doing
it sacrifices its very essence, for motion, as we said, is some-
thing essentially- obscure That is why mechanism taken as a phi"
losophy of nature involves an intrinsic contradiction. For in at-
tempting to give an adequate account of, reality by means of motion
and extension it empties motion itself of its reality. It was be-
cause Descartes failed to realize that his mathematicized motion
was not true motion that he heaped such supercilious scorn upon
Aristotle's definition:

'■At vero nonne videntur illi verba magica proferre,
quae vim habeant occultam supra captum humani ingenii, -qui
dicunt mo turn ., rem unicuique notissiman, esse actum entis in
potentia, prout e st in potentia? quis enim intelligit haec
'verba? quis ignorat quid sit motus? et quis non fateatur il-
los nodum in scirpo quaesivisse? Dicendum est igitur, nullis
unquam def initionibus eiusmodi res esse explicandas, ne loco
simplicium cornpositas apprehendamus ; sed illas tantum, ab a-
liis omnibus secretas, attente ab unoquoque et pro lurame in-
genii sui esse intuendas. (49)

It is to be noted that Descartes did not say: "Quis
ignorat quod sit motus," but "quid sit motus?" For Aristotle xhe
existence^ motion was perfectly clear ; it had all the clarity _
of a direct intuition. Descartes thought that this perfect clari-
ty of direct intuition could be extended to the very essence of
motion. That is why to his question: "quis ignorat quid sit motus?
one is justified in answering: "Descartes". Pasteur s diotum:
"Je plains les gens qui n'ont que des idees claires" , ^especial-
ly applicable to the realm of Mature where things are essentially

-U.7-.

obscure,! (50)

All this helps us to understand the solution to the
antinomy mentioned m Chapter I between the ancient and modem
oonoepteof notion. For Aristotle, as we saw, it was evident that
the continuance of a body in motion demanded a cause and without
this cause the body would come to rest. For Descartes, on the o-
ther hand, the principle of inertia was perfectly evident, (51)
and according , to this principle the cessation of the motion of
a body demands a. cause, and without this cause the motion will
continue ad in finitum,, The enigma of this striking paradox imme-
diately vanishes when we call to mind that Aristotle and Descar-
tes are talking about two different things, For Aristotle motion
means a coming into being, and since nothing can bring itself in-
to being, there must be a cause to explain the process of becom-
ing: quidquid movetur ab alio movetur, For Descartes motion! is
a state, that is to say a kind of entity which will retain its
existence until robbed of.it by some cause. The principle of iner-
tia has to do with matheraaticized motion, that is to say with a
motion which is infinitely uniform and rectilinear. This princi-
ple does not in any way involve the falsity of Aristotle's notion
of motion. They belong to two .different orders,, Aristotle made
no attempt to treat the mathematical aspect of local motion. It
is extremely important to keep in mind that this matheivcitizdtion
is not a substitute for Aristoile's definition; it is a passing
to an entirely different order 5 All too many historians make the
mistake of treating Aristotle's Ph ysics as though he was attempt-
ing to write a treatise on mathematical physics »

This question has an important corollary • in. the pro-
blem of prima via in St Thomas' demonstration of the existence
of Gqd In fhis demonstration motion is considered as a becoming,
end not as a state. And that is why it makes no senae to say that
the argument is disproved by the principle of inertia. Obviously,
if motion is conceived as a state there is no need to have recour-
se to an actio to explain it. This shows that the matheraatiz'ation
of the cosmos has a profound effect upon the problem of causality
in the universoi But before turning to this question we must con-
sider in a summary way how this matheraatization affects a notion
that is intimately connected with that of motion, namely tirne:
"tempus habet fundaraentura in notu u " (52)

Contemporary Scholastics have insisted upon the dif-
ference between Aristotelian time and Einsteinian time to the ex-
tent of denying that they have anything more in common than the
name, (53) They have furthermore claimed that, what Einstein has
to say about the impossibility of simultaneity at a distance has

-418 »

nothing to do with the time of which Ariqtotle speaks. We: feel
that this is extremely ambiguous', For the term "time" does npt
always have exactly the same meaning in Thomistic terminologjr.
In the first place,, it signifies the duration of mobile beings,
that is to say, the persistence in existence of beings whose exis-
tence is successive. But the, "time" which Aristotle defines jln'
the fourth book of the Physios does not oxactly coincide vri. th this
primary notion; although it is essentially connected with it'. For
by defining time as the measure of motion according to a relation
of; priority and posteriority he makes it clear that he is ..speak-
ing of an extrinsic determination of this duration in relation
to a chosen standard) that is to say, of a. measurement of this
duration* Consequently, in so far as both Aristotelian time thus,
defined and Einsteinian tiiae have to do with measurement they coin-
cide. And we believe that what Einstein has to say about the. im-
possibility of simultaneity at a distance applies to the time de-
fined by Aristotle in the Physics ,, For we knov/ of no way in which
the measure of motion according to a relation of before and after
can be determined so that distant events can be fixed as simulta-
neous o Of course, in so far ds time is successive duration there
is such a thing objectively as distant simultaneity even though
that simultaneity cannot be determined by us

But it would be illegitimate to conclude from this
that the time defined by -Aristotle in the Physi cs is same as the
time of which Einstein speaks. For in the time of the Physics the
notion of true physical motion is involved. Consequently, this
time can truly be said to "flow" from past to future. In Relati-
vity physics, on the contrary, the notion of motion has been empt-
ied of its proper physical meaning, There is no true process, no
becoming. Consequently, Einsteinian time does not really flow;
it la a mere dimension. It 'is studied in terms of geometry, (54)

But even before the advent of the Theory of Relati-
vity the notion of time had already undergone a profound transfor-
mation by the mathematization of nature, (55) We have already •
spoken of the symmetry of mathematical equations. The processes
of classical dynamics are reversible, that is to say, if the ve-
locities of the particles of a system should at any given moment
be reversed the motion would proceed in accordance with the same
equation in the reverse direction. In so far as the notion ot ti-
me is concerned, this means that the equation of classical dyna-
mics moke no difference between the positive and negative direc-
tions along the time axes. Professor Cunningham does not hesitate
to say that in so far as time is determined mechanically, past
and futurt \re interchangeable. (56) And Lindsay and Morgenau
write: "If equations predict future events they predict past ones

■•419!

as well. (57) Of course the physicist in order to discover "ti-
neOg" arrow my have recourse to entropy-gradient, but ever then
the irreversibility is only highly improbable and never absolute-
ly impossible. (58)

In Relativity physics the mathematical transf ormat-
ion of the notion of time becorres complete,' It is assimilated to

the notion of space united with space as a dimension in the

four dimensional continuum called space-tim which, as we saw in
the Inst Chapter, may be cut up in different ways according to
the position and velocity of the individual observer. Time then
becomes "the totality of possibilities of relative temporal posit-
ion of events „" (59) To quote Eddington once again:

In the four-dimensional world ,„ the events past and
future lie spread out before us as in a map. The events are
there in their proper spatial and temporal relation; but the-
re is no indication that they undergo what has been described
as 'the formality of taking place', and the question of their
doing or undoing does not arise We see in the map the path
from past to future or from future to past; but there is no
sign-board to indicate that it is a one way street,, Something
must be added to the geometrical conceptions comprised in Min-
kowski's world before it becomes a complete picture of the
world as we know it„ Wc may appeal to concicusness to suffuse
the whole - - to turn existence into happjsning.o heing into
be coming o But first le'iTus 'note that 'the picture as it stands
is entirely adequate to represent those primary laws of 'Katu-
re which, as we have. seen,, are indifferent to a direction of
time,. (60) ■

In this spatialiaation of time, mathematical phy-
sics has achieved the goal at which it has aimed from the begin-
ning - - the transformation of all sensuous and intuitive hetero-
geneity into pure homogeneity. The first step in this transformat-
ion was the homogenization which gradually einptied external expe-
rience of its proper and specific 'content, But even when this had
been accomplished there still remained untouched the ''form of the
inner sense" - - the process of duration which is so intimately
connected with internal experience. Through the specialization
of time this last barrier of specific experiential content was
broken down, Speaking of this transformation Oassirer writes:

This transformation of the time-value into an ima-
ginary numerical value seems to annihilate all the 'reality
and qualitative deterainateness, which time possesses as the
•fom of the inner sense', as the form of immediate experience,

-420

The stream of process, which, psychologically, cons:Situtes
consciousness and distinguishes it as such, stands still; it
has passed into the absolute rigidity of a' mathematical cos-
mic formula,, There remains in this formula nothing of that
form of time, which belongs to all our experience as such and
enters as an inseparable and necessary factor into all its
content. But, paradoxical as this result seems from the stand-
point of this experience, it expresses only the course of ma-
thematical and physical ob jecti'fication, for, to estimate it
correctly from the epis temological standpoint, vre must under-
stand it not in its mere results, but as a process, a method,
In the resolution o^ubjectivoly experienced qualities into
pure objective numerical determinations, mathematical physics
is bound to no fixed limits „ It roust go its way to the end;
it can stop before no form of consciousness no matter how
original and fundamental; for it is precisely its specific
cognitive task to translate everything enumerable into pu±e
number, all quality into quantity, all particular forms; into
a universal order and it only 'conceives' them scientifical-
ly fay vixtue of this transformation. Philosophy would seek
in vain to bid this tendency bait at any point and to decla-
re ne pl us ultra » The task of philosophy must rather be limit-
ed to recognizing fully the logical meaning- ot the mathemati-
cal and physical concept of objectivity and thereby conceiv-
ing this meaning in its logical limitedness. (61)

Once again it is important to recognize this spa-
tialization of time as an attempt of the mind to triumph over its
greatest enemy: potentiality. Designated points in space are ail
actual, and when time is homogenized with space, tl, t2, t3 : , etc.
become but a series of actual "nows". Perhaps it is legitimate to
see in this spatialization of time a striving of the human intel-
lect towards the duration of perfect actuality that is proper to
pure Intellect,, (62) But this attempt only results in the destruct
tlon of time:

„ si le devenir doit so transformer en etre (se-
lon M, Einstein), au point de vue que l'acte de se reprodui-
re,pour un evenement, devient uno simple formalxte denuee d« im-
portance (selon M. Eddington)., si la succession n est qu une
illusion(selon M. II. Morals) et si tout systemc physique cons-
titute une entite privee de changement (selon M. &annmgham) ,
cela no peut signifior qu'une chose: 1' abolition et la dispa-
rition du temps. Aussi II, Cunningham n'hesito-t-il pas a par-
ler de 1'univers non--temporel de Minkowski, [bo)

The destruction of mobility in the universe has many

•421'"

far-reaching consequences, but perhaps the most significant from
the point of view of science, which is a knowledge of things in
their causes, is its effect '_ upon causality. In the second book
of the Physics Aristotle and St. Thomas place considerable empha-
sis upon the fact that the science of nature roast study its, ob-
ject from the point of view of all of the four fundamental types
of causality: efficient, final, formal and material,

Dicit ergo primo quod cut; quatuor sint causae, 'si-
cut supra dictum est, ad naturalem pertinet et oranes cognos-
cere et "per omnes natural iter demons trare, yeducendo questio-
nem pro"pter quid in quanlibet dictarum quatuor causarum, sci-
licet formara, moventum, finem et materiam, (64)

The reason for this is fairly obvious: there is an
analytical connection between mobility, the formal object o£ the
science of nature, and quadruple causality,

Necesse est autem quatuor esse causas, Quia cum cau-
sa sit ad quam sequitur esse alterius, esse eius quod habet
causam, .potest considorari dupliciter: uno mpdo absolute, et
sic causa essendi est forma per' quam aliquid est in actu: a~
lio modo secundum quod do potentia ente fit actu' ens. Ejt quia
bmne quod 1 est in potentia, reduoitur ad actum per id quod est
actu esn; ex hoc necesse est .esse duas alias causas, scilicet
materiam, et agentem qui reducit materiam de potentia in ac-
tum. Actio autem agentisad aliquid determinatum tendit, si-
cut ab aliquo determinato principio procedit: nam omne agens
agit quod est sibi conveniens: id autem ad quod tendit actio
agentis, dicitur causa finalis. Sic igitur necesse est psse
causas quatuor „ Sed quia forma est causa essendi absolute,
aliae vero tres sunt causae essendi secundum quod aliquid ac-
pipit; inde est quod in immob'ilibus non considerantur aliae
tres causae, sed solum causa formalism (65)

The last lines of this passage throw great light
upon the effect that the mathematical' transformation of phypics
has upon causality. The student of na.taire as long as he s.ays with-
in his own field is. 'bound to reduce natural phenomena to all of
their four, causes: "In naturalibus redendum e st proper quid po-
nitus", (66) But unable to discover any universal and necessa-.
i-y propter quid for experimental propositions he is forced to ha-
ve recuse to-mthemxtica. Since mathematics, however, is a world
of imobility, the only type of prpj?ter_auid he can borrow from
it is the unique type that is proper to it: £ ro E ter_quid m the
line of formal causality.

<•&«>, .

In a passage immediately preceding the one just quo-
tccl St„ Thomas gives an example of what he raeana by formal cau<-
talitys

quandoque enim propter quid reduoitur ultimo in quod
quid est, idest in de:.;i:vi-:-.;i.onem, ut patet in omnibus imraobi-
libus, ; sior.t aunt mithiyiat^a; in quibus propter quid reduoi-
tur ai; definibionen v^etr. .ol commensurati vol alicuius alte=
rius quod demons tratur in i/arhematicis,. Cum enim def initio
recti anguli sit, ,v:od cons uituatur ex linea super aliara ca->
dento P .quae ex ufaaqua p.-cte faciat.dr.03 angulos aequaies;
si quacratur proper quid iste angulus s: ; .t rectus, re^ponde--
tur quia constituifcur ^x J-i.iea faciente dv:os angulos eaquales
e:r. utraque parte j u+, ita est in aliis, (6V) :

It is clear that the only type of causality that
can be found in mathematical physics is a Kind of formal causali-
ty consisting in an exp:.'-ossioz.>. of the metric coherence of pheno-
mena,. This metric col.ey'enee constitutes "what is known as the cau-
so.l structure- of world occurences. It ±s true. that physicists rnav
apeak in teims which may seem to indicate other ty;x:s of causali-
ty, Th:iy may for example,, u>~ the expression "officii ant causali-
ty" , but in so doing -feey merely ref or to a relation between the
states of physical sys 'con's hi different pointy of time, which are
connected in such a way t- v.'.,.. given the dete:ranination of the sta-
te of the system at any oije poim\i of . tiae ; its state at any desi-
gnated future point of time can be logically deduced, (68)
St, Thomas brings out the incompetence of mathematics in the field
of efficient causality:

Mathematica accipiuntur ut abstracta secundum ratio-
nem, cum tamen non sint abstracta secundum esse„ Unicuique
autan oonpetit habere oausam agent™, secundum quod habet es-
se. Licet igitur ea, quae sunt mathumatiea, hubeant ^ causam
agentwnij non tamei^. secu: i,fcm hatoit-juj.^ni, quara habent ad cau-
sam ageo-.em, cadunt sub consideration mathematici, Et ideo
in soientiis mathematiciia non demons teatur aliquxd per cau-
sam agentem, (69)

In pre-Relativity physics the mathematieation of
the cosmos he* already resulted in the disappearance of xrue ef-
ficiency from the concorrc of efficient causality, but in Relati-
vity physics this effacsmeuo is made even more complete, j. or now.
the concept of force, for example, is completely absorbed into
a system of determinations bound together by mathematical relat-
ions implemented by fi.o differential and . tonsonal oa-culus, -u.

-483-

In somewhat the game way, physicists often speak
of rnatcer, buo their matter is far from being the material cause
of which Aristotle, speaks, It is something that is completely ac-
tual and not a potential principle of becoming. In fact, in Rela-
tivity physics matter becomes so formalized that' i-'< is absorbed
into isotropic space „ On the other hand it must be noted that if
matter is formalized,' it is also true to say that the formal catus
is l^teriiuiaod, ^ That is to say, the formal cause that is treat-
ed of in mathematical physics is not the proper specific formal
cause which repeals, the nature of things in their heterogeneous
interiority, but a homogeneized formal cause of spatial relations
(70)

In insisting upon the necessity of studying nature
in terms of all four causes, Aristotle and St. Thomas place spe-
cial emphasis upon the importance of final cause,, "Et heac species
causae potissiraa est inter aliab cauaasj est enim causa finalis
aliarum oai).sarum causa," (71) In fact, after explaining in a
general way haw nature involves all four types of causality, they
single out only final causality for particular attention. The who-
le last part of the second book of the physics is devoted to a
study of.it, and to an insistence of it's"pHirie importance in the
study of nature. Yet of all the causes that disappear in the mathe-
matization of the cosmos, this is perhaps the type that is -lost
efficaciously and most completely effaced.,-. One looks in vain for
anything that even remotely corresponds to finality in mathematical
physics, And the fundamental reason for this has already been point-
ed out in Chapter Vis since there is no good in mathematics, there
can be no final causality} ;

Ex hoc enim quod finis non potest esse in rebus immobilibus,
videtur procedere quod in scientiis mabhematicis quae abstra-
hunt a materia et motu, nihil probatur per hanc causam, sicut
p'robatur in scientia natural!, quae est de rebus mobilibusj
allquid per rationem boni'u Siout cua assignamus causam quare
homo habet manusj, quia per oas melius potest exequi conceptio-
ns s rationis,, In mathematicis ant em nulla demonstratio fit hcc
modo, quod hoc modo sit quia melius est sic esse, aut deterius
si ita non essot„ Futa si deceretur quod angulus. in semicir-
Gulo est rectus, quia melius est quod sic sit quam quod sit
acutus vel obtusus. Et quia posset forte aliquia esse alius ^
modus demonstrandi psr causam finalem, put a si diceretur, si
finis erit, necesse est id quod est ad finem praecedere;- ideo
subjungit,, quod nnllv.s pprAva in mfAem^icxa faait mentiOEom alicuiue

qui fuit de secta Epioursorum, omnino naglexit demonstrations
ouae r-;unt ioer oausaa finales, roputanaeas viles ex hoc quod

-0.-24-"

in MuiDuB Uliheralihus sive mGchnnicis, at in arte -tectoni-
caS idest aedificatona, et 'coriarla', omnium rat iones ass-
gnantur ex hoc quod quod est illiquid melius vel doteriu". In
mathematicis yaro, quae sunt nohilissimae et certissiraae solan-
tiae, nulla fit mentio de "bonis et malis. (72)

From all this it follows that it is entirely' illegi-.
tir.zite for critics to reproach scientists as some modern Scholas-
tics have dona , for failing, to take all types of causality in con-
sideration. The very nature of his science makes it impossible for
the mathematical physicist to consider anything hut formal causa-
lity. And it is important for the scientist to he aware of his own
limitations, so that he will not, for example, confuse his subs-
titute for efficient causality with true efficient causality. There
is particular danger of this happening with this type of causality
since it is the hest known and the most manifest to the mini..

At first sight it might appear that this "banishment
of causality from the. cosmos might make the physico-mathematical
world like Malehranche ' s world, of occasionalism. As a matter of
fact f there is only a surface likeness "between the two'„ In a deeper
sense they are opposed. For in the world of Malehranche it is he-',
cessary to have .constant recourse to God, since every event! is the
occasion of His action. In the physico-mathematical world, on the
other hand, God is completely dispensed with; there is no need to
go to Him at all; nor ia it N even possible to go to Him. Because
of its rationality, its'ever increasing unity and its immut ability?
the physico-mathematical world is more like the Parmenedian' sphere,,

This analysis of the effects of the mathematical
transformation' -of. the cosmos might go on interminably. -Wo might
for example show.. that it destroys not only the "becoming of the u-
niverse, hut in a certain' sense even its "being. For as we saw in
Chapter VI, mathematics prescinds from existence, and the only mea-
ning that "being has in the physico-mathematical world is the occu-
pation of a "place" in a certain order, in a space-time schema.
In this sense, Bergson /is correct in saying that in modern scien-
ces "1' existence concrete des phenomenes de la nature tend a s'e-
vanouir , . en fumee algehrique. " (73) We might also show how
the concept of substance is transformed into the notion of persis-
tant system. (7.4) But we feel, that enough has already heen said
to show that the nature of which the i mathematical physicist speaks
is not the nature that is defined hy Aristotle and St. Thomas in
the second hook of the rhyaics as a principle of motion and rest
and as a "ratio" or *atTo^l"prinoiplo put into thing's which di-
vots them in their striving for ends. (75) The nature of the

•425-

mathematical physicist is, as Eddington has remarked, "only an empt,
shell''. (76 in other words, as we have already remarked^ in or!
der to explain nature the physicist has found it necessary to des-
troy lto

Obeissant aux deus tendances, nous avons, do theorie
en theorie, et^d< identification en identification, ' fait comple-
tement^ disparaitro le raonde reel, Nous avons d'abord explique P
c'est-a-diro nie le changemont, identifiant l'antecedent ot
le consequent , et la marche du mon&e s'est arretee„ II nous
restait un espace rerapli de corps. Nous avons constitue les
corps avec de l'espace, ramene les corps & l'espace f et les
corps se sont Ivanouis a leur toufr„ C'est le vide, 'rien du
tout', comrae dit Maxwell, le neant. Car le temps et l'espace
se sont dissous. le temps, dont le cours n'implique plus de
chamgement, est indiscernable, ineslstant; et i'espace, : vide
de corps, n'etant plus marque par rien, dispararb aussi.
(77)

It need hardly "be pointed out, of course, that the
groat loss resulting from this destruction of nature has rich com-
pensations that are daily becoming more apparent. For even though
in destroying nature we destroy intelligence that Aristotle saw
in it and roh it of its seeking for ends, at the same time we make
nature more intelligible than it is by injecting our own intelli-
gence into it The mathematical representation of nature is an
improvement of it s in the sense in which a. mathematical line is
an improvement of a physical line. We construct a model for nature,
and this construction forces nature to yield up its secrets. (78)

From all that has "been said about the nature of this
rationally constructed physico-mathematical world it is clear why
it should inevitably appear to Sir James Jeans as a world consist-
ing of pure thought, the thought of a mathematical thinker. (79)
But it should also "be clear why it is illegitimate for him to con-
clude that the objective universe, that is to say, the absolute
world condition, is nothing but pure thought and the product of
a pure mathematician acting as a pure mathematician. For even
though a physico-mathematical world may tend towards the absolute
world condition as though towards its asymptote, a pure mathema-
tician acting purely as such, neither would nor could create a
Physical universe. As Bridgman has , remarked, 'What Jeans might
have said is that Man is a mathematician, and reflected thai: it
*a no accident that he forms nature in his own image. (80)

-4-EG~.

CHAFD3R TEH

A SH&DCW V/OBLD OS 1 SYMBOLS

lo nl he Hatui-e of Symbolism..,

Having aeon how the mathematician transforms the
physical universe into a new world of his own makings we must
iij'v try to analyze "briefly the nature of this new world. All tho
best philosophers of science are now unanimous in characterizing
the physico-mathematic-'J. world as a symbolic universe c Sir -Arthur
Eddington s for example., has, as is well-known s repeatedly describ-
ed it as "a shadow vrorld.of symbols". (1) V7e holis/e that if this
phrase he rightly understood, it hring3 out with great accuracy
the -true nature of the universe constructed "by mathematical phy~.
sics. Let us try to determine what precise meaning must he ; given
to it.

In the first place, it is necessary to fix upon the
meaning of the word "symbol" And here we come upon a great lack
of unanimity. All will agree that in its primitive meaning' the
tur:.i "Symbolon" signifies a mark or emblem or index employed to
designate something,, and that consequently every symbol is a sign.
But is every, sign a symbol? Not a few authors seem to. think so.
Thus R,B. Perry writess ."Any daium may he a symbol if it means so-
mething or operates as a sign." {?-) And he goes on to explain
"that such data may include?

conspicuous features of nature, monuments, writ-
ten or spoken words, small images or familiar ohjects easily
duplicated or distrihuted. Any of these is a symDol provided
it directs expectation or interest to something other .mw it-
self. Symbolic is, then, the study of the part played in hu-
man affai-e "by all those signs and symbols, especially their

" a al_ I, , ,. c,-„y-,-," q direct and organise, record and
influence on thought. Symoc.s aj.n,»u o

«4S7'.-

communicate.. For words, arrangements of words, images, gestu-
V8s f wid such representations as drawings or miotic bw&o
wg use the term symbols,

To make the sign and the symbol coterminous in this
way is to roh symbolism of all precise meaning,, (3) And the or-
dinary usage of the term seems to insist upon a precise moaning,,
Clouds are considered to ho signs of rain, and smoke a sign of
fire., hut they are never referred to as symbols. It is necessary,
therefore, to try to press the meaning of the term a hit closer,,

In the first place „ the examples just referred to
make it clear that purely natural signs (i e„ those v/hich have a
natural and real connection with the thing signified;, prior to any
connection estahlished hy the mind) must he excluded from the no-
tion of symbolism. To apply the term "symbol" to a natural sign
is actually a distortion of language,, (4) In other words, sym-
bols are necessarily arhitrary or conventional signs, i.e. signs
in which the connection with the thing signified is net found in
nature as such, hut .cheated hy the rnind„ This does not, howbvor,
exclude the possibility of there heing in nature a foundation for
the connection estahlished hy the mind,,

Having made this . important distinction we are faced
with this prohlem: Are all convenbicnal signs necessarily syribolfc?
Once aga,in, a good many authors seem to think so"- -.at least if'
it be question of the most important type of conventional signs,
namely those which make up language. Miss Stehhing, for example,
tells us that "a word is a special kind of sign called a symbol."
(5) And again she writes; r 'A sign consciously designed to stand
for Something will he called a symbol." This opinion seems to he
shared hy Professor V/hiteheadg "Tho word symbolizes the thing,, Lan-
guage almost exclusively refers to presentational immediacy as
interpreted hy symbolic reference." (6) This tendency to make
all language and even all thought symbolic (7) makes it difficult
to attach any precise and proper meaning to the term.

Since the word is currently employed in such a loo-
se way it is necessary for us to try to fix upon the particular
moaning it is to have for us in this discussion of the symbolism
of science. Its etymology provides us with a helpful suggestion,
Tho Greek words CTU V and SaXXeJV mean "to throw together". : Now,
whatever may have heen the original historical usage of these words
'■'hich gave rise to the term we are analyzing, it is clear that they
suggest a collection of things among which there is no strict na-
'-val unity - - an agregate whose principle of unification is purely

su

"488 —

extrinsic. If we keep this in mind wo ahall be able to seo why
Saint Thomas, m hia Commentary on thejjentnr.cRn ( 8 ) gives this
description of the symbol, ",.. nomen syiriboTTiiHilitudinom et col-
loctionem import at „" It would seem that a symbol must bo defined
as an artificial sign established to signify a determined object
that is one only according to the mind, In order to bring out the
meaning of this definition, it is, necessary to see the difference
between a symbol and a name,, (9)

In his Commentary on the Pe ri hormeneiq s, St„ G?homas
explains the important distinction between the name and the infi-
nite namet

Deinde cum dici.t (Ari3totelos) "non homo voro, uon
est nomen" etc, ,exoludit quaodam a nominis ratione, Et primo,
nomen infinitum; secundo casus nominumj ibi; "Catonis autem
yel Catoni" etc e Dicit ergo primo quod "non home" non est no-
mon„ Omne enim nomen significat aliquam naturam determinatam;
ut "homo"; aut personam determinatam, ut pronomen; aut utrum-
que determinatum, ut Sortes, Sed hoc quod dico "non homo",
neque determinatam naturam neque determinatam personam 'signi-
fioato Imponitur enim a negatione hominis, quae aequaliter di-
citur de "ente" et "non ente". Unde "non homo" potest dici
indifferenter, et de oo quod non e3t in rorum natura; ut si
dicamus, "chinaera est non homo", et de eo quod est in rorum
natura; sicut cun dicitur, "equus est non homo". Si autem
imponeretur a privatione, requiroret, subiectum ad minus, exis-
tensj sed quid impdnitur a negatione s potest dici de ente et
de non ente, ut Boethius at Ammonius dicunt„ Quia tamen si-
gnificat per modum nominis^ quod potest 3Ubiici et. praedica-
ri requiritur ad minus suppositum in apprehensione, Hon autem
erat nomen positum tempore Aristotelis sub quo huiusmodi dic-
tiones concluderentur, Non enim est oratio, quia pars eius non
significat aliquid separata, sicut nee in ,nominibus compositis;
similiter autem non est negatio, id est oratio negativa, quia
huiusmodi oratio superaddit negationem affirmationi, quod non
contingit hie, St ideo novum nomen imponit huiusmodi dictiom,
vocans earn nomen "infinitum" propter indoterminationem sigm-
ficationiSj ut dictum est. (10)

It is clear from this passage that the name must si-
gnify something that is one by nature. Because of its indet emana-
tion the infinite name does not signify something that is o:_„ by
nature. Because it is a pure negation, it does not e.en have the
^termination of privation which must always be m uho , same gunus

-42'9.*-

n g the thing of which it ia the negation,,

. „. >4 . Nevertheless, in spite of the indetormination of
the infinite name* it has a significance; in some way it signi-
fies something that is one „ St, Thomas exolains this in his Dom~
manta ry on the Second Book of the Perihermo neias* '

.:-. nomen infinitum quodam mo do significat unum*
Non enim significat simpliciter unum, sicut nomen finitiim,
quod significat unam formam generis vel speciei aut etipm
individui, aed in quantum significat negationem formae ali-
cuius, in qua negatione multa convoniunt, sicut in quodam uno
pecundum raticnem, "Unum" enim eodem mode dicitur aliquid,
sicut et "ens"; unde sicut ipsum "non ens" dicitur "ens!',
non quidem simpliciter, sed secundum quid, idest secundum ra~
tionemj ut patet in IV M etaphysicae . ita etiam nogatio pst u-
rium secundum quid, scilicet secundum rationem, Introducit au-
tem hoc, ne aliquis dicat quod affirmatio, in qua suhlibitur
nomen infihitum, non significat unum de uno, quasi nomen in-
finitum non significet unum. (11) '

There is, then, a unity in the infinite name ' — a
unity that is founded upon the unity of the thing negated. It is
possible to predicate the infinite name of anything except i;he
thing negated. But it is important to note that even though, the
infinite name can he applied to any one of the things that fall
v/ithin the class which includes everything except the thing' negat-
ed, it does not properly signify any one of them. Nor does it si-
gnify the class of all those things, as a genus signifies everything
that : falls within it. The infinite a",m; :'s not a collective 'noun;
there is a class of things to which it may he applied, hut it does
not express any of them. '.

How all this has a very inpeitant hearing upon the
nature of the symbol. For we helfi.O'vathat the symbol falls some-
where "between the name of the infinite name. The name may signify
a collection, hut it never signifies a collection qua collection,
i.o. as a more accidental union. The infinite name on tho oljhor
hand, though it may he applied to a collection, does not formally
signify a collection, hecause of its indetermination. Tho sjhribol
alone signifies a collection formally as a collection. Unlike the
universal name, tho symbol does not ahstract from multiplicity;
in fact, it is precisely the multiplicity that it signifies. Like
the nama and unlike the infinite name, the symbol signifies a de-
termined ohiect- hut unlike the name and like the infinite name

"*&■,;«••■

it does not signify anything that is one by nature.

A simple example will servo to clarify the issue
la the sign "3" a symbol? That depends upon what it i s taken to
Bignifjr. If it represents the throe which is a numbering number,
Q puroa^rogftto, a collection of 1 „ •;;,, i f it la n gymto i in the
strict sense of the term, If hev/over, it ig employed to signify
nu'tber-d number, or predicament al number, which is not three o'oq
•but one three, because the three have a common physical go^us n n d
constitute an unum_po_r _se, it is not a symbol in the strict sense,
•but merely a convenient substitute for the name 'three'. In other
words, in order for a sign to he a symbol it must signify something
that possesses only logical unity; 'it must signify a collection
in its pure collectivity If i;ussell«s definition of number' as
"the class of all claaaea that are similar to it" were correct,
all numbers would he nothing hut symbols 3 (12)

The transcendent terms of ?.ogic used so extensively
in the Vriora ft. nalyt ica. are illustrations ex the symbol, for they
signify at the same time everything and nothing, Of them, St,
Albert the Groat writes; "Ideo terminis uti-.ur transcendentibus,
nihil et omnia significi.ntibus Nihil dico, quia nullam determinant
materiam Omnia voro dico significantibus j quia omnibus materiis
aunt applicabiles, sicut sunt a, b, c," (13)

It is clear that a symbol is something quite diffe-
rent from a mere abbreviation^, As abbreviation has only the 'out-
ward appearance of a symbols, and is in reality nothing but a con-
venient substitute for a naine« \ r of-3ler*s remsvrk that the language
of mathematics is pronominal, must bo rightly understood,, If it
means that the language of mathematics consists in signs that subs-
titute for names, it is true of traditional mathematics^ If it means
that mathematical signs stand in place of names in the sense, of
signifying collections which names cannot signify, it is true only
of the dialectical part of modern mathematics.

Nominalism is at bottom nothing but a denial of the
important distinction. we have just drawn between name and symbols
By a strange paradox, it is a rejection of the name in the true
sons;o of the term, for if all names signify nothing but a collect-
ion of singulars, if "being" for examples means nothing but the
Aole collection of beings, (14) all names can be nothing but
symbols,

,If names in the last analysis ware only symbols,

-431-

„nd if reality were such that it could he represented and expres-
sed only hy means of symhols, then there would he no true natures
in existence and nil things would constitute nothing more than
nn accidental^ collection without any intrinsic or essential unity.
Universal mohilism which deni.es all determined natures must neces-
sarily conceive all language in terms of pure symbolism,, That is
why Whitehead, for whom reality is a process, is logical in hold-
ing that all names are symbols, And in this connection it is inte-
resting to note that Cratylus, who pushed universal mohilism to
its ahsolute extreme, held that words should not ho employed at
all? and had recourse to the movement of a finger in order to
express himself, (14a)

And now, having fixed upon the precise meaning to
1)0 attached to the term "symbol" lot us try to see in what sense
the physico-mathematical world can ho truly called a world of
symbols,,

2o Symbol i s m and Mathematical Phy s ic s c

It has long "been customary for scientists with a
penchant towards scientism to ridicule the philosophical sciences
for their "verbalism". This attitude has heen "based upon the as-
sumption that philosophy' deals essentially with vague and shadowy
concepts which have no definite counterparts' in reality, and that
only in experimental science are things laid hold of in thsir true
objective natures- The now self revelation that ha3 occured in the
realm of experimental science 'has done much to mitigate this naive
view, it has "become increasingly evident that experimental pcienoe
in so far as it attempts to employ names, is the most verhfelistic
of all the sciences. The philosopher can define with precision the
fundamental concepts which he employs such as 3ubstance, accident,
motion, tim'e, etc., he can set forth the nature, the quod q uid
est of things, The physicist, on the other hand,, is hard put to
MM 6 define what he means hy even -She simplest and most "basic no-
tions that enter into his science, such as hody, energy, matter,
mass. As we shall see presently, every attempt to define these no-
tions involves him in an endless circle from which there is no
exit.

The fact of the matter is that experimental science
Is essentially ncminalistic in the sense defined ahove„ By its ve~

"432:..-

ry imture.it is committed to the use of symbols rather than of
namos. And nothing could fee more striking than the -contrast Dot-
neon the vagueness of scientific language when interpreted in
tarnja of names, and its precision when interpreted in terms of
symbols.

It hns taken science a long time to realize this.
Because experimental science necessarily tends towards the condi-
tion of science in the strict sense of the term, it was only natu-
ral that in its origin and development it should aspire towards
a state in which its language could eonsist of names in the proper
aanse of the word. The great mistake of the scientists has lieen
to believe that this state was already a 'fait accomp li, This was
characteristic of classical physics. It was •particularly charac-
teristic of a view that was current in the nineteenth century,
especially among such men such as T.H. Huxley, which held that
science is nothing "but organized and refined common sense, and
that its language is only the ordinary language of common sense
rendered more precise and accurate

This view is no longer popular. The cleavage between
science and common sense has 'become so profound that it has cau-
sed dismay not only in the minds of laymen who are interested in
trying to find out what science is about, but even in the minds
of the scientists themselves who desire to comprehend the meaning
of their science. How for example, can Schrodinger's oscillation
signs operating in multi-dimentional space be expressed in the
ordijiary language of common sense? We believe that this state
of affairs can be understood only by becoming conscious of the
fact; that experimental science .is essentially symbolic, that its
language is not a language of names, but a language of symbols.
Let us try to see why this is so.

As St. Thomas points out in the linos cited above
from the Commentary o n the Poriher me neias , a name in the strict
sense of the~term always stands for'a definite nature (or per-
son); it indicates something that is an unum per se - - a quod
Suid_est„ Now wo have seen that though experimental science tends
towards laying hold of natures, it necessarily falls short of its
goal, pure induction by enumeration can never of itself disclose
a nature that is strictly one That is why from the very start,
experimental science is doomed to deal with collections, no mat-
ter how it may strive to rise above their multiplicity and arrive
at the unity of a strict nature. What the nominalists taught about
knowledge is perfectly correct when applied to experimental science,
"Science, writes Weyl, "concedes to idealism that this its objectiv,

-4S

world is not given but only propounded (like a problem to bo solved)
and chat it can be constructed only by symbol a." (15) But that
in not all »

In this striving to rise above multiplicity, it is
forced to operate upon nature. This operation, as wo saw in Chap-
ter IV, never reveals the objective nature of things; its results
depend essentially upon the whole collection of concrete elements
which entered into it. Since, then, the definitions of physics can
be nothing but operational, none of its notions can stand for a
strictly unified objective nature. They can mean nothing more than
the whole collection of- elements entering into the operations from
nhich they derive; they can signify oiu.y a collection qua collect-
ion, that is to say an accidental aggregate of nature plus a mul-
tiplicity of operational elements, all of which have a unity that
comos from the mind alone* Symbols alone, and not names can! stand fo:
collections of this kind That is why all of the language which
physics uses, whether it consistsof word? or any other type' of
signs, is necessarily symbolic As a consequence, when the physical
world is identified with tho world in se it is impossible to es-
cape transcendental symbolism. Likewise to look upon these signs
as names is to confuse :.rt with nature, subjective construction
with objective reality what is one only in the mind with what is
one by nature; it is to fall into a very pernicious type of' ideal-
ism, as we shall point out in a later context.

It should be evident from the foregoing that science
is symbolic not merely in its more theoretical super-structures
but in the very results of its ur;':.m&ry eoutaci; with n<vl;Kre 4 (16)
Lindsay and Margeneau bring this point out in the following pas-
sage J

It thus appears that the symbol here is but a short-
hand expression for the results of a given operation leading to
the assignment of a number value to the symbol. Instead ; of des-
cribing in. words, the entire series of acts involved in the

• / of e M§ n icaie? h 'thI tt ^ll^^Sr''-2§ a S fe 1 med up in the one
phrase: measurement of P, Is this then all that there is to
the meaning of symbolism? If it were necessary to associate
a symbol with the results of every single physical operation
tho description of these operations mighb indeed be simplified
but it would not constitute what we now consider theoretical
physics. The real power of symbolism in physics first oecomes
clear wh^n we envisage the possibility of letting a B ymoo %
stand for a concept which is, bc to speak, the synthesis oi
tho results of a whole set of operations which may appear to

-434*-

be superficially dissimilar, but are assumed by the physicist
to have a common element, (17)

It should also be evident from the foregoing that
■tho symbolic cnaracter of sciQnce doos not consist in its abstrac-
tors, as some seom inclined to believe. The language of the phi-
losophical sciences is abstract, but it is not essentially symbo-
lic. There is, as we observed earlier in this chapter, a profound
difference between symbols and mimes which stand for abstract na-
tures,, Duhem has endeavored to clarify this distinction in ±,a The-
orie Physique ; i — -

Prenons une loi de sens coramun, une des plus simples comma
une des plus certainos:Tout homme est mortel, dette loi, fts^
surement relie entre eux des terrnes abstracts, l'idee abstrai-
te d'un homme en general ; et non l'idee concrete .do tel;ou
tel homme en particulier; l'idee abstraite de la mort et non
l'idee concrete de' telle ou telle forme de mort; o'est en ef~
fet a cetta seule condition de relier des terrnes abstraits
qu'elle peu-t etre generale, Mais ces abstractions ne sont nul-
- lement des Symboles theoriques; elles extraient simplement oe
qu'il y a dfuniversol dans chacun des cas particuliers aux- 4
quels la loi s'applique; aussij dans chacun des cas par'ticu-
liers ou. nous appliquons ia loi,, trouverons-nous des objets
concrets ou. seront realisees ces idees 'abstraitesj chaque fois
que nous aurons a constater que tout homme est mortel, nous
nous trouverons en presence d'un certain homme en particulier
incarnant l'idee generale d 'homme, d'une certaine mort par-
ticuliere iftiplicjuant l'idee generale de mort •„ „ ,

II n'en est plus de meme pour les Ibis de la Phy-
sique, Prenons une de ces lois, la loi de Mariotte, et exami-
ijons-en 1'enOnce, sans nous soucier, pour le moment, de 1* exac-
titude de cette loi, A une meme temperature, les volumes oc-
qupes par une meme masse de gag sont en raison inverse des pres
sions qu'elle supporte; tel est 1'enonce de la^loi de Mariot-
te, Les terrnes qu'elle fait intervenir, le& idees de masse,
de temperature, de pression, sont encore des idees abstraites;
mais ces idees ne sont pas seulement abstraites, elles sont,
de plus, symbolioues, et les symboles qu'elles constituent
ne prennent un sens que gr&ce aux theories physiques, Plaqons-
nous en face d'un cas reel, concret, auquel nous voulons ap~ .
pliquer la loi de Mariotte; nous n'aurons pas affaire aune
certaine temperature concrete realisant l'idee generale de
temperature, mais du gaz plus ou mains chaud; nous n' aurons
Pas devant nous une certaine pression, mais one certaine poin-
po sur laquolle on a pese d'une certaine mamore, Sans doute

-4K5--

X

a ce g Q a plus ou moins chaud correspond une certaine tempe-
rature, a cot effort exerce sur la pompo correspond unHor-
tame predion; mais cette correspondance o,t ceUe d^r.° Lo-
se sxgnifiee au Slfl e qui la remplace, d'uno realite au'symbo-
lo qui la represents. Oetto correspondance n'ost nulloment
immediate; ello s« etafcl.it au moyen d> instruments, par 1-inter-
mediaire souvent tres long et tree complice des raesures'-
pour at.nbuer une temperature determined 5 ce pvz plus'ou
moins chaud, il f au t recourir au thermometre; pour evaluer
sous forme de pression 1' effort exerce par la pompe U faut
se servir da manometre; et l'usage du thermometre, l'usage du
manoroetre, impliquent, nous l'avons vu au Ohapitro ureceWt,
l'usage des theories physiques, (13)

The symbolism of experimental science mav take on
various forms. In the first place, it may take the form of words.
But words serve the purpose of symbolism very inadequately c For
they are primarily designed to. signify natures, That is why their
use as symbols presents the constant danger of their being mis-
taken for names, and it is well known how many scientists arid
philosophers have fallen prey to this danger. It is a sign of ex-
treme naivete on the part of philosophers to rejoice over the fact
that certain terms, such as "substance";, "matter", "body"s, etc
are shared. in common by both philosophy and science, and to belie-
ve that it is legitimate for them to incorporate into their phi-
losophical system these notions as they are understood in science,,
Moreover;, there is an isolation abqut words which makes them incom-
petent to express the interconnectedness that science tries to a~
chievGo Because therefore experimental science must necessarily
speak in symbols and because words serve this purpose so inadequa-
tely, thero is a natural tendency, especially in mathematical phy-
sics, to draw away as completely as possible from words, to have
recourse to other signs, and to construct a language of its own
which defies all translation into the ordinary language of common
sonae - - much to the discomfiture of the popularizers of science,

A second form which scientific symbolism may take
is that of models, These serve the purpose of symbolism somewhat
more effectively than mere words. The danger of their being mis-
taken for natures in the strict sense of the word is to somq ex-
tent diminished, Besides this they have the advantage of giving,
'■■ direct and immediate expression of : tw;eroonnectednoss. But they
are still extremely inadequate, For one thing, because of their
direct connection with intuition they all too easily give the ^im-
pression that they represent nature in its pure objectivity, in-
dependently of the manufacturing processes of the scientist who

-43G-

works upon nature. This easily leads to tho delusion that they are
direct and immediate copies, or pictures, or at least schemas of
objective natures, That the classical physicists labored under
this delusion constantly is a matter of history, and it is now ge-
nerally recognized how great an obstacle this delusion placed in
the path of scientific progress* Models are not well adapted to
symbolize the true collections that are involved in the notions
of experimental sciences Moreover, their immediate connection with
intuitive schemas makes their capacity for expressing interconnoc-
tedness extremely limited,, For these reasons science has in re-
cent years tended to free itself more and more from the restric-
tions of these models, As we intimated in the last Chapter,, however,
since experimental science deals with the realm of the physical,
it is doubtful if it will ever he able to dispense entirely with
the sensible support that such sensible constructs provide,' But
it is extremely important to remain conscious of the fact that ,
they are mere constructs, more symbols, and to be aware of what
they actually signify,

The next step in science's search for adequate sym-
bolic forms has been the use of what has sometimes been called
pseudo-sensible constructs, (19) These constructs include, such
entities as atoms, electrons, etc Though some of these constructs
may be said to be closer to nature than others, none of the^m has
any immediate correspondence with anything in reality,, As Professor
Margenau points out, their value has no relation with their mode
of existence. There is less resemblance between them and objective
entities than there is between clues and criminals „ As Thompson
has remarkeds "We may well say of them what HoVbes said of words.
'They are wise men's counters; they do hut reckon by them, but
they are the money of fools,' " (20) Constructs of this kind
may be generated by science ad li bitum, for since they are merely
counters by which to reckon, their nature and validity is essential-
ly functional,: And their function is to construct and shape a bo-
dy of doctrine which will explain natural phenomena, though they
do not correspond to anything encountered in experience, they ser-
ve to give systematic form to the data of experience. As Cassi-
rer has observed; "thought only separates itself from intuition _
in order to turn to it with new instruments, thereby to enrich it in
Itself... They render insight into relations possible, and guaran-
tee it, although they themselves can never be perceived after the
fashion of isolated objects," (21) Ehey differ from the data of
experience by their essential interconnoctedness, Because ol this
interconnectenness they can serve to erect a coherent organism which
can substitute for the disconnected mass of experiential data and
thus rationalize it. In other words, by mapping the elements of

-437-

nature which by themselves appear as incoherent, contingent and
unpredictable) upon constructs, science is able to create a'' symbo-
lic system which is more ooii„r, )a t, more necessary, more rational
than nature. More or lass arbitrary rules of combinations x.:^ bo
employed in relation to these constructs which gives great freedom
for the mind to reason about them and which gives great pliancy
to the constructional system. The results of this rational "trans-
formation are ultimately mapped back upon nature in such a way as.
to predict phenomena. (22)

In this way science succeeds in building up a world
of its own . . a world that is rationally organisied, and intrin-
sically oohoraat,, and all the elements of which mutually imply each
other Che validity and significance of the individual constructs
which go to make up this symbolic system cannot be established by
themselves alone by appealing to experience* in so far as the notion
of verification can be applied to them, it cannot mean the esta-
blishment of any direct referendu in reality. Their validity and
significance is derived from the role that they play as members of
a theoretical complex,,

It is evident that these pseudo-sensible constructs
go far beyond the strictly physical models in their capacity to ..
serve as 3ymbols, But in so far as they resemble in some respects
these physical models they share to some extent in the limitations
attached to the letter. Both types of constructs provide the sensi~
ble support that physical speculation needs. But though thdy may
for a while stand the weight of speculation placed upon them, they
tend eventually, as Jeans has remarked, ,: to break in our hands,"
(23) That is why physics must reach beyond the limitations of these
constructs to a move perfect type of symbolism, (24)

This more perfect type of symbolism is found' in ma-
thematics, '(25). As is well knovm ; mathematics, especially^ in its
modem dialectical form, is admirably suited to play the roj.e ox
symbolism^ Its abstraction form existence, from nature, and from all
specific substances, and its. empty forms make it an apt instrument
to signify qq] lections and relations among manifolds without signi-
fying the nature of the relata. Through mathematical symbolism a-
lone can the diverse pheSSmiHa* of nature be reduced to a high de-
gree of interconnectednssse That is why physics is learning to ex-
press itself more and more ftuly in the abstract forms of mathematics
One has only to recall Heisenberg's, Dirac's and 3 chroding ? r. a re»
cent developments in quantum physics to realise how far this tendon-
°y has gone As wo have already remarked, sensible and P^ d ~"
sible constructs will never bo completely be dispensed with, but

•438.i

na Jeans has remarked, they will rersain mere parables - - me-e olo-
thing which we drape over our mathematical symbols (26) "

3o A World of Shadows,

"The frank realisation that physical science is con-
cerned with a 'World of shadows/' writes liidding-bon., "is one. of the
most significant of recent advances,, 1 do not mean that physicists
are to any extent pr.iocupied v'.j'i;. the philosophical implications
of this. Prom their point of view it is not so much a withdrawal
of untenable claims as an assertion of freedom for autonomous deve-
lopment " (27) Nothing could he more striking, than the paradoxi-
cal fact that hy attempting to introduce the hrilliance of carte~
sinn clarity everywhere in the physical world; science has made
of it a world of shadows^ We must now try to see why the world of
physics has necessarily 'become a world of shadows and what some of
the philosophical implications of this fact are.

The shadowy character of the physical world derives
principally from its symbolic nature, But even independently of the
use or symbols there are a number of reasons why the world with
vrtiich physics deals can be truthfully called a world of shadows, To
begin with, all human knowledge is oy its very nature shadowy, For the
human intellect is the lowest intellecb that could possibly esi'st;
It is essentially united with matter, and dependent upon it (at
least extrinsicallyi for its functioning., As a consequence its realm
of knowledge is at best a mere ahadowlando That i3 why Aristotle
tells us that it is like the eyes of an owl which can see well only
in the deep twilight and in the dark. And the more it attempts to
penetrate Into the realm of the sensible:, the more does its knowlod-
g$ecome shadowy,, Cense knowledge is truly an obscure knowledge,,
For it is at the utmost extreme of knowledge, where immateriality
peters out into materiality p where the light of the intentional world
is mingled with the darkness: of the purely physical world. It is
a very late tvrtljtht when darkness has almost taken over, and when
only obscure shadows can be seen. Now physics deals with everything
in terms of sensible matter. Not only that, but it is the part of
natural doctrine that is the farthest advanced in the di:;ec-=ion of
concretion, that is the most profoundly immersed in the obscurity
of matter, <Dlu>t i8 why it s object is essentially a shadowland„

-439.-

Q?ho dialectical character of physics gives" us another
roason why it necessarily deals, with shadows. For since it is a
sciontia quia and not a s oientia propte r quid, it can get at pheno-
mena alone; it is restricted to mere appearances. T>>3 nature behind
the. appearances remains in the dark. In attempting to get at this
nature, physics throws up a scaffolding against reality - ^ a scaf-
folding which is like a shadow of reality, roughly, and sometimes
grotesquely reflecting its outline, Though there is always some re-
lation between the proportions of a shadow and the reality., this
relation is not definite, particularly with regard to specific de-
tails. The relation 'between the world constructed by the physicist
and the world or reality :,s of this kind.

By the fact of its being subalternated to mathematics,
the world of physics takes on an- even stronger resemblance to a
3hadowland„ For a shadow is something that deduces the heterogene-
ity of the object it represents to pure homogeneous exteriority,
The qualitative is swallowed up in the quantitative. To he more spe-
cific the mathematical line is a shadow of the physical line, and
when the physicist studies the physical line in terms of the mathe-
matical line, he is getting at reality only by means of its shadow,.

But it is principally because of its symbolic charac-
ter that the world of physics is a world of shadows. And the reason
for this should be fairly evident. We have seen that symbols differ
from names in that they do not stand for natures in the strict sen-
se of the term, That is why when they are used as signs, the preci-
se nature of the things signified remains blurred and hidden in the
background. And no manipulations of symbols can make them emerge
from this background.

As science perfects its symbolic forms, the physical
world take3 on more and more the character of a self-authenticating
formal system in which ihe interrelatedness of nature's manifold
is seized upon and reflected. The principal criterion for the use
of these'-aymbolic forms is not that they should individually have
a direct correspondence with something intuitively given, but that
they be able to fit coherently into the. self-authenticating- system.
From one point of view the increasing perfection of the symoolic
reflection of nature's interrelatedness throws greater light upon
the relata, but from another point of view it makes them more xike
shadows.

Sir Arthur Bddin '.-ton has lnll great emphasis upon .
this point. In the introduction to ThgJTature of the Physical Worla

-440^

he writes!

Science. aims at constructing a world which ahal]
be symbolic of the world of common-place experience it is
not at all necessary that every individual symbol that is
used should represent something in common experience or even
something explicable in terms of common experience. The' man
in the street is always making this demand for concrete ex-
planation of the things referred to in science, hut of neces-
sity he must he disappointed. It is lika our experience in
learning to read. That which is written in a hook is symbolic
of a story in real life,, The whole intention of the hook is
that ultimately a reader will identify some symbol,- say BHEAD
with one of the conceptions of familiar life. But it is': mis-
chievous to attempt such identifications prematurely, before
the letters are strung into words and the words into senten-
ces,) The symbol A is not the counterpart of anything in; fa-
miliar life. To the child the letter A would seem horribly
abstract; so we give him a familiar conception along with it.
"A was an Archer who shot at a frog. " This tides over his im-
mediate difficulty; but he cannot make serious progress.; with
word-building so long as Archers, But.chers, Captains, dance
round the letters. The letters are abstracts and sooner or
later he has to realize it. In physics wo have outgrown archer
and apple-pio definitions of the fundamental symbols. To a <
request to explain what an electron really is supposed to be
we can only answer. "It is a part of the A B of physics."
The external world of physics has thus become a world of sha-
dows CI1 ■

■ ' It is difficult to school ourselves to treat the
physical world as purely symbolic. We are always relapsing and
mixing with the symbols incongruous conceptions taken from
the world of consciousness. Untaught by long experience we
stretch a hand to grasp the shadow, instead of accepting its
shadowy nature. Indeed, unless we confine ourselves to mathe-
matical symbolism it is hard to avoid dressing our symbols _
in deceitful clothing. When I think of an electron there ri-
ses to my mind a hard red, tiny balls the proton similarly is
neutral grey. Of course the colour is absurd - - perhaps no
more absurd than the rest of the conception - - but I am an
incorrigible. I can well understand that the younger minds are
finding these pictures too concrete and are striving to cons-
truct the world out of hamiltonian functions and symbol? s„
far removed from human preconception that they do notom
obey the laws of or&dox arithmetic. For myseli I find some
difficulty in rising to that plane of thought; but I am .on-

-i*l-

vincod that it has got to come.

Later m the same work ho brings out this point more
specifically in connection with his explanation of the cyclic me-
thod employed in physics. (28) All of the constructs out of which
the structure of physics is formed, such as point-events, potentials,
matter, etc* are definable and translable only in terms of each
other, not in terms of anything else, and in particular not in
terms of any underlying reality that is independent of the 'mind •
of the scientist or the physical objects of the perceptual world*
These constructs form a closed circle. By beginning at any point
of this clrclowe may define any one of the members which form it
in terms of the others, and from it deduce the others. But as we
travel around the circle at no point do we make fresh contact with
reality* At a certain point, e.g. "matter" we may think that we
are talking about something that has a direct embodiment in the
T?orld of reality, but in point of fact, the "matter" that is dealt
with in physics has no direct counterpart in nature. It is by work-
ing around this circle that we derive the physical laws,,

In this way physics remains within its own domain;
it constitutes a closed world of its own, and this world is but
a shadowland reflecting the underlying reality which can never be
made to emerge from its obscurity) >

And you can see how by the ingenious device of the
cycle physics secures for itself a self-contained domain for
study with no loose ends projecting into the unknown. All ci-
ther physical definitions have the same kind of interlocking.
Slectric force is defined as something which causes motion of
an electric charge; an electric charge is something that exerts
something that produces motion of something that exerts some-
thing that produces .' . ad infinitum ..

The supposed approach through the physical world
loads only into the cycle of physics, whore we run round and
round like a kitten chasing its tail and never reach the world-
stuff at all , . . ■ ,

However much the ramifications of the cycles may be _
• extended by further scientific discovery, they cannot form their
nature trench on the background in which they have their being
- - their actuality,, (29)

It is particularly in its use of the theory of groups
*Qt the physical world takes on the character of a world of sh a-
Aows, LB WQ saw in the last Chapter, it is POSsiDlo to give an o-
xact mathematical description of patterns, while the natuie of the

-44 Z-

3 ntities involved m them remain in the dark, «n (mathematics)
iis.uiases the individual elements "by assigning to them symhois,
leaving it to non-mathomo.tical thought to express the knowledge,
if f-n«Y> that we may have of what the symbols stand for ! „

Every path to knowledge of what lies beneath the structure is
U'.en 'blocked "by an iniponetrahle mathematical symbols" (30)

All this discussion eJbout J cho shadow world of phy-
sics calls to mind the famous shadows of the Platonic cave. In fact
tha well-known passage frcm the Republic is so relevant here that
;te cannot refrain from quoting it°

And now, I said,, let me show in a. figure how far
our nature is enlightened or unenlightened! - Behold' human
beings living in an underground cave- which has a mouth open
towards the light and reaching all along the cave 5 here they
have "been from their ehildhood s and have their legs and necks
chained so that they cannot move, and can only see "before tiiem 9
"being prevented "by the chains from turning around their heads,,
Ahove and "behind them a fire is "blazing at a distance, and
between the fire and the prisoners there is a raised way; and
you will see, if you look,, a low wall "built along the w'ay f li-
ke the screen which marionette players have in front of them,
over which they ishou the puppets,

I see.

And do you see, I said, men passing along the. wail
carrying all sorts of vessels, anci statuos and. figures of
animals made of wood and stone and various materials, which
appear over the wall? « u .

You have shown me a strange image, and they are

strange prisoners,,

Like ourselves, T. replied; and they see only their
own shadows, or the v ther shadows which the fire throws on
the opposite wall of the cave

True, he said; how could they see anything hut the
shadows if they were never allowed to move their heads? _

And of the ohjects which are heing carried m luce
manner they would only see the shadows?

Yes, he said,, „j.v„~

To them, I said, the truth would he literally nooh.n f ,

but the shexows of images o (31)

All that has heen said in the course of this study
a^out the nature of experimental Boienoo «*» it evident how much
^ scien-iot is like a prisoner in a dark cave f?™*?™^.^
to which he is committed are the chains which hind him and p.evenu

-443-

hiffl from turning his head and seeing reality in its objectivity.
A8 Plato's observer saw both other shadows and his own thrown aga: net
tho wall of the cave, so in the shadow world of physics the', scien
tlB t sees both the shadows of objective reality and his own, but
in this case the two are inextricably blended together.

The following parable brings out still further the
similarity between the physicist and the cavedweller of Plato.

An aged college Bursar once dwelt secluded in his
rooms devoting himself entirely to accounts. He realised the
intellectual and other activities of the college only as/khey
presented themselves in the bills. He vaguely conjectured an
objective reality at the back of it all - - some sort of pa«
rallel to the real .college - - though he could only picture
it in terms of the pounds, shillings and pence which made up
what he would call "the common sense college of everyday ex-
perience." The method of account-keeping had become invete-
rate habit handed down from generations of hermit-like bursars; '
he accepted the form of accounts as being part of the nature
of things. But he was of a scientific turn and he wanted to
learn more about th<3 college. One day in looking over his books
he discovered a remarkable law* For every item on the credit
side, an equal item appeared somewhere else on the debit side,,
■■Hao'" said the Bursar, "I have discovered one of the great laws
controlling the college, It is a perfect and exact law of the
real world, Ciedit must be called plus and dobit minus;, and
so we have the law of conservation of L s, d. This is the true
way to find out things, and there is no limit to what may ul-
timately be discovered by this scientific method. I will pay
no more heed to the superstitions held by some of the Fellows
as to a benificient spirit called the King or evil spirits
called the University Commissioners. I have only to go on in
. this way and I shall succeed in understanding why prices are
always going up."

I have no quarrel with the Bursar- for believing that
that scientific investigation of the accounts is a road to exact
(though necessarily partial) knowledge of the reality behind
them. Things may he discovered by this method which go deeper
than the mere truism revealed by his first effort. In any case
his life is especially concerned with accounts and it is pro-_
Per that he should discover the laws of accounts whatever their
nature. But I would point out to him that discovery of -he o-
vorlaDping of the different aspects in which the realities of
J She college present themselves in the world of accounts, is
not a discovery of the laws controlling the college; that he

-444-

has not begun to find the controlling laws. The college may
totter hat the Buraar's accounts still 'balance, (52)

However much symbols' and shadows may cut off the
scientific observer from reality their essential, purpose is to u-
nite him to it. For the nature of symbols is to signify something
and the nature of shadow is to ho a reflection of reality, (33)
That is why, after having seen the nature of the physico-mathema-
tical world, we must now try to analyze its relation to the objec-
tive world* The nature of this relation has been more or less im-
plicit in much that has been said thus far, and has, wo feel, al-
ready begun to take on fairly definite outline,, But it is of supre.
me importance for a right understanding of the validity of scien-
tific knov/ledge to endeavor to make it as explicit as possible „
That is not an easy thing to do, For it should he evident from all
that has heen said up to now that this relation is far from "being
the simple thing that the classical physicists and the majority
of modern Scholastics have imagined it to he. We can only hope
to treat the problem in its general aspects without descending
to details.

-445-

CHAPTHR ELOTEN

THE ABSOLUTE WOULD CONDITION.

lo. Isomorphism.

By the absolute condition of the universe is meant
the objective world as it is itself - - the world as it is con-
templated by supramundane intelligences which do not have to .depend
upon the manifold subjective and relative conditions that necessa-
rily accompany all knowledge derived through the senses, which are
free of the barriers that result from the limitations of the human
■ intellect, which do not have to probe the world with appliances
that are within it, and a part of it, and subject to his laws,
and which do not have to reconstruct the world, and thus remodel
and change it, in order to know it. (1)

That this absolute world condition i3 not identified
with the physico-mathematical world is only too evident,, Vie must
braare of the ambiguity of the term "physical world," (2) Ori-
ginally it was employed to designate the objective cosmos, physi-
cal science was born of a desire to lay hold of this cosmos in its
objectivity. But as science grew, it gradually evolved, for reasons
already set forth, a world quite distinct from the objective cos-
mos ~ - a world of its own making. It is to this latter world that
the J f erm "physical world" now usually refers when it is employed
ty Physicists,

Progress in soiemce has resulted, from one point of
view at least, in an ever widening gap between these two worlds,
The scientific universe has become more and more independent oi
the objective universe, more and more closed in upon itaelx, more
^Insufficient. This has come first of all. from_the steadr.xy in-
ceasing U qe of hypothetical elements logically moerwcren inso

-446-

a coherent structuro, but it has been carried to groat lengths by
the subalternation of physics to mathematics, which, ea we hrve
seen, is independent of existence and of any necessary order to
existence, and which constitutes a closed and autonomous universe
determined only hy its own intrinsic logic. In this way, physical
science has tended to become more and more a formal, self-authen-
ticating system, even the raw materials of which are no longer
taken directly from the objective world, hut are subjectively crea-
ted constructs,,

From this point of view, then, the scientific world
is a": self-contained world, distinct from and independent of the
absolute world condition. Science has become like a platonic demi-
urge, fabricating a universe out of its own subjective constructs
and rationalizing , it by means of mathematics. And in this perspec-
tive' there is a great deal of truth in Maritain's remartj "ce n'est
pas la realite qui demandera a la science d'etre vraie, e'est la
science qui demandera a la realite d'etre ^cientifique*, et de lui
presenter ses papiers " (3) In order to know that there is a vast
difference between the scientific world and the absolute world con-
dition it is not necesgary, as some might be tempted to suppose,
that we have direct knowledge of the world in itself and thus be
able to compare the two. For in the first place we know that there
is a negative distance between the two universes by our experience
with the kind of knowledge we have., which must go from the more
general to the more concrete without ever being able to exhaust
the concrete. The history of science brings out this point and un-
dorscores/our great ignorance. In a position wo may know that there ,
is a vast difference between the two universes because we know that
in order to carry on scientific endeavor wo must construct and must
inject mathematics,, (4)

But this independence of the scientific world from
the absolute world condition is only one side of the picture, and to
exaggerate it to the extent of obscuring the other side would be
to vitiate the whole meaning of scientific knowledge, (5) The
objective world is not merely a malleable matter which alloys the
scientist to make airy eonstructions he may wish. In erecting his
scientific world he enjoys a great measure of freedom, but he is
not completely free. (6) Though mathematical physics is formally
mathematical and from this point of view independent of the real
"orld, it is terminativo naturalis; its whole purpose is to get
to know objective i»^7ThIHioTontific world remains bound down
to the objective world at both ends; that is to say, the scientist
must both begin and terminate his work in contact with nature,
(?) While it is true to say that in one sense, the theory of

-445

Relativity, for example, as it pursues its constructive elrborat-
ions never returns to the world of experience tut seems to draw
farther and farther away from it, in another sense it does return,
Einstein lmow before he started that all of his mathematical cal-
culations and constructive elaborations had, in the end, to load
back into the "black - hands of the Mlchelson interferometer,, The
scientist must solve problems that are initially given in the ob-
jective world; his solutions must explain facts as found iii expo-
rianca, Vrtiile the experimental operation measures the world condition,
there is a sense in which it is true to say that the absolute world
condition measures the experimental operation As Eddington has
observed, "The study of physical quantities, although they 'are the •
results of our own operations (actual or potential), gives us some
kind of knowledge of the world-conditions, since the same operat-
ions will give different results in different world-conditions."
(8) . : Moreover, there is a sense in which, it is true to say that
the scientist, deals with familiar objects of the ob j ective worldo
A. sign of this is found in the fact that commercial companies con-
cerned with these objects always have recourse to the help of
scientists, (9) ,

All this helps us to understand the problem ^hat the
meaning of real existence presents to the mind of the modern scien-
tist, (10) If the question is raised; "Does the scientific world
really exist?" or "Does ari electron really exist?" it is impossible
to answer either yes or no 3 for we are dealing with constructs com-
posed of both reality and mind. Taken from the point of view of
the subjective elements they contain, they do not really exist c
But taken from the point of view in which they are a reflection of rea
lity, they do really exists In fact, in the latter perspective we
may say that they exist in a more real sense than the sense world
or the world of philosophy of nature, for science, in coming closer
to concrete objectivity becomes more like the knowledge that God
and the separated substances have of the absolute world condition
than any other type of knov/ledge we have. (11) That is why al-
dington, writing of Hossetti's Blessed Damosal who contemplates
the world from heaven, can says "If the Blessed Damosel see:; the
earth in the Einsteinian way she will be seeing truly - -I can_
feel little doubt as to that - - but she will be missing the point,
» is as though we took her to an art gallery, and she (with -chat
painful truthfulness which cannot recognize anything that is not
really there) saw ton square yards of yellow paint, five. of crimson,
and so on » (12) The scientific world is made up of yaruoOl
Paint taken from the objective wofcLd; but these yards of P««*
have been caught up into -a composition that is not found in nature.

-4AB-

In the light of these remarks a number of passages
in the writings of modern scientists which at first sight might av~
vaa r hafflingare rendered perfectly intelligible, A good example
is the following passage of Eddingtbnj

However, so far as I can judge the meaning of the
question, the answer appears : to he in the affirmative - -
the external world described in physics (b, & q, e, ) really
exists.

One thing can perhaps usefully he added, I do not
think that with any legitimate usuge of the word it can lie
said that the external world of physics is the only world
that really exists,, (13) , "".

There are in fact an infinite number of "physical
universes",, There was for example the original universe of Einstein
which was full of matter and static, That has now heen ahandonnedo
■There was likewise the universe of De Siti'er which was empty, There
is now the universe of Ahhe Lema3.tre s which contains matter in cons-
tant expansion. These "physical universes" may he multiplied end-
lessly. All of them can he said to exist really in the sense just
determined, hut none of them cf.n Ije considered the only one the.t
really exists.

Perhaps the central prohlem with which we are concern-
ed in this Chapter ' cs. 7 ?. "be made cleare:-: hy casting it in the fol-
lowing for.uij Is the scientific world true? Is it the truth ahout
oh j active. nature? What oxfat sense can he attached to Eddington's
statement that if the Blessed i'amosel sees the objective world in
ths ainateinian manner she see;-; it truly? (14) As is well, known 5
truth may ho defined either in terms of intrinsic coherence or in
terms of" extrinsic conformity,. Every science in so far as it cons-
titutes a hody of doctrine and takes 6--i systematic form must possess
truth in the former sense. There are some sciences in which this
kind of truth is of primary concern, These are particularly the mr..=
thematical sciences which deal wi-ch ahstract a ut ahstract a f r.no. w.nicV
prescind from any actual order to existence. But in those disci-
plines which deal with reality and which are sciences in the strict
meaning of the term it is truth in the sense of extrinsic conformi-
ty that is of primary concern.

Now from what was said ahove ahout the scientific
world constituting a closed and intrinsically coordinateo. uyaTjam
and ahout the criterion for thfi choice and elaboration 01 construct,
being not correspondence with ohjoctive entities hut then capaci^
to serve es principles of internal coherence., it would seem oo

-443-

follw that it is truth intko first sense of the term that is cha~
ractenstic of experimental science. This would seem to derive Doth
from the vast use of hypothesis and especially from the introduct-
ion of mathematics it is true that there is some connection hetween
scientific constructs and objective reality, hut it would seem that
this connection must he viewed not so much in -terms of truth as in
terras of goodness, since the validity of these constructs is judged
hy their functional role, hy their explanatory efficacity, The whole
question comes down to this, thon s can the conformity definition
of truth he applied to the relation hetween the scientific world
and the ahsolute world condition?

It should he immediately evident that if the confor-
mity definition he taken in its full and ahsolute meaning, the ans-
wer must he no, Truth in this sense has the implication of unique-
ness 'and to apply it to the ever changing scientific world would
make it an extremely protean thing,. On the other hand, it is. equally
evident that there is some correspondence and some kind of confor-
mity hetween the scientific world and the ahsolute condition of the'
universe, that some relation similar to' truth ohtains hetween them,
if for no other reason than that verisimilitude is, as wo saw in
Chapter V, of the very i.ature of experimental science. Thi3 -confor-
mit.yls found even with regard to the most theoretical parts .of scion'
og, for since theory is the. source from which the phenomena of natu-
re logically flow, and the ohjective essences of things are the sour^
ce from which they really flow, it is ohvious that there must he
some kind of correspondence hetween the two, even though theory
may not give an explanation of reality that is true in the strict
sensd of the word.\And as theory is perfected this correspondence
■becomes more and more exact,

Moreo.ver, the scientific world is made up of reality
as well as of mind, and it must not he forgotten that even the sub-
jective elements derive their whole moaning from their orientation
towards the real world.

In other words, scientific symhols lika all symhols
aro a mixture of truth and fiction, as Urban has ohservedj

It is, as we have, seen, of the very, nature of the
symhol that it contains hoth truth and fiction, hoth the real
and the unreal. This principle follows, in a sense, fromtho
two preceding. We have already seen that a symhol must sxand
for something, otherwise it would not he a symhol. V/o have
also seen that it cannot stand for anything in a wholly unam-
biguous way. If 'it did it would not ho a symhol. L fictional

"150-

clement in every symbol is made necessary by the principle of
dual reference, it is ef the nature of . the symbol thatTf ei-
ther reference is taken exclusively it becomes unreal or else
a mere, substitutional sign,,

A- relation of two domains is involved in over"- sym-
bolic function. If the symbol is taken literally, as we say,
if, in other words, the reference to tho'pr'imary' domain is
taken exclusively the symbol is a fiction and misrepr esents,.
If it is taken wholly as a sign without any reference tp the :

intuitive domain, out of which it. springs, it is again a' fiction, '
in this case a merely conventional sign, The symbolic function,
as distinguished from literal representation or description
and from the merely conventional, is not only this dual' referen-
ce but the combination of truth and fiction which arises out
of it,- This is as true in the region of scientific symbolism
as in any other. It is, in fact, one of the main issue3 : in
modern, scientific concepts is truth and how much fiction,
(15) ;.

It is clear, then, that in spite ot its self-authen-
ticating character,, mathematical physics has a definite relation
of correspondence with the real world. By the very fact that it is
terminative naturalis , it must in some way realize the conformity
definition of truth that is characteristic of all sciences which
deal with reality. (16) And if mathematical physics' appears' as
something that is from one point of view essentially, dominated by
the coherence definition of truth and at the same time from; another
point of view principally dominated by the conformity definition, ,
it is chiefly because it is a scientia media. Let us try to fix
upon the nature of the correspondence between the two worlds. .

This obviously depends upon one's theory as. to the
nature of scientific knov/ledge. For those who ! press operationalism
to the limit of maintaining that science reveals nothing hu t a set
of operations carried oil hy the scientific worker, this correspon-
dence is extremely tenuous, when it exists at all. At least this is
true if the notion of correspondence he considered from the.-: point
of view of speculative truth, , as it is heing considered in this con-
text. For many operationalists, scientific symbols do not represen t
the objective universe at all; they merely reveal how one has ope-
rated upon nature and how one must operate upon nature in order to con
*w>l it, (IV) These authors foil to realize that the art that is
involved in experimental science is purely functional and that its
whole purpose is to serve science by helping to disclogo theobjec-
Wvo logos. In other words, scientific symbols are likepoetic sym-
bols in this that they turn aside from a direct expression of ran-

-&51"

Hty only that in aomo sense they may express it more profoundly,
(18)

The majority of modern scientists and philosophers
of science hold that the physico-mathematical world must v bo con-
sidered at least a partial representation of reality. This opinion
is held by Einstein and Planck;, among others, and according to
Cassirer it constitutes the essential modern scientific standpoint,,
It adopts, a mediate position between the copy theory of the classi-
cal physicists and' extreme operationalisra. For most of those who hold
this view, the scientific representation of reality consists in a
reflection of nature's order, structure and interrelatedness, ra-
ther than in a direct representation of intuitively given natural
phenomena.,

V/e "believe that this opinion is essentially corrects
But it is necessary to try to give greater philosophical determi-
nation to the correspondence "between the scientific world and the
absolute world, Some have sought to solve this problem "by saying
that the scientific universe is analogically true„ Hoenen has "been
particularly favorable to this solution;- (19) He holds that phy-
sical theories express an analogical relation to reality and that
if all the superfluous elements in them are eliminated "by means
of experiments and reasoning, it is possible for the relation to
become univocal. We shall not linger over the latter part of this
opinion,, for all that has "been said in preceding Chapters makes
it abundantly evident how utterly untenable such a view is„. In so
far as analogy is concerned, we "believe that this opinion is extre-
mely ambiguous. It is clear that if the term analogy he taken in
a broad and loose sense it may "be applied to the knowledge that
science gives of the objective world, in that the scientific world
is partly like and partly different from the absolute world condi-
tion, But it -is extremely important to keep this use of the term
distinct from the proper use that is found in metaphysics, In true
analogy we find totum acutale that is the amlogum in which the^.;
parts are knov/n.~In"bhe~case in hand, on the contrary, the par.s
are not known well enough, The objective and subjective elements
in the scientific world are. so intimately interpenetrated and fused,
that it is impossible to distinguish between them; it is impossible
to say what is in conformity with objective reality and hat is no-,
it is impossible to determine which particular ^rt is- due vo nature
And which ib due to mind.

We believe teat the correspondence between the scien-
tific world and the absolute world °°^ition can best be explained
in the following terms. In the first place, the scientific universe

-452-

is a sign of objective universe. Tfjvery sign represents an object
distinct from itself to a cognitive power. But -because there are
too essentially different ways in which this represents ion can
•tie effected, there are.„ as is well known; two essentially diffe-
rent kinds of. signs: formal and instrumental. Since every sign is
a means hy which a cognitive power gets to know an object, oven a
formal sign is a kind of instrument. But it differs from an instru-
mental sign in this that it delivers the object it represents so
directly end immediately to tho mind that in this deliverance it
do^b not itself constitute an object of knowledge. Thus the' concept
which the mind forms of an objective entity is a formal sign of
that entity because it does not: interpose -itself as an object bet-
ween the mind and the entity, A,n instrumental sign, on the other
hand, is ono that is first known in itself as an object in its own
right; and only by being known in this way does it represent another
object distinct from, but virtually implied in itself. (20) In
other words, as Cajetan has remarked, (21) there aro two kinds
of beings; some are primarily designed to b3 and only secondarily
do they represent; others are primarily designed to represent other
things, The former are instrumental signs and the latter are for-
mal signs.; In Thomistic terminology, a formal sign is _id_in quo
aliquid cognosoitur, an instrumental sign is id per q. u p^_J' J -_i" l inid
cbgnh scTtur ~the first is a f orma i ntra potenti am informah s t the
second is "an obiectum extra p oten tial movensc

How the great error of many of the classical physi-
cists and of the majority of modern scholastics is that they have
looked upon the scientific world as a kind of formal sign directly
and immediately revealing the absolute world condition. To view the
scientific world in this light means to fall a prey to a great il-
lusion, It means to destroy the scientific world's character as a_
sign, for it wipes, out the true revelation it gives of the objecti-
ve universe,,

The physico-mathemat/ical world is not a formal sign,
■but an instrumental sign of the absolute world condition. It cons-
titutes an object in its own right, and must be known as such befo-
re it can reveal the oojective universOo Like all instrumental signs,
it hides tho object it represents at the same time that it revels
it. And it is only by viewing the scientific world in this ligho
that through it we can in some fashion come to know the objective
TOrld as it is in itself.

No otner notion brings out so accurately the true
character of the relation between the two worlds than this ™t^on
into which enters both instrumentality, mvl signification. x t ozp-an-

-453-

how the physico-mathemntical world can ho at tho same time comple-
tely closed in upon itself and completely opened to the objective
world. It reveals why the criteria. of the validity of the scien-
tific structure can be both goodness and truth, with the goodness
entirely subservient to the truth, why the scientific universe is
at once practical ,end speculative, with the practical oompletely
orientated, towards the speculative, at once art and science, wii-h
•the art entirely ordered to the science, (both in the sense in which
fine art reveals an original, and in the sense in which Useful art

serves a purpose the practical purpose in this case being found

in the speculative order) „ Neither pure instrumentality alone, nor
pure formal signification can tiring out all of these paradoxical
elements and serve to establish them in their proper relations

The physicQ-irathematical world is in many ways a
particularly perfect type of instrumental sign and it tends to-
wards the perfection of a formal sign, Even those elements in it
which are not taken directly from the obj ective' univorseimd which,
consequently from one point of view serve to hide it, are introduced
into it only to reveal the absolute world condition all tho more
In this it is similar to a work of art into which tho artist's own
logos has been injected only for the purpose of revealing the ori-
ginal with greater clarity „ '

But the scientific world is an oven more perfect sign
than a work of art in that the fabrication found in it, while inter-
posing- an object between the mind and the real world, can never cons-
titute an end in itself. The scientific world is art 3 but not sim-
pliciter c It is essentially speculative knowledge, and ns such its
whole raison d'etre consists in its orientation towards the real
world. In this it is similar to a formal sign,

John of St. Thomas assigns five conditions which _ must ■ ■
be present if one thing is to be the sign of another,, (22) First,,
the sign must be something distinct from both the' object signified
and the cognitive potency, This condition is fairly obvious and needs
no comment. Secondly, it must have the nature of a representation.
This establishes a transcendental relation between v )he sign: and tho
thing signified. Thirdly, the sign must be more inoable than the
thing signified.. By reconstructing the objective universe, by in-
jecting his own logos into it, by introducing- the rationality of
mathematics, the scientist succeeds in ren der ing it ™£ £*£££'
We, Fourthly, the sign must be less perfect than the th ing sigm-
fled and inferior. This recalls to mind what we ^^.^^J
Chapter about the physico-mathomatical world being w - thanjho
•real world precisely because it is beuteio _^

-454-

substitute for tho real world. Its Tola ia purely functional. This
moans that over and above the transcendental relation mentioned a
moment ago, there is a predicamental relation between the scientific
world as sign and the absolute world condition as thing signified,
iphis relation belongs to the species of relation that exists bet-
ween a measure and a thing measured, (in tho sense explained in
Chapter Vtll in connection with the various types of relation) .
Tho absolute world condition is the measure of the physico -mathe-
matical world. It is this predicamental relation and not the trans-
cendental relation that constitutes the latter as the sign of the
former, (23) The fifth condition laid down by John of St. Thomas
io that tho sign and the thing signified must be dissimilar. Tho
vast difference between the scientific universe of discourse and
tho objective universe has already been sufficiently stressed.

The foregoing makes it clear that in experimental
science the mind does not assimilate tho objective world diroctly }
but rather reflects it by constructing a schema of its own that is
founded upon reality. But it is important to try to determine the
nature of this schema and thus bring out as accurately as possible
tho QKao-b character of the instrumental sign. We beliovo that thio
can be done by having recourse to the notion of : isomorphism. Isomor~
phism, as the word implies, signifies identity of, structure or form,
and it is commonly defined in the following terms'. Given two clas-
oesi S, composed of elements a, b, c, .;;, and SS composed of ele-
ments a', V, o', ,,.; if the elements of S can be placed in one-one
correspondence with those of S', in such a way that a corresponds
to a', b corresponds to V, etc.; and if for every relation R bet-
ween the elements of S (e.g. a H b) there exists a relation R' bo„~
ween the corresponding elements of S< (e.g. a' R b'), the two clas-
ses are said to be isomorphic. A familiar example of isomorphism
Aa ftnmd in an ordinary map. There is identity of structure between
tho relation between the points on the map and the corresponding
points on the countryside to which the map is related It is impor-
tant to insist upon tho fact that isomorphism is not founded upon
a mai-erial correspondence between the elementr Involved, but the
identity of structural form. It prescinds f ro m the P?°P« £*"£
of the matter to which the :Torms are applied. But oh is P^oindin*

is not a negation. In fact if the heterogeneity of the «te f
tlw different systems were destroyed, the isomorphism would aUo
be destroyed,

^w this rotion of isomorphism orirgB out tho nayara
of the rolation^Xeon^he P^ioo-mathem^c.l wor d and ^ f -
solute weld condition. *or ^^ama-oxcal phy^c^ "^^
system au\ ordar. As wg havo aeeup i« -ons^u^ . -

-455-

aystom, but in so doing it is determined in its every move, either
directly or indirectly, by measurements made upon the real world,
in spite of the arbitrary elements ir/neasurements. the absolute
world condition remains the measure of the measuring process, in
such a way that although different codes of measurement employed
in relation to the same world condition v/ill render different re-
sults, as long 'as the same code is employed in relation to the sa-
me world conditions, the results will tie identical. (24) That is
why,- after the physicist has constructed his schema he is able to
map it hack upon nature and predict natural phenomena. (25)

As Duhem has observed, the relation between the
scientific world and the objective world may he compared to the
relation "between the form of a suit of armour and the form of the
'body of the knight who wears it t (26) There is always a similari-
ty of structure in this, relation no matter how imperfect the suit
of armour may ha. This similarity grows as the suit becomes more
perfect, as the number of pieces of metal which compose it increa- '
3os, and as its structure becomes more complex. At the limit the
form of the suit would be identified with the natural form of the
"body. This limit can never be actually reached, obviously; but it
can be indefinitely approachecL And as the artificial form pf the
suit gets closer to the natural form of the body, it is at the same
time drawing farther away from it in the sense that it is constantly
becoming more artificial.

2. Logical identity.

The gap that exists between the absolute world con-
dition and the structures manufactured by the scientist is some-
thing that the mind must seek to bridge,. It must seek to go beyond
the relation of isomorphism of which we have bsen speaking and ar-
rive at some kind of identity. In order to see how this may be ac-
complished, how what is at once both reality and artifice can be
erected into a unified object, it is necessary to have recourse to
the notion of predication of identity.

Aristotle and St. Thomas speak of this notion :-n e
>ral places, notably in the fifth book of the Metaphysics .
id the fourth book of the Physics, (28) £n the latter tex* we

Cenus potest cum additione unitatis vel identitatis
praedicari de plunbus mdividuis existentibus in una specie,
et similiter genus remotum de plurilms specieMs existentibus
sub uno genera propinquo; neque taraen species de individuis,
nequo genus propinquum de spoolers diversis potest praedicari
cum additione unitatis vel identitatis,,,. 3b huius assignat
(Anstoteles) rationem; quia oum idem ot diversum seu differons
opponantur, ibi possumus identitatem dicere, ubi differentia
non invenitar, sod non possumus dicere- identitatem ubi inveni-
tur differentia,,

In order to make npredication of identity of things
that are different it is necessary to ascend to a genus that is
not divided by their proper differences. Aristotle and St. Thomas
explain this by having recourse to examples taken from mathematics,,
Thus it is possible to say that a scalene triangle and an equila-
teral triangle are the same figure, But it would ho incorrect to
say that they are the same triangle, The reason is fairly obvious,
For the one condition for identity is the absence of difference.
Now the scalene triangle and the. equilateral triangle divide the
genus triangle by a difference that is proper to the, triangle, sin-
ce they are different species of triangle. That is why wo cannot
say that they are the same triangle without falling into a contra-
diction. But they do not differ by a difference of figure, since
they both fall under the same difference that divides the genus fi-
gure, namely triangle^ And that is why we' can say that they are the
same . figure >

Aristotle and St. Thomas give another example taken
from the realm of number, Kven though it is impossible to say that'
ten cows and ten dogs are the same ten, it is possible to say that thej
are the same number. In other words, there are two different species
of teji. but the same number. The same number is neither the ten cows
(for :then either the dogs would not bo ten or they would be identi-
fied .with the ten cows), nor the ten dogs (for similar reasons). It
is neither the one nob the other determirtately, but different from
both; It is not different, however, in the sense of being non-ten,
as three or twelve. It is' ten, but indifferent to the particular
species of ten„

Prom this example it is clear that the relation of
identity is something created by the mind. For to be the same num-
ber does not mean to be identical, otherwise the ten cows and the
t<m dogs would be the same, Hence the identity in question in this
"holo context is constituted by a relation of reason added to that
*ich is oredicable as genus of individuals or as remote gonus of

-457*

gpOOJ. GS a

V.lmt has been said of figures and numbers may bo ap-
plied to the ratio enti s, Both real being and logical being may
be said to bo the same being, provided that the ratio entis in
question be not identified with either the one o'rTlTcTother, In
othor words, the ratio entis can be snid to be the "same" only on
condition that it be "other", that is to say, it can be "3ame" only
if it is not identified with any of the terms in relation to which
it is said to be the samo. It must bo like the ratio "ad" of rela-
tioii, which is indifferent to "inesse", or "Non-inesse", or like
mathematical quantity which is indifferent to real or logical being,

It is to be noted that this oredication of identity
is not tautological. When we say that a scalene triangle and an
equilateral 'triangle are both the same figure, we do not merely
wish to say that figure is predicable of both of them in so; far
as both of them are figures. For the samo could bo said of jrian-
gle/slnce both of them are .triangles, predication of identity doea
not merely have to do with what is the "same" in the spociejs, na-
mely the genus, or with what is the -'same" in individuals, ijamoly
the species. It has to do with the differences in their very diffe-
rence - - not in an absolute- way, of course, for that would; make
them absolutely identical, but in their relation to tho genus that
is predicated of them by identity. Thus, this predication is not ma-
de after tho terms in question have been stripped of their diffe-
rence, for any genus may be. predicated of its inferiors in this way.
On the contrary it presupposes the differences. It is this, in fact,
that gives it its special significance.

That in relation to which the differences are said
to be the "same" is purely logical, namely the logical genus in so
far as it takes on a potentiality that derives from our mode of _
conception. The indetermination in question is not found either m
the terms themselves to which identity is attributed or in that
which is attributed to them, for both a scalene and an equilateral
triangle on the one hand and figure on the othor, are in themselves
definitely determined things, The indetermination is found in me
figure in so far as it is considered as a predicable genus In othe.,
words, predication of identity can exist only because it involves
logical intentions,

t- i, niP^r ihon, that by withdrawing into tho po-
tentiality of ^lo i^rS'where /fences oan be blended
It is possible to predicate the "same" of thing th t e essen^
tially diverse, to unite into ono things that aib uiv

rem.
tion

-458-

And this is of extreme importance for the question of the relc-
between scientific constructions and the absolute world.

In order to see why this is so, let us take a simple
example. When after an ordinary process of measurement we declare
that the proper length of a certain -body is two meters, this sta-
tement can he taken in two way 3. It may, in the first place, moan
simply that a meter measure has "been placed twice end to end along
the body; in other words that the length of the body is equal to the
length of two meters. As a matter of fact, however, when we say that
a certain "body has a length of two meters, we are not speaking for-
mally of the relation of equality "between the "body and the meter
placed twice end to end along its surface. We are not speaking
fonnally either of the absolute length of the "body, nor the absolute
length of the meter placed twice along its surface, nor of r the rela-
tion of equality "between the two, though all this is presupposed.
In qrder to "be able to say that a certain body has a length of two
meters it is necessary to go beyond a mere relation of equality and
arrive at identity. If the length of the body is equal to the
length of two meters they are the same length, but the^ire 'not the
same length of two meters, just as ten cows and ten dogs ar^e the
same, number but not the same number ten. '■

In other words, we have seen that operational defini-
tions do not allow absolute attributions, since the practical ope-
ration involved separates us from the terminus from which it is or-
dered,, Now when we say that a body has a length of two meters we
have! in a certain sense surmounted the gap created by this 'separa-
tion, for merely to describe the measuring operation and to say that
the : body has a length of two meters are not the same thing. This
has been done by ascending to a logical genus to which we have added
the relation of identity. In this way it has -become possible to
predicate the "same length" of the body in question. But, we re-
peat, the same length is not the same lgngth of two meters. In other
words, we have attributes to the body a logical genus which cannot
be identified with it. We are not in the real order, but merely turn-
ed towards it. If in this predication we actually reached the real
order there would be contradiction, for the length that is said to
*e the same for the body of two meters and for the '^Placed
twice along its surface would be. identified with both of them and
one. wo.uld be two.

It is clear that this identity adds something to the
u x ,v. j. a i,,r +hf» nnpmtional experiment and

unura secundum quid constituted by the operational A *

^b^iute-^ndition of the world By arriving *^£* ™
though it be merely logical, v/e have in some un,y

-459-

diversity involved in the umjmsoouradum quid, and have achievpd n
kind of counterfeit unum peFTe^ ~— ' achieved a

— ■ ■ \

What has just been said about the simple process of '
measurement/can be applied in a general way to nil of the construct-
ions manufactured by the scientist. Mathematical physics deals
neither .with the world of its own constructions as such, nor with
the absolute world as such; it deals formally with a world that is
a logical identity of the two.

But this logical identity is not an end in itself; it
is only a moans. And its purpose is to draw the scientist closer to
the absolute state of the universe. In so far as it keeps the scien-
tist 1 in the logical order, and' in so far as, the goal sought- for is
the world i ii so , experimental science must never strive to escape
from this purely logical identity and to draw ever closer to the
real world. In other .words, logical identity is not sufficient,
Science must seek to surpass it by tending towards real identity.
Ve have seen that mathematical physics is dialectics and that "ornnis
dialectics est tentativa" From the construct which is the physico-
mathematical world it must ever strive to roach the real wprld. To
this dialectical movement we must now turn our attention.

3=, Movement towards Seal Identity,

That the scientific world is constantly in movement
is a fact of history. But. there are two things to be noted about
this movement. First, it is something that is essential to the scien.
tific world, \7ithout it science would loose its meaning, In' this
experimental science differs radically from all the sciences in the
strict sense, which, though caught in the flux of history and in
some; measure subject to it, are intrinsically independent of all
movement. The reason for this characteristic property of experimen-
tal science has already been emphasized, the scfe ntific universe
is essentially a dialectical construct which must ever seek to go
beyond itself; it is a vehicule of progress and not a mansion of
residence,

'The second thing to be noted about this movement is
that it has a very finite direction. "It is P^»"> ^^^f '
"that when regarded as a whole, all the changes m the different

-460-

viowa of the world of Physics do no.t constitute a rhythmical swing
of the pendulum. On the contrary,* we find a clear course of evolut-
ion making more or loss steady progress in a definite direction."
(29) Frqm this point of view it is interesting and instructive to
contrast the history of experimental science with the history of
philosophy* Though philosophy in its inaer essence is independent
of movement, as we pointed out a moment ago, it appears to .he much
more a prey of the irrational flux of history. When viewed in its
entirety, the history of philosophy presents no definite direction;
it is constantly repeating and refuting itselfo

As Poincare has observed, to those who are unacquaint-
ed with the true meaning of experimental science the ephemeral cha-
racter of scientific views' and the constant succession of new theo-
ries may seem to have the same aimlossness. A.s a matter of fact,
however, these views and theories are continually tracing out a de-
finite pattern, "Sans doute, au premier ahord s les theories nous
semblent fragiles, et l'histoiro de la science nous prouve qu'olles
sont ephemeres; 'elles ne meuront pas tout entieros pourtant, et
de chacune d'elles ii reste quelquo chose." (30) The follpwing
comparison of Duhem "brings out with great exactness the existence
of a definite direction in the movement of science underneath a
superficial appearance af aimlessness,'!

Celui qui jette un regard de courte dur6e sur les
flots qui assaillent une greve ne voit pas la maree monter;
11 voit une lame se dresser., courir, deferler, couvrir une
etroite hande de sahle, puis se retirer en laissant a sec le _
terrain qui avait paru conquis; une nouvelle lame la suit, qui
parfois va un oeu plus loin que la precedente, parfois aussi_
ji' attaint meme pas le caillou que celle-ci avait mouille. Mais
sous ee mouvement superficiel de va-et-vient, un autre mouvement
Be produit, plus pro fond, plus lent, impercoptiole a lfohser-
vateur d'un instant, mouvement progressif qui se poursuit tou-
jour dans le meme sens, et par lequel la mer monte sans cesse.
to va-ot-vient des lames est l'image fidele de ces tentative^
d'explication qui ne s'elevent que pour s > ecrouler qui ne s a-
vancent que pour reculer; au dessous, se poursuit le progres
lent'et constant de la classification naturelle dont le flux
conquiert sans cesse de houveaux territ^es, «J ^ "sure
aux doctrines physiques la continuite d'uno tradition. (31)

fc«i science advances it often happens that ihe new

-461-

hecoraing more and more a suhjective construct. But no matter hov;
divergent new theories may he, they are nover torn in P vpouuib-
there is always a continuity with the past, "it happens", 'saysVeyl,
"that hroadened or more precise experiences and new discoveries' do
not overthrow old theories hut simply correct them. One looks for
the least possihle change in the historically developed theory
that will account for the new facts." (32) The Bohr atom did not
destroy the Hut her ford atom, hut merely corrected and developed it.
And the same is true of other changes through which physical .science
has passed* This does not refer merely to the gradual changes that
take : place in physics. Hven in the' so-called revolutions there is
always continuity with the past, The formulation of the Quantum The-
ory,, as Planck himself admits, was prepared hy summer's, pringshein's
Ruhen's and Kurlhaum's measurements of the spectrai distribution of
energy, hy Lenard's experiments on the photoelectric effect \ and
hy Franclc and Hertz's experiments on the impact of electrons. In
the same way, the Theory of Relativity was prepared hy Michelson's
experiments on optical interference. But more than that, it' is a mis-
take 1 to helieve, as many do, that the theory of Relativity and the
theory of Quanta mean a complete destruction of Classical physics,.
For it is necessary to assume the Classical theory in order to de-
fine 'the experimental conditions in which the theory of Relativity
ohtains to a higher approximation. And that is why Einstein hegins
his first paper, on the special theory of Relativity with the State-
ments "Let us have given a system of coordinates, in which the equa-
tions of Newtonian mechanics hold to the first approximation."

Like tho system of Euclid, or Ptolemy, of Newton which
have served their turn, so the systems of Einstein and Heisenherg
may give way to some fuller realization of the world. But in each
evolution of scientific thought new words are sot to old music,
and that which has gone hefore is not destroyed hut refocussed.
Amid all our faulty attempts at expression the kernel of scien-
tific truth steadily grows; and of this truth it may he said - -
The more it changes the more it remains the same thing. (33)

It is clear, then, that the development of. tho scien-
tific world does not take place in a haphazard fashion hut follows
a very definite direction. At the end of Chapter V we noted that
there is a similarity hetween experimental science and the type of
knowledge descrihod hy Russell in Mysticism and Logic in which de-
ductions are drawn from freely chosen hypotheses. How it is neces-
sary to see that there is also a vast difference hetween them. For
in the typo of knowledge considered hy Russell, there is no direc-
tion; we may, as he says, take any hypothesis which seems amusing.
Experimental science, on the contrary, is knowledge that is essen-

-462-

Unlly ordered towards a definite goal.

Now the relation between the scientific world and
the absolute world condition cannot he properly grasped unless it
he viewed m terms of a movement that "is essential to the former
and essentially orientated towards the latter. And we know of no wa;
of hringing out accurately this dynamic relation except hy having
recourse to a notion which plays its most familiar role in mathe-
matics and especially in the calculus, hut which can ho fruitfully
applied to other fields as well, We have in mind the notion of a
variahle ordered towards a limit. A hrief analysis of this notion
will throw groat light upon the orientation of the' scientific u-
nivarse towards the ahsolute world condition, (54)

This, notion in its most simple and generic form, is
usually expressed in terms similar to the following" k variahle
quantity x is said to tend towards a determined limit if the suc-
cessive values of x approach a certain fixed number a in such a way
that' the difference x - _a hecomos less than any give"n number e, no
matter how small it may he, Thus, for example , the number 2. may he
defined as the limit towards v/hich the following series tends;

1, 1. 4

s »

In the same way, a circle may he defined as the limit towards which
tends a regular inscrihed polygon whose sides increase indefinitely,
Applying this now to the question in hand, we hold that the scienti-
fioworld may he considered as a variahle quantity which hy passing
through the successive stages of its evolution approaches the ahso-
lute world condition as its limit.

An analysis of this notion reveals that it involves
toth a heterogeneity and a homogeneity, hoth an otherness and a
likeness, The heterogeneity, the otherness, consists in the fact
that there are necessarily two terms which helong to different
orders or to different species} e.g, discontinuous-continuous; ■
point-line; line-surface; polygon-circle; curved-straight, etc.
Heterogeneity is essential to the notion of limit, even under the
aspect in which the limit is considered as a value of the variahle
term- it is precisely in its heterogeneity that it is the Unit
value of the variahle. It is not a polygon (no .flatter of how many
sides) that is the limit of the polygon whose sides increase inde-
finitely, hut a circle. On the other hand, oven though the polygon
becomes more and more like a circle, it is not changing m its spe-
cies (in which it remains essentially a polygon), ^™*f/ ™
its values. No* this heterogeneity is found m the relation hetwocn

-463-

tho scientific universe and the absolute universe. A great derl
of emphasis has already been laid upon thoir essential otherness.
It ib not an advanced stage in its development that the scientific
world i8 attempting to reach in the movement that is essential to
it, hut something beyond itself and essentially other than itself,
namely the absolute state of the universe. On the other hand, even
though the scientific world in its development comes ever closer
to this absolute state, it does not in airy degree lose the other-
ness which derives from the fact that it is essentially a construct
On the contrary, this otherness increases, just as the polygon be-
comes, in a sense, more of a polygon, i.e. a many-sided figure,- the
more its sides are increased,,

But along with the heterogeneity there is an essen-
tial homogeneity involved. This is evident by the very fact that
one term is said to be the limit of the otber^ When we say that
x has a as its limit (lim x a), the fixed term a is considered as
the limit value of x, in such"a way, that lim (x - a) 0. From this
point of view the heterogeneous terms are consTdered as belonging
to the same order, that is to say, one is considered as a value or
a case of the other, A polygon with a hundred sides is considered
as a case of the polygon; a circle is considered (in a hypothetical
way) as another case - ~ the limit cases if the limit could bo rea-
ched the case of the polygon which is the circle would differ from
all the other polygons in that it would have the greatest number of
sides possible, From this point of view there is an order of conti-
nuity between the variable and the limit. And it must be noted that
the"more" or "less" of the formal order of the variable quantity is
not merely quantitative,, That is to .sa5 r , a certain given value of
the variable is not merely greater than any preceding' value; it is
at the same time more like the limit. In other words, by running
through its values the variable is related to the formal structure
of the limit. The increasing structural similarity tends towards
structural identity.

A homogeneity of this kind is found in the relation
■between the scientific world and the absolute world. The former
tends to issue into the latter. If the limit of scientific develop-
ment could be reached there would be identity of structure between
the two. And as the limit is approached the likeness of structure
which we explained above by the notion of isomorphism, increases.
While at any given stage of the development there is a certain
likeness of structure between the two worlds, it is inadequate
and often extremely misleading to consider this static relation
independently of the dynamic relation that the movement which is
essential to the scientific world involves. This is suggested m

-464-

tho following' passage of Sir Arthur Eddingtonj

_ Scientific discovery is like the fitting together
of the pieces of a great jig-saw puzzle; a revolution of scien-
ce does not moan that the pieces already arranged and interlock-
ed have to he dispersed; it means that in fitting on fresh pieces
we have had to revise our impression of what the puzzle-picture
is going to he like. One day you ask the scientist how he is get-
ting on; he replies, "Finely, I have very nearly finished this
piece of hlue sky," Another day you ask how the sky is progres-
sing and are told, "I have added a lot more, "but it was 1 sea, not
sky; there's a "boat floating on top of it," Perhaps next time
it will have turned out to "bo a parasol to "bo upside doy«i, hut
our friend is still enthusiastically delighted with the pro- N
gress he is making. The scientist has his guesses as to how the
finished picture will work out; he depends largely on these
in his search for other pieces to fit, hut his guesses are mo-
dified from time to time "by unexpected developments as the fit-
ting proceeds. These revolutions of thought as to the final
picture do not cause the scientist to lose faith in his'handi-
work, for he is aware that the complete portion is growing
steadily. These who look over his shoulder and use the present
partially developed picture for the purposes outside science,
do so at their own risk„ (35)

There i3, then, in the notion of limit the paradox
of heterogeneity. And the key to this paradox, as has just heen
intimated, is found in movement. For one term is ordered towards a-
nother as its limit, not in its proper specific character, "but only
in so far as it is considered as a variahle whose successive values
approach the term which is the limit. These successive values must
he indefinite; "between any given value and the limit there must
always he an infinity of other possible values in potency. But this
potential infinity is not sufficient. It is merely the foundation
of something more, namely a progression, a movement, a hecoming.
Because of this movement the difference hotween the two terms de-
creases indefinitely,, In this way the variahle tends to enclose the
limit as its own final value, Heterogeneity tends towards homogeneity
The variable tends to hound itself hy going heyond itself, that is
to say hy going heyond any value actually given within itself; it
tends to hreak through its own form and thus destroy itself hy ta-
king on the form of the limit. (36) In other words, hoth the .
variahle and the limit have a douhle state-, an absolute state , which
consists in their irreducihle otherness, and a state of hecoming
*y which they tend to reduce this otherness to sameness, The valla-
te is always essentially other than the limit, hut at the same time

-465-

it is always becoming the limit. In the same way, the limit has
an absolute state by which it is essentially different from the
variable, but at the same time it is a state of becoming - - a
stata of "coming from" the variable. i; The limit must bo coming from
the otherness that is the variable, asJLf it were precontained in
that otherness. The variable - - whoso proper values are being more
and more actualized, so that the variable itself is becoming more
and more the self that it over can be - - must at the same time be
moving away from itself and becoming identical with what is other-
ness to it, viz. the limit." (37) In so far as the limit is consi-
dered as coming from the variable it may be said to be generated
by the progression of the variable^ Thus this progression triumphs
over the givenness of the limit and in this sense rationalizes
the irrationality of this mere givenness.

This movement of which we have been speaking is nui
genqris, for by its very nature it is a movement that can never ar-
rive. Whereas the terminus of every other movement, such as the beco-
ming of a house, is defined by the possibility of its actually being
reached (whether it actually will be reached or not) the limit of
this movement is defined by the impossibility of its being reached.
Whereas the terminus of movement in the ordinary sense can still be
considered the terminus even though the movement towards it has ac-
tually ceased> the limit of this movement ceases to be a limit once
the getting closer ceases. In other words, the notion of limit sup-
poses an actual and indefinitely prolonged movement. Just as all re-
lations consist in an "esse ad', 30 all movements are towards some-
thing other. But just as some relations are of such a nature that
they cannot "be in" that "toward" which they are, so this movement
cannot actually reach the limit towards which it tends. From one
point of view this movement seems to be an end in itself* since it
can arrive never at anything beyond itself. But from another point
of view, it is not an end in itself, since it must ever tend towards
the limit which is beyond itself.

How all this has an application to the relation bet-
ween the scientific world and the objective world Both .of them have
an absolute state by which they are essentially heterogeneous. But
they also have a state of becoming which tends to reduce this ^re-
ducible heterogeneity to Tamogeneity. In so far as the scientific
world is concerned this state of becoming co ™™\ZZZZZ£T
development by which it draws ever closer to the 0* ,ective world,
in so far as L objective world is concerned ^J ^° ^
ming does not, obviously, mean a real change 1* m j
as the scientific world draws closer to the <*jootxve world, the
latter may be considered as corning from the forme*. In thxs way,

-466-

the absolute world condition may be viewed as being generated by
the construction of the scientist; thus its pure givenness i3
triumphed over and the irrationality of this glvimiess rationalized.
As we remarked in Chapter IV, if the scientist could roach his goal,
man would he God. But there ia one difference to lie noted here bet-'
ween the movement of a variable towards a limit and the movement of
the scientific world towards the objective world. In the former case
the limit is already known ."before the movement towards it begins.
In the latter case, this ia not trues the absolute world is an
unknown quantity that gradually reveals itself as. the movement to-
wards it progresses. In this way. the state of becoming of the objec-
tive world has more of the nature of a generation,

: It is clear that the objective world as a limit can-
not be reached by the progress of science. The aim of science, writes
Planck, "is an incessant struggle towards a goal which can never bo
reached. Because the goal is of its very nature unattainable. It
is something that is essentially metaphysical and as such is always
again and again beyond each achievement. 1 ' (38) . The very method to
which experimental science and especially mathematical physics is
committed makes it impossible for it to over reach the objective
universe as it is in so. And yet by a strange paradox, it is only
by remaining faithful to this method that it can bo carried closer
and closer to this goal, (39)

All this brings us back to what was said earlier in
this study; experimental science is essentially a vehicule of pro-
gress and can never become a mansion of , residence. And to consider
it as a mansion of residence is the most effective way of destroying
its true relation to the absolute world condition. From this point
of view, the movement, of the scientific world may be considered as
an end in itself, and in this sense we may accept the dictum of
Gotthold Leasing to which frequent reference is fomid in the writing,
of modern scientists: "Not the possession of truth but the efiort
in struggling to, attain it brings joy to the ^searcher. J£0)
But from another point of view it is obvious that the movement of the
scientific world is not. an end in itself, ffl^end ^^"f^
the absolute world condition. The scientist who lose s himse If in
the development of his own subjective constructions is not true to
his science. It must be noted, . moreover that while it is bettor
to be able to move towards truth than not to be able uo ^^ ^
at all, it is absolutely speaking far better to be m the full posses
aion of truth than morely to be approaching it.

Tt 1 , obvious that the reason "while tho vr.riabld can-
It is obvious *n. ivnl WO uld involve a contra-
ct arrive at the limit is that tnis a"««

-467-

diction. The limit of a polygon would be both a circle and o poly-
gon, that is to say, Tooth a circle and a non-circle, Tooth aono-si-
dod and a many-sided figure-, hoth an unbroken and a: broken line,
This contradiction is an essential condition for terms to he related
as variable and limit. V/'hen it is stated that a polygon and a cir-
cle meet at infinity, this merely means that they would meet if per
jinp ossibile "at infinity" could ho. The variable tends tovzards its -
ultimate value and at the same time at something that is essentially
other than any of its values. In other words, tho tendency to rea-
lize itself is a tendency to destroy itself. But this does not mean
that the dialectical movement towards the limit is in itself contra-
dictory and meaningless. The contradiction that would Toe is only at
the limit, which cannot Toe attained. The movement itself . cannot he
considered contradictory simply "because it cannot attain a contra-
diction. The possibility of this movement does not depend upon the
possibility of attaining the limit but upon the possibility of con-
sidering the term toward which the limit tends as the limit of this
movement » The movement in itself is meaningless precisely "because
it never goes "beyond the 3tage of "being towards",.

Now the movement of tho scientific world is a move-
ment towards contradiction. This has. already "been alluded to on seve-
ral occasions throughout our study. We have seen in a general way
that' the scientific universe in seeking to posit itself more fully
tends to negate itself and to vanish into emptiness. Several parti-
cular forms of this tendency towards contradiction have already
bee^ indicated. But it is of extreme importance to examine this quest,
ion more closely here, for nothing could bring out more clearly and
fully the poetic structure of the scientific world. A.nd this can
best he done .Toy showing that the most fundamental and the most pro-
per principles of experimental science are such that they could
not he really true without contradiction, that is, they could not
be true without being false. These fundamental principles are the me-
thodological principles such as the principles of definition, of
identity, of unity, of order, of induction, of simplicity, etc. Lot
us consider a few examples in detail*

The first example to he examined is the principle _ of
definition. We have seen that in mathematical physics all d ^ini-
tions are in terms of operations of measurements. Now both from the
point of vi,w of measurement and from. the point of view of oration
this principle of definition i-olvos^mathe-tica phy ic in a mo

v^S a Uȣ tnTattaii^of which would imply^ con-
tradiction, in so far as """^Va^nSTJeSca foV a^ma
from all that was said m Chapter VIII about th ___

mensura„ Progress in measurement must consist

-468-

greater precision and certitude. The limit of this movement would
■be an absolutely minimum measure. But such a measure is a contra-
diction since it implies a quantity that is at once continuous and
non-cont inuous .

A similar movement towards contradiction is discovered
when the nature of operational definitions is analysed. We saw in
Chapter IV that these definitions express a mixture of nature and
art, of a quod and a quo, of suhject and ohject. The thing defined
is neither a pure operation, nor a pure ohjective quantity,, hut an
inextricahle mixture of the two. In other words, the definitum is
only an u num per acciden3 and not an unum per se. The unity is con-
ferred upoii it "by the mind. If it were a p"er~3lT ~unity , the world
would ho at the same time nature and a human work of art. This is the
position of the Marxists.

It is clear, then, that 'while operational definitions
are destined to help us to know the real i n se (for operations are
not carried on for their own sake, and physics doea not consist in
mere descriptions of what physicists do), a reality which could he
known in se hy means of operational definitions is an impossibility,,
By means of operational definitions we tend towards a limit which^
cannot he attained hy means of operational definitions. The practical
operation involved separates us from the terminus towards which it
leads us. Arrival at the limit would involve a complete arrival at
the limit and a complete, separation from it at the same time.

Another good example is found in the principle of
induction. Poihcare's statement that all generalization is an hypo-
thesis is true of the type of induction that is characteristic of
experimental science - - induction hy enumeration. When a general
proposition "Every A is B» is founded merely upon the enumeration
! "4 is B", "A is B", "A is B", etc., itf cannot he true. For it
"Every A is B" is true, "Some A is non-B" is false. But in so far as
"Every A is B" is founded merely upon, a collection ox particular
cases, it cannot he said that "Some A is non-B" is xalae. qenoe,
"Every A is B" is logical proposition that tends ^f^g^
without heing ahle to attain it. It is, so to speak, a /^ r ^ f
without "inetse". If "every A is B" wore true in so far ^found.d .
upon a collection, all A's would not only he ^f^^Jvae-
identified - - They would he the same A. Eor ix in d ^J 10 " ^ ™
ration could give a universal i » ^^tloSlar ; aSes would ^ the
would ho the particular, caso, and the parcicuiax u
same particular case. Hence there would he contradiction.

The principle of causality as employed in physics

-469-

offers a third example' for our analysis. Events are knowablo . by us
only in so -far as they are determined. Hence the future can tie ade-
quately known by us only to the extent in which it is already de-
termined in the present,, G?he future is, of course,- of great im-
portance in the fluiduniverse that constitutes the object of phy-
sics. The future is a part of our world, for without before and after
there would be no time Now it is evident that there must be a certain
amount: of determination in the relation between present and future,
since the universe is not run by pure chance. The question is, how-
ever, is this determination abaolute?

It is the purpose of science to get at the determina-
tion in the cosmos,, In order that no real determination may escape
it, it must consider all the indetermination that appears as merely
provisional. It is- necessary for science to act- in practice as though
the determination in the world were absolute and without limitation.
It must, as Laplace said, consider the present state of the univer-
se a : s the effect of its anterior state and as the cause of the state
to follow. "In this sense, then, science must take determinism as a
methodological principle.

But this methodological principle cannot be made a
real principle without a contradiction being involved, For ?.n order
for this principle to be real, it would have to be verified' in expe-
rience. Such a verification, however., is impossible. More than that,
even if it were possible the principle would be absurd. For; this prin-
ciple has to do with the future, and with the future in its. enter ity.
Hence the verification of it would have to mean verification of the
whole future. The. known, present cannot serve to confirm i.^IJow it
the future in its entority were present to us, this principle v/ould
be useless, for it wouldPS- pure tautology. In order for it xo have
any meaning at all, it is necessary that the present truth pf «£
future be future in relation to us. If the truth of the ^£« were
present, the principle would not need to be confirmed m thp present
experience, , '

In other words; on the one' hand, the validity of the
principle' which af firms the present truth, of the future depend- u- -
pon the future as non-present. But the truth °°^ d ^°J h J y ^ e is
principle is not known as certain exoepVxn s far as he f uture^
effectively present. On the other hand ^ * h £ mna th0 condi _
confirm the principle is Present ^ w ^% fltal)liBh the validity
tions required for a confirmation that woiu . s0 far as

of the principle for the whole future I ^haf the future is
it is future and not in so ; far as it is p dQ terminism would

not certain. She verification of the pxinoipi

-470-

lmve to consist in rendering tho future evident in the present by
means of the future which is non-evident, in other words, to make
■the future certain by means of the uncertain future.

It is clear, then, that the principle of causality
in physics has a meaning^ and can he said to he true, only in so
far as it is not really true. It is mereiy a methodological principle j
it tells us how to proceed and not what things are objectively. In
so far as it tells us how to pro'ceed it is true. In' so far as it
attempts to tell us how things are in themselves, it involves a
contradiction.

These examples suffice to show that the movement of
the/scientific world tends towards a contradiction. The meaning of
this tendency will he made clearer if we return to the notion of
predication of identity discussed earlier in this Chapter, yie saw
that in this predication we consider the terms which are either spe-
cifically or individually different not merely in what they have
in common absolutely, hut in their very formal differences. It is
this, in fact, which characterizes predication of identity. ; polygon
'and 'circle, for example, have an identity in their very differences
(considered of course in relation to their remote genus). Npw in our
discussion of the notion of limit we saw that it supposes two terms
which are at once the same and different. That is to say, thelimit _
must he comprised in tho variable; since it is the limit of the varia-
ble it' must be considered as comprised in the or Aer of the variable.
Now this tendency which the limit supposes is accomplished by pas-
sing to the genus that is predicable by identity. Consequently, the
notion of limit is founded on a predication of identity of the dif-
ference So

Now the dialectical movement consists precisely in
the tendency of one difference towards another diff ^e with in
their abstract identify. This identity in the <li«°renoo is a prin
ciple of dialectical movement, but it is not a ^™3 .^ *^£
dency of one difference towards another diff a . r ?"° e "^^^J^,
tract identity, is a tendency towards an idont ity of »»°^ °™er,
namely real identity. It is the realization of .his real identity
that is impossible.

All this makes it evident onca again how much truth the
All this "^ K ° b t if goionce/could arrive at

re is inMeyerson's central « ie,ra *£"* rG3U it would be a vast
the goal which it is constantly Peking the resui comeotiorl

tautology, and how correct De Broglie is in ^^ of Vnllery
with his description of I/Ieyerson's J" ™ 1 ™ * la r . n al)SU rdity.
to the effect that what science seeks to achiev.

-471-

It is clear, however, that this absurdity is not merely that of a
vast tautology, "but that of an intrinsic contradiction. If the iden-
tities which we posit in science were real identities, the logical
and the real order would he identified.

In spite of the contradiction at the limit, science
tends to emerge from mere logical identity to real identity, <'-'e
find this tendency on every level of the scientific structure. In
the definitions we tend to pass from logical identity to the abso-
lute world condition, even though an arrival would tie contradictory.
The same is true of scientific laws; generalizations tend towards
a universal nature, even though if such a nature were achieved it
would "be contradictory. The case of hypotheses is very much the
samet they are destined to make the truth known, hut they cannot
provide truth ex propriis . The truth which they help to reveal does
not depend in any way on them. If hypotheses could "bo identified with
their terminus (which is known "by experience) they would destroy them-
selves as hypotheses. Finally, scientific deduction is orientated
towards a true conclusion. But it cannot provide this true conclusion,
that is to say, the conclusion cannot "be true qua conclusion. Bet-
ween the conclusion- taken as such and the truth that it permits us
to discover there is only an accidental connection, since another
deduction could serve to reveal the same truth.

Since therefore, the initial definitions cannot give
us the real as it exists in itself; since physical laws are ; only
generalizations which are never really founded in any absolute sense;
and : since deductions cannot be true as such, it is evident that the phy
sical world cannot be identified with the absolute world condition.
It js, consequently, merely a construction of the mind - -^cons-
truction which imitates more or less the absolute world , 1B *£™f d
towards the absolute world, and can approach it indefinitely without
ever being able to reach it.

The Marxists have sought for a proof of _ their dialec-
tical materialism in this characteristic nature of science Their
line of argument may be reduced to this, The ™<^° d °l°f 1 ^ ?t is S
are true. But if they are true, the world is cent ^^'^^
the same time affirmation and negation of itself , at the *™*J>™»
true and false; there "Is no absolute truth Oonseg.«ontl y since
this state of things cannot satisfy specu a iv, | p ^J^^*
made for thaught, but for action. (41 ine ° llu "true" in

consists in an exploitation of the ambiguity of the term true in
the proposition "the -the do logical g^^^oSTa- logical
"ig analysis has made it clear that they aro 3 \ mifY the

principles and not in the sense in which they would signify

-472-

truth of tho world _in^G 1 In other words, there is a confusion
hGrS between the logical and the real order. Logical possibilities
have greater freedom than real possibilities. In the logical order
it is reasonable to build structures with elements that are not
capable of realization,, Nor does the lack of this capability pre-
yen* the possibility of drawing closer and closer to the real. It
is possible for logical constructions to comprehend being nhd non-
being at the same time. "Non homo", for example, is an indetermi-
nation which comprises at the same time both being and non-being,

Because the scientific world is a logical construc-
tion, because it is dialectics, there is deep within it an essentia]
conflict from which it ever seeks to deliver itself. Ii^he first pla-
ce, there is the conflict between being and non-beingl Experimental
science tends towards being by means of the impossible. It tends
towards the real by means of the purely logical. There is, moreover,
a conflict between tho one and the many; it tends towards the one
by means of the many,. There is a conflict between the speculative
and the practical, between science and art. Because of its opera-
tionalism, 1 mathematical physics tends in its experimentation to-
wards the res in its physical, entitative stutust at the same ti-
me it tends towards pure science in the intellect. For this rea-
son it tends to issue into two contrary directions* on the one
hand pure' science, independent of phys.ical operations of things
in their entitative status; on tho other hand, pure operation by
which things are mastered through action. That is why there will
always be two fundamental tendencies in mathematical physics- one
towards a kind of Platonic mathematicism, and the other one towards
a kind of dialectical materialism whose ultimate aim is to master
things through 'and for practical' action.

Perhaps "the general drift of this whole Chapter can
bo summed up by saying that, the scientific world is a structure
composed of both the subjective and the objective and that if the
goal towards which it strives .could be^ reached it would be at the
same time completely subjective and completely objective. For this
reason it is necessary befor.e bringing this study to a close, to
turn our attention to the question of the subjective and the ob-
jective in mathematical. physics. It has been customary for scien-
tists to claim that philosophical and theological knowledge are
essentially subjective and thai only experimental science is ca-
pable of giving true objective knowledge, f/e must try to see why
JU-st the opposite is, 'the case.

-473-

CH/lPTTTR TV/3LV®

OBJECTIVE SUBJWIVITY

1. Subjectivity and Objectivity.

As we. explained earlier in this essay, all knowled-
ge is by its very nature objective, since to know is to -become a-
nother thing in its very otherness. But not all knowledge Is, equal
ly objective, for there is a direct proportion between objectivi-
ty and the perfection of the knower. In God alone is perfect objec
tivity found.

Now the word subject can be taken in two ways. In
the first place, it can be understood to mean simply a knower.
In this r.ense, all knowledge, in so far as it implies that a known
thing is in a knower (cognitumest in cognoscente) involves both
a subject and ah object. In its proper meaning, however, the term
subject implies a. state of subjection., Ehia involves passivity,
and consequently limitation and imperfection.

When the term is understood in the first way there
is no opposition between it and objectivity. In. th " " n ^ 1 J h ^_
be applied even to God, in Whom knowledge is _ so Perfect and there
fere so objective that there is.no real distinction between the^
knower, ■ J the knowledge and the ^ J"^ ts P op-mean
ing, however, there is an opposition between i* an •>

In fact, a pure subject in this sense is an object that
know at all.

la in some. measure a subject in the piopoi

-474-

o

all creature a recGive thoir knowledge from without and their state
of being recipients involves passivity and subjection. This is true
even of the angels, for thoir intelligible species aro impressed
upon them by God. An object, in its full formality ns object, is
above every created intellect, for in so far as an intellect is
a subject in the proper sense of the term it is measured by the
object, and a measure, from the point of view in which it is a' mea-
sure, is always more perfect than the thing measured. Creatures
cannot be the measure of objects because their being is not the
source of these objects. Their cognitive powers cannot reach the
very root of these objects because they are not the root.

This subjectivity (in the sense of the term in which
it is opposed to objectivity), already found in the highest angel,
increases as we descend the hierarchy of created beings. It is found
in the fullest measure in which it can bo found in sense knowledge,
for here a material organ, which in itself is a pure subject and
hence absolutely opposed to objectivity, enters into the very in-
trinsic structure of the cognitive power. But already in the human
intellect (which is the lowest type of intellect that could possi-
bly exist) a large measure of subjectivity is found. For the human
intellect has this in common with the senses that it receives its
species from things. This involves a greater measure of subjection
and passivity than is found in angelic knowledge in which tho spe-
cies though coming from tho outside, do not come .from thing3 (they
are, in fact, prior to things) but from God, How the obscurity a-
rising from this passive subjectivity forces tho human intellect
to have recourse to a kind of active subjectivity, That is to say,
it can know only by constructing logical beings, by composing and
dividing in its judgments, by fabricating formal discourses in its
processes of reasoning. This active subjectivity is also an obsta-
cle to pure 6bjectivity. For all of these reasons it is necessary
to agree with "Eddington that "it is the inexorable law of our ac-
quaintance with the external world that that which is presented
for knowing becomes transformed in the process of knowing." (1)

But this subjectivity of the human intellect must
not bo exaggerated there is a sense in which it is true to
say that thf mind is capable of a kind of pure objectivity. In its
ordinary processes and in the way in which it functions m the
Philosophical sciences it is able to disengage the quod -quid est
of things - - their objective essence %fe Tjjere ^a^aySg a MffiW subjectivity
a»unt of subjectivityfe V °tehls° 2oV& fech o that rtuofc is
known as to tne way in which it is known or the a ***°l+Z\l?£t
is known. To use Scholastic terminology, it is a ^f °^* £*
affects rather the n E ^s^uo_cc^no^citur than i d quod cogn,sciuur .

-475-

Bharo is of course , a kind of subjective element entering into the
object known, but it is more of a negative than a positive thing,
That is to say, in comparison with the object in se the object
as known is always imperfect and inadequate. BuT~this does not trans
form the object in the' sense of making it a now object, Definitions
of the kind can apply with perfect truth to things as they are in
_se. In other words, the mind does not project a now positive ele^
ment into: the essence it knows in such a way that this essence is
reconstructed into something different. In this the intellect dif-
fers essentially from the senses which in knowing their object ne-
cessarily transform it into something different because of the phy-
sical interaction which takes place between object and organ.

How, as we have seen, physics deals with sensible
thing's under the aspect in which they are the most profoundly im-
mersed in sensible matter,, That is why the obscurity of sensible
matter and the subjectivity and anthropomorphism attached to sen-
sibility are of major concern for it. We have seen what means it
has devised to triumph over these obstacles and how great has been
their success. V/e have noted that Planck was correct in writing
"that as the view of the physical world is perfected, it simulta-
neously recedes from the world of sense; and this process is tan-
tamount to an approach to the world of reality." (2) But we have
also insisted upon the fact that this movement away from the world
of sense and towards tho world of reality is at the same time a
movement away from the world of reality towards a subjective world
in such a way that if it be asked which of the two famous tables
of Sddington, (3) (the familiar table and tho scientific table)
is tha more objective and which is the more subjective, it is ne-
cessary to lmke a very important distinction! the scientific table
is at once more subjective because of the essential subjectivity
of scientific method, and more objective, i.e. more like a table
as it is known by a superior intellect.

Tho profound subjectivity of the phygico-matheiwvti-
cal world 1 is now generally admitted by all the bettor scientists.
(4) But it is important to try to determine tho nature of this
subjectivity. By a strange paradox, the movement of science awny
from tho sense world towards the world of reality is at the samo
time a movement away from the world of reality to a world that is,
from one point of view, subjective in essentially * h ° ° a ™ "f "°
the sense P werld. What we mean here is tha t. ,us as he onae v rid
^subjective in a way that pats a P« s ^ S ^ angforminK lt
''he object and reconstructs it vo miu oxi^nu _

«„ ,L ttl „« ai „«»t 3 o » rs£ Sr - S S y-^

sitivo subjective olomont into its oojeoi, ,.n

-476-

somothing essentially different. There is therefore, a sharp dis-
tinction to be drawn between the type of subjectivity that 3a cha-
racteristic of experimental science and the type mentioned a few
moments ago that accompanies other kinds of intellectual knowledge.
In tho latter case, art merely surround s the object, whereas in
the case of experimental science art enters ' intrinsically into tho
object and constructs it. And just as in the case of sense know-
lodge the objective and the subjective are so intorpenatrated that
it is impossible for tho knower to draw a lino between them and
thus set forth the object in its pure objectivity, so in mathema-
tical physics the subjective and the objective are so fused that
it is impossible for the scientist to distangle them. (5) In or-
der to do this he would have to have direct intellectual intuition
of the real world.

In the course of this study we have endeavored to in-
dicate the most important ways in which subjectivity enters into
scientific knowledge,, (6) All of them s as has already boen sug<-
gosted, may be traced back to two sources First there is a. phy-
sical intrusion of the subject in tho experimental operation in
which the object known becomes irretrievably confused with tho way
by which it is known. Secondly, there is an intellectual intrusion
consisting in a priori hypothetical construction. Mathematical phy-
sics has no other means of getting to know reality except by re-
fashioning it in these two ways. It cannot assimilate reality di-
rectly; it can only reconstruct it„ It is, as Einstein and Infeld
have suggested, (7) in a position something like that of a man
trying to understand the mechanism of a closed watch. Since! he has
no way of opening the case, he cannot know the insido of the watch
as it is in itself. All he can do "is construct something that will
account for the moving of the hands and the ticking. As Meyers.on
has remarked: "nous voulons le reel cpnformo a la rnison, ma is nous
comprenons en nieme temps que s'il l'etait, la raison devrait pou-
voir le recreer," (8)

Since, then, the scientific world is formally a
subjoctive construction, it follows that its constitution is pre-
determined by the methodological principles employee in construct-
ing it. "Operabilia. sunt quorum principia sunt in nobis. A also
follows that to the extent in which it is so predetermined it can
De known a priori by a close analysis of those principles and their
implicate This, it seems, is the gist of 3 clding-ton'sJhe|hi-
Igsg ghy of Physical Scie nce, the substance of which he has expiessed
in the following passages t

Let us suppose that an ichthyologist is exploring

-477-

the life of the ocean. He casts a net into tho water and "brings
up a fishy assortment. Surveying his catch, he proceeds' in the
usual manner of a scientist to systematize what it reveals.
He arrives at two generalizations:

(1) Ho sea-creature is loss than two inches long.

(2) All sea-creatures have gills,.

These are both true of his catch, and he assumes
tentatively that they will remain true however often he re-
peats it.

In applying this analogy, the catch stands for the
"body of knowledge that constitutes physical science, and the
net for the sensory and intellectual equipment which we use
in obtaining it, The casting of the not corresponds to obser-
vation; for knowledge which has not "been or could not ho ob~
tained by observation is not admitted into physical science

An onlooker may observe that the first generalization
is wrong, "There are plenty o.f sea-creatures under two inches
long,' Only your net is not adapted to catch them," The ichthy-
ologist dismisses this objection contemptuously, "Anything un-
catchable by my net is i pso facto outside the scope of ichthyo-
logical knowledge, and is not part of the kingdom of fishes
whichi has been defined as the theme of ichthyological know-
ledge. In short, what my net can't catch isn't fish." Or -• -
to translate this analogy - - "If you are not simply guessing,
you are claiming a knowledge of the physical universe disco-
vered- in some other way than by the methods of physical science,
and admittedly unverifiable by such methods. You aro a meta-
physician. Bah.' "

The dispute arises, as many disputes do, because the
protagonists are talking 'about different things. The onlooker
has in mind an objective kingdom of fishes. The ichthyologist
is not concerned as to whether the fishes he is talking about
are from an objective or subjective class; tho property that
matters is that they are catchable. His generalization is
perfectly true. of the class of creatures he is talking about
- - a' selected 'class perhaps, but he would not be interested
in mak S generalizaUons about any other ^-' n'scfencT
logy, if we take observation as the basis P^^f^^fcm
and insist that its assertions must be ver ifiaole by Ration
we impose a selective test on the knowledge which is admitted
as physical. The selection i^J^ £ ^L^url™
on the sensory and ^ el \ e0 ?™ wl °2£ P ™£ is to such subjectively
of acquiring observational knowledge. , v formulated

selected knowledge, and to the ^ TarB ° ™ io "g "_ the so-
to describe, that the generalizations of physics

called laws of nature - - apply •

•-478-

It is only with the recent development of opistono-
logical methods in physios that we have come to realize the
far-reaching effects of this subjective selection of its sub-
ject matter. We may at first, like the onlooker, ho inclined
to think that physics has missed its way, and has not reached
the purely objective world which, we take it for granted, it
was trying to describe. Its generalizations, if they refer to
an objective world, are or may he rendered fallacious through
the selection. But that amounts to condemning ohservationally
grounded science as a failure because a purely objective world
is not to be reached by observation. . „

Suppose that a more tactful onlooker makes a rather
different suggestion;, "I realize that you are right in refusing
your friend' s hypothesis bf uncatchablo fish, which cannot
bo verified by any tests you and I would consider valid. By
keeping to your own method of study, you have reached a gene-
ralization of the highest importance to fishmongers, who would
not be interested in generalization about uncatchable fish.
Since these generalizations are so important, I would like to
help you. You arrived at your generalization in the traditional
way by examining- the fish. May I point out that you could have
arrived more easily at the same generalization , by examining
the net and the method of using it?"

The first onlooker is a metaphysician who despises
physics on account of its limitations; the second onlooker is
an epistemologist who can help physics because of its limitations
It is just. because of the limited - - some might say, pervert-
ed - - aim of physics that such help is possible ....

Generalizations that can be reached opistomologically
have a security which is denied to those that can be reached
empirically,., some laws of nature may have an epistemological
origin. These are compulsory; and when their epistemological
origin is established, we have a right to our expectation that
they will be obeyed invariably and universally. The process
of observing, of which they are a consequence, is independent
of time or place. (9)

It would take us too far afield to analyze and assess
the validity of the development and applications "which Sddington
subsequently, makes of the principles laid down in.; those passa-
ges. But after all that has been said, about the subject iv? ins-
truction of the scientific world we do not see how the Prin-
ciples themselves can be called into question, ho reove , we
feel the implications of these principles are so ^ leachin
that all of the laws of physics without except ion ^ ^
cognized as subjective. (10) later m the same voik Aldington

-479-

,. R ys great stress upon a point that is vital for tho question which
forms the. subject of this Chapter- tho scientific world is not sim-
ply discovered, it is manufactured -by the scientist.

The question I am going to raise is - - how much do
we discover and how much do we manufacture by our experiments?
When the late Lord Rutherford showed us the atomic nucleus did
ho find it or did he make it? It will not affect our admiration
of his achievement either way - - only wo should rather like
to know which he did, Tho question is one that scarcely admits
of a definite answer. It turns on a matter of expression, like '
the question whether the spectroscope finds or whether it makes
the green colour which it shows us. But sinco most people are
probably under the impression that Eutherford found the! atomic
nucleus, I will, make myself advocate of the view that he made
it, The tendency of writers on quantum theory has been perhaps
to. go farther than I do in emphasizing the physical interference
of our experiments with the objects which we study. It is said
that the experiment puts tho atom or the radiation into the
state ; whose characteristics we measure. I shall call this
Procrustean treatment, Procrustes, you will remember, stretched
or chopped down his guests to fit the "bed he had constructed.
But perhaps you have not heard the rest of the story. He mea-
sured. them up "before they left the next morning, and wrote a
learned paper "On the Uniformity of Stature of Travellers"
for the Anthropological Society of Attica, •. .

Suppose that an artist puts forward the fantastic
theory that the form of a human head exists in a rough-shaped
block of marble. All our rational instinct is roused against
such 4n anthropomorphic speculation. It is inconceivable that
Nature should have placed such a form inside the block. But
the artist proceeds to verify his theory experimentally - -
with quite rudimentary apparatus too, Merely using a chisel
to separate the form for our inspection, he triumphantly pro-
ves his theory. Was it in this way that Rutherford rendered
concrete the nucleus whichhis scientific imagination had creat-
ed? ■

It is difficult to see where, if at all, a lino can
be drawn. The question does not merely concern light waves,
sinco in modern physics form, particularly wave form, is at
the root of everything. If no line can be drawn, we have the
alarming thought that the physical analyst is an artist in dis-
guise, weaving his imagination into everything _ - - an d unfor-
tunately not wholly devoid of the technical skill to realise
his imagination in concrete form , , ,

The question is raised whether the experimenter

-480-

roally provides such an effective control on the imagination
of the theorist as is usually supposed. Certainly he is< an in-
corruptible watch-dog who will not allow anything to pass which
is not observationally true. But there are two ways of doing
that - - as Procrustes realized. One is to expose the falsity
of an assertion. The other is to alter things a bit so as to
make the assertion true.. And it is admitted that our experiments
do alter things. (11) ' :

All this undoubtedly conjures up the dreadful spectre
of idealism in the minds of many and particularly the neo-Scholas-
tics 'for whom the stigmatizing phrase "ducit ad subiectivisum" is suf
ficient to demolish every doctrine which does not propose the uni-
vocal type of realism which they consider inseparable from all
knowledge ? As a matter of fact, however, it is only by recogni-
zing the essential subjectivity of scientific knowledge that one can
■be a true' realist. It is for this reason that we have entitled this
Chapter "Objective Subjectivity". Most of the critics v/ho have
■belabored with the redoubtable club of accusation of idealirm ]5d-
dington and other modern scientists who have tried to bring to light
this subjectivity are far more idealists than their victims, For
they project into the objective world something that is essentially
the product of the mind. They are in many respects worse than the
Platonists of whom St. Thomas writes; "Ex hoc in sua positione
erravit (Plato) quia credidit quod modus rei intellectae in suo
esse sit sicut modus intelligendi rem ipsam," (12) From what
was said in the last Chapter about logical identity it is evident
that they i identify the logical and the real in reality, and that
is essentially idealism. Nor can the subjectivity of the scientific
knowledge ;be considered a falsification of reality, as professor
Do Konlnck has pointed out:

Ne disons pas quo les concepts de la science repo-
sent en definitive sur une distorsion- du monde et que des lors
les documents du physicien sont par avnnce truques ot trahis-
sent : la realite. Mais justement il ne faut pas so^ laisser a-
buser ; par cette distorsion. Les documents sont fideles a leur
facon et ne nous trompent que lorsque nous leur protons une
signification a laquelle ils ne pretendont pas. Est-co que la
lumiere est un mechant. genie qui se joue de nous lorsqu'un ba- ■
■ ton plonge dans l'eau paralt brise? Pas plus que monposte
de T.a.F. n'Qst responsable de ce que mes enfants croient qu'il
y a un monsieur cache dans la boito. (13)

It is futile to try to rule out the subjectivity of
mathematical physics as some modem Scholastics have done* by a P -

-481-

pealing to the Thomiatio doctrine that ideas are not id quod sed
i d qao co gnoscitur. (14) For while it is true that in non-refl e-
xive knowledge an idea ia a mere ^ which carries the mini to a
quod and not just to itsolf known as an idea, the quod to which
the mind is thus carried may he either , objective reality or a cons-
truction of the mind. We hold that in mathematical physics the
quod to which the mind is carried is formally something thai; is
manufactured by the mind - - though not without dependence upon
objective reality. . . "•

If the subjectivity which wo have attributed to tho
scientific world be rightly understood there is no reason to fear
idealism, While insisting upoi^this subjectivity "Bddington likewise

insists upon tho fact that .it can never be more than partial

that objectivity is also essential to physical science. (15)
Meyerson has shown how great and how constant is the concern on
the part of all the greatest scientists to remain in as close a
contact as possible with an objective universe. This is true even
of physicists/ like Binstein and Schrodinger whoso theories seem
to have the greatest likeness to idealism. (16) . Whereas idealism
begins with a denial of the objective universe, -physical science
begins by; postulating its existence. (17) All through its; deve-
lopment the contact with this objective universe remains unbroken.
And even though science constructs its own subjective world- as so-
mething distinct from the objective world, the latter is reflected
in the former and is grasped in some way through it. Whereas i-
dealism seeks to arrive at a maximum of ideas with a minimum of
experience physical science tends towards a maximum of experience
in order to arrive at a minimum of ideas. Meyerson has shrewedly
pointed out that having started with sensible reality, it is the
sensible rather than the reality that physical science tends to
dissolve and that this dissolution of the sensible actually results
in a reinforcement of the reality. Idealism does just the opposite
- - the sensible remains but the reality becomes nothing apart
from the ego. (18) It may readily be admitted that as physics
advances in its theoretical elaborations it seems to take on more
and more the character of idealism. (19) But the likeness is only
superficial; For in idealism subjectivity is an end m itself. _
In physics, on tho contrary, it is only a means; its character is pu-
rely functional. Because the whole purpose of tho subjectivity oi
Physics is to carry the mind to avgreater measure of objectivity,
it is essentially different from the subjectivity of idealism.
There can be no doubt that Helativity physics for example is much
subjective than Classical physics was. But at the same *™° i* ™
far more objective, for it has purged physics °^™ e ™ D ^.^-
jectlve elements that were lurking unsuspected m the Classical

-482-

systera. It delivered physics from tho subjectivism of individual
observers and made all systems of coordinates equivalent fr-r the
expression of the general laws of nature.

There is another side to the theory of Relativity,
Y/e have pointed out in the beginning how the development of
science is in the direction to make it less subjective, to
separate more and more in the observed facts that which be-
longs to the reality behind the phenomena, the absolute, from
the subjective element, which 'is introduced by the observer,
the relative. Einstein's theory is a groat step in that di-
rection, V/e oan say that the theory of Helativity is intended
to remove entirely the relative and exibit the pure absolute,,
(20)

2, Mathematical Physics and Kantianism.

Eddington sums up the, substance of his philosophy
of Physical Science in the following terms;

The subjective laws are a consequence of the concep-
tual frame of thought into which our observational knowledge
is forced by our method of formulating it, and can be discovered
a prior i by scrutinizing the frame of: thought as well as a pos -
teriori by examining the actual knowledge which has been forced
into it, (21)

It is impossible to read these lines without finding
them reminiscent of Kantianism. And as a matter of fact, as wo noted
in Chapter I, TUddington himself draws explicit attention to the
remarkable affinity between Kantian epistemology and the modern
developments of physics. Let us recall his words once again,

If it was necessary to choose a leader from among
the older philosophers, there can be no doub hat our choice
would be Kant, we do not accept the Kantian label; but
■totter of acknowledgement, it is right to ^ ™f f £ a ££
cipated to a remarkable extent the ideas to which we «ue now

-483-

being impelled by the modern developments of physics. (22)

Nor is TJddington the only one who hp.s drawn attention
to this affinity. From the start the theory of Relativity has seem*.
ed to have profound philosophical implications and it has "been a
natural tendency to attempt to associate it with some philosophi-
cal system. And, as Meyerson has remarked, (23) the philosopher
whose name has been mentioned the most frequently by the relati-
vists themselves (Einstein seems to he an exception) has been
Kant,

As is well-known, Kant was perfectly conversant with
Newtonian physics, and had a vast admiration for it. This admirat-
ion led him into two serious errors, First, he considered Newto-
nian physics to, he definitive. For him it was not merely dialecti-
cal; on the contrary it had the supreme certitude of science in
the strict sense of the word. Secondly, not only was it a perfect
science, '/but it was the perfect science, In other words the pro-
perties of physics "became for him the criteria of all speculative
science. And that is why the Critic of Pure Reason is in the .last
analysis nothing hut a critique of physical science, or more exactly ;
a critique of speculative knowledge in terms of physical science.
These two fundamental errors necessarily compromised the validity
of the whole epistemological structure of Kant, "but they did not
prevent him from seizing upon the proper nature of physical science
- - at least in an obscure way. That is what we must now try to
seo. And-: our "brief analysis will consider two points: first we
shall try to see how Kant seized upon the general nature of phy-
sical science; secondly we shall consider the relevance of his doc-
trine for mathematical physics in particular, and especially with
regard to its' object. It is this, second point that is of greater
interest -to us

It is well known that Kant erected his philosophi-
cal system as a reaction to the empiricism of Hume in *i<* Jo
recognized the utter destruction of true science. B *\f" paction
did not blind him to the essential role that exp er lence P^ l *
science. In his introduction to the 0viti^^L^2^m^ emke3
it clear that all speculative knowledge is reducible to ob jocts

of experience, alone. (24) ^^™^^>^o^
upon the fact that experience alone is not sui^i d ^

scientific knowledge, 'that the mind cannot simply be mea ured by
external reality but must in some way ^cobb^s ~^™° r
vords, that true scientific knowledge ™ s ^°fJ^^o i? evl-
His intimate acquaintance vrittytho physics of ^\f™.^ ° fic con _
tart to him that the universality and necessity of scienti

-484-

oep ts could not be derived from the singularity and contingency "
of experience and consequently had to" he a contribution of the
mind.

We have already intimated, particularly in Chapter
IV, to wlm extent Kant was justified in arriving a t this conclu-
sion, We have seen that experimental science by- its very nature
demands that the mind by means of hypothetical constructions of
its own making supplies for the universality and necessity which
experience cannot provide, and even predetermines experience. We
have seen that he was correct in maintaining that in experimental
science the mind cannot know reality as it is in itself; it can
only approach it provisionally. And in getting to know reality,
it necessarily fashions and forms it according to its own precon-
ceived ideas. Kant's great mistake as we said a moment ago consist-
ed in making experiHlQW;al science the pattern and norm of all spe-
culative knowledges-. This mistake did not derive from the fact that
ho conceived all speculative science as necessarily composed of
an a priori element as well as an element drawn from experience,
for that is perfectly true, hut rather in the fact that ho failed
to recognize that there are two essentially different kinds of
a priori elements. For in so far as philosophy of nature, for exam-
ple, is universal and necessary it contains an a priori element
in the sense that this universality and necessity rises above,
and hence is independent of singular contingent experience. This
a priori element, however, does not consist in something posited
ty the subject,, hut in something revealed hy the object, namely
an analytical and hence necessary truth concretized in the singular
contingent experience.

As we saw in Chapter IV, it is precisely because the
mind is unable- to discover truths of this kind in experimental'
science that it is forced to have recourse to another kind of a
priori/ element which is conferred "by the mind. And in so far as this
type of knowledge is concerned, Kant was justified in making syn-
thetic a priori judgements the pivotal point of science. It should
be recalled that for Kant synthetic a priori judgements were those
in which there is added to a subject a predicate that is essential-
ly extrinsic to it. As a result such judgements were a purely arti-
ficial synthesis consisting- in an accidental composition whose u-
nity derived from the mind. Their truth was not founded upon the
Principle of contradiction as was that of analytical judgements,
but on the possibility of experimental verification.

Now all this is a fairly accurate' description of the
tTOe of judgements that are characteristic of experimental science.

-485-

ye have seen that experimental science is based essentially upon
induction by enumeration. If it were to limit itself to the indi-
vidual cases of the enumeration ("This A is B ») its judgements
would he purely synthetic, and it would he completely deprived of
the character o A _ science. On the other -hand, induction by enumera-
tion can never give true universal natures and hence analytical
judgements with the a priori knowledge that is characteristic of
such judgements. ThaTiiwhy experimental science must necessarily
have recourse to synthetic a priori judgements in which the a pri-
ori element is nothing' conferred by the mind. When, therefor e, ex-
perimental science declares; "Every A is B", this judgement is at
once synthetic, hecause based on purely synthetic judgements (This
k is B", That A is B", etc.) and a priori , hecause the form of
universality is conferred hy the mind without adequate foundation
in nature. However, hecause of the regularity found in the multi-
plicity of cases, it must he noted that such a judgomont is neither
purely synthetic nor purely a priori .

Because judgements of this kind are not founded upon
the principle of contradiction hut upon the possibility of experi-
mental verification they can never he anything more than hypothe-
tical. Because of its helief in the definitive character of New-
tonian physics Kant failed to recognize their hypothetical nature
and attributed to them perfect necessity that derived from absolu-
tely fixed forms of thought which were his categories. The disso-
lution of the Classical system has shown how unwarranted his as-
sumptions were in this regard. Nevertheless it must be, noted that,
in spite of the essentially transitory character of the hypotheti-
cal constructions of the experimental science, Kant was not 'wholly
wrong in attributing a fixed and necessary character to the a prio -
ri element found in it. For earlier in this Chapter we saw that
the construction of the scientific world is predetermined and sha-
ped by the methodological principles which constitute the very es-
sence of the scientist's approach to reality, and that as a conse-
quence a close examination of these principles makes it possible
to know a priori the fundamental lines of this construction, just
as the examination of the fisherman's net; makes it possible to know
ajpriori a great deal about the nature of his catch. Because those
methodological principles do not change, because they are fixed
forms which are essential to the very nature of experimental scien-
=o, the laws which are known in this a priori way have a necessity
that those deriving from experience do not have. And in all this
there is certainly a striking' affinity with the Kantian categories.

But of greater importance in this study of the rela-
tion between Kantianism and mathematical physics is the consider atior,

-486-

of the similarity between Thomlstic doctrine with re^rd to th P
object of mathematical physios and Kant's doctrine of sensible in-
tuition. (25) Let us recall the substance of what Kant has to sav
a bou„ sensible intuition. Early in his Critique of Pure Reason ho
explains what he means by intuition in ielSTaT; — [26 ) He defin es
it as the necessary means by which all knowledge is related' to
objects and which all thought uses in order to attain them. Kant
agreed with iiristotle that all our knowledge begins in the senses
and he hold that all intuition as found in man is necessarily sen-
sible - - it has to do with an object furnished by sensation, ne-
vertheless, he felt that sensible intuition could not consist me-
rely in the reception of physical data coming from external reali-
ty, For his whole purpose, as/is well known, was to save science
from the devastation it had received at the hands of both the ex-
treme rationalists who had followed in the wake of Descartes and
of the extreme empiricists such as Hume. And he thought that this
could be accomplished only by considering the whole structure of
science as determined by a kind of noetic hylemorphism in which
the matter would be a posteriori and furnished by physical reali-
ty and the form would be a priori and provided by the subject.
That is why in setting out to disclose and analyze the a priori
forms of cognition he felt that such forms should be found even
in our sensible intuition of the external world, in such a way
that even our direct experience with nature would consist in a
fashioning of physical reality by the subject.

And in order to explain how this is possible he dis-
tinguished between two aspects of intuition- pure intuition and
empirical intuition.. The former is sensible intuition considered
from the point of view of pure sensibility, that is to say, from
the point of view of the capacity of the knower to receive objects
coming from the sensible world, prescinding from actual sensation
and from any particular objects that such sensation might furnish
Hie latter is sensible intuition considered from the point of view
of actual sensation of physical objects. In pure sensibility he^
discovered certain forms or determinations which were a priori in
the sense that they were prior to all actual sensation and hence
completely independent of it. Those a priori forms of sensibility
which constituted pure intuition wore space and time.

. Wow it is extremely significant that for Kant space
and time were the object of mathematics. Ho defined mathematics
as the science which considered these two a prior i forms of sen-
sibility in abstraction from all concrete sensible data. Space
constituted the object of geometry which deals, with lines and fi-
bres; time constituted the object of arithmetic because it deals

-487-.

with numbers which are a succession of units.

It is evident from what has just boon said that for
Kant sensible intuition involves something moro than just sensi-
bility in the ordinary sense of the word, It is in fact riot mere-
ly sensible knowledge, hut intellectual knowledge. It is called
sensible "because of its dependence upon sensation which provides
it with the matter to which the a priori forms are applied.

Now the two a priori forms of space and time which
when taken "by themselves, in abstraction constitute the object of
mathematics, when applied to actual sensation caused "by physical
reality constitute something that Kant calls a phenomenon ., This
phenomenon is a composite made up of two elements; a material ele-
ment which is a posteriori and derived from nature through actual
sensation, and a formal element which is a priori and consists in
the forma of pure sensibility, Only "by the application of the lat-
ter to th'e former can the raw materials of knowledge coming from
nature "be unified, ordered, rationalized, made significant, and
rendered 'capable of entering into the structure of science.

Co qui, dans le phenomene correspond a. la sensation,
je l'appelle matiere de co phenomene; mais co qui fait que
le divers qu'il y n en lui est ordoime suivant certains rap-
ports, je le nomme la forme du phenomene, Comme ce en quoi
soul les sensations peuvent s'ordonner, ou ce qui seul -lour
permet de les ramenora une certaino forme, ^ ne saurait etre
lui-meme sensation, il suit que, si la mat ib re do tout phe-
nomene ne nous est donnee qu' a posteriori, la forme on doit
etre a priori dans l'esprit, touto prote a s'appliquor a tous,
et que par consequent, on doit pouvoir la considerer indepen-
dammont de toute sensation, (27)

It should he evident that pure intuition and the
form of the phenomenon are merely two aspects of the same thing.
Pure intuition is the a priori form in so far as it is considered
as a determination of p5?rii5ail>ility. The form of the phenomenon
is the same a priori form considered in relation to the manifold
of sensatio ns to whi ch it is applied and to which it gives order
and unity,

t+ iq to "be noted that in the passage just cited,
Kant, in speak ng^f the^on .f the priori form with the jtftex
of sensation, uses the word "aPPH°^ion». This is significant
For it brings out the fact that in this union the form .xb osson
tidily extrinsic to the matter. If the very being of the phono

-488-

menon arises from the extrinsic application of one of its compos-
ing elements to tho other, it follows that it can ho nothing but
an artificial composite whoso unity is purely accidental.

How tho close affinity between this object of sen-
sible intuition and the- object of mathematical physics as rrnly-
zed in this study should be immediately apparent. This affinity
is found both in the fact that tho two objects are accidental
composites, and in the very nature of the elements which enter in-
to the composition. In so far as the composition itself is concer-
ned, it is clear, that in both cases there is a union of two ele-
ments one of which plays the part of matter and the other that of
form, In both cases- the form is something essentially extrinsic
to the matter, and as a result the union consists merely in an
application of one to the other effected by the knowing subject.
Consequently, the union is in both cases something purely acci-
dental, something due to the mind rather than to nature, and hen-
ce the resulting composite is an artefactum,

k similar affinity is found in tho very elements
which go to make up the' composite. For in both cases the material
element is a sensible datum, something deriving from physical na-
ture, and the formal oloment is something drawn from mathematics.
In both cases the mathematical form orders and rationalizes tho
physical datum and gives it scientific significance.

It is easy to see why for Kant tho application of
mathematics to nature is not only possiblo but even necessary.
Without this application no true knowledge of physical reality
is conceivable. That is why physics for Kant is necessarily ma-
thematical' : physics„ But more than that, since tho whole Kantian
structure of speculative science is based upon sensible intuition,
the speculative reason is, in tho last analysis, capable of no-
thing but physico-mathomatical knowledge. Kantianism is the most
radical form of scientism.

But in spite of this profound epistemological a-
Dorration there is much to bo said for Kant if his Critique be
Utaited to the realm of mathematical physics. For in mathematical
Physics the mind does form and fashion reality, in an a prion way;
it does become the lawgiver of nature. And that is why we , can find
no better way; of summing up the general theme of this Chaptoi than
ty quoting the following lines of Eddingtonj

Wo have found that where science has progressed
the farthest the mind has but regained from nature that which

-489-

. the mind has put into nature,

life -have found a strange foot-print on the shores
of the unknown. We have derived profound theories, one after
another, to account- for its origin. At last we have succeeded
m reconstructing tho cronturo that made the foot-print And
lo! it is our Own. (28)

-490-

CHAPTER THIHT]33tl

THE NATURE OP MATIIWATICAL PHYSICS

1. The Essence of Mathematical physics.

By way of conclusion it will bo well perhaps to gi-
ve a brief resume of some of the important points in this study,
and thus to try to fix upon the specific nature of mathematical
physics in. the light of the foregoing analysis. And we know of no
tetter way of going about doing this than by returning to some-
thing we 'saw in Chapter I. 'After presenting the various opinions'
proposed by philosophers of science with regard to the fundamental
meaning of the mathematization of the cosmos, we pointed out that
in a general way all of them may be reduced to two extreme posi-
tions. In the first place,, there is the opinion of those who, li-
ke Pythagoras, 'bring the mathematical world and the physical world
into so close a union as to' arrive, in one way or another, at an
identification "between them. In this position the object of mathe-
matical physics is simply and perfectly one. At the other extreme
there is the position of those who, remove the mathematical world
so far from the physical, world that in mathematical physics the
former remains a pure instrument, a pure logical or linguistic tool
in relation to the' latter. In this position the object of mathema-
tical physics is also simply and -perfectly onej that is to cay,
?-t is a pure physical object to */hich mathematics remains comple-
tely/extrinsic.

There is something highly significant in tho wide
divergence of these two opinions. For.it brings out the fact that
the mathematical world is at once extremely close to and extremely
distant from the' physical world. V/hon this is grasped, it becomes

-491-

SLS^rSfi^fo? ST^t — ? n hGve divided

Physical world, and the other Sif^ J ^ ^"ch
consists m purely formal knowledge based on free creation! of the
mind and schematic concepts devoid n f all content, and the seconf
in a natural science known as practical geometry. 'as we noted in
Chapter VI this is actually a denial of the true nature of geome-
try, since the frrst branch seems to he nothing hut dialectics
and the second nothing- hut a physical science. The distance^ hot-
ween the physical world and the mathematical world and the close-
ness of them was also a problem for Plato, as we saw in Chapter I.
On the one hand he drew, them into a union that was extremely inti-
mateln the sense that he made the physical world indefinitely ame-
nable to mathematization and conceived of this mathematization as
a revelation of a logos that is proper to nature. On the other
hand, he created an immeasureably wide gulf between them byi con-
ferring upon the mathematical world an ontological existence' that
was independent of the physical world. There is this to be noted
immediately about the distance created by Einstein between ihe
two worlds and that created by Plato i in the first case the! gulf
can be bridged in the sense that the dialectics can be success-
fully and fruitfully applied to the physical universe as an instru-
ment, even though it must ever remain essentially extrinsic to the
object of physics, whereas in the case of Plato, as we intimated
in Chapter I, in the measure in which the mathematical world is
conceived to have an oniological existence of its own, not only
must it remain extrinsic to the object of physics, but it cannot
even be used as an instrument in relation to the physical world.

We bel-iere that.it is possible to hit the very heart
of the problem of mathematical physics .by saying that both Plato
and the moderns have erred by making the mathematical world at once
both too close to the physical world, and too distant from it. In
the Thomistic solution of the problem thoy are brought together
without identification and separated without the creation of a
gulf between them. And. once this has been understood it becomes
possible to see how mathematics can enter intrinsically into the
object of 'physics and at the same* time remain extrinsic to. it and
serve as an instrument. It also becomes possible to see that the
object of mathematical physics is not something simply and. per-
fectly one, but rather something that is under one aspect one , and
under another dual. Because it is one, Aristotle and St. Thomas
could conceive of mathematical physics as a science. But because
" is at the same time dual, they found it necessary to conceive
of it as a scientia media . Lot us try to analyse these points and
nee how they fit together.

a

-492-

In the first place, /.rlstotle and St. Thomas make
definite and clear-cut distinction between tho physical world
and tho matnomatical world by means of their doctrine of tho dif-
ferent degrees of formal abstraction. The physical world must ho
studied in the light of the first degree of formal attraction.
It is a world of motility and everything in it must he defined
in terms of sensible matter. The mathematical world is the result
of the second degree of formal abstraction. It is a world of im- ,
mobility and everything in it must he defined without sensible
matter. Once we have made this initial distinction and turn to e-
xamine the nature of the abstraction by which the mathematical worlc
is set off from the physical world something very significant im-
mediately : strikes us. For there is a peculiar quality about mathe-
matical abstraction that is not found in either physical or meta-
physical abstraction. In both of those latter cases there is a
correspondence between the way tho object concerned exists outside
the mind and the way it exists inside the mind. The object of phy-
sics depends upon sensible matter both for its being 'and for its
''■being known", the object of metaphysics is independent of sensi-
ble matter both for its being and for its "being known". But tho
object of mathematics is on the one hand dependent on sensible
matter for its being, that, is to say, for any existence it can be
said to have outside the mind, and on the other independent of sen-
sible matter for its "being known". In this dichotomy between the
way mathematical objects are conceived and the way they exist lies
the secret of the distance between the: mathematical world and the
physical world and their closeness. But before attempting to see
why this is so, it is significant to note that both Pinto and the
moderns conceive of tho distance between the two worlds in a way
that puts mathematics in a state which can in some sense be said
to correspond to 'the third degree of abstraction. Y/e explained in
Chapter II that both metaphysics and logic fall within the general
category of those sciences whose object is free of all matter.
Metaphysics arrives at this state by means of positive abstraction,
logic by means of negative abstraction. Now in so far as Plato
attributes an ontological existence to abstract mathematical forms
he conceives of them as though they were separated substances.
And that is why, as we noted in Chapter I, his metaphysics is a
kind of mathematical metaphysics. On the other hand, m so far as
the moderns identify mathematics with dialectics they make of it
a kind of logic. To put mathematics into the third degree of abs-
traction is fo separate it too far from the P^ical worl d and at
*he same time not far enough. It is only by ^^n Sue mtu-
nature of the second degree that we can understand the true natu
re of its separation. But before insisting upon * h " ^'»™£° n '
let us try to see why mathematics ever remains in close contact

-493-

with physical reality,

,. „ ,- , „ ^he niathe.natioal world is intrinsically and essen-
tially linked to the physical world, k3 we remarked in Chapter VI,
if the material world wore impossible, the mathematical world
would also lie impossible. Since prime matter is the principle of
homogeneity, and since homogeneity is the fundamental postulrte
of all mathematics, there is no possibility of mathematics without
an intrinsic reference to prime matter. In other words, it is only
in a world of composed essences, in which formal oppositions are
incomplete "because of the common matrix of prime matter that the
mathematical world can originate. All mathematical notions are
drawn from the physical universe, and even after the separation
of abstraction has taken place, they still retain a necessary con-
nection with the world of matter. ?or unlike the case of meta-
physical abstraction, the separation effected by the mind in sim-
ple apprehension cannot in the case of mathematics he transposed
to the second operation of the mind, The essence of the judgement
is the copula, and this expresses existence, and if mathematical
entities are to exist at all they must exist in the physical world,,
In the universe of matter there are linos and circles and triangles
which may "be considered the physical counterparts of mathematical
lines and circles and triangles, evei)/£hough/;he realization of the
latter in the former is not perfect, since they lose what is pro-
per to them as abstract entities through their realization in the
material universe. >

Che fact of this loss suggests how far the mathema-
tical world is from the physical world in spite of the neariiess
upon which we have just been insisting. In a sense the mathematical
world is. farther removed from the physical world than- is the world
of metaphysics. For while mathematical being has a necessary re-
lation with the real physical world, it never retains the onto-
logical essence of the thing with which it is connected. Metaphy-
sical abstraction does. Arid that is why the communia entis can bo
said to be realized directly in the physical world as well as in
the world of separated substances. Mathematical entities are not
realized directly in the physical world. In other words, by the
very fact that metaphysics deals with sensible beings m so far
as they are beings, its notions can be predicated of the physical
diverse. Mathematical entities on the other hand can be predicat-
ed directly of nothing existing- in physical reality, precisely be-
cause they are defined in a way in which they cannot exist, tluu
is, as separated from sensible matter.

rfhllo all sciences deal with the abstract, the ma-

-494-

thematical_ sciences are the only sciences which deal v/ith the at-
tract precisely as attract. Their world is an autonomous world,
sot apart from reality, and governed by its own intrinsic laws,
in it the .rand is eminently free, it deals with notions originally
drawn from the physical world, hut notions which have been trans-
formed into a condition that is especially congenial to its 'own
nature. Though dealing with things originally connected with sense
matter, it is not hound down to the necessity of having its pro-
cesses terminate in the external senses. Though its notions always
retain some kind of physical reference, they acquire a pliancy and
a capacity for manipulation that are utterly foreign to the phy-
sical world.

All this is at the hasis of the doctrine of John of
Saint Thomas that, the mathematical world prescinds not only from
the actual exercise of existence, hut also from any intrinsic or-
der to existence, and that as a consequence .mathematical being is
different to oither real or logical hoing, just as the essence of
relation consisting in the esse ad is indifferent to either real
existence or purely logical existence. And this explains why it
has heen possihle for modern mathematicians to huild elahorato
dialectical superstructures upon mathematical foundations - -
dialectical superstructures which, while essentially distinct from
mathematical structures, are nevertheless hasod upon them and in
some way patterned after them. These dialectical superstructures
have immeasureably increased the pliancy and instrumentality of
F^thematlds, ,

The foregoing makes it clear that the mathematical
world/is an intermediary world between tho purely material and the
purely immaterial worlds. And this explains why mathematics ;can
at the same time enter intrinsically into the object of mathemati-
cal physics and at the same time remain extrinsic and serve as an
instrument. And while heing a medium hetween the material and the
immaterial, mathematics is at the sarno time a medium botween tte
objective and the subjective, as is evident from the last paragraph.
Ihls immeasurably increases its effectiveness as a scientific ins-
trument, because it gives freedom to the mind to elaborate its own
rational schemas, and at the same time provides the possibility
o" these schemas being applied to cosmic reality.

Having in this way solved the problem of thedistan-
co and the closeness between the mathematical and the physical worlds
and explained in a general way how it is possible for ■mthem.tics
in mathematical physics to enter intrinsically into tho object and
at the same time remain extrinsic as an instrument, it remained

-495-

for Aristotle, St. Thomas, Cajetan and John of St. Thomas to wot*
oat this possibility in fuller and more spocific do^il This thov
did in thoir doctrine of subalternatlon and sctontS ^ '

* 4-v, ■+■ ^ m * thGi f tical Physics, physics is subalternated
to mathematics in the fullest sense of the word; that is to say
there is subalternation by reason of the object. This moans that
the object of the subalternated science contracts the object of
tho subalternating science by adding something tolt. (She addition,
however, can be only an accidental difference, for otherwise there
would be no formal "distinction of sciences. This is an important
point because it means that the matter of the subalternated scien-
ce remains extrinsic to that of the subalternating science even
though the two Onter into composition.

As soon as we examine the nature of the elements
entering into mathematical physics another reason for this extrin-
sic character presents itself. For mathematical entities are united
with physical elements in the state of idealization that i3 proper
to mathematical abstraction. This union is, therefore, not a direct
concretion of mathematical entities in sensible matter. It does
not consist in something that would bo morely the reverse of ma-
thematical abstraction - - the mere putting back of mathematical
entities into the sensible matter from which they were drawn. This
means that the composition of the two can never bo anything more
than the application f the former to the latter. In other words,
it is a composition that is not discovered but created by the mind}
it is a logical composition. It is something remarkably similar
to the Kantian "phenomenon", and from this point of view as well
as from the point of vi ew of the innumerable predetermining a pri-
ori elements that the mind contributes to reality in all exp'ori-
mental science, many concessions must be made to Kantianism by a
realistic philosophy of mathematical physics.

Now tho union between the two worlds is effected by
tho mind principally through a process of measurement which lays
hold of tho quantitative determinations in nature directly, and
indirectly of the other determinations in so far as the former
can serve as surrogates of the latter. But our processes of measu-
rement can never be anything more than approximative, and heroin
"o find a third reason why the mathematical world remains essential'
ly extrinsic to the physical world. If it were merely a question
of tho first two reasons, mathematical physics could still bo a
science in the strict sense of the word. Tho third reason, however,
Prevents it from being a true science and makes it dialectics.
In fact, at this level it has already become doubly dialectical.

-496-

For by the very fact that it is experimental science, physics is
without a true propter quid and has to have recourse to a more pro-
bable reasoning; and the attempt to find a propter quid in Mathe-
matics only results in an approach to nature which i s so extrin-
sic that it provides nothing bettor than a substitutional and ap-
proximative propter quid

In. so far as the mathematical element which enters
into composition with the physical element always remains extrin-
sic to it, the object of mathematical physics is dual. But from
another point of view it is one, For in the first place, even
though the composition in question is logical, it is not completely
logical. The elements involved are brought together by the mind
- - hut fbr an objective reason. Jiiven though the mathematical en-
tities applied to nature retain their abstract and idealized sta-
te, the fact remains, that they do have physical counterparts in
nature. And the union "between, the two elements is so intimate that
mathematical phy3ics employs a unique type of abstraction, an in-
termediate abstraction which participates in the nature of both
mathematical and physical abstraction at the same time.

i ■
But the most important point in connection with the
unity of the object of mathematical physics is that, a scientia me-
dia does not have as its object simply and directly the composite
of the two elements considered as an accidental being. In mathema-
tical physics, only the^hysical element is considered directly,
the mathematical element is .considered obliquely, in so far as
it is connoted by the pliysical element and in so far as it informs
and modifies it and thus tiiakoa it scientifically fruitful by pro-
viding a source of new properties. In this way, even though there
is no res^ media , there can be a s cientia modia.

In this notion of connotation we touch the very heart
of the Thomistic philosophy of mathematical physics. For it ex-
plains how the object of the science can bo at the same time one
and dual, how mathematics can be brought into intimate contact
with physics and yet retain its distance, its autonomy and freedom,
and how it can enter intrinsically into the object which specifies
mathematical physics and at the same time remain an instrument.
The very fact that it is the physical element that is considered
directly and per se, whereas the mathematical element is brought
Into the consideration obliquely and connotatively makes the role oi
the latter essentially functional. Moreover, while this gives wide
scope to the exercise of the functional role by leaving ™£ho^° s
the autonomy that is native to it and by thus making it o

for it to exploit all of the conceptual richness and viituosity

-497-

that is intrinsic to its nature, it keeps the mathematical elabo-
rations completely subordinated to, and always essentially orien-
tated towards, the; physical element. :

One gets an idea of how wide is the scope granted
to mathematics m Thomistic philosophy of mathematical physics
when one recalls that in the structure of a mixed science an ac-
cidental element taken from the lower science is added to the~ob-
ject of the higher science. This moans that from the point of viow
we have in mind here the physical element is merely an accidental
addition to the mathematical element. Moreover the latter plays
the role of form in relation to the former. This meansthat in ma-
thematical physics the illumination and conceptual determination
comes from mathematics. As a result, even the things that are most
proper to the study of nature 'lose their puroly physical status
and are mpithematicized: motion is transformed from a becoming in-
to a state; the flow of time becomes a dimension; the four causes
are reduced to the formal cause} etc.

In taking advantago of the freedom that all this
gives to mathematics, the mathematical .physicist is not obliged
to have a direct and immediate/physical counterpart for every
mathematical element he incorporates into his conceptual structu-
re. The notation of connotation keeps the mathematical elabprations
essentially orientated towards physical reality, but this orien-
tation must not be understood in too narrow a sense. It is possi-
ble to maintain the essential contact that connotation implies e-
ven though mathematical, elements which have no direct physical coun-
terparts are introduced in order to enhance the theoretical power
of mathematics 113^0 far as it is employed as an instrument. In e-
laborate physieo-mathomatical theories the essential connotation
is maintained by means of the text or dictionary.

■ / ■ ■ ' ■'

The mathematical physicist, thereforo, is free to
push the pliancy and instrumentality of mathematics to the limit.
In doing so, ho may, if he wishes, go. on beyond the limits of ma-
thematics in the strict sense o'f the word and construct dialecti-
cal superstructures which will give greater scope to this theore-
tical explanation of physical reality. Bven though the application
of these dialectical constructions to physical reality does not
constitute mathematical physics in the strict Thomistic sense of
the word, it is governed by the same general P^iplcs and follows
the same, general pattern as the latter. Through the use of tne^e
dialectical constructions mathematical physics, which is already
doubly dialectical, becomes triply dialectical.

-498-

The objoctu m formale quo d of mathematical physics
is the phyaioai oonsidered-M-ooSSHSe tho mathematical! IA hen-
ce from this point of view it is more physical than nathomatical
(•'magis naturalis quam mathematical • its whole aim is to get to
know the physical world and not, the mathematical world, ha ob-
jectum formale quo is the special type of abstraction that impro-
per to it, which, while it participates in the nature of both ma-
thematical and physical abstraction, is more mathematical than
physical, since mathematics gives the propter quid and plays the
part of form 5 hence from this point of view, mathematical physics
is more mathematical than physical ("magis af finis mathematicisV ).
Though formally mathematical, it is not specifically mathematical.
For in it mathematics is applied to a physical object in order to
constitute a new subject and new principles proper to a science
concerned with- physical reality. Consequently' it is specifically
distinct from both pure physics and pure mathematics. Since it is
not a science in the strict sense of the word, but dialectics, it
has no habitus that is proper to it. The habitus that rectifies
the intellect in it is the habitus of logic. However, mathematical
physics is, not pufce dialectics. It proceeds per modum scientiao.

2. The Tsxistence of Mathematical Phy3ic3.

Having seen how in relation to tho problem of the
essence of mathematical physics Thomism steers a middle course
between the two extreme positions indicated at the beginning of
this Chapter it will be helpful in order to round out this summa-
ry to explain how it likewise steers a middle course in relation
to a problem which in a general way can be called the problem of
the existence of mathematical physics. Y/e.have intimated that for
someficholastics the grounding of physics upon mathematics is an
error which should never have been committed or at least a mere
historical accident. At tho other extreme is the opinion of those
who hold that this grounding of physics upon mathematics is so ne-
cessary that no other valid way of studying reality is possible.
Frue Thomism accepts neither of theso opinions.

A-ainst the first opinion it holds that the subal-
ternation of physics to mathematics is not only logitmate, but

-499-

necessary and inevitable. In4;he course of our analysis we h.vn

ln ff°2 e ™o,rS ° f rGaS ° nS ^ thlS " S °- ^ * "would ,e
well to recall the more important reasons. The very definition

of scxence itself _ cognitio certa P er causas, gives us the central
reason, For experimental science is neither certain knowledge nor
is it knowledge of things in their proper causes. Hence physics
has a double reason for reaching out to a scientia propter quid,
i,e„ mathematics, in order to obtain for itsel f at least a subs - '
tituto certitude and a substitute propter quid . Boxa naturally
aspires to the status of episteme ; the "infirmusliodus demonstran-
di" that is characteristic of the study of material nature,' par-
ticularly in its concretion, seeks support in the more sure, typo
of demonstration that is found in mathematics.

Moreover physics is inevitably lead to abandon the
attempt to treat nature in terms of the proper sensibles and to
substitute the common sensibles for -them. For sense cognition is
to some extent necessarily subjective, and at the same time extre-
mely limited, and as a consequence knowledge of nature 'in its con-
cretion that depends Upon the proper sensibles is necessarily an-
thropomorphic. Hence it lacks the objectivity and intersubjecti-
vity that : all science seeks to attain. Moreover the proper sensi-
bles are iri many respects irrational) they cannot be defined^
they are incapable of analysis; they are deficient in comrnunica-
bility; they can neither be demonstrated nor be the principles
of demonstration; they are isolated. For all those reasons physics
is lead to treat nature in terms of the common sensibles. And sin-
ce these are all reducible to quantity, this inevitably results
in the subalternation of physics to mathematics. For only the con-
sideration of quantity in the light of mathematical abstraction
has sufficient rationality to carry physics forward towards, its
goal.

Physics becomes subalternated to mathematics becau-
se through this subalternation the mind is able to realize its
natural desire to triumph over the heterogeneity of reality through
homogeneity. The mathematization of the cosmos provides a homoge-
neity which while it breaks down the barriers isolating the. speci-
fic properties of nature and thus triumphs over their pure givennoss
at the same time makes it possible to maintain contact with these
specific properties through their quantitative surrogates. In o-
ther words it affords at the same time both a unity to provide for
"hat is lost by the emergence of physics from generalities, and
a distinctness to enable the mind to follow its natural movement
towards concreteness. The mathematization of nature makes it pos-
sible for the intellect to realize its instinctive desire to know

-500-

reality in terms of what is most knowable for ^ | m ,i +v ,
up for what is lost by drawing away C g r n ^^ JfShe
same time' in terms of what is most knowable in s S W'Lke
up for the deficiencies of purely ,eneric knOTg Tin "phy-
sics there is always a n opposition between what is most knowable
for the mmd and what is most knowable in se. Hence the inevitable
tendency to ground physics upon the one-^iince in which what is
most knowable for the mind is at the same time most knowable in se.
And this grounding OliQbles the intellect to realize its natur^r""
desire .for deduction. Since the universals found in pure natural
doctrine are merely universals in praedicando , natural science if
left to itself cannot become a purely deductive system. Hence the
inevitable turning to mathematics which is the deductive science
par excellence 'because its universals are similar to universals
in causando.

As natural doctrine moves towards concretion it is
getting farther and farther away from the knowledge of nature that
is most in conformity with the human intellect. In this can : be found
another reason for its turning to that science which is of all the
sciences the most in conformity with the human mind. The least ra-
tional of the speculative sciences reaches out to the most- ratio-
nal to supply for it.s deficiencies. In this way the mind is ahle
to study its most natural object (the essence of material things)
through the science which has the greatest connaturality for it.
The mathernatigation of the dosmos enables the mind to fulfill its
natural tendency to dominate its object, to impose its laws upon
it, to become prior to it, to triumph over. its givennoss, to cons-
truct it, and to get at its most profound aspect- the order of the
whole »

A final reason for the subalternation of physics to
mathematics must he added here. \.'e have seen that by its very na-
ture experimental science is- led to express itself through symbols
rather than through names. Mathematics provides the most perfect
symbolic system for this expression.

Because of these reasons and many others that might bo
added, it is manifestly erroneous to consider the grounding of
Physics on mathematics an accident or a mistake. On the other hand
it is equally erroneous to make this grounding so necessary that
no other valid approach to reality remains possible. Thomism avoids
this opposite extreme by situating mathematical physics accurate-
ly in the whole epidemiological scheme. When this is done it beco-
mes evident that not only is mathematical physics ^he only approach
to reality in Keneral. since metaphysics is a valid science and

-501-

the most important of all tho purely human sciences, but it is
not even the only approach to physical reali+v JL V -,

tho part of natural doctrine thatYs liZ £ tow^ d cotreUo?
thnt_ requires subalternation to mathematics, pSS^£1S£o
remains a valid approach to the cosmos, and ono which L man™-

T*- " I T^ ^^r 59 thaU the broach of mathemaUcal
physics, since it deals with the most fundamental problems of the
universe and since it provides Pledge of the most noble natural
form - - the human soul.

Thomism recognizes tho worth and importance of "ma-
thematical physics. It "believes that the most profound knowledge
one can have or reality is knowledge of it in its proper causes,
and from one point of view at least mathematical physics come clo-
ser to this type of knowledge than philosophy of nature. Thomism
even goes; so far as to hold that in mathematical physics the mind
possesses a knowledge of the cosmos which in many respects is like
the knowledge that God has of nature, since it Carries the mind
far along the road towards knowing reality in its specific con-
cretion. At the same time Thomism insists upon the many profound
limitations that are inherent to the type of knowledge that mathe-
matical physics provides. In the. first place, it is not science
in the strict sense of the word, hut merely dialectics. It is not
a mansion of residence, hut a vehicule of progress - - a vehicule
of progress that must travel over a road that' has no ond. Thomism
-believes that even though it is hetter to make progress than to
stand still, per se a mansion of residence is more perfect than
a vehicule of progress. By the very fact that it is experimental
science mathematical physics can never arrive at universal and ne-
cessary propositions, and must remain in probable ' reasoning. Its
definitions are operational and cannot give the quod quid est of
things. It can get at the' objective logos only by projecting a
subjective logos into nature, in such a way that the two become
inextricably intermingled. Because it is subaltornated to mathematics
the only type of knowledge it can give of nature is that provided
Ijy measurement. The data out of which it's whole structure is built
is, in the last analysis nothing but pointer readings. Now metric
knowledge is at best an extremely meager kind of knowledge. For
it comes to grips only with the quantitative determinations of na-
ture; it is utterly blind to all the" determinant properties of
things in their specific essences, to tho very inner nature of
thing's, to all that is of greater significance to philosophy, for
nrt, and for human life itself. But it cannot even get at the quan-
titative determinations of reality in the sense of bemg able to
tell us what these determinations are. By the very fact that it
is "quantitative" knowledge it is not "quidditativo" knowledge.

-502-

It cannot answer the question "what", tut only the question "how
much?" And it cannot answer this question in any absolute wry, sin-
ce a minima mensura in continuous quantity is a contradiction in
terms. It can give us only knowledge of ratioH determined by ar-
bitrary standards. Hor is it possible to progress indefinitely in
the direction of a minima mensura . And besides all this, oljhor in-
numerable limitations of metric knowledge result from the maze of
hypotheses in which all measuring processes are involved, from the
physical interactions between the measuring instruments and reali-
ty, from all the cosmic influences that enter into every measure-
ment, etc,

For all these reasons the physico-mathematical world
can be nothing more than a shadow world, ii/spite of (or rather
precisely because of) ■ all the Cartesian clarity with which it
becomes suffused in the light of mathematical intelligibility.
The true, natures of things remain in the background. As a matter
of fact, mathematical physics does not get to know the objective
world in its absolute state directly; it knows it indirectly by
constructing an imitation of it - - an imitation which is better
than the objective world because more rational, but at the same
time worse, because its whole purpose is to load to the objective
world in its absolute condition. The physico-mathematical world
is not a formal sign, but an instrumental sign of the absolute world
condition. Between the two there is a relation of isomorphism.
The mind must ever try to bridge the gap between the two worlds
by bringing the scientific world ever closer to the absolute world.
But in (Joining continually closer, the two continually get farther
apart. The reason is that, the scientific world is at once essential-
ly subjective and essentially objective, and the more objective
it gets, the more subjective does it become. This subjectivism of
the scientific world does not favor idealism, since its whole pur-
pose is to orientate the mind towards the absolute world condi-
tion. As a matter of fact it is only by admitting this subjectivity
that it is possible to escape idealism, for otherwise one inevi-
tably mistakes one's own mental constructions for objective reali-
ty.

While rejecting the exaggerations of scientism which
have tended to make physico-mathematical method the only ™1 id ap-
proach to reality, Thomism recognizes the truths which B"ont«m
has exploited for'its own ends, and the source from which h n s come

-503-

oonstruotion in which the intellect posits its -own objoct, At the
same time this speculation brings it closer to the ottot that is
most proper to it - - the essence, of material thiLs /nd Sis
intimate knowlodge_ of material things reveals the^LSity nd
malleability that is native to them and thus gives the mind the
poser to refashion nature to its own image and likeness. Because
Nan is composed of matter and spirit there are two fundamental ten-
dencies in him, to draw everything from matter, and to draw every-
thing xrom spirit. The quantitative homogenization of the cosmos
and the study of it in the light of the abstract rationality of
mathematics makes it possible for him to realize both of those
tendencies simultaneously. Or to put the thing in a slightly dif-
ferent ways the combination of the first and second degrees of
formal abstraction enables a man to be at once an idealist and a
realist. The induction of experimental physics satisfies his de-
sire to know cosmic reality; the deduction of mathematics satisfies
his desire for perfect . rationality. The first without the second
lends him into, impenatrable obscurity) the second without the first
cuts him off from reality. The combination of the two provides a
way out of obscurity and a way back to reality. More than that,
it provides man with a kind of wisdom — not the divine wisdom
of metaphysics which is so far above him, which is only loaned to
him in a very inadequate way and never really given to him, and
in which he must make his way with continual strain and effort,
but a human wisdom - - one to which his mind is particularly attuned
and in which he can move in comparative ease and security. It ia
a wisdom whose ideal is to see the whole of cosmic reality in the
light of a few fundamental methematical formulae. Already the T5ins-
teinian system has brought us far along towards this ideal. And
ifa as it is only natural to hope, Relativity and Quantum physics
can eventually be integrated into a unified system, man will have
come near to realizing his ideal. This is the wisdom to which Des-
cartes dedicated himself a wisdom that is not restricted; to

an elite, but one in which all men can share on equal footing, a
wisdom so 'connatural to man that he tells us in his Hegulae , if
a student only follows the right rules "there is nothing, gene-
rally speaking, that any other man is able to know that he himself
will not be capable of knowing." And this wisdom not. only satis-
fies the minds desire to dominate its object in the speculative^
order, it also satisfies its desire to dominate it in the practi-
cal order, for, as is well known, technological fruitfulness has
inevitably followed in the wake of every advance in theoretical
Physics. Small wonder then that this type, of knowledge has-been
transformed into a philosophy of life, that it has become ohe light
of the world.

-504-

. ,. , ^ STQa l 9rr0r ° f B °ion*i8m has toon to Relieve
that the knowledge most connatural to man is also the knowledge
most essential for him. u

k P P 3 N D

PAST I

. NOTES
Chapt er I

(1) Mayme I.. Logsdont A Mathema tician Explains, University of
Chicago press, Chicago, 111771935, p, 158,

(2) University of Chicago Press, 1944, pp. 3-4.

(3) LaTheorie Physique , (deuxiemc od, Paris, Mnrcel Riviere
& Cie., 1914, p, 158, Cf also p. 166,

(4) Q?he Mew Bac kg round of Science , Cambridge University Press,
1933, ps 296* ~~

(5) Cf» Pierre Duhemj Le Systeme du Monde, Paris, Librairie
Scientifique A. Herman et Fila, 1916, I, pp„ 128 - 129

(6) Met . I, oh. 5; 985 b 23 - 986 a 10 „ Trans, by Vf.D. Eoss,

(7) Cfo Hoy Hack; God in Greek Philosophy T o The Time of So -
crates, Princetown University press, 1931, pp 4 47 ff„

(8) T he Mysterious Universe , (second ed„ ) Cambridge University
Press, 1937, o 117o As we shall see later, there is p.
sense in which it is true to say that mathematics enters
into physics from above? mathj^t^a_J^tj[ie_ - lim^_,tgwards
which fatter) is jlravijn. But JeansTaa something quite
different in "mind here„-

(9) Ibid, p. 122, Cfo also N ew Background of Science, pp,
296 - 297 It is interesting' to note in passing that
this view is vigorously contested by Eddington, Of. The
Philoso phy of Physical Science , Cambridge U. Press, 1939
p7~137~, etc. We shall consider aldington' a opinion later,

(10) Science and the Mo dern Wor ld, ,pp„ 36, 47 - 40 Cfo Sir
vTmToTn Danroierj Hi^tory_5d^cience S Cambridge University
Press, 1943. "In our own day, Aston with his integral
atomic weights, Moseley with his atomic' numbers Planck
with his quantum theory, and itdnstein with his claim
that physical facts such as gravitation are ex ^^

of local-space time properties, are revivii^ ideas «,at »
in older, cruder forms, appear in Pythagorean philosophy,
- - n.o 20 o

20)

(21)

[3)

(ID fj^^-^Wtm, Yale University Proas, 1944, pp. 210,

(12) Y/hitehead. Adventures o f ideas, Cambridge University
Press, 1933, pp. 194 -"195",

(13) Je 1 'Explication dans los Sciences . Paris, p a yot, 1927,
Po 133,

(14) O p. cit , p. 101.

(15) Cf. St. Thorn. In I Mot. Lect. 10, no. 143 s "Unde et
Plato tamquam eius auditor, recipiens Socratem, idest
sequens, suscepit hoc ad inquirendum ■ in rehus naturalihus,
quasi in eis hoc posset evenire, ut universale in eis
acciporetur de quo definitio traderetur, ita quod de-
f initio non daretur de aliquo sensihilium, quia cum
sensihilia sint semper 'transmutantium? , idest trnns-
mutata, non potest nlicuius eorum communis ratio as-
signari. Nam oranis ratio oportei? quod ot omni et sem-
per conveniat, et ita aliquant immutahilitatem requirit."

(16) Of. Aristotlej I Mot., ch„ 6 387 a 30. After discus-
sing the 'position of the Pythagoreans he goes on to say;
"After- the systems we have named came the philosophy of
Plato, which in most respects followed these thinkers,
hut had peculiarities that distinguished it from the
philosophy of the Italians. For, having in his youth
first hecomo familiar with Oratylus and with Heraclitean
doctrines (that all sensible things are ever in a state
of flux and there is no knowledge ah out them), these views
he held even in later years."

(17) I Met. oh. '6, 987 h 23 - 33.

(18) Philehus, 55,56, Of. Fields "Plato and Natural Science",
in Ph ilosophy , Vol. VIII, Ho. 30 p. 139.

(19) Suhstanoe and Function a nd ji^tein^JThgory of Relativity
Chicago, The Open Couut Co., 1923, p. 134.

A. ■*. Taylor, Forms and Numhers", in pMlosophical stu-
dies, London, Macmillan and Co., 1934, pp. 14, - 150,

t m«+ qq? n <50« Of, St, Thorn, Leot, 17, no. 259; "sod
Piato^fpraetermiuentihus huiusmodi -sis J^nt
natural!* ac si es^e^^hmaatica "nemoou dum pi.n

(4)

quod mathematics, detent tractari non solum propter
seipsa, sed aliorum gratia, idest naturalium, inquara
turn passiones mathematicorum sensibilious attribua' J baiit " a

(22) "Number and spatial magnitudes cannot exist apart from

things." Met. XIII, 1085 h 35.

(23) Procedures and Metaphysics , University of California

Press, 1936, pp. 24 - 25; 27- 28.

( 24 ) Substance and Function and ainstein's Theory of Relativity ,
p. 135.

(25) Cf. pp. .40. ff.

(26') Pp. 184 - 186 -, 214 - 215, 217.

(27) Cf, F.S.C. Notthorpi Science and First principles, Hew
York, The Macmillan Co., 1931. p. 16° "The third conse-
quence of the mathematical theory is methodological in
character. Since mathematical forms are not observed in
nature, "and, as Plato, said, are suggested hy, not con-
tained in the world of observation, it follows .that one
cannot proceed, as did the physical theory of nature, from
the facts of observation to one's scientific principles

hy the necessary relation -of formal implication. The facts
merely suggest the mathematical forms; they do not imply
or contain them. Hence, as Plato maintained, the funda-
mental scientific method in this theory is the method of
hypothesis. Since this method always commits .he logical
fallacy of affirming the consequent, Plato triedto intro-
duce the dialectics, which is not the vicious thing its
modern connotation suggests to certain minds, hut the sim-
ple sound idea that all hypotheses must he traced to their
common presuppositions and unified in -co a °°™»>°"* J°~
ductive system. ■ V/hen wa attempt to do this for psychology
and epistemoloP-y as well as mathematics and astronomy we

It was the mathematical «nd rat lonal cna
organic universe that made man an idealist m

t-t 17^ nited hy Handnlls The

(28) O pus Majus , ed Br ^°*> U >™^ „ H oughton mffliiTco. ,
^kinjj_ofJAe_Itoder bJi». Boston, n u

1940', P. 211.

lb)

29)
(SO)

(32)
(33)

, ! nntnre Philadelphia, Univor-
sity of Pennsylvania i^s ,

BGjfcnte, OH. VI.

+ frm So, 259).
-qq possost ^p. a-> 1 I f

Xndivi^^ii--- — - — 1027, pp. j-*s 1J '

(34)

(35)
(36)
(37)
(38)

■EiSHgTOmBr,^.^ 0Wed , y Band.ll } .op J

,, 85 _ .as, iia- etc ° cxte y
ottTaai. 236 °

cf . Sampler, £E°^ *' ^

H^J^- U " IV ' CaP ' ' vTT> .^od and wt^-lQ-
'"'~ - ~^ ' „< + Chapter VII t »■

of . S «, <*.£*•

in Kepler". - --?■»'•

(41)
(42)
(43)

(44)
(45)
(46)
(47)

«*. Cited ty B^rt UJ op _
Opere, IV, *»■ p . 134.

2^B2i^- -— ^; pari8f Bditions B.iw_
-> fle Descartes, k»

^^ and Boss ed. I. *' U *

Sescartes, Hal— •* nI , p . 121,

Q ed. Cousin,
* Mersenne, eLL '

TI, 64.
prlnciPiS.' iif .

• tt 4|9.

principiS.' iJ -'

(48) De Potontia , III, 1 ad 17. .

(49) Cfo Begulae ad Mrection em Ingenii, ed, Adam et Tannery,
pp. 426 - 427. ~ ~~

(50) L'Sxperienco Humalno , Paris, 1922, p. 185, Cited "by
Meyersons La_Deductl'on Relativi ste, Paria, Payot, 1925,
p. ,254. *

(51) La Theorie Physique , pp. 169 - 170,

(52) Principia, IT, 199,

(53) Whiteheads Science and the Modern V/orld , p. 69. It v/ould
he difficult to exonerate the importance of the discovery
of the calculus for the mathematical interpretation of the
physical universe, Cf. Handall, op. cit. p, 258» "Such

a method of measuring movement and continuous growth New-
ton discovered; he. had arrived at the most potoni instru-
ment yet found for 'bringing the world into subjection to
man. Since any regu lar motion , ho it of a falling hody,
an electric current ,~or the~cooling of a molten mass, can
he re presented "by _a curve, he had forged the tool hy which
to~attack, not onTy~the~figures, hut the processes of na-
ture the last link in tho mathematical interpretation

of the world. By its meana a Lagrange in tho eighteenth
or a Clerk-Maxwell in tho nineteenth century could hring
all measurable iDhonomena into tho unified world of mathe-
matics, and. calculate, predict, and control light, heat,
magnetism and electricity."

(54) Op. cit, 255.

(55) Cited hy Burtt, op. cit,, PP» 204 - 205,

(56) The Limitations^ ljcienco, New York, The Viking Press
1933, p. 6. ~ '

(57) Science and Humanj jxpjglgnce* *<° nd ° n > * ilX ™™ «"* ^^
Ltd,, 1931, p. 38.

(58) Tho Philosop hyj>lHiyii££L^Sg£> PP- 18 ° ' 189 '
pr, 10, pp. 44 - 45,

(7)

(60) Cited hy Meyerson Do L' Explica tion dans los Sciences,
p. 458 „ " " '

(61) Throughout this study the phrase "classical physics"
refers to Newtonian physics and not to tho physics of
the Greeks,

(62) The Mysterious Universe , pp. Ill ff. ■

(63) Some authors have. attempted to press this continuity

to the' extent of seeing in the distinctive achievements
of recent physics the realization of the main trends
in the history of mathematical physics,. Cf. Juvet- La
Structure des Nouvelles Theories Physiques , Paris, Alcan,
1933, p. 177s "Les ITombres de Pythagore,, les iddes de
Platon, la mathernatique universelle de Descartes, la
caracteristique de Leibniz sont de helles anticipations
metaphy3iques que la nouvelle philosophie naturelle
fondoe sur les travaux d'Binstein, de De Broglie,
d'Heisenberg, la mathernatique moderne creee par Galois,
Lie, Cartan, weyl confirment et precisent avec l'aide
de la mothode axiomatique d'HiHiort. Les Nombres et les
Idees sont las groupes, le symbolisme des axiomes, c'est
la Caracteristique Leitmizienne, et lessucces sans fin
de la mathernatique, qui ne se justifient que par la co-
herence creee grace a l'emploi. des groupes, font du reye
de Descartes la realite d'aujourd'hui," We helievo that
the true nature' of the continuity hetween contemporary
mathematical physics and the past is something quite dif-
ferent from what is indicated here hy Juvet.

(64) I ntroduction a 1'Btude do la Medoci ne gxperimentalo, pp»
94 - 95

(65) The T,or,-ic of Modern Physic s, New York, The Jfe°«»iilon Jo,,
1932, p. 61. Cf, V/oodhridg e. AnJSss^onHature, Few York,
Columbia University Press, 1940, p. 124,

(661 The Method of T heoret^,Physics, Oxford, Clarendon Press
l^sTlpTTI^n^rCfT^eJTorld^lA^l^i' pp - 32 34 -

(67) The Mysterious Univ erse^ P. 113«

,,„, „ ., , 1P1 rf .whe Mathematical Aspect of the Uni-

(68) Op, oit. p. 121. °*« !£° L Philosophy, Vol. VII, No.
verse " hy the same auth « « *j;^ ld0lrt how this

doct^of" Joanf fit, in ««* *»* *» «*» ^ ^^

(8)

77)

and Philosophy. "The same is true of all the dis-
coveries of the pure mathematicians they nro uni-
versal in the sonse that they would bo true in any
world, and so cannot tell us anything ahout the
> special properties of this particular world." -
■bridge University Press, 1943, p. 49,

Cam-

(69) Op, cit . p. 62,

(70) Op. cit, p, 176,

(71) "Mathematics For The Doctor in The Million," in
Philosophy of Science , Jan. 1944, Vol. -II, v "No. 1,
pp„ 48 - 49. Cf. also Lennrdj Great Men of Science ,
New York, The Macmillan Co,, 1933, pp, 220 - 222,

(72) The Principles of Quantum Mechanics , Oxford press,
1935, Preface, p. vii.

(73) P, 137.

(74) Cf, also New Pathways in Science , Cambridge University
Press, 1935, Ch» XII.

(75) Cf. Dantzig; Aspects of Science , Hew' York, The Mac-
jnillan Co,, 1937, p. 74*

(76) An Essay on Man , New Haven, Yale University press,
1944, pp, 211 - 212. This position is developed at
gi?oat length in his Philosophie der symholoschen
For men, and in Substance and Function .

Since a numhar of Historians have seen fit to consider
the doctrine of St. Thomas as a perversion of the philo-
sophy of Aristotle, it is worth while noting perhaps

• that we consider Thomism to ho in the strictest peri-
patetic tradition. It would take us too far afield, how-

Vevor, to attempt to estahlish this point here.

i.evor
(78'

Our Knowledge of the Sxtornal Vforld , p. 240.

(79) P . cit , p. 40. Cf. Eandall, op* cit. p. 236.

(80) Cf. Science and tho_j fodern World, PP- 15 - "' '***J or

of the °7--^. h o 1 f;; fli relLct thought was implan-
ted rtheVS afmtnd" y the long dominance of scholas

(9)

tic Iobj.0 and scholastic divinity. The habit remained
after the philosophy had been repudiated, the priceless
habit of looking for an exact point and sticking to it
when found. Galileo owes mora to Aristotle than appears
on the surface of his ■ Pial ogu.es ; ho owes him his clear
Uiead and his analytic mind, I do not think, however,
that I' have yet brought out the greatest contri-
bution of medievalism to the formation of the scientific
movement, I, mean the inexpugnable belief that every de-
tailed^ occurence' can bo correlated with "it's accidents
fe'a'perf^ctly' def Inite mamior , exemplifying general
pr incip les T . . My explanation" is' that' the ""faith" in" the poo
sibility of science s generated antecedently to the do-
vej : o^^it_£f_modern scientific theory", Ti'an "unconscious
derJ^^^n_from_ifedieyal~thGor6ey.

(81) Paris, Librairie- Felix Alcan, (42 ed„ ) 1932, p, 40.

(82) Paris, Librairie Felix Mean, 1931, p. 149, Cf, ibid 695;
Je 1'ilixplioa bion dans les Sciences , pp<, 489, 528, etc, etc

(83) ''Physique Ancienno et Physique Modornej Leurs Concep- \
tions de 1' Intelligible", in Travau x du IXe congres in - \
t ernntional de Philosophic , T 5 Partie. II,, Paris, Her- J
mann et Cie, 1937, pp„ 197 - 198,

84

(85

Science and the Mod e rn Vforld , pp, 36 - 37, Cf, Adven -
tures of Ideas; Aristotle's Logic "entirely leaves out
of account the interco nne ctions between real things . ...
(It) renders an interconnected world of real things unin-
telligible, gho universe i ^ghi vei-od into a multitude
of disconnecte d substantial things . . , But substantial
I thing cannW^all unto~i5bstantfal t hing. , . But the pla-
/ tonic doctrine of the interweav ing_ of harmony w ithjija-
thematical relati ons has been Triumphantly vindicatod.
TheTristotelian classifications bjised/upon qualitative
predicates have a very restricted application apart from
thTintJoduction of mathematical formulae. Indeed, Aris-
totelian logic, by its neglect of ma-thouatioP.1 notxons,
has done almost as much harm as good for the advancement
of' science, Wo can never get away from the^uostions^

How much Inwhatj?raportions -

gT^^^mi^rrhj^Jij^ other things , » PP.

and, In wha t pat tern
TST- 170," 196.

P hilosophy of Histo ry
ppo 61 ff ,

tory, Oxford, Clarendon Press, 1936,

t'86) pp. 71 - 74,
(87) P 73o

(10)

(88) In Philosophic et Science , (Journees d'Etudes do
la Societe Thomiste, louvain, 1935), Lea aditions
du Cerf, Juvisy.

(89) In Philosophie et Sciences , pp. 26 - 28,

(90) "La Conception Scolastiquo do la Physique " in phi
losophie et goie.ncos , pp. 48 - 49.

(SOa) Tbid, pp. 55 - 56.

(91) Cited "by Riezleri Physios and Reality , New Haven, Yale
University Press, 1940, p. 119. Cf. Strongs op. cit,
pp, 161 - 162> "Galileo treats Aristotle in The Two
New Sciences as a predecessor in the inquiry into pro-
blems of vacuum, infinity, and continuity, and mechanics.,
He introduces principles advanced by Aristotle, in order
to agree as well as disagree with them, In the earlier
work, The (Two. Great Systems of the Vforld , Galileo pre-
sents a more patient Salviati and a Simplicio who is
less of a student and more of n controversialist than '
in the latter treatise,, The "Peripatetics" receive the
■brunt of an attack launched against' those who will not
receive the evidence of the telescope and the demonstrations

.of mathematics; tut this is not done to convict Aristotle,
since Galileo believes that he should have changed his ;

lopinlon in the light of the new evidence. In neither
of his two major works does Galileo present an opposition
between Platonic and Aristotelian metaphysics. Aris-
totle is rather presented in The_Two New Sciences as
more closely affiliated with mechanical questions than
is Slato!" In the preface to this E Judes_s^J;eonard_o
da v'nci Duhem shows that it is a mistake to believe
thaV the science of Galileo is a victory of modern science
Sfr medieval philosophy. It is rather a vie W -^_

(mechanics that was .horn in Pans in «^f/*™ r ^ 3
tury and, based upon the critical treatment of doctrines

[derived from Aristotle and Averroes.

of Modern Knowledge , Aaisea oy
part 15, p. 142b.

j. 17 tin 2- I. 32, 1, ad 2.

(93) Cf. Do Coelo , II, lect. 17, no. *, i,

(94) Op. cit., P. 74 (footnote);

(95) Ch. VIII, 1073 h & 7 - 18.

I J. J.)

(96) Loo. oit.

(97) Of. St. Thorn. Lect. 9, Ho, 2566. "Quot r.utem sint
motus planetarum, no a nunc dicomua on. quae circa
heac mathematics i dicunt, ut ci rca haec' roddgmur at-
tenti, at aliqaa pluralitas(^ erminata Vnente con-
oipiatar a nobis. " ' "

(98) Philosophical Essays for Alfred North Whitehead ,
Clv.pt er I j Tho Mathematical Background and Content
of Greek Philosophy, p. 39. Cf. A.I!, Taylorj "ijum-
hers and Forms" in Philosophical Studios .

(99) Op. £it. pp. 25 - 27.

100) Do 1 'Explication dans los Sciences , p. 174.

101) 78 h 33 - 79 a 15.

102) 193 h 23 - 194 a 10.

103) That is to say, scientific in tho Aristotelian sense
of the term.

104) Cf. Mot. Ill and XIII.

105) The Nature of Physical Theory , New York, John Wiley and
Sons, 1931, pp. 39 - 45.

106) Cf, In II Phys , lect. 11, no. 3.

107) This notion of the intrinsic incorruptibility of tho
heavenly hodios has, of course, heon rejected hy mo-
dern Thomists. But the question of the_£Q S3iblo exi s-
tence , of such ho dies still remains open.

108) In this connection it is interesting to note that in
some respects the physics of the medieval Scholastics
was more mathematicized than modern physics; "Los mernes
conceptions mathematics se trahissent d'aillours dans
le detail des theories particuliores. C'est «xn« quo
la doctrine arithmotique classique dos cordes vihran os
se completait au moyen age par une «^iffif-^ C ^°
ojvtierLejrtJ^^^

th^oriciennes etaient mi - - ; euv e danj * s^or.es

(12)

de Wore le noir par centre le minimum (Da Sensu,
VII, 97 - 102, 104, 115. Gf. ibid. 95, la dWHST
qui fait du noir le minimum et non lg privatio n).
Toutes les nut res' couleurs resultaient ensuite d'un
melange en pro portion d e finie de ces dame composantes
aimpJLe3,(lMd., 102) pour les belles couleurs, dont on
ajoutait mele.ncoliquement, qu'elles etaisnt aussi rares
que les beaux accords musieaux, lo rapport cles composantes
sera une fraction^ particulierem ent simple. _pour_leg
vilai nes un incommens urablG (TbTd. mi, in/i, avec la
thoorie mathematique aux 98 - 100) Les qualites percues
par le gout resultent pareillemont d'un melamge en pro-
portion definie de d eux composantes simples le doux ot
i l ' amer . ( He Sensu XI, 148). St l'esthotique se faisant,
cette fois gastronimie, on ajoute que Ids gouts ngreables
correspondent aux fractions simples, les mauvais gouts
dux incommansurahles, (Ibid 149) Les odours, enfin, qui
sont definies par le gout auxquelles elles sont conjugueos 3
suivent la meme loi, ( Jo Sensu , XII, 174 - 176$ XIII, 177 -
178 ) Bt sur tous cos points la ph. y3iq.ue ancionno 3e montr e
manifestement hoaucoup plus mathematiaee que oelle des
modernes . " B.P. Salmans "La Conception Scolastique de
la Physique" in Philosophie et Sciences, pp. 46 - 47.

[109) We believe that M. Maritain has thrown some confusion
u pon this point ; "Cet usage des principos mathematiques
Sana la connaissance de la nature pout ou bien restor
kecidentel et representor un emprunt fait^aux mr.theraatiquas
par lo' natural is , ou bien etre essentiel a la science
consideree, qui- est alors proprement une s cientia media ;
ot il est clair que divers degros do •mathematisation' ac-
cidentelle doivent conduiro progrossivemont do la science
puroment physique a la scientia media . La physico-ma--
thematique des modernes realise lo typo de la _Bcxentia
media d'une facon par fait o. Au. contraire nous pensons quo
i^slge des mathematiques en biologie par , example ou en
psychologie, n' arrive ra jamais a subordon ne r typxqueme nt
co ^sciplineT^Tl eiles^^x^lica ti^nnkthonBtique s:''

Degres~~da Savoir, p."" 284 footnote.

[HO) Of. V/.H. Thompson, Scxe i}£ e_a^^^ L°W

Green and Co,, i^TTTW^Ip"^ q X°is
maintain that the perfection of sc xentxixcsynt he.xs
depends on the extent to which x has re ; oiv ^{^
into mathematical concepts. There are, no^v ,

(13)

•° f i CXe " CQ HL w ^°h the (£a ta )seem QrtremolyrrSft^ tHra to
I^G^tiotaurterprottrtion, where wo cannot CO i^ uf~
that they could he any process of distillation or foi-
ling dowrvwe reduced entirely to measurable elements.
Science, ( fin Its present fornQ thus presents itself, not
I as a deliquescent arid temporary construction melting - "
\ to a dead level of mathematical materia I, I hut as a so-
lid and ordered hierarchy of different kinds of Knowledge .
all true, all sci entific," hut divorse.."T "i '

(111), "Reflexions sur le prohleme de l'indeterminisme" in Re-
yuo T ho miste , novo -doc, 1937, p. 394, - ~~

(112) Le Cosmos (revised edition) p. 7.

(113) '"Et c'est aussi en oheissant a cette.memo tendance
Iternelle de 1' esprit humain quo la chimie s'etonne. de N

. la diversite des substances ot que cot etonnement,
selon le temoignage autoriso de M. Joh, constitue le
point de depart de ce-bte science touyentiere." Meyer--
sonj De 1' Explication dans les Sciences , p. 181 - ■182 c
Cfr, also Essais, Paris, Lihrairie J. Vrin, 1936, p.
18-19. ' ' .

!ll4) Cfo Oours de Philosophio Positive , Vol, II, 281; III,
idc Comte taught that tho extensive use of mathematics
in physics was simply an inexcusable prejudice*, Cf» _
Pours de Politique Positive , 3e ed. ; IV, Appandice Gene-
ral, pp:„ 1' - 3. It is also interesting to note that
J. S. Mill considered the hope of applying mathematics
to chemistry and physiology as_ someth i ng chimer ical - -
Cf„ Heyerson* Du Cheminement^dT'la Pe nseo, p. i9~0T~

115) Critique of Physics , London, Kegan Faul, Trench, Truh-
ner and Co., 1931, p„ 142.

116) For a good summary discussion of this point see Vf.R.
Thompson: S cience and Common Sense , Chapter VI.

117) One of the simplest examples of how mathematical for-
mulation may serve to throw light upon hiolog ical phe-
nomena is indicated hy \7.B. Thompson m the following
lines, "in the simplest cases, such, as the ^"^ical
or spherical form, which we find ma >^"itude of or
ganic formations, the mathematical formation is n-
stantly made. It might seam, at first signt, tte t *ho

(14)

recognition of a raindrop or an ischinoderm egg rs a
sphere, a plant stem as a cylinder, add nothing
to our knowledge of these forma. But the recognition
of a sphere, for example, implies the knowledge of its
mathematical law, as expressed in the equations in-
dicating the order "between its quantities, such v.a
S = 4 r2 and V = 4/3 r3. By the knowledge' of the
law of its quantities, thes e are r educed from mul -
tiplicity to unity so thalPbhe form of the sphere "be-
comes in tel lig ible and we have, in the formal order,
a cognitio per causas, Furthermore, the rnathemr.tical
definition of a sphere is the "basis of a true deduc-
tive argument. Knowing that a cell is spherical, we '
know that it is something whose surface is smaller for
the volume it encloses than any other possible figure j

/and we' see also, looking at the matter from another
angle, that it exemplifies the law according to which
'any system of "bodies arranges itself in such a manner

^ that_jthe _potential energy of tho s yste m is a minimum? '.
(sT~Sdser., General Physics, 1913, p. 290)" - - Science
and Common Sense , pp. 76 - 77,

118) Cfo Thurstonj Vectors of Mind , Chicago 1935; Spearman;
ghe Abilities of Man , New York, 1928; Sdgeworthj Mathe -
matical Physics , London, 1881, etc,

119) It 'is important .to keep in mind that whenever intel-
ligence is studied "by any method of this kind, what is

nriflp^hnnjljTy,^!!^^^^^ 6 ^ — ^ e tolWln 0nly °P erp ^ H-

n 77TT77^h^ris~To~""s^yT Tt can he defined only jy^ajles,-
cTj jtion of t h9_ooncrate_j3rooedji ros of measurement h y
which t hgj^juU3__hav o heen arrived a t.

,120) Introduction to Mathematics, The Home University Lib-
rary, 1931, p. '223.

:i81) The Physical Princ ipj 1 e_ £ Ou £ mtum^heory, Chicago Uni-
versity Press, 1930, p. 62.

[122) Pp. 4-7,'

;i23) L'ldee de Lo i V^^^J^JSJ^SL^^^^
Paris, Lihrairi e^hllb^phique J. Vnn, 1925, p. 13b.

:i24) Philosophical Ag EgoWJBawnBolgsSQ, ^,-York, The
Macmillan Co., 1932, p. 63,

115)

.25) ■gheJW ysteriogs Un iverse. pp„ 122 _ 123,

,2b) ^^g££H^of Sciance, p, 298. Cf, also pp. 296
2S7. Of. "Iho l*athenr.tical impact of the Universe "
in Philo sophy , VoI„ m, Po 14, '

(18)

Chapter II

(1) Cf. Thompson, An Introducti on to Science, New York.
Henry Holt and Co,, 1911, ChnpteFlvT -

(S) The P hilosophy of Physio s. London, George Allen
and Unwin, 1936, p, 81„

(3) The. Three Reformers, London, Sheed and v-fard, 1936,
P. 64.

(4) Of, Gilsons "L'Idee de(] >unite da corps des sciences^ ),
est inseparable, |chronologiquement et loft i quement J do
1' extension de la methode mathemntique a la totalite
du domaine de In connaissanco ." Disoours de la Methode,
Texte et~Commentairo, Paris, '1925, p. 214.

(5) Cursus Phil , I, q„ XXVII, a. 1.

(6) Of. Charles De Koninclc; "La diolectique des limitos
comme critique de la raison", L aval Theologique et
jphilosophique , Vol. I, No. 1, pp, 177 ff

(7[ Of. Ill, II, 6; Of. also John of St. Thomas, Cur,, Phil.,
I, Part II, Q. XXVII, a, 1. (Reiser ed. pp. 819 "b and
824 h).

(8) Of. Contra Gentes , III, Chap. 100,

(9) Of. Henri Pichettej "Consideration sur quelques prin-
cipes fondamantaux de la doctrine du speculatif et du
pratique", Laval Theologique et Philosophique , Vol I,
Ho. 1 pp. 52 ff,

10) Cf, III De Anima , c. 10 433 a 10.

11) II Met. I, 99313 20; jny3p^._de_ji^in. , V, I, o.

117)

[12)
(13)

115)
116)
(17)
(18)
(19)

I - II, 5V , 1. ad X '
X „ II, 3, 5 ' ad 2 *

of, i» 14 ' l6a

'pejer, III. 5 > ad 4 "

Be VGritate> I, 2. °'

^ , i disp. 2. ^ 10 ' nD ' 18 *

' 9 a 10, WO. »•

Our 3^3221" i8 ^

i9 no. 25b. i/-«-« i.

(20)
(21)
■(22)
(23)

(26)
(27)

iS-ii-^V a. 3, so 1 * 4 * Ttl. 3, °«

Cf. S. Stomas, ^J£ii^ — * z _^__——

*■*• "• "• "' - 48 „„» ««««• wrW> *• 76 '
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< p cf. ijlS-S^'

* " I J-Lr I ?nys. I, cy *

(28)
(29)
(30)
(31)

tart, LiHi^^ ^ , 10 .

Pf. Poat^iii^'

° ^^^x. 177, 3, oto.

XnPosii^' 1 ' .corruptioi-^^"'

(18)

(32) In I, 1, 3, no, 3, 4, 5, Cf. Also John of St Thomas
loc. cit, p. 819 and 260,

(33) This objective light must, of courso, he distinguished
from the interior light by which the cognitive potency
is actualized.

(34) I, 1, 3. ■

(35) Of, St. Thomas in Boeth. Be q'rin . V, 1.

(36) Cf. St., Thomas I, 14, 1;' In Be Anima II, lect. 24;
De Veritate , II, 2, etc.

(37) Cf. In VI 3th . lect. 3 f no, 1145 - 1149, etc.

(38) Be Trin . V, 1; Cf. Post, Ana l. I, lect. 41; In VI Met .
lect. 1; I, 85, 1, ad 1 and 2; In Be Sensu et Senaato ,
lect, 1, etc.

(39) In Be "3nte et Essentia , Proem.

(40) In Boeth. Be Trinitate , V, 3, ,

(41) "...propriam et determinntam rationem praodicati in-
feriors (non) accipiat," John of St. Thornasj Ars
Logica , p. 31 a 22.

(42) Cf, John of St. Thomas, Curs. Phil, p. 265 a 30 - 35.

(43) V, 1<,

(44) Cf, St. Thomas: I, 85, 1 ad 1; In VII *fet . t Be Trin. V,2,

et^. I.; is hardly necessary to pSInToSTthat the student of nature

uses individual observations and experiments

but o nly as a . point of departure and as a means to arrive

at the common s ensible matter. ^

(45) Lest confusion .arise, it must be pointed out ^at modern
authors use the W ord»metaphysics'< xxa a muclx brondei sense
than the traditional Thomistic acceptation of the term £
is now generally employed in such a ™V w! shaS uso^he
losophy of nature as well as metaphysics. Vu shall use
word in its strict Thomistic meaning.

(46) Cf. In^hys^, lect, 1," InV!**. loot, 1; luXIM*. loot.
4, etc.

(19)

(47) E.g. Maintain: Mij£eres_du_Savoir, Paris D esclee do
Brovver et Cie„ri9tfa, pp . 77> 797-^0. Besc lee de

(48) De Subjecto Naturalis Phllosoph igo.
(4-9) Cur 3. Phil . II, q. j ( a> 1#

(50) I, lect. 1„

(51) In Phys . I, lect, 1.

(52) Qursus . Phil. I, 1. This point may give rise to a diffi-
culty; a metaphysical definition (actus entis in potential
seems to he employed in philosophy of nature . The answer'
is that the word potency has a different meaning when the
definition is used in metaphysics and in philosophy of
nature. In the latter case, it means the physical potency
o f matter . In the former it is considered in its general
meaning as( a principle of beingy ) avery; act of- a being in
potency is necessarily the act of a material thing, but
while in reality there is identification, the aspect under
which this reality Is considered is different.

(53)/ '!...lta quod ens mobile, licet complexionem nominum contineat
I incomplexum tamen et per se< unum quod quid est significnt,
\ sicut ens per se," D e Subjecto Naturalis philosophi ae, La-
Vval edition, 1939,. pp. 9 and 10. [

(54) "'Sins mobile non suniitur complexe pro agregato ex ente et
mobilitate ut duobus , sed incomplexe pro quidditate, cui
convenit mobilitas," Cursus Phil . II, q. I, a. 1.

(55) In IV Met, leot. IS, no„ 683. Of. In III Met - lect, 3,

(56) Cf„ In Boeth De Trin . V, ' 2; In_Vi_5th. Lect. 3, etc.

(57) In VI Bth „ lect. 1, no„ 1123.

(58) In Anal Pos t. I, lect, ' 16. C'f. 1, 86, 3.

(59) ih Be Trin ., V, 2„

(60) In vi Met , loot. 1.

(61) Cf. St. Thomas; In II P hya., leot. 3; In^e^rinltato, V, 3o
' 6 S) Cf. St. Thomas; In De Tri nitate , V, 3,

(20)

(53) .In Boo th. Je _Trin., VI, 2 Of. De Coelo et Mundo . m.

(64) Of. Be Anima I.

(65) DaJTeritate XII, 3 ad 2.

(66 Curso Phil . I, Pars I, q„ 27, a. 1,

(67) S>So 11-11,1, 1; I -II, 54, 2, ad 2; otc.

(68) E.g. In VI Met , lect, 1; In I Poster. Anal , loot, 41, otc.

(69) VI, leot. 1,

(70) Of, e.g. L.M. Begis. "La Philosbphie do la Nature i Quel-
ques aporios", In Philasoph ie, Cahier I, Btudes et Re-
cherches, College Dominicai'n, Ottawa, 1936.

(71) "La Physique Aristotelicienne et la Philosophio," In Phi -
losophie et Science s, pp. 25 - 26.

(72) Ibid, p. 24.

(73) It nay Toe true that Aristotle himself never brought out
this hierarchical structure as explicitely as St. Thomas.
Yet the latter merely clarified what was already implicit
in Aristotlo's doctrine. That is why we see no point in
Mansion's argument when he writes t "On no voit done pas

foe qui justifie, en bonne doctrine aristotelicienne,
I'abstraction mathematique entendue comme un degre special

^'abstraction. II faudrait pour cela que les notes quan-
titatives possedassent vis-a-vis des autros notes auxquelles
elles sont unies dans, la realite physique, une antoriorite
logique ou metaphysique, qu'Aristote n'a point cherche
a etablir. Les efforts faits dans ce sens par las soolas-
tiques - - Saint Thomas ou d'autres - - ne peuvent pas entrei
en ligne de compte pour formuler une appreciation concer-
nant la position doctrinale du Stagirite lu-meme." Ibid,
p. 25,

(74) Ibid. 25.

(75) Ibid , p. 25.

(76) Ibid, p. 23 - 24.

(21)

(77) Ibid, p. 24,

(78) In Be Trin . , V, 3, ad 5,

(79) ibid. p. 27 - 28.

(80) Loc. oit.

(81) Chap. 27, Trans, by G.H.G, Mure; McEeon edition.

The Basic Works of Aristotle . New York,. Bandom House,
1941, p. 1537 :

(82) lectio 41, no. 5.

(83) La Theorio Physique , p, 167,

(84) Qurs. Phil , II, Q. I, a. 2.

,(85) Loc, Oit.

(86) Meyerson makes it clear that the superior immateriality
of arithmetic has been quite generally recognized, Of..
Iiu Cheminement de la pensee , p. 302« "Telle etait deja
1'idee de Gauss c 'Nous devons admettro humhlemont,
ecrivait-il a l'astronome Bessel, que, le nombro est
uniquement (le produit de notro e spr ifT) 1 ' espqce , meme
du point do vug de notre esprit, uons-titue une rdalito
a laquello nous ne pouvons a priori dieter complbtement
ses lois'. Dedekind, dans la preface de son fameux
opuscule sur la nature du nombre a vivoment insisto sur
cette idee deCT' autonomic d e l'arithmotiquo ^ a I'dgqrd
du reel . Le nombre est une 'emanation immediate dos
lois pures de la pensee' et 'en'tierement independent des
concepts de temps ot d'espa co'; le's nombres sont dos 7
•creations libres de.l' esprit humain, ils servent de moyen
pour saisir plus "aisement et avec plus do precision la
djversite des choses' ( Was sind und was sollen die Zahlr.i?
^2 ed., Brunswick 1923, p. 111... "ISaia Locke, deja
jugoait que 'le nombre °°^r P 1 "" aimnlo et la plus uni-
verselle)de toutes nos idees" ( Basal Philosophique , II,
ChTxvT^ no. 1), et Hume considerait la geometric comme
moias assureo que 1 • arithmctiquo et l'algebre au point do
vue de la valeur apodictique de ses affirmations, (poy-
\o hologie , tr. Henouvier et Pillion, Pans 1878, p. 98)-.

I Q 7) Cf. H, P. Salman, "La Conception Scoiastique de la Phy-
sique 1 ', in Philosophic et Sciences p. 37j 'Coux-ci (ios

(92)

(93)

(94)

(22)

anciens) connaissaiont Men los distinctions- les
sciences et les arts, les arts liheraua et les arts
serviles^les sciences pratiques et speculatives,
oes_ derniores diversifi es selon lour degre d' abstraction-

mais jamais dans aucun domaino, ils n'ont oppdsFune '

J^science" a une "philosophies

Cf. Marltain- La Philosophic de la Natur es "Toutefois
cette verlto capitale etnit payee che.z lis anciens, • chez
Aristote lui-meme et chez les anciens scolastiques
egalement, au prix d'uno grave fauto de precipitation
intellectuelle. . . pour, l'optimisme des ancians, qui se
portait tres rapidernent a des raisons d'etre quolquefois;
tres hypothetiques quand il s'agissait du detail des
phenorneae s , (philosophie et sciences exp eriment al os etaiont
mi soul et irieme savoi r),[ et toutes les sciences du mondo
materiel etaient des subdivisions d'one soule et unique
science specif iqu e qui s'appelait 'x^h iloso phi a naturalis*"")

6 o « ~* ~" P a «3JL o

(89) Cf. Reflexions sur h' Intelligence , Les Degre3 du Savoir,
La Philosophie de la Nat ure, Science et Sagesse , etc.

Cf„ also Yves Simons Marltain's Philosophy of the Sciences,"
in The Thomist, Vol. V. pp. 85 - 102.

(90) Art, cit, p. 95. (Italics ours)

(91) Curs Phil. II, q„ I, a. 2.

C f . In Phys I . (f eet. 1, nos. 6 - (^ ) Be Sonsu et Sensato ,
•leot. 1, no. 2s gg Gonerationo et Corrupt ion Oj, Proem,,
In Be Anima .Clect^Jpno, 1; In Meteo rolog<lect<, Jj De
eoelo et Mundo se lect, X ) etc,

The full significance af this statement will he hremght

put in the next Chapter^hich the question of suhalternation

will he studied in detail.

In VI JB3t. lect. 1, no. 1147,

(95) ■ Ihid . no. 1165.

(96) |iMd. no. 1149. Even on ^is ^oir^V^i^LS^JS.

he antJThomistic , for he writes s »Je note ent e ?«en-
th eses, quo 1' e tude des premiers fondements o ^logi^es
Us makLatiquos, la philosophie du nomhro ot du oonuiim,

(23)

rentre dans la sphere de la philosophic de la nature,
i%ZX r i * raath ^ ati ^» - Portanb pas do so sur

Until " V° C °T T ' e PaS d ° SQgGsse dQns s °* ordro ^
propro," -- De^oBj^Spjoi^, p. 92 - 93, of. p. 345.

Cf. also laP hilosophie de . la N ature: p. 91. It is' dif-
ficult to see how the philosophyTTinathemtrticB, tho pro-
hlems of number and continuity fall within the sphere of
philosophy of nature s which is the study of things in
tgrm_s.^f mobility. Philosophy of nature is, indeed~
kind of wisdom within its own realm, |in the sense tha t^
the general principles of rnohile heing which it studios

( give order ^tqfthe entire study of natural thing s, hut it
is a wisdom fohly in terms of mohility .') May~not tho source
of Maritairfs confusion he, at least in part, his sub -
stitution of sensible heing for rnohile heing ? The phi-
losophy of mathematics pertains to metaphysics not only :
for the reason given at>ove, hut also "because, hoing wisdom,
metaphysics has as one of its functions not only the cri-
tique of its own nature, hut also of that of all the other

^sciences. '■

(97) Pegres du Savoir , p, 352, footnote 1.

(98) Ihid . Of, La Philosophie de la Nature, pp. 88. ' "

(99) V", 1, ad 5,

!l00) Art, cit., Of. St Thomas: In Be Sensu et Sensat o, loot. 1,
no. IS.

101) Cf. Les Pegres du Savoir , pp. 77 f f . , 94 - 95, .352, etc.

102) After consistently assigning sensible heing as the formal
ohject of the study of nature throughout Les Pegros du
Savoir , M. Maritain notes in La Philosophie do la Nature

(p. 113) that as Cajetan explains in his opusculum, Pe
Suhjecto Katuralis Philosophiae , the expression ens son -r
sihile is less .apt than ensjnohile. He still insists,
"However, that 1 ^ legitimate to assign ens sonsihile _
as the formal ohject. For reasons indicated earlier in
this Chapter x^eJeeljtt^JhSI^B^mS^WJm^moldo^
here than a question of^aptness j

103) "Maritairis Philosophy of the Sciences" in MwThomiat,

Vol. V, pp. 90 - 91.

(24)

[104) Va prescind hero from tho important difference that
the _ first definition in so far as it has to do ^7ith r
living being, is based uponcbo'tii* internal and oxter-'

/ !^i_2S££i?fa e » whereas o^erim'enTaI''"'s'c"ientists

witiythe exception of thjs^experimontalCps'ycn'oioeist-s
in some cases, have adopted tho method "of 'drawing" only
from external experience even when dealing with

V living beings ) We shall return to this poInTTn Chap-
ter Till. For the moment it jLs„sufficient to note
that t hg_ difference in the ( sou r£es^of experience employed

^cannot, ob viously, constitute a^speciYi^'nfftrence""""'"
beJweeji_s_cjJncVs._J™ " ~ " '"*"' -—■*»—

[105) In I Post. Anal , lect 16 no, 5.

[106) Art, cit. p. 94.

[107) Ibid. p. 32,

[108) Begre3 du Savoir pp. 74 - 75, Having made this admis-
sion he illogically holds ^ that tho experimental sciences
come before the philosop h y of nature in pedag ogical or-
der. J )

[109) Of, Degres du Savoir , pp. 344 ff.

[110) L a Philosophie de la Nature , pp. 91 - 92.

[ill) The Sophist 219 a ff, Of. Aristotle's criticism of this
doctrine: Prior Anal . I, 31; P03t, Anal . II, 5 and 13.

[112) Morceaux Choisis , pp. 44 - 4£.

[113) In I Phys . lect. 1, no, 8,

!ll4) Cf. La Philosophie do la Nature , p. 23.

1115). Cf. Wos, 1147, 1151, 1165.

ill6) Degres du Savoir , p. 77.

!ll7) Ibid. p. 351.

ills) Cf. Fulton J. Sheens The^i^^^!"^; Mil-
waukee, The Bruce Publishing Co, > 19 ^ P ?°Z; S ia3 '
Father Vhittaker, O.P. ! "The Position ox Mathematics

(25)

In the ^Hierarchy of Speculative Science," in The
Thpmist, Vol. Ill, M . 3, p. 503, —

(119) Cf. Cajajan, Do_SubJQotojaturali B Philosophiao
ed. Laval, pp. 19 - 20.

(120) Ibid , p.. 91.

(121) 639 b IS -'640 a 9.

(122) St. Thomas, lect 15.

(123) Lect, 15, no. 5.

(124) Jo Part. A n., loc. cit.

(125) Cf. Phy.s . II.

(126) In VI Met . Lect. 1. no. 1149.

(127) Ibid , no. 1146.. Cf. I, 46, 1 p.d 3.

!l28) We prescind here from the special case of man, whose
f uture existence could h av e been demonstrated with
apodictic necessity once gi ven t he existen ce of a ma -
terial cosmos ,! f pl " horoTHe basjsTo f demons^rability
was^ something extrinsic^

129) Cf. St, Thomas. In Do Trinitate , VI, 1; Bx hoc autom quod
considerntio naturalis est circa materiam a pluribus
dependet, scilicet a consideratione materiae et formao,
et dispositionum materialium et proprietatum quae con-
sequuntur forniam in materia, Ubicumque autem ad aliquid
cognoscondum dportet considerare plura, ost_d ifficilior
cognitio: unde in I Posteriorum dicitur, quod-minus
certa~scientia eat quae est o x additiono , ut geomotria
ad arithmeticam. Bx hoc vero quod oius con3ideratio
est circa res mobiles, et quae non uniformiter se habent
eius cognitio est minus firma, quia eius domonstrationos
ut in majori parte sunt ex hoc, guod^contina it aliq uando
aliter se hafeore: et ideo qunndo aliqua sclent in magis
appF opinquat ao f singularia, sicut oparativae, ut modicinn ,
alchimia.re t moralist mirm.EOsgu nt habore de certitudxne
propter multitudinem oorum quae c on sideranda s unt_in ^
taTibus 80 i9ntila ,(im03^jolJpggl^Z^iaag^ r
erratur, ot propter eorujnj^uj^jjjjjj^g^

(26)

(120) Cf. St. Thorn. In I Bth. Nio . loot. 1.

(133) The uniqueness of the method proper to each science
does not, of course, exclude the possibility of a
general treatise on scientific method, forlogic,
writes St.- Thomas, "tradj : t__cgmmu nem modun Tprocodendi
j.n omn ibus aliis aoientj isTTJodur nnt.nm ^PmTTT
singularum scientiarum, in sciontiie singulis circa
principium tradi debet." ( in II Hot , lect. 5, no
335) In view of this distinction of St. Thomna, the
following assertion of Gilson is at best extremely
ambiguouss "An' Aristotelian discourse on method is.
an impossibility; it is possible to speak" only of a
discourse on methods ." (Op. _cit. p. 71) Far from
being an impossibility, a discourse on scientific
method was actually written by Aristotle, namely the
p osterior Analytics .

(133) VI, 2, '

(134) La Philosophie de la Mature p. 24,

(135) Lect. 21.

(136) Ibid, no. 2.

(137) Ibid, no. 6.

(138) Lectio 22.

(139) Lectio 15,

(140) Ibid, no. 4,

(141) Du Chi

(142) No. 4,

(143) No, 4.

(144) Ibid. nos. 5 - V.

, no. <t,

lemlnemont de la Pensee, p. 481.

(27)
Chapter in

(1) In maintaining that philosophy of mathematics is rn \
intermediar y_ gcience -botwoor. tim second and third~7le- \
groe of abstraction, Father V/hittaker has confre od 1
the kind of application just mentioned with true
suhalternation. Cf, "Tho Position of Mathematics in the
Hierarchy of Speculative Science," in the Thomist, Vol
III, No. 3, p. 496,

\

(2) Cf. John of St. Thomas: Cars. Thool o, I, q, 1, d.2, a. 6.

(3) Cf. Gilson; L 'Esprit de la philosophic Mediovalo , Paris
Xihrairie J. Vrin, 1932, p. 4> "Alors que le rationa-
liste pur place la philosophic au sommet ot l'identifio
it la sagesse, le neo-scolastique la suhalterno a la
theojogie , qui reste soul a meritor pleinement le nom
de sagesse; mais pourquoi certains neo-scolastiques
penserit-ils que rneme sutialternee a la theologie, leur
philosophie demeuro identiquo en nature a cello qui ne
reconnait aucune Sagesse au-dossu3 d'olle."

(4). Cf. Ars Logica ; P. I, Q. XXVI, a. 2.

(5) In De Trinitate , VI, 1»

(6) Tho ancient Thomists sometimes culled this typo of de-
pendence suhalternation secundum quid , hut denied that
it was subalternation simpliciter , Cf. John of St,
Thomas, Curs. Phi l. Ars Logica . P, II, 1. XXVI, a. 2,
pp. 798.

(7) Loc. cit. p. 796.

(8) V, 1, a. 5.

(9) Ihid

;i0) Loc. Cit. p. 796 D 43. Philosophy for John of St. Thomas
means the science of naturo.

Ill) Cf. Fulton Sheen, "Furthermore, the more developed the
■ • -, • ,-= +im lintter is the raw material upon
empirical sciences tho hotte is sciGnt ia medic

which metaphysics may speculate to duixu

(28)

or the Philosophy of Nature." The philosophy of
Science, p. 189. "Just as the TcTelJ^e-bTnEthlErtlcal
physics is forraod hy the application of mathematics
to physics, so too, the scionco of the philosophy of
nature is formed hy. tho application of tho fundamental
principles of motaphysics to the natural scioncos. »
Ihid. 164. Cf. 'Thitakorj "Tho philosophy of nature is
tho intermediary science between the physical and ncta-
physical orders," Op. cit. p. 503.

(12) Cf. ;„nnibaldus: In I Sont . dist. 1, q. 1, a. l f "primarum
sciontiarum proximum principium est intellectus, enrum
vero scientiarum, quae sua principia ah allis supponunt,
proximum principium est credulitas principiorum ah aliia
suppositorum; primum vero onrum principium est intclloctus
Perficitur tamon certitudo istarum scientiarum cum

per viam rosolutionis in ipsum intellectum primorum
principiorum perveniunt."

(13) John of St. Thomas, Curs Theo l. I, q. 1, d. 2, a. 6.
p. 369 t>.

(14) Curs Thool . I, q.. 1, d. 2, a. 5, p. 364 a.
(15)' In 1, 1, 2.

(16) Curs. Thool , I, q.. 1, d. 2, a. 5, p. 364 a.

(17) Curs phiU Ars Logica , II, q. XXVI, a. 3. pp. 799 ff. .

(18) Do Veritato , XIV, 9.

(19) De Veritato , XIV, 9, ad' 3.

(20) Curs Thool . loc. cit. p. 369.

(21) In Be Trinitate , V,' 3, ai 6.

(22) Cf. Vassily Pavlov: "Mathematics for tho^octor in the
Million " in Philqs^phy_of_Scionco, Vol u, no. i, P.
48. "...an Q ff^tHf^^olTSde77.to a pretense ot ap
ilvia- the concrete sciences to the abstract ones, *t
plying tlio concr^.o ao . w hybrids in inverse
has eono to the exteno of naming noa V

order as physical - « ° ', ft S CE \, „„d the lite...

Bioal geometry :)... hi °J°^°?J lik0 th0 application

...To this writer it still l° oK3 ^ reverse."

of mathematics to hioloey rather than tho

[29)

I, loot. 5, no* 7.

In Do Trinitate , V, 3, ad 7.

In I Post Anal ., loot. 25, no. 2.

Ars Logioa , Par3 II, q, XXVI, a. 2, p. 797.

Ars Logica, Pars II, q. XXVII, a. 1,. p. 827.

"La Conception Scolasticjue de la Physique" in Phi-
losop hie et Soionces , pp. 48 - 49.

In I, 1, 2.

Ars Logica , Pars II, Q. XXVI S a 2, pp. 798 - 799.

Lest tnis definition soom to exclude a posteriori know-
ledge "by which we know things through their effects,
it is necessary to note that the' term "cause" in the
definition rofer3 to the. cause of science.

Ars Logica , loc. cit. p. 798 h.

Cf„ James A. McWilliamss "Idealism in Science," in The
Itodern Schoolman , Vol 14, p. 7j "These scientists are,
in their turn, victims of the initial error of grounding
their partial, science on another partial science, on
mathematics instead of metaphysics."

Loc cit . p. 799a

In I Post . Chap 12.

I Post Anal , lect. 25. no. 4.

I Post. Anal , 'oh. 13, no. 6.

In II Phys . loot. 3, no. 6.

Substance and Function , p. 117.

Of. Physique et Philosophie-' in *^%£g^^
p. 86- "Dbs quo nous entrons en phys^ u» n
r Oe rapport- ejtje ^S^L n-y a ici
deviont pas pour cola ao ^ ^ r ^ aciencc

set-s^s sz r='; *«i ««• —

[30)

pathomatique a In physique. Lea relations entro
grandeurs variables sont donneos iraplicitement quand
cos grandeurs sont donnees; ranis il faut les oxpliciter
et los synthetiser." Of. also p. 81j "puis tout se
traduit en nonibres concrots. Les nomtires fournis par
les instruments ne sont pas des nomtires a'bstraits ni des
etre^ do raison math.6matiqu.os; ce sont des nomhres 'qua-
liffi'ds par 1' instrument qui les a fournis'. 7 volts et
7 degres no sont pas la memo chose pa'ree quo lo premier
s'ohtient avec un valtmetro et le second avec un ther-
mometre."

(41) in I Post. Anal i, lect. 25, no. 4,

(42) Phys. II, 2.

(43) In Da grin . , V, 3, p.d'6.

(44) Curs Phil ., Vol. II, Q, I, a. 1.

(45) In II Phys ., lect. 3 S no. 8, Many modern authors hold
that in this. passage St. Thomas is guilty of misreading
Aristotle. Maritain, for example, has th'is to sayi "Ici
j'ouvre uno parenthese d'ordro historique. Aristote, en
realito, n'n pns dit cela expressement, e'est Saint
Thomas qui l'a dit en s'appuyant sur un texte d'Aristoto
pour notre plus grand profit mal compris. Aristote, au;
livre II de la Physique, chap. 2, 194. a, 7, parle do la
connaissance mathematique, et il parle des partios des. _
mathernatiques qui sont plus physiques quo los autre?., qua
concerned d-avantage les choses physiques, e'est co qu'i
appello <r& (puaixw-uepa twv iiaeriu&jwv ,

les traducteurs modernes traduisent a hon droit, "les
parties les plus physiques des mathematxques". Sninu Tho-
mas, au contraire, dans aa troisieme lecon sur 1 Lxvre ,,
de la Physique, entend'qu'il s'agit non pas des ai les
plus phy^es'des .^ematiques ma^e -ox -^^
physiques que mathematiques, myi s nnuuxt ^

^ansion, op. cxt. p. ^ £ *pit ; ° mritain might ,

ficial reading of the GreeK ja . Relievo that

seem to favor his i nte ^ ro ^™\T r ^roct g, hora a rc
St. Thomas- reading of An atotlo » --^omas ^ ^
three reasons. First, the renui^xu phrase,

incompatible with the const n ictxo, of the Gro.k P* «
Secondly, the Latin translation which .t. Thon, orod

was that made hy William do MboiDoKe, an

(31)

Dy the most competent of modorn critics as extremely
accurate. Thirdly, the exactness of the version of

the S:™t 3 ed 3 -oh lGar ^^ th ° V ' h ° le **°^ in Sich
the disputed phrase appears, f or in lt Aristotle Bhowg

precisely that optics, for example, is rnoro physicnl

than mathematical. In order to Taring out this last

point wo give here the whole paragraph in Greek,

together with \7illlam of Moorbeke's translationi

At]Xo~ 6e xai ia cpoaixarcepa tuv iiaernicrtcov, oi5v
6iiTixfi xat AptxovLxf) xai aaTpoAoyfa' iv&7taAi v yap

TrpoTiov tiv' e'xouat it) yEWfieTpia. ' AAA' f) p.£v
yew(iSTpfa nepi ypa[i\iy]c, cpuaixfic oxotcsi', 'aAA'
oux f] (puaixf]' f]'5' oTcrixf] jia9ri(j.aTi xrjv |i£v xpa\i[i.f\v
' aAA' o6x t\ liaerpatixf], aAA' t) (puaixf].

"Domonstrant autem ot quao magis physica quam mathe-
matica, ut perspoctiva' et harmonica et astrologir.s
o contrnrio enim quodammodo so hahont ad geometrir.rn.
Geometria quidem enim physicam intendit lineam, sed
non inquantum est physica! sed perspoctiva quidem
mathematicem lineeim, sed non inquantum mathomatica,
sed inquantum est physica.

[32)

(46) IX - II, 9, 2, ad 3.

(47) Ours. Phil . II, Q. I, n . i,

(48) Of. Fulton Sheen, "ivory science is constituted of a
material and formal ohjoct. Tho material otgect ia
what ia studied; the formal ohjoct is the aspect or
the how it is studied. The now mathematical physics
ij.3, from the material point of view, a science of the
?oal world, hut it soon loaves that concrete, roal
world to manipulate it in terms of mathematical sym-
bols." The Philosophy of Science , p. 83.

(49) Op. cit . p. 28

(50) On the Method of Theoretical Physics , P. 12. Cf. The
vforld As I 3ee It , hy the same author, pp. 32 - 34.

(51) Ihid. p. 7.

(52) Op. cit . p. 173 footnote.

(53) Du Ohominement de la Ponsee, p. 482. In spite of tho
great name that Meyorson has won for himself in tho
philosophy of science, and especially in the historical
■background of science, we find it necessary to remark

[ that throughout his many writings he has consistentl y
misinterpreted Aristotelianism and Thomism.J

(33)

Chapter IV

(1) Lect, 1, no. 1.

(2) Cf. I. Met. c. 2, 982 a 17.

(3) Cf, Be C oelo et Mund o, Prooemium; Be Gen erationo et
Qorruptiono, P rooemium; Do MeteorologiciV , i; p e Anima.

I, loot. 1; Be gensu ot "gensato , lect, 1; Bo pnrtibus
Animal ium , II, c. 1; De Generationo Animalium , I, c. 1, etc.

(4) Lectio 1, no. 2.

(5) It is worth while noting in this connection that the
scholastic manuals which make the study of nature a part
qf metaphysics are perfectly logical in placing the study
of general metaphysics before that of cosmology and psy-
chology.

(6) Of. Harold H. Smart: The Logic. of Science , New York, B.
A : pploton and Co,, 1931, p. 80.

(7) The. Analysis of Matter , New York, Harcourt, Brace and Co.,
1927, p. 130,

(8) Cf. Whiteheads Science and the Modern V/orld , p, 41 •
''Nothing is more impressive than the fact that as mathematics
withdrew increasingly into the upper regions of ever greater
extremes of abstract thought, it returned back to earth . ^
with a corresponding growth of impo rtance for the analysis
of concrete facto"... . ,. .
Che paradox is now fully established that the utmost a*» Tac-
tions are the true weapons with which to control our thought

\of concrete fact."

(9) Translation by H. P. Hardio and B.K. Gaye.
,10) Lect. 1, no, 6.

(11) Cf. St. Thomas: In_ i _Post^Anal. loot. 4, no 16, Ir t omni
onim gonerationeTTu^Te^i^otentxa est prius tempore

et posterius natura, quod autem ^V^nTutL Sn^ris
prius natura et posterius tempore Cognxt^ute^ge^riS

^^LVS^i^l^^^m££^^ Undo

K2£i£i. in ^ a ? tU + rnostrar-priu S est cognosoore magis

in generation scientiao nostrae pixu 5homa3 ., Log ica

k commune quam minus commune," Cl. Jcmn u

II, q. I, a. 1.

(34)

[12) I, lect. 1, mo. 1.

[13) De Pot. Ill,- 2, ad 2.

[14) pp. 128 - 129.

(15) "Que si, cependant, on considere lea limit os de co <iu'on
entondait ainsi deduire, on s* apercoit que ill-gel est
resto 'bien on doca de son medolo. Aristote et 30s soc-
tateurs au moyen age, nous l'avons vu a propos do Ger~
sonide, limitaiont sans douto la deduction a l'universol,
mais ces unlvorsaux oompronaient tout co ' qui constituo

la science, puisquo celle-ci ne pout traitor quo du genre.
Hegel ne declare deductible, que certains aspects trbs
generaux de la science, tout lo reste etant issu, do
I'arbitrairo, de la nature et justiciable seulement de
savoir empirique, O'est qu'en dopit de touto son 'arro-
gance logique', la philosophic hogolienne 03t obligee
de tenir compte de ce fait qu'un enormo acquis scion-
tifiquo s' interpose entro olio et les dorniers soctateurs
de la physique peripatetique, et que cotto evolution
lui interdit de poussersa regression au dd-la de certaino3
limitos." pp. 476 - 477.

(16) No. 8,

(17) In II Phys ., lect. 6, no. 3; Do Tr in. V, 4, etc.

(18) Mothodologie Sciontifiquo ,xLaval. p. 26.

(19) Cf, for example Reys "La Physique scolastique uvait

la pretention d'atteindre diroctorAont los propositions
genorales dont so deduisait le systomo complot do lr.
nature. Conta?e cotto pretention s'61cva la Physi^uo
de la Renaissance." - - La Theorio Phys_iq uo, p. 344.

(20) Be Partifrus Animalium , Ch. 5. Transl, by Ogle.

(21) I, c. 2, 316 a 5 - 15.

(2?) Of r qir,™. The Story of Living Thing s, pp. 9 -44;
l«| 01. G. bingor; ino oi>u j — rrpoks" in 3nc. of J.lodorn
"The Birth of Science Among the Groeics m _ i _ L _

Knowledge , pp. 1415 - 1427.

(35)

(23) De 'Jrin .. VI, 2.

(24) The Philosophy of Physical Scionco. p. 10.

(25) Leot. 14, no. 8. Cf. In XII Met , lect. 12.

(26) Cf. Phys. II , loot., 2.

(27) Loc. cit. Cf. also page 133: "Ainsi le panlogisme peri-
pp.tetiq.ue et 1g panmathemntismo et panm6cnnisme plntoni-
cien et moderne se rencontrent dans cette foi en In
rationalite complete, et, partant, en la deductiMlito
de la nature,"

(28) The Mysterious Universe , pp. 123- 124.

(29) Cf. Spinozaj JSthics, Part I. prop. XXIXs "Things
oould not have been produced by God in any other way
or in any other order than the way and the order in
vvhich they have been produced."

(30) ."None of even the relatively gross structure that the
microscope has revealed was suspected to exist 'before
it was seen." - - Yves Delnge. Cited by V. r .H. Thompson,
Science and 8ommon Sense, p. 45,

(31) Paul V.dlery, Cited by Louis de Bioglio; Hatiere et
Lumiere , p. 318.

(32) Cf. Meyerson. Id3ntite^t_Raalit£, p. 368. "on ne Pout
mieux oaractdriiSF-liiT^its distinct if b de ceUo methode
que ne l'a fait Paul Tannery t 'D'une part, ten-

dance a s'attacher aux phenomenes tels duo les sons les
revelent a 1'ohservation superficielle et groaaxero, on.

dT^mG^ToVSilii ne sent pas visxblem ent JM » ,
part, tendance a remonter le P us f u P« ^^ ^
plus tot possible dans la S ^ G f ^ ro £ our nouven u
simple analyse du concept et sans * uc x m _

\ 1,e 7 6ri r e ;:t L e" s i B n r SnS: SiSS i9oo > ™-

ture choz. Aristote" in o°"b"-°

IV, p. 214.

, , „r, 1 Cf. Lettro 'a Moraonno (1632)

(33) principia , III PR", ° h - /• ° ' Q , „ j0 suis devonu si
Bd. Adam et Tannery, t. I, P« ^ •

(36)

hardi quo j(oso maintenant chercher l r . cauao do In
situation de chaque etoile fixe."

(34) Science et Hypotheae , .p. 168.

(35) De Coolo , III, c. 7, 306 a 5 ff. Trans, by J.L. Stocks.

(36) Do Coelo , I, c. 3, 270 1) 10.

(37) Iiect. 7, no. 6. Cf. lect. 3; "Unde hoc non est demons-'
tratum sod suppositio quaodam.^

(38) Oh. 12.

(39) Lect. 17, nos. 1 and 2.

(40) I q. 32, a. 1, ado 2. Cf. In I Meteor , lect. 11, no. 1;
"Postquam Philosophus roprohant opiniones aliorum, hie
incipit ponore opinionem propriam'de comitis. Et primo
ostendit modum cortltudinis qui est in hnc materia es-
quirendus. Bt dicit. quod do tnlibus quae sunt imnianifosta
sensui, -non est exquirenda certa demons* oat io et neces-
saria, sicut in mathematicis et in his quae subincent
sensui; sed sufficit per rationem demonstrate et ostendere
causam it a quod quastionem solvamus per aliquam solutio-
nem possibilem, ex qua non sequatur aliqaod inconvemens,
per ea quae hie apparent secundum sensum. Unde hoc modo
in poroposito ad habendam causam est procedendum. CI.
also In XII Met. lect. 10, no. 2586., etc., etc.

(41) Cf„ lex Theoi-ie Physique , pp. 54 ff.

(42) prodromus n-i n.^rta tionum Cosmogr aphioarum,

Continens

ICYSterium ^ ^^^^ ^12^^^^^^
Opera Omnia , t. K, pT TTa - 153 - - Cited by Duhem, op.

pit , p. 58.
(43) Op. cit . p. 59
<«> 0?. person, ^^^M^^^^^

/exemple, la loi de Mariotte) , doiv ^ ieg 6brnnlor>

jamais. Toute rechorcheulturieuie ^^ lg c01lteuu , oat
ou seulement a en modifier ou . ^^ r i g0 ureusement
jugoe parfaitementoiBeu.se e comtQ Qgt r0V6ntt fc

Vproscrite. Cost la un theme b ^ ^ exprim - p . vec

rnaintes reprises et au sujo"

(37)

l'energie la plus grande. Accumulnnt los termes de re-
jrobation, il Q declare 'incoherents ou steriles-? pro-
codant d'une -curiosito toujours vaine et grovement per
turbatnceS d-uno .puerile curiosito stimuli par u^o
yaine amotion' , les travaux ou 1-on SQ sort d- instru-
ments de mesure trop precis; il a proteato hautement
contre 'l'abus des recherches microscopies et le cre-
dit exagoro qu'on accorde trop souvent encore a un moyen
d* investigation aussi equivoque'."

(45) Posterior Analytics , I, c. 27, 87 a 30 ff. - - Trnna by
G.R.G. Mure.

(46) Qfa Meyerson: "L'irrationnel scientifdqueressemble done,
a, certains egards, a celui que, selon Renouvier, oons-
tituerait un acte de libre arbitre ; il reprosente aussi
dans un ordre de considerations tout different il est
vrai, un 'commencement absolu. •" Be L'Sxplicat ion dans
les Sciences , p. 543.

(47) Of, infra Ch, VI.

(48) Lect„ 41, no. 3.

(49) Met. II, C. 1, 993 b 3. Trans, by Yf.D. Ross.

(50) Cf. St. Thomas, In De Trin . V, 1.

(51) Curs, Phil ., I, p. 767 b 28 - 41.

(52) "le Probleme de i'indeterminisme" in L'-A-iadomio Canadionne

Saint Thomas D'Aquin , Sixieme Session, Quebec,. 1937,
Typ. 1 'Action Catholique, p. 67.

(53) Curs . Phil ., I, p. 200. Of. p.- 790.
154) Cf. Infra Chapter VI.

[55) Cf. John of St. Thomass Logicaj?. 60.

156) Cf, e.g. Bertrand Russell* "The general V™^f Q3 °\ hQ

science, such as the belief in the reign of law, «" e

belief that every .vent must have a ™™*>*™J» l™g£ a *

dependent upon the ^n^S^l s a e ZlTelT
of daily life. All such general principles

[38)

because mankind have found innumerable instances of
thexr truth, and no instances of their falsehood. But
this affords no evidence of their truth in the future
unless the inductive principle is assumed. » problems

of Philosophy , London, Thornton Butterv/orth. Ltd

1912, p. 107. ' '

(57)1 Cf, e.g. .Mansion: "Be ces donnees - - encore incom-
pletes - - il faudrait conclure: pour Aristote, la science
I - - prise au sens moderne du mot ou dans un sons analogue
V- - n'existe pas," Op. cit . p. 19.

(58) Of, e.g. textsof Maintain considered in ch II.

(59) Ch. I, 642 a 5,

(60) Ch, 5, 645 a 5.

(61) Essay Concerning Human Understanding , IV, ch, 12 sect. 10,

(62) Cf, St. Thomas: In I 13th . lect. 4, no, 52:1 .. .

St quia nos ratiocinando notitara acquiriraus, oportet
quod procedamus ab his quae sunt magis nota nobis; et si
quidem eadem sunt magis nota nobis (efr) simpliciter ,\ tunc
ratio procedet a princiviiis. sicut in mathematicis.J Si
autem alia magis nota sint simpliciter, et alia quoad
nos, tunc_o- portet Q conversoCprocedere sicut in natur q-
libus et moralibus.^ j Cf. also In I Poster Anal , lect 4,
noTT6. "This is just one of the several points in which
/>' modern scholastic s' _ho v^ made philosop h y a kind of mathe r
matics,

(63) Lect, 6, no. 10.

(64) Philosophy of Soienco p. 167.

(65) : Cited by Thompson: introduction to Science, pp. 124 -

125, Cf! also CohenT^OH^eTTAnl^l^ c ^^/° g s "
'and Scientific Method , New York, Harcourt, Brace, and Co.,

1934, p. 149.

(66)

•„.,„. tun vr>lativitd des connais-
Cf. Hey: La Theorie^hygio^, ^f^^ « a cot , d e
sances physico-chimiqu s leur permet _— de3quolles

rv^H cniiTin flsnncos n 0.03 ai-i- J- iu " lui

cjs connaissancos

(39)

la physique est incompotonte. alia no lour permet pas
d'en connaitre l'objet." p. 367. P

(67) Physios of the Twentieth Century , New York, The Philo-
sophical Library, 1944, p. 142. Of. Dingle; Through
Science to Philosophy ; "If, without violati ng the p rin-
ciples on which physics and biology have developed
science can extend its correlations over ( the whole of
experience ) ! t will -become philos ophy." p. 34. C f. a i 30
Whitehead; Process and Reality , Cambridge University
press, 1929, pp. 12 - 13,

(68) La Philosophie de la Nature, p. 141. We heliove that
Maritain's error is at least partially due to the fact
that he looks upo n the whole of philosophy as peda^o -
gioally ( posterior) to the experimental sciences . He
writes; "II y a, c'est bien stir, une forte dependence
materielle de la philosophie a l'egard des sciences.
Tout d'ahord, dans la hierarchie des connnissances la
philosophie est commo le termc dulminant ,C gt qui par
suite vient p eda gogiquomerit on dernier lieu .J- - Les
Degres du Savoir , p. 101. In our opinion the correct
pedagogical order of the speculative sciences is as fol-
lows; Mat hematic 3 , f philosophy of nature ,; the experimen -
tal sciences , Metaphy sics. J We shall try to explain in

IChapter VI why mathematics is put in the first place-
/ of all the speculative sciences it has the greatest ha r-
mony with the human mind . That is why there are child
prodigies in mathematics and not in the other speculative
V gciences.

(69) /The simplicity and commoness of the experience that is

sufficient for philosophy of nature has led some authors
Uo make of philosophy a kind of logic. Thus professor
Watson writes: "The student of philosophy already knows
how to speak in a manner that is understood^ his fel-r
low in every-day affairs. When he begins philosophy, .
Questions are asked ^^^JS^SSSL^S^J^^
new facts abou t the ^^rg nTirSIiFta^nnture of

l -^ n nTT„+ timir solution does not; nave

philosophical problems that their sola vnnt to

/to await th^_be^pming_0iJact3 . it mu»o ,

Philosophy what actually happens in the w rfA. If JJ*-^
were not so how would P^ 30 ^ ^ Qn Under3 tanding Phy-

j/hoso business is with facts. "

sics, Cambridge University Press, 1938, p. 6.

21. ■ ' .

(40)

(70) Introduction to V/hat Man Has M ade of Man, m xi - tH
Of, Dottarar, Phi^ophy By way^Flhinsciences "if
then, while including the^eTcriptions arrived at hv !
common sense within the class of. genuine descriptions
of reality we should deny this status to the results
attained hy scientific research, our attitude toward
science would, seem to imply the principle that the more
pains we take in trying to discover the nature and stru'c-
ture of the actual world, the loss likely we are to suc-
ceed in the attempt.'' New York, the Macraillan Co., 1929,
p„ 297,

(71) Les Sciences Sxperimentales sont-elles distinctives de la
philosophie de la nature?" in Culture, 1941, IV, p. 473,

(72) The Philosophy of physical Science , p. 8.

(73) In Met . I, lect. 1. Cf. In I Sent . ,d. 38, q. 1, a. 5, c.

(74) Cf. Duhemj La Theorie Physique , p. 248s "Si done ^inter-
pretation thoorique enleve aux resultats de 1'experiemce
fie Physique la certitude immediate que possedent les don-
nees de Conservation vulgaire, en revanche, e'est l'inter-
pretution theorique qui permet a 1' experience scientifique
de penetrer hien plus avant que le sens commun dans

1' analyse detaillee des phenomenes, d'en donner une des-
cription dont la precision depasse de heaucoup 1'exactitude
du langage courant." Cf. also Pp. 246 - 247.

[75)

[76)

[77)
[78)

An exception to this last statement is found in divine ;
knowledge.

De L'Explication dans les Sciences p. 214. Cf. Chemine-
ment da la Pensee , p. 52.

La Theorie Physique , p. 29. Cf, p. 497.

Cf. Meyerson: Tde ntite et Realito , p. 20, "La loi qui
rogir^action-du-lelFiir-n^vI^ge quo le -levier matho-

7. ± aclilon uu f . M n (, ue n ous ne rencon-

matique.; or ™» 3 ™"f°f ™* ? ft nature . Do memo .
trerons jamais rien de pareil dans ^ ^

nous n»y re ncontrerons £™" ^^ montren t los mo-
sique ni les enstaux tale que nous v en3eMe for _

deles cristallographiques. . . On connai ^

midahle de travaux auxquela Stas a mi se i f

tenir do !• argent a pau praa clumiquement pur, on

(41)

d'ailleura qu'il avait choisi ce corps comme point de
depart de ses determinations parce qu'il lui paraisaait
offrir lo plus de facilitos, et l'on salt aussi que
1' argent ofrtenu par lui n' etait pas reellement pur, do
sorte qu'il a falludepuis rectifier los rcjsultats aux-
quela ll etait parvenu. On peut voir, par cet exemple
topi que, combion le substrat memo de la loifle concept
' gjnorajj^ep^ejt^chos^dj jriotre pensee ." Cf. also De
L'Sxplioation dans les Sciences , pp. 23, 26, etc.

( 79 ) Substance and Function and The Theory of Relat ivity.
p„ 130, ~ ~

(80) Prui'ace to the second edition,

(31 ) Theoretical Biology. Introduction.

(82) "La seule science qui merit e propreraent ce nom est celle
dont la certitude est apodictique; la connaissance qui
ne peut contenir q'une certitude empirique est ce qu'on
n'appelle qu' imprqprement un savoir." Uh savoir theorique
"ne merite le nom de science de la nature que dans lo

cas ou. les lois do la nature qui en sont le fondement sont
connuQS a priori et ne sont pas de simples lois de 1' ex-
perience." Kant: premiers Principes Metaphysiques ■:■.
de la Science de la Mature , French transl. fry Andler et
Chavannes, Paris, 1892, p. 4. Cited by Meyerson: De
L' Explication dans lea Sciences , p. 458.

(83) Science et Hypothese, p. 170. Cf. Whitehead. P rocess
and Reality, p. 22 s "Every scientific memoir m its re-
cord of the 'facts' is_sjiaii thron g and through with m -
terpretation." And Schrodinger writes, "\7e cannot close
thidJoTTTthe entry of subjective factors in determi-
ning our scientific policy and in giving a definite di-
rection to our line of further advance." - - Sciance and

. the Human Temperament , New York, V/.W. Morton and Co.,.
.1935, p. 87, ~ '

(84) Pp. 39, 44, 53.
Pierre Duhem is Particularly goquent on th ^point .«o

has shown with great P on f ra ^°* ™ ience n pplies_with
is true of hiological ^d medical c ience ap^^___

infinitely ^re S bev^QTaS^2J^m^' in wni

(85

142)

that theory plays is so great. C f. La Thoorie Physique ,
pp. 276 - 278, et passim. 'Venselg^ement de la Physique
par la mothode purement inductivo t elle que l'a dofinie
HOTton,Ce 8t one chimera^ Celui qui protend suisir oette
chimere se leurre et leurre ses dleves. II leur dome
pour faits vus dea faits simploment prevus; pour procodos
realisahles, des experiences purement ideales; pour lois
experimental, des propositions dont les termes ne
peuvent sans contradiction, etre pris comme exprlmant des
realitds. La Physique qu'il expose est une physique
fausse ot falsifiee." p. 309.

(86) Du Cheminement de la Pensee , p. 463.

(87) Cf. Meyeraonj "De plus, exposes sans hypotheses, les
resultats experimentaux nous apparaissent comma quelque
chose de definitif, d'achevo, sans que nous aporcevions lr.
voie qui y a mene, ni celle qui pourra nous conduire plus
loin; car la science n'est pas haconienne , et l'exporienco
seule, sans le secours de l'hypothese, ne saurait y mener

Jjien loin. C'est ce qui fait que l'image de la science
ou d'une partie de la science quo 1'on nous offre ainsi
sera en quelque sorte statique, alors que la science se

trouve en realite dans un flux perpotuel et dynamique.j'

I dent it 6 et Seal it e , p. 468.

(88) Where is Science Going? p. 97 Cf. Jeans- The Hew Back -
ground of Science , pp. 46 - 47,

(89) TJu Cheminement de la Pensee , p. 45.

(90) Cf. Lindsay and Margenau: Foundations of Physics, New
York, John Wiley and Sons, Inc. ,1936, pp. 4-5.

91) /Cf. Cassirerj Nevertheless it would he a mistake to as-
' sume that exact science, owing to this ^oterxgio
feature of its concepts, withdraws more and more from the
Suffered hy concrete empirical existence Precisely
in this apparent turning away from the real ity °* «^»
science ij directed upon them in a new w y ,hey only go

heyond the given m order |o^r^ tner^^___^ t
systematicstructural^el^tip^oO^ ■ _

229 •■ . 128„

(43)

(92) Of. Jules Tannery, Scienco et Philosophie, pp. 32 g _
333: Je pourrais Men ne pas parser do la Sis se puis -
que c'est one unite fondamentale ot non una unite dorivee,
pais je no vois auoun inconvenient a ce qu-on definisse

la masse d'un corps en docrivant l n faeon dont on fait
one pesee (une double pesoe si l'on veut) au moyen d'une
balance. Si 1'on equilibre avec des grammes on dim que
la masse en. grammes est exprimoe, ou encore, qu'on a
pris le gramme pour units de masse; c'est la meme chose.
Quant au 'gramme' il ne me generait nulloment que les
eleves pensassent aup;etit cylindre decuivre que l'on
sait, mais il est Men entendu que pour satisfaire tout
le monde on parlera du morceau de.platine irradio, etc...
en n'oubliant pas, si l'on veut etre dans le train, dei
dire que ce morceau de platine est depose au pavilion
&e Bellevue, et non aux Archives,, Je sais Men que
l'on criera au cercle vicieux. . . "

(93) Cf, Eddingtonj The Philosophy of Physical Science , p.
J70.

(94) Cf, aldingtons The Philosophy of Physical Science , p.
JL89j "Reference may also be made to another general phi-
losophical system, namely logical positivism. Our insis-
tence that physical quantities are to "be defined in such
ft way that the assertions of physics admit of observa-
tional verification, may suggest an affinity with logical
positivism. The meaning of a scientific statement is to
he ascertained by reference to the steps which would be
taken to verify it. This will be recognized as a tenet

pf logical positivism - - only it is there extended to al]
• statements. When it is limited as hero to items of phy-
sical knowledge, it is in no sense a jhilosophica ^. tenet;
it is only a bringing jinto_Une of the l R nguage_of_bheo-
garf^T^ir^flBir g5irt5rihyBl08 , so that we may .not
^liSETEi-i55porror^iiHaCIo5^r assertions which
have no observational foundation. If it were a general
characteristic of knowledge, it would not be so useful to
us in discriminating physical ^°*&.**™ °f ° rG Ss-
of knowledge. We are therefore not ^° tt ^_£°J 1B
posed to favcurthe more a^" 1 ^-^^;^,^.

(44)

(95) pp. 1-4. Cf. Thejjatureof- the Physical World m

not he divorced from the nature of the appliances with
which we have ohtamod the knowledge. The truth of the
law of gravitation cannot he regarded as subsisting
apart from the experimental procedure hy which we have

Ascertained its truth. The conception of frames of
space and time, and of the non-emptiness of the world
descrihed as energy, momentum, etc., is hound up with the
survey "b y gross appliances . When they can no longer he
supported "by such a survey, the conceptions melt away
into meRninglessness." "3b passim. For a detailed analy-
sis of the meaning of operationalisra and its implications
cf. Percy Bridgman: The logic of Modern Physics , Mow
York, The Macmillan Co., 1932., and The Nature of phy -

. sical Theory , Princeton University Press, 1936.

(96) Methodologie Scientifique , p. 16.

(97) principles of the Quantum Theory , p. 3,

(98) jioo. cit.

(99) Cf. St. Thomasj "Sunt enim quidam, qui veritntem intel-
ligihilem capere non possunt, nisi eis particulatim per
singula explicetur. "Bt hoc quidom ex dehilitate intellqc-
tus eorum contingit." I, 55, 3,

(100)

Of.. Mnsteinj Introduction to vmere is Science Going?
p. 13;- "In every important advance the physicist finds
that the fundamental laws are simplified more and more as
.experimental research advances. He is astonished to no-
tice how suhlime. order emerges from what appeared to hq
chaos." Cf. also Hermann V/eyl: The Open World, p. 41, ^
"The astonishing thing is not that there exist natural laws,
hut that the further the analysis proceeds, the finer the

/ details, the finer the elements to which the Phenomena,
are reduced, the simpler - - and not the ^ ^ » J
as one would originally expect - - the » m ^ c ^f 10nS
hecome and the more exactly do they descnhe the actual

\ occurrences."

llOl) Physios , Blc. II

(102) Cf. Boutroux: ^eJ^UeeJ^P^^^

dan3 In

ui, isou-croux: ua " ^- " ,,, Tot , | -j a m e-

Science et danTT^hil£l°SL5> p. 42: Les low

(45)

(103)

camques ne peuverii done etr'e considered comma
realiaees telles quelles dana la nature d e B eheaea,
Los concepts dont eiles' se compoaent deviohnoht itt~
telligihles quarid on en fait ilea ^-bi'ea,**

Lea Prlncipea de la PhyaiqueV (adaptation of Phyeiee.
The Blementaj, Pafls, Lihrairje Felix- Aloan, '1939'^ } 22.

(104) La Theorie Physique^ p, 252Y

(105) $he ; Philoaophy of Physical SsienCeV ppv S£ " 67,

(106) 0f,« Mnstein- and Infeldi Evolution of phyaica , Bp» ll
™. 1.2 .Bdaihgtohi The Nature, of the Physical World, pp,

126 ff„ .-',-. .... •. . . • : — : — "

(107) For other .examples cf. Duhem, op, oit . pp» 325 - 387.

(108) Cited hy Duhem, op. pit. 'p.- 3l8,

(109) The Nature of the Physical World , p. 234.

(110) L'Bvolution des Ideea en Phyaigue , Paris, Flammarion,
1938, p. 286. Cf. Meyerson, Do L'Sxplication , etc. p.
60. •

[ill) C-fi Loui3 de Brogliej Matiero et Lumibre , Paris,
Editions Alhin Michel, 1937, pp. 319 - 320.

[112) Cf. Mnsteinj The World As- I See It, pp. 35 - 36 1

,c Tho natural philosophers -of those days were, on the con-
trary, most of them possessed with. the idea, that the fun-
damental concepts and postulates of physics wero not in
the logical sense free inventions of ihe human mind Hut
could W deduced, from- experience *Djf 'abstraction' - - that
is to say % logical taoahs, fc clear recognition of the , .
erroneousnosa of this' notion Really only came With the .
generar.theory^oforelativity, which showed th «J' ^ jouid
take account of a wider range of. empirical facts, and t hat
too in a more satisfactory and, complete manner, on _ a faun-

it the same tileThat^^Tattompt at a log

(46)

of the hasic concepts and postages of mechanics from
elementary experiences is doomed to failure."

(U3) Of. Arist: I Anal. Post . , 22. "fern ah offectu qui a

plurihus oausiB procedere potest, non potoat una illarum
concludi,

(114) Cf. Meyersonj "Una science privee de theorie apparaltrait
en quelque sorto comma entierement achevee, statique,
alors que la vraie science, nous le sentons, doit etro
en flux, evoluer, progresses - - Do L'T*xplicntion , etc.
p. 55.

(115) For an excellent example of how science advances hy suc-
cessive-theoretical syntheses, see Louis Do Broglie,
Matiero et Lumiere ,' pp. 15V - 177.

(116) Cf. Duhem, op. cit . pp, 268 - 269; Hey, op. cit . p. 194,
etc,

(117) Cf, Cassireri "It is only owing to the fact that science
ahnndon3 the attempt to give a direct, sensuous copy

.of reality, that science is ahle to represent this rea>
lity as a necessary connection of grounds and consequents.
It is only through going heyond the circle of the given,
that science creates the intellectual means of repre-
senting the given according to lav/s." op. .cit. pp. 164 -

165. Cf. p. 280.

(118) /Cf. Petit, CS.Ct Methodo logie Scientifique , p. 10.
"Au-dessus du sujet, au dela de l'ohjet immediat, la
science se fonde sur le projet . Dans^laje nsee , . s ci e n-
tifique, la mediq1;i anJ^JJ^Jelja£ig^BJgLH2S^

^ toujours lq form g_du_E£2J^°"

(119) L'Intellectualisme de St. T homas D'ilquin, Paris, Becu-
chesno, 1924, p. 146.

(120) Cf. Curs. Phil . I, ed. Raiser, p. 86 a 18; 87 a 49 - * 23.

(121) Cf. St. Thomas, "A forma quae est in anima nostra pro-
cedit forma quae est in materia m nrtifioialiDus,
naturalihus autom e contrario. I n Hot. V lx.

(47)

(188) V, 1, ad 3. In the context of this passage St.

Thomas mentions explicitly only logic ^ ^he-
matics, but irom the examples he cites it is
evident that he includes mathematical physics under
mathematics.

(123) I, 93, 1.

(124) Of. Goethe: "Hypotheses are the scaffolds which are
erected in front of a tmilding, . . " In Maxims and
Reflections , Cited t>y Franks Betwe en Physics and
Philosophy , p. 30, Cf also von Uexkull, loc, cit.
p. x.

(125) Cf. e.g. Sir James Jeans; The New Backgro und of
Science s "The, history of physical science in the twen-
tieth century is one of a progressive emancipation
from the purely human angle of vision," p. 5. Cf, pp,

-227 - 228,

(126) Op, cit . p. 445.

(127) Cf, Niels Bohr: "The present day situation in physics
hrings forcefully home to us the old adage that we are
actors as well as spectators of the grand drama of exis-
tence," Cited hy Tohias Dantzig- Aspects of Science , p,
135, '

(128) Evolution Oreatrioe, pp. 131, 101, 356. Cf, Matiere
et Memo ire , 3e ed. Paris, 1903, p, 201.

(129) Les Prinoipes de la Physiqu e, p, 166,

(130) "L'Outillage Mental," in Sncyclopedie Franqaise , Paris

1937, Cited Tsy Supple: Dialectics and Experimental
Biology , p. 20.

1131) Cf, C. de Konincks P 2 _ II _^™»tP. Bu Bien Commun Contre
Les Personalistes , Idltio^Oe l'Universite Lavax, Que-
bec, 1943, pp. 159 ff.

1132) Experience and Natu re, pp. 357 - 358. Of. ^° J%J&
F or Certainty , P r202, "The doctrine that nature w in
he^STTStional was a cos ly on^_^ ^
idea that reason in man is an Jpff^Tird^rived rea-
rationality SS^SSM-SS^LS^^^,^ * tB Uisine3g
son in man of an »°*« a "J^ ^olioaily. to view
was simply to copy, t0 / " P °MUtv to mn i M a transcript
a given rational structure. Ability

148)

of h! structure xn mathematical formulae gives great
dolxght to those who have the required ability. But
^ dpea nothing; xt makes no difference in rn'oure. In
effect, it limits thought in man toretraversing in cog-

<nitxon _a pattern _fix8d_ and complete~iFTE¥ elfrThe

doctrine was both an effect of the traditional sepa- '
ratxon between knowledge and action and a factor in per-
petuating it. It relegated practical making and doing
to a secondary and relatively irrational realm."

[133) Onze Theses 3ur geuerbach , pp„ 87, 95. Of. Jean Lan~
gevin: "Science et Industrie" in A La Lum i ere Du
Marxisme, p. 114 • "La methode experimentole, est veri-
ta'blement active, puisqu'e'lle consiste prlcisement a
maltriser ou a modifier les circonstance3 naturelles.
C'est a olio, eri premier lieu que s'applique la phrase
de Goethej ^Au commencement do tout, il y a l'action.'"

[134) Freedom Versus Organization , New York, 1934, p. 192. Nor
was Hitler unaware of the essential meaning of Marxism ,
as the following passage clearly indicates; Co qui
reste du marxisme, c'est la volonte de construction re-

(volutionnaire, qui n'a plus beeoin de s'appuyer sur des
bequilles ideologiques et qui sa forge, un instrument de
puissance implacable pour s'imposer aux masses' populnires
et au monde tout ontier. D'une t!5ledbgie a base scienti-
fique, il sort ainsi un vrai mouvement^ revolutiormairo,
pourvu de tous les moyens necessaires a. la conquete du
pouvoir. Et quel est le hut de cette volonte revolu-
tionnaire? II n'y a pas da but precis . Rien qui soit
fixe une fois pour toutes. Aveg-vous tant de peine a
comprendre cela? Nous so mmes en mouveman t.tJoiltLiS
^V^irTiT^^t.^ SaTi^ous savo-HiT^s qu'il n'y a
iF ^r^etat defiTiitif, qu'il n'y a rien do durable qu'il y
a une evolution perpjtugUe-LOe_a. ui ne se transform e

nas c'est ce qui es Tl^Q Lgj3]gsent_e8t_dgjajaMe.
SSI' ■S-^1-^J^-^-- !S ^—^t^Su nis^le dTs possibilites-

feis U^e^T^rirmm% inepuisable des P° s
^nfinies d'une creation toujoursnouvelle/ in Hermann
helming, HitlerMVA_Dito pp, 211 - 21 * " " C ^° a * y

De Koni'nok.-^^SIo^S£l22*i£i3!3£» 1939 ' p< ? '

[135) Morceaux Choisis, p. 222,

"'"^r^sSi^cT^Hy^otheso, p

(136) of. Poincare* ^J_es 1 laS0^3S^^^r^P SQ f. 258 ,

l'action quiest_lo_m°I2Bj

(49)
Chapter y,

lj Physics and Philosophy , pp. 179, i 8 i„

(2) The Mysterious Universe , pp. 138, 111. c f. The Now
Background of Scienc e, p. 60.

(3) Cf, The Scientific Outlook , p. 88.

(4) Cf. Beg , II and III.

(5) , Wew Pathways in Science , p. 24.

(6) Gf, John Of St, Thomas, Ars Logica , p. 839.

(7) I, lect. 20.

(8) Lect, 20, 'no. 574.

(9) Cf, St. Th. In Met V II, lect ,2, no. 1280- "J3t quia
posset alicui videri, quod ex quo philosophus ponit
omnes modos, quibus dicitur substantia, quod hoc suf-
ficerot ad sciendum quid est substantia; ideo sub-
jungit dicens, quod nunc dictum est quid sit substantia
'solum type', idest dictum est solum in universali,
quod substantia est illud quod no 11 dicitur de subjecto,
sed de quo- dicuntur alia; sed oportet non^ solum ita
cognoscere suhstantiam et alias res, scilicet per ^ ^
definitionem universale m st logicam £ \ hoc enim non est
sufficiens ad coanoscendum na turam rei^ quia hoc ipsum
555r^iii^a^3Tplo~HiH5rtion5Tili, est manifostum;

Hon enim huiusm odi definitione tangu ntur principia
rei. ex auib us cognitio rei dep onflgt; sed tangi.ur

L; -i — ■ — r-r-r-i ™= ~_„ „.,„« 4-nlia nnl.lfl

a]

\ds

aliqua communis conditio reT por quam talis noti.icatio

datur^

:i0) For other examples cf. Post_._Anal . . l ect - 27 > n0 ' 7; leot
33, nos. 1-2; lect. 38, no. 6.

Ill) .Cf. m^hys. lect. 8. nos 1 - ^/^Jf o^r-Is-

sage St. Thorns ^^%°l^fJl%^%AialeoUo^^-

! cause they P^^iS-^^ second type of
mon principles such are founa " ^^^^rojmm&y
dialectical reasoning, butjwpcipi^ .

^accepted Cand henc e_JPgojgPig^

(50)

(12) Cf. St. Thomas lect. 2, nos. 24 - 28; of. also St
Alport the Great, De Anima, I, tract 1, cap! 7
"Pliysicus autem et dialectics diffiniunt differenter
unumquodque i.storum quae diximus esse animae opera

et passiones. Si onim quaerimus quid est ira, int'en-
dentes de diffinitione quarere, dicetC^ialelkclIs)
quod est appetitus contrarii doloris, out" aliquid
huiusmodi diffions per intentiones communes formales,
quae non sunt vara causa rei propria, sed intentiones
Voommunes inventae in pluri"bus et nulli propriaej et
ideo diffinit per formam quae forma de se communis est,
et non appopriatur ad esse i -ei nisi per propriam materiam
uniuscuiusque rei.( jphysicus )autem dicet quod ira as-
census vel ascensus ef calefactio sanguinis circa cor,'
tangens propriam causam officientom quae. est ascensus
et calefactio sanguinis, et propriam materiam quae est
sanguis cordis hulliens, et subjectum quod est cor.
1st ho rum quidem alias reddit materiam propriam scilicet
physlcus; alius autem reddit speciem et intentionem
formae simplicem, et communem quae est rei ratio com-
munis,, Hie onim considerat rationem sive intentionem
communem rei, eo quod non descendit ad propria: ille
autem considerat principia realia.quao dant esse rei.
Necessariu.m autem est quod ista realia principia sint in
materia huiusmodi 1 quae determinata et propria est, si
er.unt ot hahent esse in natura,"

[13) Lect, 5, no. 9, v . • ■

(14) Cf. Cajetan> in I, 17, 3, nos. 7-8.

[15) Cf, St, Thomas In I Sent , d. 38, q. 1, a. 5, c- "In
istis causis ef'fectus futuri non habent certitudmem
ahsolutam, sed quandam, inquantum sunt magis determmatae
causae ad unum quam ad aliud; et ideo per is.as causas
potest accipi scientia conjecturalis de futuris, quae
tanto magis erit certa quanto causae sunt magis deter-
minatae ad unum; sicut est cognitio medico. de ^^.f
morte. futura, et judicium astrologi de pluvixs et ventis
futuris."

116)' Topics , I, 1, 100 h 21 - 23,

!l7) 121 h 2 - 3.

(51)

(18) I, 1, 1355 a 14,

(19) ma,

(20) In I Top , c. XII, no. 4,

(21) In his Commentary on the I'opi cs. St Albert the Great
brings out the meaning of -probability and its connection
with dialectics' "Probabllia autem sunt verisimilia.
Dupliciter autem verisimilia s aut enim in so sunt ve-
risimilia, eo quod ipsa habitudo praedicati ad sub-
jectum verisimilis est, eo quod nee praedicatum est

in subjecto per se, nee subjectum in praedicato por se,
ned utrumque in utroque, nee praedicatum hecessariam
et essentialem inhaerentiam habet cum suhjecto, sed
verisimile est in signis non in causis necessariis ac-
ceptum, Aut quia necessariam habet inhaerentiam, sod
non accipitur nisi per signam; et hoc est probahile se-
cundum modum acoeptionis, quamvis in se sit necessa-
rium; sicut solem esse majorem terra (eo quod ubique
unius quantitatis apparet) probabiliter acceptum est.
Solem autem esse majorem terra per quantitatem diametri
acceptum est necessarium et non probabile, secundum quod
probabile et necessarium opponuntur. probabile autem sic
dictum verisimili est quod per suiipsius veritatis fi-
guram videtur omnibus aut pluribus aut sapientibus, et
his sapientibus videtur omnibus aut pluribus aut maxxme
notig6t probabilibuss ita quod sapientibus et his vel
omnibus sapientibus vel pluribus vel maximo notis vel
probahilibus, totum pro uno membro ponatur.
"Signn vero verisimilitudes, aut occurunt statim in su-
perficiae et in exterioribus rei quao accipxt sensxtxva _
potentia coraparens sensata ad invicem, et si talxa sunt sig
na, probabile est quod videtur omnibus, sicut nivemesse
albam per hoc quod nix est parvao partes perspxeux in
parva conjunct!, in duius partibus undique lux dxffun-
Stui, hoc enim signum sensui est medium. Sx autem sxgna_
indisium facientia de verisimilitudxne sunt ™ £ ™P£ d
ficie, sed aliqualiter profundata,. non " c °^ a ^ a '^f
nee in superficie extrinsocus manentxa .- tunc est xd^uod
videtur pluribus: quia sensui alxquxd nascent ™^ 9 >
. . , ._,-,_ j„ „„„rfn minoris ursae sxu polus, oo

sicut quod Stella in -"^^motus : hoc enim
quod non deprehendxtur exus sxngu arx^ ^^

rationis judicium sensux est permit

(52)

verisimilitudinis profundatur in essentialium et con-
vertibilium causas quae sunt convert! Vn i V- ;
tunc est quod videtur sapientibus lieu Is TXLT^ 1
luna moveatur m epiclolo, quia profundi et altuis
transit per umbram tarrae, hoc enim non est sausa sed
signunu

"Ideo illud quod videtur sapientibus gradus habet, quia
aut videtur omnibus, aut pluribus, aut mavime notis vel
probabilibus . i^uia signum convertibile cum causa, vel
apparet mixtum sensui, et tunc videtur omnibus, ve] in
ipsis substantialibus profundatur, et tunc non videtur nisi
probatis et probabilibus sapientibus vel medio est ac-
cepting et hoc dupiiciter. Si enim plus esb inclinaturum
ad sensums tunc est quod videtur pluribus sapientibus.
Si autem plus est profundatum ad necessaria essentialia
in intellectualias tunc est quod videtur maxime notis,
qui ex potentate scientiae et arti3 hoc deprehsndere
novorunt. Hoc igitur est probabile;, ex quo fit syllo-
gisraus dialecticus,, quod tali et taiiter divor3i.ficato
deprehenditur signo. Haec est sententia commenti A.ra-
bicig et. sic scientia demonstrativi et etiam dialectici
syllogismorum determinata est." - - Lib. I, tract. I, Ch.2.

22) Iiidetermination in things may. of course, he a cause of

the indet'ii.-minatiqn in the mini that is proper to opinion,
as St., Thomas points out in numerous oocasions, hut this
latter indetermination may also he had 'when things ere
objectively determined, Cf. DjLY?^-.*??. 6 .' :w » Zf ad 3i
"Sunt autem quaedam in quibus non os'o possi'bile talem
resolutionem facere ut pervsniatur usque ad quod quiu^
est, et hoc propter incertitudinem sui esse; sieut est in
contingentibus in quantum contin^entia sunt; unde talia
non cogrioscuntur per quod quid est, quod erat proprium
objectum intellects, sed per alium modum scilicet per
quaedam conjecturam de rebus illia de quibus plena cer-
titude haberi non potest. u»da_aa_hog_ sUa p otentia
requiritur. «Jt quia haec potentia non potest reducere
■ ' rT^ioHii-inquisitionem usque ad suum termnum quasi ad
quv.tom, sed .consistit in ipsa inquis ™°™^f™
motu, o^inionem solummodo ,inducons de J" ^ ^ rit '
ideo quasi a ternv.no suae operatioms hn o o tentxa
ratiocinativum vei. opinativum nominatur. Of. also _Do
Anima III, loot. l6o

(23)

(53)

Of. St. Albert the Groat, In^er^hermeneias, i, tract.
II, .o. 5. p. '493 h vives TTXT^Tatt^de quod
licet nomeu infinitum nihil ponat ot nihil signiflcet,
non tomen est vox non~significativa ut hrriaTmf. quia
vox non-significativa non excitat intellectum ad aliquid
de aliquo mtelligendum, sed non-homo oxoitut ad in-
telligentiam de hoc quod est privatae qualitatis, quod
tamen in rerum natura nihil est, quamvis sit in appre-
hensione: et hoc modo opinio est circa id quod est in
apprehensione tantum. Bt est simile sicut quando di-
cimus ihnominahile, hoc nihil est, et tamen prout ca-
dit in apprehensione per suum oppositum quod est no-
mlnahile, ad aliquid excitat intellectum."

[24) In I Post. An al, "lect. l p no. 6. Cf. St Albert, In I
Physicorum , Tract, 1, cap. 5j "Dico autem quod omnis
scientia quae hahet prinoiuia sic procedit, et ilia
sola est vere scientia; quia est demonstrativa, et ef-
fectus solius demonstrationis est scire. Si autem ip-
sa non haheat verum nomen scientiae, tunc ipsa erit
scientia topica dialocticae vel rethoricae, et ei'fectus
eius non erit scientia, sed opinio. . . "

[25) Ars Logica ., II, Q. I, art. 5, pp. 278, 280,

[26) IV, lect, 4', nos. 576 -577.

[27) The Philosophy of physical Science , p. 1.

[28) . Ars Logica , Pars I 3 p. 5.

[29) Ihid , Pars II, p. 250,

[30) I, c. 12, 105, h 10.

[31) St. Thomas In I Posterior , lect. 9, no. 4.

[32) Cf. I Post. Anal ., lect. 5, nos. 7 - 8; lect. 19 nos.
4 - 5-.

[33) H. Poincare, La^cienc^li32£«^' p ' **'

[34) In I Post. Anal., loot. 5, no, 4.

(54)

(35) I, leot. 21, no. 3,

(36) Cf, Top, Ch. 14. '

(37) Matlere et Lumiere, p. 177, M. Jean Perrin writes-
■'Tout concept fmit par perdre son u^ilite, sa si-
gnification meme, quand on s'ecarte de plus en plus
des conditions experlmentales ou il a ete formule." - -
Cited by Petits Me thodologie Soientifique . p. 18,

(38) Philosophy by Way o f thejjoiences. p. 124,
(89) Op. cit. p. 6S.

(40) (When experimental science is. made the only valid '
typo of knowledge, and when it is applied to social
and economic problems, it is easy to see that radical
and revolutionary social doctrines are 'bound to he the
result. (JMarxism__ i3 a proof of it?)

(41) Top. I, tract.' Ill, c. 1.

(42) The New Background of Science , pp. 46 - 47.

(43) Op. cit. , . pyeface p. x,

(44) Of. I Post. Anal , lect 1; I Top.., c. 2,

(45) Loc, cit. It must he noted in passing that the state-
ments "In nature everything is certain" is at best am-
biguous. In relation to subjective probability it is
true,' hut in relation to objective probability as de-
fined above it is false.

(46) Op, cit . preface p. ix.

(47) Cf. for «w«mnTa.' T De Generation et C orruptions, c. 2 P
316 a 5 - 15.

(48) Topics , I, C .1.

(49) An Outline of Philosophy , P» 16S "

(50) Myatioism and Logic, London, Allen i Unv-n, 1927, p. 75,

(55)
Chapter vi.

(1) Bertrand Russell. Mysticism and Logic , p. 91.

(g) It is worth noting that the Thomlsts are not the
only ones who insist upon the essential relation
■between mathematics and quantity, p, number of
modern thinkers are beginning to realize that the
only adequate solution, for many of the problems con-
cerning the nature of mathematics is a return to
this traditional notion. Cf. Harold H. Smart: The
Logic of Science , Chapter III.

(3) Cf. Burtt, op. oit.. p. 43; "...the orthodox Aris-
totelian school minimized the importance of mathe-
matics. Quantity was only one of the ten predi-
caments and not. the most important."

(4) In Met V, lect. 15, no. 983.

(5) Some modern Thomists erroneously make quantity a
common sensible. Thus Marltain, who, after asser-
ting that quantity precedes the whole sensible or-
der says; "3311e (la quantite) est un 'sensible
commun'." - - Les Degres du Savoir , p. 281.

(6) Cf. also I, 40, 3, c. .

(7) De Trinitate , loc. clt.

(8) In I, V, 3, ad 4 , no 4.

(9) Op. cit.

[10) Loc. cit.

[11) Curs. Theol ., la, Q.V and Vi, disp. 6, art. 2.

[12) John of St. Thomas, Loc. cit. no. 17.

[13) John of St. Thomas, loc. oit. no. 20.

114) Science and The Mo dernJVorld, p. .44. ffrltor in go
same work he writ is, ^^^l^^S^l
in the sphere of o«^a.*J^ f SS £ l8
ticular instance of what it is *£*£*£ the pttr

is perfectly true, hut it does not bring

(56)

ticular character of mathematical attraction, for
the Bamo stateme nt could be made of n+.h OT - tycosTf
a bstraction , " — — —

(15) The Nature of physical The ory, p. 67.

(16) In Phys. II , leot. 3, no. 6.

(17) Substance, and function and Sinstein's Theory of Be- '
lativity , pp. 19 - SO. Later in the sa¥e work (pp.
229 - 230, footnote) he writes- "The 'concrete uni-
versality' of the mathematical concepts has also in-
cident ly "been recognized and emphasized from the "
standpoint of Hichert. 'The gap for conceptual know-
ledge between the universal and the particular, ' says
Lask in his work, Fiohtes Idealismus unde die Ge3chichte ,
and the consequent irrationalit y (is bridged in the ma -
•Ehematical view t hrough the possibility of construction? )
The individu al cases r ealizing the mathematical con-
cept can be (ginarate~d)by the concept itself . From the
concept of the circle, we can attain by construction

the mathematical individuality of the particular cir-
cle, and thus go from the universal to the individual
in its individuality ... in mathematics, also, the in-
tuitive object is an individual concrete and given ob-
ject-) but it is given a priori , not a posteriori jliko
thematsrial of sensation it is a logical unique, some-
thing individual, but at the same time capable of being
construed a priori; We see here also that Richert's
criticism would have taken another form if he had con-
fceived the concepts of natural science decisively and
from the beginning as products of constructive mathe-
matical, rather than as results of 'abstractive' pro-
cedure} The insight once gained for ^matics would
Whad to be transferred to physics; for precisely her
lies the r.al problem - - that mat hematics is «° ^i™
unique ', but that it yv osmSil^y--BJmi^J^-^f^

ihainatical 'deduction' is ai

c^c~ept. TheTb^Tof mathematical '^ -" " ~

-r-^— a. . a ,-v, +v,A form of physical 'induction , u,y

ready contained m the form oi game

which we grasp' the empirically real, an

method of mastery of the particular by the universal

118)

^achi eved. _ ) (
I a, a. VandVt, disp. 6, art. 2 [T. i. PP

(67)

(19) Pu Cheminemont de la Penaee , p. 694,

(20) Loo, oit ,,, no 9 20.

(21) Physios and ■Philosophy , p. 16.

(22) III Met. Ch. 4.

(23) I, 5, 3, ad 4-.

(24) Loo, cit ., no. 29.

(25) Cf„ In I Poatg Anal , lect. 19, no. 6.

(26) I, 5, 4, ad 1, '

(27) Met. XIII, ch. 3, 1078 Ta, 1 - 5.

(28) Pp. 120 -121.

(29) P.. 121,

(30) I, 85, 1, ad 2.

(31) In De Trin . V, 3, c„ -

(32) In De Anlma III, lect. 8, no. 708. ''Quaedam ergo _ sunt
formae quae, mater iam requirunt sub determinata dispo-
sition sensibilium qualitatum; et huiusmodl sunt omnos
formae naturales; et idcirco naturalia conoernunt ma-
teriam sensibilem. Quaedam vero sunt formae, .quae non
exigunt materiam aub determinata dispositione sensiDilium
qualitatum, tamen requirunt materiam sub quantitate

• exlstentem: siout triangulus, et quadratum, et huiusmodl-
et haec diountur mathemat ioa, et abstrahunt a mateiia
senaibili, aed non a nmteria intelligib Hi, inquant urn
in intellectu remanet continua quantitaa. abstracta r.
aensibili qualitato. "

(33) In De Anima III, leot. 8, no. 714 - 715.

(34) Cf. In VIII Mot , loot. 5, no. 1761.

(35) I Mot. Ch. 6, 987 b 15.

(58)

(36) I, laot. 41, no. 5., Of. De Verit ate. H, 6, ad i.

(37) 2 P ^'„? 1 ;», C " ed ty BUrtt! Th^Ifetophysicol F oun-
dationa of Ph ysios, p, 57. — i

(38) 'Of. Iljtot. lact. 5, no. 336; VI mil., lect. 7, no

1209; De frin, VI, 1, otc.

(39) II Met ., Ch. 1; St. Q?h. leot. 1, no. 281.

(40) In II Mot ., lect. 5, ho. 336.

(41) Leot, 7, nos. 1209 - 1210,

(42) "Movet circa hoc quaestlonem scilicet quure puor po-
test fieri mathematicus non autem potest fieri sapiens
idest metaphysicus vel physicus, idest nature.lis. Ad
hoo respondet Philosophus, quia haoc quidem, scilicet
mathematical ia cognoscuntur per abstractionem a sen-
siMli'bus quorum eat experientia; et idoo ad cognoscon-
dum talia non requiritur temporis multitudo. Sod
principia-naturalia quae non sunt abstrncta a sensibili-
Tms, per experientiam considerantur, r.d' quam requiritur
temporis multitudo. Quantum autem ad sapientiam', sub-
jungit quod invones sapientialia quidem scilicet meta-
physicalia non credunt, idest non attingunt monte,
licet dicant ore; sod circa mathernatica non est immani-
fostum ois quod quid est, quia rationos mathomaticorum
sunt rorum imnginabilium, sapientalia autem sunt puro
intelligibilia. Juvens autem facile capere possunt

ea quae sub imaginationo cadunt. Sed ad ilia quae ex-
cedunt sensum et imaginationom non attingunt mente, quia
nondum habent intollectum exercitatum ad talos consi-
derationos, turn propter paucitatem temporis, turn proptei
plurimas mutationes naturae." Ibid .

(43) In II Met ., lect. 5, no. 334.

(44) Ibid no, 336.

(45) Of. Gerard Petit, C.S.C.: Mgbhodp^ogie^c^^ PP.
72, 78 etc.

(46) Cf. Timaaus 35 a.

(59)

Esprit -o-bl. -i^S^^^ST
que nous sommos embarrasses pour determiner co que
noua dovons attrite r o. 1'uno pu Wutro source? ot
que nous pouvons, on fin de compto,. solon dos rai-
sonnemonts qui s'appuient plus ou moins sur des' ox-
penoncos, modifier cette attritiuUon. »

(48) Da Chominemont do la Ponsee, pp. 658 - 659,

(60)

_Cha pter v ii.

La 1'heorie Physique . pp 158 - 159,

IT) id .

The Na ture of _tlie^h^J£g^orij ; pPo 251? 253<>

Saunderson, author of a treatise on optica, was
Wind from tho first year of his Ufa.

T he Philosophy of. Physios , p. 16. Cf. alsoj Theore -
tical Physics , Colunbia Univ., 1915, pp. 4 ~^~5~.

Cf. Opticks, Bk. I, Pt, II, 1931 ed., pp. 124 - 125.

Leviathia n, p. 3.

:33 3ay Concerning Human Understanding , Bk. II, ch. 8,
par, 9 ff.

Tho Universe in the Light of Modern P hysics, p. 14.

Cf. Lindsay and Margenauj Foundations of Physics , p. 20;
Norman H. Campbells Physics - f Jho moments ; Stehbingi
P hilosophy and the Physicists , p. 80; Bortrnnd Hussell:
The Sciontific Outloo k, p. 67, etc,

Cf. Med, VI; Principi a,IV, 198? 199. etc.

Cf. I Phys ., .lect. 2, no. 7; TLJJngl* > l° ct ' l > n0 < 8 '
VllfPhys .', lect. 1, no. 3. etc.

Dominique Salman. "La Conception Scolnstique de la Phy-
sique," in Philoso-ohie et Sciences , p. 54.

Methodologie Scientifique . Laval Univ. 1939, pp. 18 - 19

I, 27, 5, c.

In Be Anima III, loot. 2, nos. 592 - 593,

In IV Met , lect. 12, no. 673.

In Do Anima II. loot. 10, no. 350.

161)

(19) Science and The Modern World , p. 113,

(20) How Pathways in Science, pi 4,

(21) Mot. IV, 5, 1010 h 33 5 Of. Content, of St. Th. lect
14, no, 706; Jq A nimg, III, lect. 2.

(22) The ambiguity of the word "physical" may give i-ise to
some confusion on this point. Ve understand it here-
in its primitive meaning in vrtiich it signifies some-
thing pertaining to objective material nature. In the
passage which we are ahout to quote from Siding-ton it
has an entirely different moaning; it designates the

. world constructed hy science. That is why there is no
contradiction hetween Eddington' s position and ours,
"Writing this chapter on an autumn day, i f e ol myself in
a familiar world whoso most prominent character-istic is
colours There is no colour in the physical world. I
think that that is the right %ay to put it, It is true
that each colour is represented in the physical world by
a number supposed to indicate the longth of a wave of
some kind. Similarly I am represented at the telephone
exchange hy a numher indicating a hole in a switch-hoard;
hut it would not he correct to say that I inhahit the tele-
phone exchange. To put it anothor way, there is nothing
in the accepted description of the physical world which
owes its acceptance to the fact that we have a sense of
colour. Everything that we assert can ho verified hy
a colour-hlind person; and indeed most of our accurate
knowledge has heen ascertained through the medium of a
colour-hlind photograph plate." - - New Pathways^^ijcignce,
pp. 11-12.

(23) P. 152. Cf. Whitehead! ghojogcgpt of Mature, p. 29s "For
natural philosophy everything perceived is in "^e.^e
may not pick and choose For us he > r £"£ ^sun^
set should he as much part oi nature , a « Piro i P in.

(84) Dissertation of 1770.

(25, in IV Met ., loot. 14, nos. 705 - 706. Of. J**** III,
loot. 2, no, 596„

161)

(26) Of. I, 78, 3, ad 2.

(2?) In v Mot °» lQ rt, 15, no, 985. ■

(28) In I Met ,, loot. 2, nos. 5-8.

(29) Methodologie Solent iflquo _. pp. 19 _ 20,,

(30) Lect„ 1, no. 6.

(31) Loots 1, no, 6.

(32) Of. I, 69, 2, ad 1„

(33) Of. IV Phys . , loot. 20, 22,

(34) In De Anima I, lect. 2.

(35) Of. Plancks The Philosophy of Physics , p. 17s "Once
the specific perceptions of the senses as fundamental
concepts of physics had been eliminated from that science
it was a logical stop to substitute suitable measuring
instruments for the organs of sense. The eye gave way

to the photographic film, the ear to the vibrating mem-
brane and the skin to the thermometer. The introduction
of self-registering apparatus further eliminated sub-
jective sources of error. The essential characteristic
of this development, however, did not consist in the in-
troduction of new measuring instruments of steadily
growing- sensitiveness and exactitude', the essential point
wis that the assumption that measurement gave immediate
information about the nature of a physical event - -whence
it followed that the events were independent of the in-
struments used for measuring them - - now became the foun-
dation of the theory of physics."

[37)

Les Principes de la Physique , p. 16 - 18.

... j „ m nf Tie L'15xplication dans
La Deduction Eolatiyiste, p. 11. 01. jgJiJ ^ . . ni|lfl „ ■

lea aclenc-eTrTTTeaT^Il est manifesto, on ^ ^
TZZT&S&L&e physique ne^auraxt M^g^ » G
reelloment motivee par la .raison suff^nte, 0011

^iretoat,^lit^ont ™^^ZZSS^,

Sui™ ss: S-;- c ^f£Sm— SSS

am exigences denotre^ ^"? °°™men rationnollo ne
Cast done que la matiero veiitnbiemen
peut etre au fond que de 1'espace," Cf. also Hoy. __
Theorie Physique, p. 214,

(63)

(38) Sir James Joans. TJioJ^Background of Science, 29 -

82* /'Thus a colourblind p^oT^a? lIoTl^e to ap-
preciate tho full subtlety of Swlnburno's observation,
•Thoao eyes, the greenest of things blue, the bluest
of things grey," hut give him the spectroscope and ho v/U]
discriminate colour hy wave -lengths a million times
as finely as the oyo of tho keenest artist. A deaf per-
son cannot distinguish tho horrors of modern dance music
from the sonatas of Beethoven, hut by tho use of Lis-
sajou's figures he can detect differences of pitch of
which the ear of the most sensitive musician would be
unconscious,"

(39) Mind and M ature, p. 15,

(40) Idontite et Rea lite, p. 392,

(41) Cf, Hoys "pour Poincare, commo pour lo mecanisto, la
matibro du physicien implique une cortaino homogeneite.
Ce n'est pas 1' homogeneite simple ot ab3olue que la
mathematiquo reclame de son objot, mais elle s'on rap^
proche indefiniment comme do sa limito naturollo,

I Cetto marcho vers 1'homogeneite expliquo la possibilit p
\ pour la physique de prondteo la forme mathematiquo, " La
Theorio Physique . p= 186, Cf. also pp. 263 - 265.

(42) This dooa not moan that quantity is strictly t h e subject
or' the sop* of "the other Occidents,, : but the medium J by
JhT^h thev are roo ted in the substanc e^ "Accidens non po-
test per se esse ^M-eotum accidentia, sed unum accidens
per pkui-raoipitur in substantia quam aliud, s i out quan -
titasqufonqualitas." I, 77, 7 £ ad 2.-'., .subioctum re~
C -Iplj1ln7Sn^cIde"ns alio mediante, sicut corpus raoipi|
calorem modlfliitg_au £ grfioLo, ot sic unum accidens cll^itm
altori inesse," I - II, 7, 1, cd 3„

(43) Cf. St.Tho.nas: "Ex volocitate ottarditate motuum ^con-^
tingit gravitas ot acuita. in soms." ,n_XJfet, '

no. 1948,

(44) I, 42, l.fld'lj cf. I C. G. 43, etc.

(45) A, 11. ad 10,

(64)

(46) Of. Meyerson: 'C'est , encore une fois, l-accord entro
la realite et la mathematique, pl ua particulilrlent
la geometne, dent nous avons traite au chap it re pre 1
cedent, en t ant quo fondement du panmathematismo. Mais
ce quo nous devons constator ici, ou il 8 ' a glt do con-
cepts du sens coramun, c'est qu'il n »y ., pas seuloment
accord, mais union, union immediate et, au fond in-
dissoluble. Tout ce que notre perception nous presente
comrne reellement existant assume aussitot la forme
spatiale, et cette forme, nous no pouvons 1'en depouil-
ler sans atteindre par la l'existence elle-memo . . .

|B ri.stence et. S Batialite sont done ici 3ynonjir.es ou, du
mo ins, inseparables, et e'est la encore un aspect do
cette superiority du panmathematisme en tant quo. con-
ception metaphysique applicable p. la science, que nous

\ avons constate." Be Ii'Bxplication dans les Sciences,
pp. 576 - 577.

(47) The Ph i losophy of Physical Science , p. 122. •

(48) Hddington, op. cit . p. 124.

(49) Op. cit . pp. 123 - 125.

(50) P. 464. Cf. also Hey, Op. cit., pp. 214 - 215. profes-
sor Dewey in the Quest For Oortainty explains the sig-
nificance and fruit fulness of this homogonization of
nature from the point' of view of instrumontalism- "Phy-
sical science disregards the qualitative heterogeneity
of experienced objects so as to make them all members m
ono_com 2 rehonsj ; TC_jipm^^ capable
of translati on or conve rsion-One_into another^ This
homogeneity of subject-matter over abroad range of
things which are as disparate from oadhothe^m direct
experience as sound and colour, heat and ligno friction
and electricity, is _the source oJ _tto i 7idejind_free_Mn-

knl^aedge-clSrWe^tl^^^

here ant there by isolated couples. But it cannot pos
sibly join them all up together so *^ ™ °i"ntifio
W one to any other. Tha(^pmo|£^ ■» °"°^°

Bj SB SO s _±ij!S^J^^^ B n f A e l^ Ql l ex± - ble scheme of
makes this indefinitely broad and nexio

55

(56)

(57)
(58)

(65)

uransitions possible. The meaning which one event
has is translatable into the meanings which others
possess. Ideas of objects, formulated in terms of
the relations and changes hear to one another, having
common measures, institute broad, smooth highwryTby - ^
moans/. f which we can travel from the thought of one
part of nature to that of another, in idea at least,
we can travel from any meaning -.- or relation - - found
anywhere in nature to the meaning to he expected any-
where else. !t I n John Dewey's phi losophy by Batner,
p„ 33? „ ~

(51) De L'Sxplioati o n dans les Sciences , pp 25-27.

(52) De L'Explication dons les Sciences, p. 14. Cf. La De-
duction Eelatixzisto, p. 258, etc.

(53) Matiera et Lumiere , p. 316.

(54) Cf. Boutrouxj La Cont ingenoe des Lois d o la Nature ,

p. 71: "Pour que la loi mecaniquo puisso otro consido«
ree comme la traduction do la loi physiquo propromont
dite, il faut que 1' equivalent oxiste, non soulamont
entre los deux ordres de fait, mais entro les doux or-
dras de rapports, entro l'enchainement des faits phy-
siques ot l'enchainement do leurs conditions mecaniques.
tOr cette seconde equivalence sembie inintelligiblo piu-cc
quo, tandls_jueJ^_iarwUa^_esjLJ^^
qui do it en o t re la fonoti on_ost heterogbne.y

"C'est la qualite meme, non seuloment 1* etude ou olio
apparalt, qu'on reussit a meuurer." - - ja Ponsee et
Quant it e, Paris, Lihrairie Felix Alcan, 1927. p. .54.

'-La quantite n'est rien d» original, pas plus mat iere,
etendue ou duree, que grandeur pureme nt , lo Siaue: ello

.est une construction conceptuelle fondee a la fois sur
la diversite et 1-homogeneite ^^^J^lf^Z
de pensee. " rbid. P- 379 « "Olos^fLiESHP-Jili-Sl^^.

VquantiteV ' - - p. 273.

Cf. Sir James Jeans, Phy^ juji^hUospphy , p. 197.
Cf. Benjamin, ^^O^S^B^^^J^SSSS.' *• 31?i

(66)

methodological postulate is given the status of' a meta-
physical judgment. Tho quantitative aspects of tho
. oorldj.ro soon looked upon as representative of its
essential nature. To explain qua lities is to explain
them away. To understand them iTtoH^- convmcod th at
they nro more a/pearances. To rationalize them is to
construct a system in which they do not function at nil
as explicit elements. To talk about them is to talk
about them vicariously. To grasp them is to realize
\that they cannot bo grasped."

(59) Cf. Tannery, Science ot Philosophic , p. 38.

(60) W.R. Thoiupsom Science and Common Sense , p. 69.

(61) "And so we have our schedule of pointer readings ready
to make tho descent. And if you still think that this
substitution has taken away all reality from the pro-
blem, I am not sorry that you should have a foretaste of
the difficulty in store for those who hold that exact
science is all-sufficient for the description of the
universe and tint there is nothing in our experience
which cannot lie brought within its scope. " aldington;
The Hature of the Physical World, p. 254„

(62) Curs . Phil . T. II, p. 764.

(63) Ramsperber, Philosophies of Science , Kew York, F. S.
Crofts, 1942. p. 233.

(64) The Quest For Certainty , in John Dewey's philosophy, by
Eatner, pp. 338 - 341, '

(65) Cf. Max Planck, W*J*n°*°£^^
characteristic oTlhTTdeTeloim^trhowever did not

consist in the introduction of new ™» s ^.""*~* ' _
of steadily growing -nsiU = an -^^°»
sential point was that the assunipuiui nhysi-

gave immediate information about ^ ^'^nts J* in _
- o*l event - - whence iu follovoa ™ th _ _

dependent of tho i*f™ts ^ « ^rtos . 0n
now became tho foundation oft he theo * \> n

this assumption a extinction must fee

.3

(67)

physical measurement takes place, between tho obiec
tivo and actual event, which takes „i aoe ooSioSly"
independently and the process of measuring? which i
occasioned by tho event and renders it Por ; , J * X
Physics deals with the actual ovoirta, nS "tT^
is to discover the laws which these events obev "
pp. 17 - 18, *"

(66) Planck: The Philosophy of Physics , p. 95,

(67) Ibid , -p. 104.

(68) Eddington; Space, Time and Gravi tation, pp. 15. 31.

(69) Of. Bddington; Few Pathways in Sci ence, pp. 12 -13,
"When we have olirainated all superfluous senses, what ■
have we left? We can do without taste , smell, hearing,
and oven touch. Wo must keep our eyes - - or rather one
eye , ffo r there is no needto use our fac ult y of sto -
r^os^ojp_ic_jrisionP The eye need mot have tho power of
measuring or graduating light and shade; I think it is
sufficient if it can just discriminate two shades so as
to detect whether an opaque object is in a certain po-
sition or not. . ,

In 1915 Sinstein made another raid on their sonsory
equipment. He removed all the retina of tho oyo except
one small patch. The observer could no longer roco-

■ gnizo form or extension in the external world, but_ho
could tell whether two _thi ngs wore in ap paren t coincidence

< jar not.'J j

(70) iilddingtons New Pat hways in Science; pp. 2-3. Cf. The
Philosophy oT WsIcal science , p.' 77, etc. Cf. Bertrand
Bussell; The A nalysis of Ma tter^ p. 6j "AH empirical
evidence consists in the last analysis of perceptions;

thus the world of physics must ige^J^ss^mi^o^^?

which su^liS^ nilM^J^

(71) /Pb 69 - 70 Cf. Lenaon, '«Bw qualitative phase of physics

' ^esse^a -stituefeven ^g^^^
Sloped system of physical ^^'J^f^^m^^
mediate qualitative oxporioncL. if .

(68)

Theory, p. 46.
(72) Introduction, viii.

(69)
Chapter Till

(1) Curs. Phil . <S. II, q. I, a. 1(

(2) L'Svolution Creatrioe , p. 360.

(3) Cf. Planets The Universe in tho Light of tho Modern phy -
jjics; "The occasion of this development was that extreme
refinomoht in measurement which is an essential con-
dition of tho progress of science." pp. 87-- 88. Cf.
also pp. 58 - 60; 73 - 74.

(4) "Je crois que la predominance de la physique est duo
principalemont a. sa methode. KLla a l'avantago sur les
autres .sciences d'introduire la mesure lo plus loin
possible dans .bgs raisonnements. Tout lo secret de sa
valeur et de son influence est dans le fait qu'elle est

(la. science de la mesure. 'p Les Princ i pes de la Physique ,

p. 19,

(5) Cf. "jdaington; Space, Time and Gravitation, prologue,
p. 2 S "Physicists *I really cannot tell you anything
about it, if you wil^iot let me make measurements of any
kind. Measurement is my only means of finding out about
nature. I am not a metaphvsicist. '"

(61 In Spite of the numerous criticisms that certain aspects
of aldington' b position have evoked, we believe ohnt his
fundamental ideas on the relation between physics and
measurement are quite correct, and ^VdTnTcoTyears
opinion that has become generally aooep to J ™ ™?° n * r £ nr8!
at least among those most compete* ; to a th tiua

meaning of physical ^^°; f £™£ of the scope of
.like to make it clear thao the ™™ . t (l ^ilo-
physics to pointer ladings and th 1 ke is_^_p__

^^«m^M^^Z^ot « tendency dis-
Vsciontific doctrine. It is une "" formulated

cernible far back in tho - ^X^lnSity theory."

(7) The Nature of tta rtgiSSlJB^' PP " ^ _ ^'

. • T>v,,ra-iniiQ» iii Hovuo Heoscolns -

(8) Cf. Henoirtes La Theorie ^ff^^-^T^^T^^
tiquo, Nov. 1923, p. 363t ii

(70)

pas s'arretor nux mots- log n oms quo ii on domie n , h

tri.uts etudies en ^ w ont £ raL™t ^l^"
immedia* avec dos hypotheses sur la nature de ces at-
tribute. L'oxpression -longueur d'ondo d'une lumiere' a
un sens ohvie daae, la theoric da l'cndulution at olle ne
repondrait a rien dans la theorie de 1' emission. Mais
olle correspond™ toujours a un procede oporatoire par
lequol on trouvo un noabre-raosuro. Quoi qu'on imagine
sur 1\ nature 1 do ,.la matiero, lo procede fera trouver
lo memo nornhre; on continuera sans doute a le represen-
tor par,\ , mais on praferora I'nppolor autremont qua
'ol-devant longueur d'onde.'"

(9) L a Science et l'Hypothese, p„ 193

(10) InJ^Jfe^.. lect. 2, no. 1938.

(11) Ours, Theol . Prima Pars, Q,X, disp. 9, a<, 1 (Ed. Solesmos,
p. 50 h).

(12) Qfo "Quarta Via", I, 2, 3, c.

(13) "Beflexions sur le prohleme de l'Indeterminisme p ' ! in Ho
vue Thomiste , novembre-decembro, 1907, pp. 393 - 394.

(14) Hat. X, Ch„ 1,

(15) Ibid, no. 1938.

(16) Cf. John of St. Thomas. £arsus__Theoi , , loo. oit., p. 53 a.

(17) Ihid no, 1938. Of. In T Met, loot. 8, no. 872, "Hatio unius
^iF-in hoc, quod sirpHSotpium alicuxus nunrcri. Qu od ex
hoc -oatet quia uxinm est pri ma mensura nu mori, quo omms
kuLrts mens^tur, mensukn^^

quia per, mensuram res mensurntae cogn cuntur es autem

pognoscuntur persua ?™P™ »^ ^ ^iUlis circa 4 «od-
(.0) if. Cretan, ,ns autem minima unum est ^— J^^

rSo^fo JLE. d S.'i?SS ***** ™ -« -»■*•

(VI)

oompabitur." Be Ente ot Bsaentln, oap TI , odlt Imi _

8, no. 875s "Sciendum est autem quod-TisTlSnsurr.ra
est propria ratio unius secundum quod est principium
numon. Hoo_aut em non est idem cum uno quod conver-
•MJH--gH2_girtgjI«t - ln qaarlo dictum e~s t . Ratio enim
illius unius in sola L indivision0i consistit. Huiusmodi
autem unius mensuratione,"

U 9 ) I n X Mot, lect. 2, no. 1945. Of. John of St. Thomas,
Curs. Thool . , In Prima Partem, Q, X, disp. 9, a. 1, -
(ed, Solesmes, p. 49 )• "Perfectissimum seu magis in-
divisihile in uno genere, est mensura cetororums eo
quod quanto eat magis indivisibile , efet magi 3 certum i
Si quidem minus illi additur, vel aufertur; et sic quod
est simplicius in ctliquo genere deservit pro. mensura,
quia ad mensuram pertinefc certificare de suo monsurato."

(20) Ibid,, no. 1953. ,

(21) I - II, 91, 3, ad 3.

(22) In V Met ., lect. 8., no. 875.

(23)

(26
(27

"Et hoc mnxime dicitur in quantitate, et inde derivatur
ad alia genera ratio monsurae.. " Ibid. ,,no. 1938o

24) Ibid , , nos 1939 - 1940

25) Ibid . , no. 1944.
Ihid . ,. nos. 1945 - 1946.

It is interesting to compare this *><*rxno of St^h °- 3
on the difference hetween the measurement of number and
of magnitude with what has heen wr * e " ^J^f sir"?-

It iS th, ^^Ot^^B^^^rBBSSTB

Vlate_OEeration. If two men coun one_of_them

in this room, and reach dlft( ™ ± not nn

must he wrong. "5?he measurement of distan

(72)

, absolute operation. It is possible for two men to
measure the same distance and reach different results
land_.yet neither of them he wrongV l results,

(28) In Met, V . , lect. 17, no. 1007.

(29) Ihid ., 1935,

(30) In Post. Anal ., I, lect. 36, no. 11,

(31) In De Coelo II , lect. 6, no. 4,
(32] Ibid.

(33) Phys, IV , lect. 19.

(34) Cf. Sir James Jeans« physics and philosophy , pp. 7-9.

(35) Cf. St. Thomas, loc . cit . no. 1950s "Nam sonsus non
percipit differentiam valde parvorum, sed eorum differen-
tia percipitur 'in rationibus 1 , idest secundum diversas
ration es proportionum quae ex diversis proportionibus
flumoralihus causantur .

(36) Cf„ St. Thomas, Ihid . No. 1953 s "....sicut mensura po-
dalis , quae quidem indivisihilis est proportione, sed
non natura. "

(37) Cf. also Bridgmanj The Logic of Modern Physics , Chat). I.

(38) Of. St. Thomas i n V. Met , lect. 15, no. 978, ';Si ossot
longitude infinita, non esset lines. Linea onim est
longitude mensurahilis. St propter hoc m ratione lmoae
ponitur, quod eius extremitates sunt duo puncta.

(39)

John of St. Thomas: C^s^Theol., loc. cit. p. 58 a;
rf ^id t> 92 as !f Do rationFWsurae est quod sit
L,X„ iDia, p. »<5 a: u" " „„*;„« mnioris '

unir
con-

ifX. iDia. p. a<s as » m p nqu ratae ratione majoris
notificativa quantitatis mensuratae in m „ nqur „ u
# -a. a.- „ 0,-,-in qriHcet quantitas mensura^i

formitatis suae, quia 3 ": 1C ^ J aduotQ SGU comparnta
fusior vel inaequalior * J' 9 ™S- tatQm notificatur
ad uniformiorem ot simpliciorom quanoi a

ad uni formic rem

et explicatur eius confusio."

(40) Curs. Theol ., loc oit. , V- 49 *•

(73)

(41) Cf. Do Koninck, Methodologie Scien+n>im,„ T

i n f^t m dSnir l^t f r° ti0n PM rnpP ° rt **» iSSe ro o
11 taut defmir l'etalon par sa fonction. Aiors que le

phyaxcien clas 31(1 ue,croyait s-assimilor 1-univers on

1'ahsorhant do face, supposant tout droit dovant lui la

limite qu-il voulait attoindra, . 1 physlcion rnodome avance

a rooulons, las yeux tournes vers 1 'ombres du monde, la-

quelle se precise a mesuro qu'il recule."

(42) I - II, 97, 1, ad 2,

(43) John, of St. Thomas, Curs. Theol . Loc, clt ., p. 50.

(44) In X Met ., no. 1954, Cf. John of St. Thomas : Curs. Theol ,
loc, cit„ p. 49. "Nihil enim mensuratur mensura propria -
et adaequata, nisi por aliquid quod est sui generis. Undo

I distinguuntur diversae mensurae secundum diversa genera;
/ et per feet issimum seu magis indivisihile in umo genero, est
i mensura ceterorum; eo quod quanto sat magis indivisihile,
V ast mngis oertum. "J X

(45) No. 1955,

(46) I, - II, 19, 4, ad 2.

(47) Cf. St. Thorn.. I, 3, 5, ad 2; "Ojectio ilia prooodit do
mensura proportionata, hanc enim oportet esse homogeneam
mensurato. Deus autem non est mensura proportionata ali-
oui; dici tur tamen men3u ra_om nium ex eo quod i m umqiiodquB
tantum habet de esse , r quantum si appTOpJ^guat^Cf. Coram,
of Cajetan, nos, 9 ff«

(48) Lect/ 15, no, 978,

(49) Cf. Da.Ver. I, 1, 5, c; Iljeat., d. 2, q. 1, p.. 2, ad 1.

(50) Curs The ol., loc. cit. p. 50 a. Of. CH^Bii- *■ ^ P "
I, q. lB,~a. Ved. Reiser, pp. 381 - m*.

(51) Curs. Theol ., loc £il' » *><• 53 a#

(52) Curs. Theol . log. £il» P° 67 '

^•1 rn t p I, B. XVIII, a 3,

(53) John of St. Thomas: Curs^Phil. 1. -,

(74)

ed. Reiser p, 382,

(54) Curs. Thool . loc. oit. p. 92.

(55) Of. e.g H Dalbiez. "Dimensions Absolues et Mesures
Absolues" m Hevue^homiste, i 925 , pPe 147 f f , 6S

(56) V Met , Chap S, leot. 17,

(57) II) id, no. 1003. .

(58) Science and th e Modern Worl d, p. s 37.

(59) Cf. Benjamin, The Logical Structure' of Sc ience, p. 326.
:"The fifth objection is the general incapacity of a quan-
titative system to represeht(d ichotomous i divisions? i.e.
to handle two -value systems . Where qualities manifest
themselves not by degrees, "but by complete presence or
complete absence ,r there can. be no -quantitative represen -
tation,) Thus, it is impossible to show quantltavely how.
two qualities may be at the same time similar because '
Species of the same genus and yet contradictory because
implying contradictory differentia,"

(60) Cf. Mdingtonj "Distances are linkages whose intrinsic
nature is inscrutable;, we do not deny the inscrutability
when we apply measure numbers to them - - 2 yards, 5 miles
etc. - - as a kind of code of distinction ." - - The Mature
of the Physical World , p. 81. ,

(61) New Pathways in Science , p. 224.

(62) !»Le Probleme de .1'Indeterminisme" in l' Academie Canadienne
Saint -Thomas J'Aqui n, 1935, p. 100.

(63) Substance and Function and the Theory of Relativi ty, p. 358

(64) Critique de la Manure ,, (Actualites Scientifiquos ot Indus-
trielles Paris, Hermann and Cie, 1937, p. iu.

(65) Hew Pathways in Science , p. 229.

<,.,! .j. • i/i ff Frank- Between P hysias and phi -

(66) Beneze, -op. oit. p. 14. C±. iran*. d^ _ _

_10£ophy, pp. 94 - 97.

(67) The Philosophy of Phygi°£> V- 26 «

(75) •

(68) Physios and Philosophy , p, 142,

(69) Cf. "Reflexions sur le profrleme de l'indeterminisrn ..

1937, alSo:_ Lg Profrleme de l'Indeterminisme." in Aca-
lIS! ^^^ -^i^^i^ll^uin, Sixieme SesH^n,

(70) This seems to he the opinion of Osfcwald, for example: -When
every magnitude appearing in the formula is itself mea-
surable, then we are concerned with a lasting formula or
with a law of nature;. . . if, on the contrary, magnitudes,
which are not measurable, appear in the formula, then we
are concerned with an hypothesis in mathematical form, and

the worm is in the fruit." Yorlesungen uher Ifaturphiloso -

phie, Leipzig, 1902, p. 213, Cited by Cassirer, Substance
and Function , p. 141.

(71) Cf. Planckj Where i3 Science Going? pp. 92 - 93; "Every
measurement first acquires its moaning for physical science
through the significance which a theory gives it. iUr/bcjdy
who is familiar with a precision laboratory will agree that
even the finest and most direct measurements — such as
those of weight and current — have to be corrected (93)
again and again he fore they can he employed for any prac-
tical purpose. It is/obvious that these corre ctions cannot
he sugg ested fry the mea surement proces s_ltself> They must
first be discovered through the light which some theory or
other throws upon the situation; that is to say, they must
arise from an hypothesis." Cf. Also p. 95.

(72) Substance and Function , pp. 357; 365.

(73) The Logic of Modern Physics . p„ 10 e

(74) Cf. Bridgman, The_Log^LM ? de £ nW}y^> PP- 9 - 25 -

(75) Ibid. pp. 17 - 18. .

(76) The Philosophy qf agalonlJS^agi* p ° 81,

(77) Sddington, The_^hi^Pj^ll^£^L^i^ PP ' ?3 " ^

(78)

The philoBophyofJhjrsioap p. 22 -

1-

(76)

(79) Of. Lindsay, 'Where is Physios n n1 „ rt •

Monthly, Vol. 30, p. 246? ^ ln S^^llll

<80) StSt^*' ^^^S1^ S J^ S ^ S ^ 3 p . 142 ,

(81) "Reflexions sur le prohleme de l'indeterminisme," in
Reyugjrhomiste^ novemhre-decemhre, l 93 7 pp . W5 . ^6,

(82) Ig| gg hways in Scionco, pp. 224 - 225 . C f. The Ilathemat
calTh^^o^Relativity, pp . 5 _ 6 , cf< Ifax pi 555ErfH5 -
Philosophy oi Phys ics, pp. 95, 104,

(83) 0f. Lenzon, Procedur es of aSmpi rioal Science, pp. 15 ~ 19,
etc, . " ~

(84) Where is Scienco Going ? p. 95.

(85) Matiere et Lumibre, p. 312,

(86) Cfo Eddington^ S pace, Time and Gravitati on, p, 3 sqq.

(87) Of. Henoirte, "La Critique Binstelnlenne des liesuros
D'Espace et de Temps," in Revue Neo-soolastiquo , 1924.

(88) Cf. aldingtons It is perhaps not superfluous to add that
no/question arises as to whether the standard of length de-
fined in this way is re ally constant at all times and placoa
The question implies that we have in mind some more ulti-
mate standard ( inv^abM_wlt h 'reality ') by which to define
the delinq uencies of the physical standard . The conception
oTphyTical quantities having to conform to some particular
role alloted in advance in a vaguely imagined realm of ._
reality, is not recognized in physical scienco; quantises
such as length and time-extension are introduced solely tor
the purpose of succint description of observ ntional mea-
surements actual oxM,ypothetical."Th ± Jh^so^ L of_Pby"
sical Scien co, p. 76 «

(89) Space, Time and Gravitation , p. 11 »

(90) Space, Time and Gravitation, p, 12.

(77)

(1) ^J^iv^^tjioj^ po 58o

(2) Cf.Renoirte, l^mo nt s__d e Critique des Sconce, et do
Oosmologio , p. 135, — - i oo-enoojjo^ oo

(3) Of. Boutrouxj L^Id^^eJ^jatMoUo ) Po 33o
(4.) P. 120.

(5) Of. Duhernt La Theorio Phy sique. pp. 249 - 269; Henoirto.
"La Theorie Physique 1 ' in jtovuojffeo-soo^^i^uo, nov. 1923.

(6) Cfo Eenoirte's Critique des Sciences et Oosmologi o,. p. 148.

(7) Cf. G-eometrie ra id B rfahrung , Berlin, 1921.

(8) "La science ne se contente pas de forrauler dos lois d'expcrien
ce elle cherche "bien plutot a construire un systerne lo-r
gique, roposant 3ur un rainiraum de premisses ot comprenant
dans pes consequences toutes los lois do la nature." - -
Einstein: La Theorie de la Relativit e ed. Rouviere, Paris,
1921, p. 109.

(9) Op. cit. .p. 26„ In this connection it is interesting to
note'That in Le Sys terne ;du M onde,. Jiuhom claims that vhe
Aristotelian doctrine of homocentric spheres vms the first
physical, theory in the modern sense of the word- "pour la
premiere fois, en effot, dans la constitution de cette
theoria, on vit le geometre partir d'un certain norobre do
principes simples qui lui etaiont donnes d'ailleurs et, con-
formant a ces principes, construire un systerne mathematique
liypothetique, retoucher, compliquer co systerne jusqu'a co
qu'ii: sauvat avec une exactitude suffisante les appnronces
decritQ3 par les ohservateurs. • „„„,-„.„„„
"Lorsque Conservation eut fait . centre des f^oraeaes

que tout systerne de spheres homocentrxques etaxt a tout
• . . • 4- -„ amnrrr lf>s astronomes geometros ac-
jamais, impuissant a sr.uycr, l-a a n0UV eaux

coptbrent d'autros principes et, a •"■ n1 ^ .^ . ,

i , . » j. ^i„ nnn-5-elle3 hypotheses; maia i>\

principes, comhmereno J 8 "^"^rulra do nouveaux sys-
methode ^'il = , SM1 ^. / f celle qui avait sor-

temes astronomiques ne diffeia pas ontri<luDa ... - i,

vi a edlfior le systerne des spnores
p. 128.
10) Suhstance andJAmction, etc. p. 135.

(78)

(11) Critique of Physi cs. p . i 59a

knowledge^ tho physicStrVS ^^oVL?"
single science which will per]lQps { Q ■ ^ £'° ^

™+f Z 11°*} °\^^S^^rioal conception Why
not? All the knowledge is derived from measurements mr.d~
with various instruments. T he lnateunrantB U30d lntho na -
different fields of inquiry are not fundamentally unlike.
There is no reason to regard the partitions of tho scien-
ces made in the early states of human thought as irre-
movable. " - -Thejat u.ro of the Physica l TEorld. p. 137. of,
also Einstein: The World / L s I See It , ppT~3g~T 34,

(13) Op cit„ p. 24. Cf. also Garrigou-Lngrange • Le Sons Com-
mun, La Philosophio de l'etr e, p. 70s "Los Tciancos po-
sitives ne peuvent jamais quo classer des faits generaux
par des hypotheses proviso:' ros (hypotheses representatives
et non, explicatives) , . ,"

!l4) Cf. Dr.ntzig; Aspects of Sci ence , p. 231* "The continual
use of such terms and phrases has finished by converting
them into so many now patterns, and to tho extent that
thoy conjure up in the mind of the exports definite phy-
sical situations, those weird patterns fulfill their pur-
pose as fully as did the classical mechanical models, " Cf,
Diracj Quantum "Mechanics, p. 10$ "One may extend tho moaning
of the word 'picture' to include any way of loo king at the
fundame ntal laws which ma kes their self -coi^sj^on^yj^bvious.
WitlTthis extension," one may acquire a picture cf atomic
phenomena by becoming familiar with tho laws of tne quan-
tum theory,"

15) Cf. Tho principles of Quantum Mochanic s, by Dirac.

16) The Mature of Physical Theo ry;, p. 62.

17) Cf. Shrodinger Scieno^r^t^^^^^^, P- ^P- '

io.

18) Op. cit. pp. 63 - 64 «

„■? +hn Phvsioal World, pp. 198

19) Cf. Bddlngton, <sheJk^±±iJ^2-Il^-±

(79)

llptic orbit. This is only "Si L ^T 1 ^- ^ Gl -
tains nothing of the sortf a^rca! ton ! iV? ™ ^
thing which it has not entorSd iȣ V, C01 , ltalns S0MO "
conceivG, which h, 3 ho t le mind ° f rten t0

by Sohrodingar.no?: Si" lT^ 30 ™ « iCfaly

^Cotte hypotheso de Bohr) olt o X ceptiom W nL n r^ve.
3110 est on offet, on contradiction formellc avec l es
lois do l'eloctrodynamiquc. Do co chef, m.lo peat 6tre
qunliflee d'absurde. Si done les theories otaient ft-ites
pour expliquer loa phenombnes et donnor ainsi a 1' esprit
In satisfaction de les comprendre, on aboutirait a ce
resaltat singulier qu'il faut recourir a 1'absurde pour
faire una theorie coherento du monde, Mais, comme uno
pareille maniere do voir est tout a fait inticceptable, on
doit conclure que les theories mecaniquos n'ont d'uutvo
fin que de creer des modeles coioinfes. Toutos loa hy-
potheses relatives a ces modblos sont acceptable3, car,
d'une part, le monde ideologique auquel ils appartiermont
ne saurait etre que convent ionnolloment astroint aux lois
du monde sensible; d' autre part on no saurait oxiger. dos
modeles quo de scheiaatisor des faits et do pormottre a
l'esprit de p.revoir d'autres faits par los raisonnoments

felativeinent simples qu'ils pouvent suggeror," Cited Toy

Senoirto, Op. cits, p. 158.

(20) Pp„ 189 ff.

(21) Ibid. Louis de Broglie has brought out tho truo relation
between models and mathematics in physical thcor'-i "Uno
autre conclusion 9 'impose "a nous. Les representations, con-
cretes ont souvont aide et aideront souvent oncore los
theoricions dans lews rochorchos, mais olios constituent
en renlite la partie fragile ot passable des theorxosp ce
qui subsist! Soo sont los formes abstraites au^o Uo, cos
representations ont conduit. Fresnel etai. parven a
liquation des Ondos en imaginant un <*h« elnoti* 1 ^
brant. Maxwell et ses continuatours remplacon^ cot ,th,r

elastique par ^.^^mSs S—
moins concret. Sinstein et los ^latx vil) „tion eloc-

compretoment 1'ideo d- ether et redu xsent 1
tromagnetique a n'otro qu'une piuo gr ndo g oMOro

dirigee. La nouvelle Mecaniquo, enf n, n p i ^
attrihuar uno nature phyBiguo P^ 010 " " "

(22)

(80)

envisage, et cola ne 1'oirroechP «>,n« M * ^
"Le veritable but de la Eil^, ^ S ° (16tc1o PP^'«

etre da decouvrir of d'etudier los fT^ ^ d ° n °
dans lGsauollpq i„' k . ? lQS fomos ^thematiquos
aans xesquelles les phenomena physiques pouvont vonir
so loger Assigner ce role a la physique Loriquo? e'est
aans doute faxre participer ootta science a la rigour do s
SS 1 ^* ltols »'"* ««Bi lul marquer SGS 1"
derrxere 1 'harmonic quo nous revble la possibility do
coaler des f aits dans des moulos analytiquos so cacho one
Realite dont l'ossence nous dernoure prodigieuscment incon-
nue . Bocuoil d ' exposes siu- les Qndea et Corpuscu los, Paris ,
1930, pp. 24 - 25. Cited by Hono'irte, ^p _cit. pp. 162 - 163,

Bouasse; He la Methodo dans las Scien ces, Paris, Alcrn
42 ed., 1915, p. 124,

(23) Cited by Duhem, op. cit . p. 116i

(24) Sciences et Hypotheses , p. 174.

(25) Op, cit. pp. 60 -61„

(26) "L'O'euvre d'Binstein et l'Astrononrio" in L'Astronomie ,
juillet, 1931. Cited by Maritaint Le s Degre3 du Savoir ,
p. 305 footnote.

(27) Cf. Duhemj "En exigeant que les operations mnthemntiquc3
par lesquelles les postulats produxsent lours consequences
aiont toujours un sens physique, on impose uu geomotro d'xn-
supportables ontraves qui paralysed toutes sos demarches;
a.vec B. Bobin, il en vient jusqu'a redouter l'cmplox du
calcul differentiel; eniait, s'il se piquait de _ satis fair o
sans cesse et scrupuleusement a ces exigences 1 ne pour-
rait presque plus developper aucun calcul; dosses pie.uois
pas, la deduction theorique se trouveraxt ar reteo. gideo
Slue exacte de la ^hode phys i.uo uno p « ^^ c<m _
cation entre les propo itions qui o nt « . rondront

trolo des faits de c^f 3 ^^ p ormQ ttront d'user, pour
au geometre toute sa liDerte et lux pe t du Woo

la plus grand developpement de a theo.i P
les ressources de l'algobre." op. cit. p. .

„i n 105t "Comment nier que la
,28) Cf. Mllhaudt lojatipnnoli ^ A e noU ons fictivos, in-
solence tire le plus f tt >* /J°£ m turo, aux condition,
verifinbles. echappant, par lour

(81)

ordinaires de determination des choses
thematiques, chaquo syraholo nouvoau in^rodJt °" -~
ralisation precisement dan- loo „ \nuroduit par gene-
ditions premieres, Uo.^l-L'^ -^^ ,«-

rLr s ir 0nt ce P :o S nrS^ nt f* "^ ***** ^"potent
^Ta I' ° Parfcua des vuos manifostement rtsurdos '

Cited hy Meyerson: Dujn^emont de la pon8 . ' ^° s « -
Of. W.B. Thompson Science a nd cS^Tst^rj; 87. oh-
viously fictitious entities-^n-Te^e-IF-a-n explanation on-
ly if the explanation ho conceived as not heing definitivo
nor ontological nor proper. Cf. Meyerson. "II nous somhle
aller en quelque sorte do soi quo la veritahle explication
soit en moDie temps une explication reollo, par co qu'il y
a au-dossous de phenombne, par oe qui est . Souls, les '
habitants d'un aaile d'alienes, dit avec raison Hartmnnn,
pourraient tonir des explications physiques a l'aido de
concepts sciernment irreels. " - - Be L'lgxplication dans la s
Sciences, p. 61.. Cf. Whiteheads The Concept of nature, p p.
44 - 46 „ "~

(29) .Cf. Space, Time and Matter , (French ed. ) p. 280.

(30) A. 3. aldington: The Mature of tho Physical World , pp. 161 -
162. Cf. Cassirer: Substance and Function , p. 116.

(31) I'Tho feeling that all the steps in a mathematical theory
must have their counterpart in tho physical system is tho
outgrowth, I think, of a certain mystical feeling about the
mathematical construction of the physical world. Some sort

of an idea like this has heen flitting about in tho background
of the paraphernalia of the thinking of civilization at
least since the days of Pythagoras, and every now and then
perhaps after some particularly striking mathe^tical success,
it buLts forth again like a crop of mush rooms a ter a rain,
as in the recent fervid exclamation of Joans that -God is
a mathematician. '" - - Op. cit- P- 6V -

Suhstance and Function, p. 116. (Italics ours)

Of. Prop. 0. Castomuovo, Sciontia, Vol. BSIII. PP. ™ " "<>.

132)

!33)

M) Th e New Kackgrounj ^flgig^ W' 60 ' U4 ' 136 '

!« pf ijiddington:
!35) Cassirer: Substance__and Function, P- If". • •

(82)

"Wo must seek a knowlnrWi ,-,t,j i •
nor of actions? S of whicK S/ 3 T'^ ° f nCt ° rs
a vohloule. The knowledge can ""J" 8 ™ d notioilB
of structure or patter/conL SdTS act SuV^
New Pathwa ys .ofjojgnnn f p. 25 6.

r.ro

It M _

(36) I, 18, 1, ad 1,

(37) Physics II.

(38) Physics III, 1.

(39) Cf. Phys ., IV, lect. 1,

(40) Loot. 15.

(41) No. 985. ' '

(42) V. 3.

(43) The Nature of Physical Theory , p. 73,

(44) Cf. Moyerson; "Nous avons vu qu'h 1' origins lo concept
'do la Vitesse n'est qu'un rapport ontro deux tormoa li-i
mites et que le mouvement apparait comma un changemo nfc
analogue au changement de couleur. II n'on ost plus
ainsi pour nous; le mouvement nous apparait ' commo un etafc,-
annloguo par consequent non au chr.ngoinont do couleur, mnis
a lr. couleur ello-raemo." - - Idontite ot Realite , p. 159.

(45) Space, .Time and Gravitation , p. 51.

(46) Cf. Phys. IV, lect. 20* "Non onine quod non rnovotur, quios-
cit; sed quesciens ost privatum motu, quod tamon aptum na-
■j;um e3t moveri,"

(47) Cf. SWin^on, Th^Naiure^fJh^Jh^iciaUo^, pp. 132 -
135.

j -^ nt„ n 91- "The world of

(48) Cf. Hiozlor: physjxsjind^al^y, P- • _ _ ft000 mpliahocl
your physics TTinS&XT**™ £ * og»

work extended in time a rea £ °£ °™ of , otunlit y. YoUF
. naturata. . . Your world is *^ verified

lHwT^eTato actualities to one another.

183)

(49)

*£ ax ^rlon OQ i n a 3tratum det

observer from tho totality of ^T y e an °nymous

is the behavior of classes °Jj henomo ^- Its content

certain conditions prevail ne f °^s. So f„ r tta

your large scale inorganic wn^n 5 d ° P rG ™il in

is governed by your sort of X/si^i" f° pl(ine of "duality

straight -lino causality. But the °i ™l' H ° nCG ^

not tho entiro body of re^lL/V ™ of Rotn «l«y is

actuality and potential!^ SaS TZ"^ '• „

ff^^^^^^ edxt. * ot .e.nnery, ' "

(50) Of. Bertrand Russellj Mysticism and Logic , pp. 80 - 8A-.
'Veierstrass, by directly banishing from mathematics the'
use of infinitesimals, has at last shown that we, live in
an unchanging world, and that the arrow in its flight is
truly at rest. . . As regard motion and change, we get
similarly curious results. People used to think that when
a thing changes, it must be in a state of change, and that
when a thing moves, it is in a state of motion, This is
now known to be a mistake,.. we may now indulge tho comfor-
table belief that a body in motion is just as truly whore
it is as a body at rest. Motion consists merely in the
fact that bodies are sometimes in one place and sometimes
in another, and that they are at intermediate places at
intermediate times. Only those who have waded through the
quagmire of philosophic speculation on this subjoct can
realize what a liberation from antique prejudices is in-
volved in this simple and straightforward commonplace."

(51) 'Qf. Prinoipia , c. 37 and 39.

(52) I Sent , d. 19, q. 10, a. 4.

(53) Of. e.g. Maritain, Theonas, p. ^-^/V"^
la philosophie parleSra-te-J- r so ^ ^ ^f u^Stein

^^^r-So^-i^or- - p& - e «rdans

chose, d.uno entity mathematiauequ, ^2\Z^l™U«
une equation, et quin'aquo J. a nom ub

(») »Y0« define .«»». But t M. -«- «» — L7.S."'"

(84)

toots is extended and stands qtin m- • x.-
living time you are familiar wiS* J \ ^ " n0t th °
and respite, turning future into VT T B W " h0ut r ° st
and devouring yoirSlvee^^^Sj "" ^^
Physics and Reality, pp. 54 - 55! PrGSGnt ' " " Hiezlor,

(55) Of; Eddington: "Objection h,s sometimes been folt to the
relativity theory because its four-dimensional picture of
the world seems to overlook the directed character of timo
The objection is scarcely logical, for the theory is in this
respect no hotter and no worse than its predecessors. The
classical physicist has been using without misgiving r. sys-
tem of laws which do not recognize a directed time; "ho is
shocked that the new picture should expose this so glaringly,"
The Hnture of the Physical Yforld , pp. 68 - 69,

(56) The Principle of Relativity , p. 213.

(57) Che Foundations of Physics, p. 76, Of. Sddington- "So if the
laws of Haturo are indifferent as to the doing and undoing
of an event, they must he indifferent -as to tho direction of
time from past to future. That is theii^ommon feature, and
it is seen at once when (as usual) tho laws are formulated
mathematically. There is no. more distinction between past
and future than between right and left. In algebraic sym-
bolism, left is - x and right is x ; past is - t and future
is to" The Nature of the Physical V'orld , p. 66.

(58) Of. Aldington, op. cit. Chapter IV and ?.

(59) Lonzen: The Nature of Physical Theory , p. 75.

(60) Op, _crt p. 68. ■
Substance and Function , pp. 449 - 450.

"Cette vue do l-uni,ers est . . . la ^l^jT^T
qui serait capable ^ra-aer d-un ^.^J^do Lin-
tout do I'espace et du to n.ps Itoi ohr , ngera ont en sos
tolloct hurnain resolventce tout P 1 " . Qt i:QVenir du
aspects temporel et spatial, f* ^ d0 x , intelligence
monde physique est le P^° ^ The Principl^O^iiliH

[61
[62

qui le perQoit. ■

T^^^^i^S^^^^^

(85)

Cambridge, 1914, p. 213 .. _ n * + n/ , ,, ,

HolntiviBto; p. 101. ty M °y° r 3on. L r. Seduct ion

(63) Moy arsons La Deduction Rolativisto' r, inp , ■

known, Ber^T^-fHCT^F^^ 1 - fc t: ££;_
za.ion o timo at great length. y /o do not S^Ly
of nis views on the problem, hut at le, a t ho has offoc^
tively demonstrated th,.t modern science has destroyed the
true notion of time. Of. Los Donnees Immediates do la Con-
science .JPureo e^jimnltaneite, LprT^a^eWTTToTvTnt—
Hatiere at llemoiro, etc. : '

(54) Lect. 11, no, 1. Cf. In I phys . lect. 1, no. 5.

(65) Ibid , lect 10. no. 15.

(66) It) id lect. 11, no. V.

(67) No. 14.

(68) Cf. JTorthorpj vmitohe ad's philosophy of Sciono o, p. 187, Cf.
also Riezlor, op, cit. p. 4:2s "If I am not mistaken

there is some confusion about causality, Many of you, it
seems to me, mix up the prineipium rationis with the lav;
of causality. 3ach ought to be kept distinct „ M10 prin-
eipium rationis binds reason and consequence. \7hon you s
draw your conclusions in the realm of your mathematics you
are inclined to call reason the thosis to start from - -
say, a given triangle having a right angle. From this you
proceed to the Pythagorean proportion of the saaares, spea-
king of consequence. In doing, so you refer merely to «">
process of your thinking. You may also start from the squares
and conclude that the angle is a right angle. Then rea-
son and consequence exchange places.

(70)

I, 44, 1. ad 3.

■Musi, il est impossible d-en f g^^^l-Sonne-
LoBpaoo constitue pour -«^ £ £,„
ment d< essence smon. iden.ique, a .^ dftM l0 torrrp3 ,

celui que nous decouvrons dans if. ^^ conclusiotl qu o

et des lors nous ne pouvons et ™ ^ vorg loquo i ten-
si nos raisonnomonts sont oxac "» ^ollemont a romplacor
dent explication ot thcorios consis dQ vidon _
ce mondo infiniment divers qui nous

no

(86)

tiquo dans le temps et l'osninn i„ n -

pout etro que 1-ospace LiS'" Ti w WldM,0rt '

cation dans 1 93 _ JotogngH r p. ™6. Il0y01S0n > D^L'Scpli-

(71) In II Phys. loot, 5. no. H. C f. In I P h YB la( .+ ,

^rSL? PMpW *— f ™ -- « - « '

(72) In III Met, lect, 4, no. 375.

(73) Bssai sur los •domeesimmediat es de la conscionoo. Prris
.1906, 5e ed; p. 157. ''

(74) Cf. Lonaon- "Bodies and processes are represented by numbers
or "by symbols which may tie represented by matrices. Hence
the search for substance becomes tho search for constants
and invariants. Chore aro functional relations between num-
bers. Thus the permanence, objectivity, and self- determi-
nation of substance are replaced by the constancy, impa-
rlance and functional relationship of numerical mer.3urqs " - -
The Nature of Physical Theory , p. 277. Of. ^ddington; The
Philosoph y of Phys ical Science , pp. 129 - 130.

(75) lect, 1, no. 5; lect. 14, no. 8.

(76) Space, Time and G r avitation , p. 200,

(77) Meyerson I dent it e ot Bealite , p. 285. Of. La Deduction 3o-
l ativiste , p. 258,

(78) Cf. Meyersons Du ChendnonOT*_dfl_la_PonBdo, p. 707, N'y-R.-
t-il-pas la, v-eTitabiTm^ntT^T^ a l-emervexllernono le plus
profond? Comment, en s'ecartant ainsi du reel concroo, on lo
fellaS auTpiods intentionnellement («^ ^ ^ c ion
certes pas trop forte dans ce °« P"* 10 )*. 18 *'' *° rd ivlc
a-t-il pu neanmoins rester aussi intxmement en accord avec

son rythme profond?'.'

!V9) Cf. TJioJvIysJerj£UsJhT.iX e 3-l'- pp ° 113 " l ^°

67 Cf. Stebbing: philos o-
180) The Natu re o f^hysj^ajjaegg' p ' ', Qiven this exclusion,

phy and the Physicistss p. ^ " „ ntp CO uld bo replaced by p.

then tho. soSdToTTBacrthoven a °™^ iwl formalf , . By studying

series of curves or a seo oi ran - Boe!th0 ven v/ns a ranthe-

these formulae we might discover that

(87)
matician. Wg should

« UJ -^""' >»u snouid not ^ , ,

a musician hecause wo have ronl.,™^? di3c °vor that ho was
thematical oxpr a8BlonB « <l f, ! 3 - 3 * «"> **~
thoimtionlly hut not musical h , th ° y Could ^ Ifr -

musician wo need further it Tc°l 1', *° diSCOVOr ^
musical concopta. But it I on £"' V '° Uld no douM c * u
that tho univLso i^^^^T^ H "***»
Bod thinking mathematically. "Perhaps the sourc of tho
coniusions into which joona falls li QS in the fnc t h(1
holieves hoth that a mathematical description of r pheno-
menon can .givu complete knowledge of tho phenomenon and
also that the phenomenon is indeed an appearance of an
unknown reality. "

(88)
_ Chapter x.

Cf.

e.g. WwJatjmMofJha Physical World, mtro
et passim, z±±!±> j-m,ro r

(2) X Gonoral Theory _qf_Valuo, p. 408.

(3) Of. Urtam Langungoj^ no allt y London, Goorgo Ulon
and Unwin, 1939, j5pT40F. tj

(4) Cf. Urhmij op cit , , p. 405 t ' "From recent psycholo-
gical literature I gather the following 'gora'. 'I4y
hehaviour symbol relative to at owning foods may ho a
reacting of the salivary glands. » To say that the
reaction of my salivary glands is a sign of tho pre-
sence of food is entirely appropriate, hut to call
it a symbol is a, linguistic distortion which is not
only in. itself inexcusable, hut hars the way to any
proper use of the concept of symbol."

(5) A Modern Introd uctio n to Logic , p. 13.

(6) Process an d Reality, p. 263. Ogden and Richards make
syrnholism coterminuus with- all usos of language with
tho exception of the emotive and the evocative. —
The Meaning o f Meaning, Chapter X. "A. symbol", thoy
tell us, "symbolizes an act of refarenco, that is, arnuig
its causes in tho speaker, togothor no doubt with do-
sires to record and communicate, and with attitudes
towards hearers', are acts of referring. Thus a symhox
hecomes when uttered, in virtue og heing so caused, a
sign to a nearer of ah act of reference.' - - p. <*io.

(7) Cf. Delacroix: LoJ^^lJ^ii^^Vr'also^^ert.
p. 580 8 "Toute fcSS^ est symholiaue-. Of. also Lamhert.

O rgano n, part II.

(8) III, d. 25, q., 1, ?•• !•

(9) in this connection the term "name" include the vorh.

!l0) In I perih . lect. 4, no. 13.

ill) Loot. 1, no. 3.

,,„, llHr „l philosophy, Kr.cmillan,

112) Cf. l J it 1 r£ducti^i i j£j^. 1 }£ u ilii C -^ i --

1938", d. 18.

189)

!13) Priora Analgtlpa, I, tract. I, c g (vW , „

p, 472 b) ' c# 9 IVivos-Borgnot,

[14) Of. John of St. Thomas i Curs. p hi i m , ■„

Q. Ill, a. 2, pp. 3l5 ff. -~-~ L ' x > Pnrs -

14a) Of. In IV ; Mot ., loot, 12, no. 684,

15) Mind and Nature, p. 38.

16) \7e believe that M. Maritain has misconstrued Edding-
ton's doctrine on this point s"M, 3ddington parait
oublier ici quo non soulemont los mosuros rocuoillios
dans la nature par no s appareils nous livront quelquo
chose de reel (qui pout sombler uno 'ombre' nu re-
gard do nptre univors familier, le philosopho copoudant
sait quo ce sont autant de points d< emergence '
par ou un, aspect des choses oxistant en sqi nous np-
parait), mnis encore quo lo proraior degre ou le premier
temps de conceptualisation, parfois trbs olaboree,

ou nous degngaons da ces raesuros une description du
comportement observable des choses nous mot ainsi en
presence de renlites - - jo dis obsorvablos et mesurables
et prises pr6ci3ement oommo tolles, - - nous introduit
dans un monde de faits, de causations obsorvablos et
do structures observables que lo physician theoricion
a tendence a tenif pour une simple matiere offorto a
son genie construoteur, raais dont le physicien de la-
i)oratoire n'est pas dispose a laisser meconnaxtro qu'ils
font deja authsntiquemont partie de la science phy-
sique ello-meme. Ces faits peuvent otro etablis d one
maniera plus.qu moins certaine ou plus ou moins hy-
..othetique, ils pauvent impliquer a un degro ou r un
autre un achievement ideal du rool par la raxson
n'enressortissont pas moins a l'ordro do 1'otio reel
Dos notions comme celle de la constxtu, on do y* par
des molecules individuals on agxta on *n^f ^o
de la structure ^oticulaxre dos ^x^ux e " ftutre
de notions semblables, doiven e revenues po^ ^ ,

chose que des symboles, - - je nljl0> G t avant

traductions du mosurable et d °\ t h r . ppro fondir

que l'effort theorxque, on s app i^ ^ oxplicatl on
leur signification et a dooouv , porra etto "

complete, de quo! olios nous V^, savonS

de comprendre qu'on dernxoro annly.^

(90)

que symboliquoina.nt da quol el1 „ a „

Cost procisement ce second J "f^' ^

do conceptualisation scientifS quo M "SSL?!"* 8
C. on vug; ot la ii sGm-H- + /,„- • "Ming-ton

temoignago." - - L Sgre T"n vn • ° ■ rtou -« »on

^gios au Sr.voir, pp , 314 _ 316 _

(1,7) The goii ndntion ^HVgi'l:^ PP* 12 - 13.
(18) Pp. 250 - 251.

(19) Cf.
in

H. Ifergenau, Methodology of- Modern Physj ca"
Ph ! ilo80gij L of_Scionoo, Vol. %%, Hoa , 1 nnd 2 *

Jan,

and April, 1935.

(SO) Introduction 'to Science ,■ pi l37„

(21) Substance, and Functi on, p. 2.29.

(2'S) Cf, H;- -.Margo'nUu; "Metaphysical moments in physic.s'%
in Review of Modern physics , Tol. 13, no. 3., . • : '"

(23) The Univer se Around Us, pp, 133 - 134. '• "■■'.; ''

(24.1 .Cf. "Sddingtorii "In short, the "physicist drays .up
vvari elaborate plan of the atom and then proceeds- cri-
... tie ally to erase each detail in turn. What is left
Is the atom of modern physics. I want to explain that
...If the erasure is carefully carried out, our conception'
' of the atom need not "become entirely blank. There
is not enough left to form a picture; hut something
is. left for the mathematician- to work on." - - llo;.v_Path-
vwys in Science, p. 259.

125)

.26)

Some modern authors reserve the Urm "Symbolism to
this perfect type that is provided by mathematics, ahd
they describe the evolution of physics from tho use
of sensible, and pseudo-sensible — ct «e

of parenthetical signs a a pjogo.^ ^ ^
tism to synbolasm. See in -<iiu.a
.of Brnst 'Cussirer,

*he Hew V/orld Piotur^ofM2^2^^i ^"W
iodISil»rl5Flho^dvanooi»onT; <u ocionce,

(91)

(87) ^^FJL^tih^^^cj^om, introduction

(28) Cf. also ^l2I}9±i^^l±Sj.oj^n^onl^.

189) -^^^E^U^Kiy^ica^orW, pp „ 264> 28Qj ^

(30) «dingtom ^eJ^Uo^^^^^^^ pp>

[31) Book VII, Cf. Joansj The Jtysterious Univorso, pp. 11]

— 113 ~*

[02J Eddlngton: gjj.oJfotuxo o f the Phy sical \7orld, pp. 237

- 238. Cf. Mazier, Physics and Roalily 7"p.~ 34."

[33) Cf. ^aldingtons Science a nd tho Unseon V/orld , p, 37.

198)

Chapter' xi

(1) Cf. Sddington. Nov; Pathwaya in scienco t, ,h

in Rasetti-s ^ST^iwJiSoTSSl'l^toi Ln
from the golden balcony of n Gr . VG n !;C ro 3 °2 vo d
as low as where this earth spins li'-e .,%,.:+? T ,
Looking from the abode of truth orfoct tuth^nf' *
can enter her mind. She must so, the world as it
really is../' Of. TJp Mathematical 1'hoory of Relati-
vity, p. 1 5 Kw_Ifeturo_of Jfojjphy si oHTwbTlT ~5~S25
Cf. also M. Brunschvicg, Le_Kro^rdG~lR"55HBoionoo!
702; "Jamais n'ost apparu aussi chimerTque~l < espoir"
quo l'hommo reussisse a forcer la barribro do l'oxpe-
.rionce humai.ne ot que uno fois do 1 'autre cote il apercoivc
los chosos a la manibro dont on suppose quo Diou les

contempla dans son etornite," Cited by Hoyorson,

Du Chondnoiimnt do la Pongee, p. 689,

(2) Cf. Mdingtonj Th- i Philosophy of Physical Sciohoo, pp.
157 ff.

(3) Lo Songe do Descartes , p. 63.

(4) Cf. Bddingt.on: Mew Pathways in Scienco , p. 45: ,h .7o must
concede therefore that 'the universe a3 it is conceived
in modern physics', is not identical with what a phi-
losopher would call 'the objective physical universe, •
When we come to think of it there is no reason why it
should be. The task of physical science is to disclose
the scheme of the recurrences in tho combincd^cporionco
of conscious beings. IVe have seen that the ™rso
which constitutes the solution of this prob lorn mu.t ne-
cessarily have the characteristics 01 regularity and
externality; we said nothing' about °^f^'Jf J J° .

it happens, that _ the ^ °^™™ ^ r Z u^oYccr- .
an objective universe follow the a,.nu-
tain point and then part company.

1 • +>■,., Tifi-ht of Modern Physics, '

5) Cf. Plancks Th2j*»il2£2°J£4^^^

lost, physical law ceases to Be r . scort ainod

tween a number of magnixudes which m ^ ^ . „.

(93)

be do f mod far mora exactly by m Q n lla of
than by moans of measurement But 11 J ™ ° mtim
this method amounts to a ronunci.tf.n I ff™ tim0>
moaning of magnitude- while it nSt V , th ° trU °
that confusion and mlsu^ers^ r Si ^to* ^

(6) Cf. Cohens ^aaonjiMNoturo , p. 277 , » 3 ut this f rl i
to explain; why phenomena soom to ooonr as if tho lru
of gravitation with its inverse squares were truo 'or
why the properties of circular functions have proved
most -potent instruments for tho discovery of important
facts in almost all branches of physics. Doubtless
equations are not vibrating strings; but is it not
straining the dualistic dogma to assort that they havo
nothing .in common with each other? Do not lot us be
misled by the terms 'expedient' or 'invention', u map
or a chart is an expedient or invention. Yot if it
fairly represents its objects, is it not because cer-
tain relations between its parts are precisely those
between corresponding parts of the objects represented?'
Cf; also "The Logic of Fictions/ 1 J ournal of philosophy ,
1923, p. 447.

(7) Cf. De Sitter; Kosmos, p. 6.

(8) The Mathematica l Theory of Relati vity^ p. 3.

(9) ^ddihgton rightly objects to Professor Stobbing's con-
tention that physicists are not concerned with chairs;
Physicists are not concerned with chairs.' Are wo really
expected, to. 'take this sitting down?. . .Why « - £ that
a Transport; Company, wishing to improve 1 1 a r ang m nt s
for seating, ; consults a^hysicis o ^^
with the chairs we sit upon, in "°;- u ". l 159 _
who is?-' IJie^Wlojsophyj^^ pp>

160.

10) Cf . e.g. .ddington: **Ll!«*^^

an d the Phi losj^pj^lPj^^" 1 -- -^^ '

U) Cf.. Joans: Th^Jfi2lH^^ U -^S s ?on in advance. It
"It may be weTTTo^taT5 our oonclu.io.

(94)

the w-o^Lht™ t h :ti^ ^.^ntion.

• 4. 4. ■ . H-ouu.aaj.ng cnotn; r.ncl something must

exist out oido our mind to put this or any other con
cept into our minds. T o this southing % nay t em
porarily assign tho nao.- reality' , and it i3 this
reality which is tho object of science to study.
But v/o shall find that this reality is something vary
'diff Grant from wh it tho scientist of fifty yonrs ago
moant by ether, undulations and waves, so much so
that, judged by his standards and speaking' hi3 lan-
guage for •, moment, the othor3 and their waves are not
ro'ilitlos at all. And yet thoy aro tho most real
things of which wo have any knowledge or experience,
and so. are a3 real' as anything possibly can "be for us/'

12) How. Pathways in Science , p. 315.

13) Mow Pat hways in Science , p, 26. ("3, & 0,73, moans;
errors and omissions excluded),

.14) islsewherQ TSddington writes: 'We asked why tho story telloi
should he believed whoh he talks about galvano-
meters, although he is untrustworthy when ho talk3
of familiar oh jects. I think the answer is that
tho truth of the story ij^tJhajp.oiiitj£^
the . physic ia~iTTo"ndornod only with tho scraps ox
cipher contained in 'it. The galvanometer is a device
for leading the story into situations in which tho
underlying ciohor hecomes loss offline to interpret;
£ is noTa "ridge on the story teller's, imagination. - -
How .Pathways in Scie nce, pp. 10-11.

15) Op. cit. pp.424 - 425.

•„,,n n ^09- "sin un raot

16) Of. Duhom, 'Lrijhior^jhjtsisufl. P- ™- sor(lit
lo physicien^Torce do reconn tro^i^ ^^
deraisonnablo do travailler^au p ^ ^^ ^ plug
physique, si cotto thoorio n ° l - ci d .uno Kota-
on plus net et do plus en juub » tr;maocm dant a la
physique; la croyance est un ^ ^ 1r th - orio
Physique, est la soulo rai^on

physique.'*

(95)

|17) Of. John Douoy, ^L^^jor^,^^ ^^

[18) Of. Urban, op oit. p. 514. ^horo is, as wo havo
30,n no poaaiUo typo of sy m ,„i v,hich doos nut c n-
t-un some elomont 'of fiction- (of tho factitious?
to uso Descartes' torras), ,.ncl v/hich dooTifoTTF 1
somo way and to soma degree 'distort' roality m
tho case of tho aesthetic symbol tho artist socks
to achieve deviations from roality in order, para-
doxically, to represent roality hotter or to pone-
t rate- mora deeply into it. In the case of tho
scientific symbol tho scientist also deviates from
the intuitive, phononsnal reality - .. in this cnso p
however, to explain and ultimately to control rea-
lity: and predict happenings. 1 '

[19) Cf. De Valore Theoriartua- Physicarum" in Acti primi
Congr. Thorn , Roma, 1925, pp. 61 - 74; 269 - 275;
"Inquisitiones criticae in Theoriam Atomicam physico-
chimicam 3iusque Vnlore Pro Philosophir. Naturr.li"

in Gregoriarum, 1925; pp. 248 - 265; 1927- pp. 229 -
242; 1928: p.p. 417 - 460. .

[20) Cf. John of St. Thomas; Ars Logion , p. 681 b; "... non
alitor signatum representat quam prius. ao ut obiec-
tuin repraesentando, ulterius extendendo repraoson-
tvtionem sui ad aliud in so virtualitor implicitum

et contention. "

[21) In. I, 55, 3, no. 13.

[22) Curs. Phil., Ars_Logica, p. 692 b.

[S3) c-fj John of St.. Thomas, loo. oit. p. 647 £' ^ j£
signi fomalitor loquondo non oonsistit xn
secundum did, sod secundum esse.

.. „, mhnnrv of Relativity,
Of. -mii^omTboJ^bsm^^l^S^

Introduction. ,

. £52- "pourtnioi done

Cf. Duh'em; l£L^!^^--^ i ^tor'a'riro, nffirmor quo
le physicion peut-il, sans , pro , ^ pnrce quG sr

1' experience dooouvrira ^ ° a * ^ ^^ qu0 lG

thcorio reclame la reality ^ -

,24

[2b

(96)

conchyliologisto aorait ridicule ^ i
sonco dM,, ? case vide danstftir^rc: lu-
cres ,:uuc dieses coulours du spectre, lononait
a conclure qu'il y a dos coquillos ^^ ZnT
l.ocoan? C'ost quo visihlomont, l ( , cSssifSlon
do cocollectionneur est uu systbmo purement ,r-
hltraire, qui no tient aucun conpte dos nffinitos
reelles ontro los divers groupos do mollusquos-
tandis qu'on la theorio du physlcion, trnnsprrait
comma le reflet d'un ordro ontologiquo,"

26) Op. oit . pp, 265 - 266. "

27) Chapter 6, 1016 a 25, loot. 7, no. 863.

28) Chapter 14, 224 a 2, loot, 23, no. 13. Cf. St. Al-
to rt tho Groat. Ibid., Tract, III.

29) The Universe in tho Light of Modern Phy sics, p. 13.

30) Soionoo et Hy po these , p. 6. Cf, Joans; Ho w Background
of Science, p. 51s "Tho layman soes Science, as it
seomg to him, forever changing hor mind, hesitating,
turning hack on hor track s , and repudiating hor ear-
lier opinions. The scientist sees hor ever progres-
sing through a successi-'n of theories, each of which
covers mora phenomena than the predecessor it dis-
placed, towards the goal of a single theory which
shall Giribrace all the phenomena of nature.

131) La Thoorie Phys ique^ p. 53.

1 32) Mind and Nature, p. 46 „

33) Aldington: ^oj£^±°l^^^^^[. V '
Cf. Planokj-lhrS^SOi^S!^' P '

34) For a study of this notion and its philosophical

353.

unia ™— --- "notes on

Vol. I, Hoa 1 nnd 3.

135) o-ho Nature of thej^hysicnlj^ W . «* " 353 '

implications soo

tho Limit of a Vari. n --- r ,

, r -i -r una 1 anci •-*.
Philosophiquo, Vol. *., u° b

197)

[:-;S) Cf, Lalor, op. cit . Ho. l, p „ 143t
[37) Lalor, op. cit. p. 146^

("3) Sl ora i3 Scionco_ Going? p . B8 . cf , ffllQ Univor3Q
in the L ight of Modornjhysios, p . 15"~lhTphT
losophy of Physics, p. 31. of. also Do BroilioT
opo cit , p. 319.

[39) Cfo Duhom; La Bheorio Physiq ue, p. 450.

[40) Cfo Plancki Where is Science Going? p. 200, The
Universe in the Light of Modern Physios , p. 57~-
58, Cf. also I3ddingtons S cience and tho Unsoon
World, p. 23s 'Wo seek tho truth; hut if some
yoico told us that a few years would see tho ond
of our journey, that the clouds of uncertainty
would -do dispersed, ,and that we should percoivo
the whole truth ahout the physical universe, the
tidings would he hy no means joyful. In science
as in religion the' truth shines ahead as a hoacon
showing us the path; we do not ask to attain it;
it is hotter far that we he permitted to sook."

[41) Cf. Morcei'.ux Ohoisis sur la Harxisme , pp. 51, 52,

08)

(X) New Pathways in 3o ionne r p. 7,

(«) The ^^-^^in J ho J1 g^ oJJi0 d^^ zg ^ > p> ^

(3) ^ho_M!E£_£l .«ilWiZ 3 j£ilLJ{£LW. Introduction.

(4) Cf. "Interviews with aJininont Scientists" in The 01)-
£?£Y££» A P ril 13 > 1930 byj.ff.N. Sullivan. "TfoHHd
that not only Mnstein, Hut also Planck and Schro-
dinger fully recognized the subjective element in
scienoo. Planck in particular... regards science as

a constructed work of art, expressing a certain side
of man's nature. "

(9

;io
In
:i2

[13

[14

Of, Hoys La Theorie Phy sique, p. 350 o

Cf. Joans- The Hew Background of Science , pp. 2-3;.
67, etc.;' Physics and p hilo sophy , pp. 143 - 144.

ghe T3volution of Physics , p. 33.

Bu Chominemont de la Penseo , p. 654.

Pp. 16 - ,20.

Cf. Ibid, p. 57.

Pp. 108-112.

In I Mot , lect. 10. no. 158.

'•Reflexions sur le Precoma de l-Indetorminismo" in

Reviie ■Thomiste , 1937, p. 396.

7fi • "The
Cf.- Fulton Sheen: V2*±2*2&L°2J$^ U much like
problem whether soienoo tos a 1 ■ of v , hothe r

a modernization of the SGhoiasux modGrn language

an idea is an id que or .an id quo ^.^ t0 rQalit y
this means, do mathematics m 1? ^ mo dorn

or are they only a -f-'t'oSifio pledge is 'That
idealist would hold that f^% nhloh .. reality «
which is known" inato.d of th at y 3ub . ectivG th0 ory
. known. St. Thomas; on M «» jf ^ ^ ^^
of knowledge ia tnereioru 4

know-

09)

(1,5) "Wo must thorefore remember that not all n ,„ ,

lodg. of the physical universe is oomprLod £T'
ledge , of the laws of nature. Tho ^ Z T
superfluous as xt seems. I. have often found nn im-
pression that to explain away the 1 P , 7S of nature
as wholly subjective is the same thing as to explain
away „ha physical universe as wholly subjective.
Such a view is altogether unfounded." Op. cit t> i>5
Cf. also pp. 104, 178, 217, etc. ~

[16) Cf. e.g. Du Ohomin emont do la Ponsoo.

[17) Cf. Meyorson; La, Deduction Solativist e, pp. 134, 143.

[18) Do Impl i cation dans los Sciences , pp. 526 - 528.

[19) "La science est realiste; mais nous Savons cepondant
quo d' explication en explication, olio ne pout aboutir
qu'a l'acosmisme, a la destruction de la realito. Or,
dans/le relativisme, procisement parco qu'il consiituo,
une forme trba avancee,. tree parfaite, del' explication
theorique, ces doux extremes de 1' existence et do la
non-existence se trouvent tres rapproches Pun do

1' autre. D'ou. une sorte do conflit douloureux dans

la conscience du physicien." - - Meyersonj La Deduction

Relativisto , p. 205.

[20) De Sitter; Kosmos , p. 108.

[21) p„ 104.

[22) Ibid , pp. 188- 189.

[23) .La Deduction Rolat ivistg, p. 209 - 210.

a orii + ion (French version) p. 26.

[24) preface to the socond odition \^

[25) Cf. 13ugbno Br.bin, Op. oit,

,„,, : ■ , cn (Trri-nch edition).

[26) Cf. Tome I, p. 61 (H^on ..

[27) Op. clt. PP. 61 - 62.

3 . 201.

(28) Space, T iimarA&B^-^^' P '

(100)
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