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Full text of "Thomism and Mathematical Physics"

No, d'ordre: 







Thomism and Mathematical Physics o 
JULY 1946 




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 


«*> *■ >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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"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 

' 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 


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 


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 


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 


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 


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 


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- 


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 

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


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 


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, 


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 


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 


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 


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 •. 


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-±) 
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) 


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 


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. 


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 


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 ■ 


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 


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 


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 



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 


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) 


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: 


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 


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 

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 ~ 


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 


\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 


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 


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 

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 


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 


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? 



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 


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. 


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 


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 

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


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 


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. 


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- 


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 ,> 


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 


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. 


.. 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- 


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 


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, 


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 


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 


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 


+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 


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- 


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) 


+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^ 


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. 


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 


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 


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- 

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) 


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- 



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 


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 

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} 


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 

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 

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) 


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,, 




.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 


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 


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 


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. 


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 


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 


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 


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- 

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 


^^^^^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 

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 


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 


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 


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 


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 . 


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 


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 

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 


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 extensive. 


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- 

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 


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 


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 


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 * 


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 


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 


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 


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 


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- 

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, 


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, 


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 


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 



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 


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. * 


•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 


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 " 


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 


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 » 


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 


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 


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 


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


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 


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> 


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 , * 


, , .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- 

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


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 


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 ^ 


.y move fvom somg_ 

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


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 



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 


I This study begins with the consideration of ^"f 

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


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 


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. 




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 

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 


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. 


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 


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 


•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, 


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;; 


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 



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^^^^ 


: 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 


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 



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 


■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 


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, 


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 — 


\/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 _ 

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 


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- 

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 


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 


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 

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 


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 , 


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__ 


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- 


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. ) 



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, 


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 


(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 


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- 

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.^ 


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 


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 . 


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, 


(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 


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^ 


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; 


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 

„„. 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 


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_ 


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 


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 


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 


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 


[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 ; 


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 

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 


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 



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 


/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 


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 


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-- 


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 


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 


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. 


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 


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 , 



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 ; 


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^^!^^^^ * 


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 


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 


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- 

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. ^) 


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 




'-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 


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 


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 




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. 


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 


^^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 


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 


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^-^ > 


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 


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^; 


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 



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 


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 

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 


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 , 


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) 


.,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 

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 



„. ;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 


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. 


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 


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 


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. 


. There is, then, great wisdom in Aristotle's remark Jbhat 
is noble to soil one's hands in experiments because by so doing 


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"*^ 


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 


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 


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() 



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 


'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 

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 


'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 ^ 


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 

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 


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 


!« . & 


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 „ 


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. 


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 


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 


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 


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-^ 


/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 


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 


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 


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- 

- ...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 


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 . 


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 


/ 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; . ...-._,. 


| 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 

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 


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 

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 


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. 


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. 


• +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^ 


(ba suggested tint the second step consists in an intellectual fil- 
tration and purification of the elements entering into the 


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 , 


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 



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 


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- 



. / 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 


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. 


. . 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- 

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 . 


| 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^- —— — ~ 


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- 

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 


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 


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 



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. 


\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 


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^ 


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 


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 


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 



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 


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 


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. 



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. 




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 


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, 


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 



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) 


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 


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 


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-^) 


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 

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 




' 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." ^°> 


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 


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 


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, 




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 


*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 


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 
answer to both of~~these questions . Ana . 


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 


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 

' 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 


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 ' 


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^ , ^ 


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 



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 _,^ — 


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 _ . 


„ „-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 


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 

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 


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) 


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 


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 


• 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 


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 


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 


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 

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 


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 


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 

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 

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 


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 

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 


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^.» 


(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, 


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 


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 

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. 




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 


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 


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 

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 . . 


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 ^^ 


ta^iSTiSSBd up wixn B— — 

„a ™+heimtica non sue- 


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 


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 


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 


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 


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 


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 


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 


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, 


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." 

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 


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 * 


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 



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 


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 


, 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 °™ 


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 


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 ' ^ ^ 




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: 


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 


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 


by virtue of mathematical abstraction ^ 

,so experience. They are stripped of °?L become . lnde Pendent 

;^w*— ™~-*--°°^ red - and ? ^ d 5t^ e ^^LS xt 


o? aense 

in \ 


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, 



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 


(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, 


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 


-..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* 



,:,; 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." 

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 


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 


■ 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, 

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 


: ;,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 


, : ,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 




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 




BY ;'.'." 




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 



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 


2) ^li22il£2i^^ chap, viii 

</ 1 Science and Measurement. ....,...<,„.„„,;.„. 339 

V 2 The Nature of Measurement. . .......................... 345 

3 The Limitations f Measurements .,.........,.,....„, 3S3 

( (JVvf ) 


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 


2 Logical Identity, ...„»..,., 

3 Movement towards Real Identity 


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 




Io NotOS. •o»oooo«i«..ooom....'»co«e< 

EI» BiMiogrnphy......... ...... .............. (98) 




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 


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 


^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 


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 


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 


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 


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. 


.,, , 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 


^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 


"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- 

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. 

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 


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 


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, 


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- 


%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 


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, 



<^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- 

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 


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 


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»» 


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 


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 

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 


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 


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- 


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 


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- 

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 


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 


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 


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,, 


<,„„ -, 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 


»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 


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 


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 


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 


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 


» ^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, 


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 



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, 


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- 


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 


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 


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, 

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 


ion TtiZoln^^T 11 ^ ^ n0t thS ^^tative homogenisat- 


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 


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. 


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 


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, 



(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 


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. 


«°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 


( 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 


sis is subs- 


/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 


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 ) 




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 


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 


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 , , „ . 


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.. 


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 

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- 



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 ^^ 

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 


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) 


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 - 

A thing can be a measure, then, only to the extent 


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 


^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 


^^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- 



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-^ 


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; 


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 


^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 


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- 


' 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, 


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 


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, 


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. 


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 


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 


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 

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, 


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 


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 

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 


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 



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- 


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- 


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- 


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 


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 


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 


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 


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- 



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 


ZTa^L'lln^Ze:^ ° f ™*?^ °°» never give us .ore 


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 


by merely pointing out an individual r, 


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 


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, 


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 


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. 


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* ■ 


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: 


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 


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." 

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 

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- 


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 


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. 


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 


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) 


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 

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 


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