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gift of
Mrs. Clarence I. Lewis
STANFORD UNIVERSITY LIBRARIES
SCIENCE AND EDUCATION
A SERIES OF VOLUBIES FOR THE PROMOTION OF
SCIENTIFIC RESEARCH AND EDUCATIONAL PROGRESS
Edited bt J. McEEEN CATTELL
VOLUME I— THE FOUNDATIONS OF SCIENCE
UNDER THE SAME EDITORSHIP
SCISNCE AND EDUCATION. A series of volumes for
the promotion of scientific research and educational
progress.
Volume I. The FonndationB of Science. By H.
PoincarA. Ck>ntaining the authorised English
translation by George Bruce Halsted of "Science
and Hypothesis," "The Value of Science," and
"Science and Method."
Volume n. Medical Research and Education. By
Richard Mills Pearce, William H. Welch, W. H.
Howell, Franklin P. Mall, Lewellys F. Barker,
Charles S. Minot, W. B. Cannon, W. T. Council-
man, Theobald Smith, G. N. Stewart, C. M. Jack-
son, E. P. Lyon, James B. Herrick, John M. Dod-
son, C. R. Bardeen, W. Ophtds, S. J. Meltier, James
Ewing, W. W. Keen, Henry H. Donaldson, Christ-
ian A. Herter, and Henry P. Bowditch.
Volume m. UniTenity Control. By J. McKbbn
Cattbll and other authors.
AMERICAN MEN OF SCIENCE. A Biographical
Directory.
SCISNCE. A weekly journal devoted to the advancement
of science. The official organ of the American Asso-
ciation for the Advancement of Science.
THE POPULAR SCISNCE MONTHLY. A monthly
magasine devoted to the diffusion of science.
THE AMERICAN NATURALIST. A monthly journal
devoted to the biological sciences, with spedid refer-
ence to the factors of evolution.
THE SCIENCE PRESS
HBW TORK OARRISOIT, IT. T.
THE FOUNDATIONS
OF SCIENCE
SCIENCE AND HYPOTHESIS
THE VALUE OF SCIENCE
SCIENCE AND METHOD
BT
H. POINCARE
AUTHOBIZED TBANSLA.TION BT
GEORGE BRUCE HALSTED
WITH A SPECIAL PBEFACB BT POINCAB^, AND AN INTRODUCTION
BT JOSIAH BOTCE, HABTABD TTNITEItSITT
THE SCIENCE PRESS
NEW YORK AND GARRISON, N. Y.
1913
Ck)pyright, 1913
Bt The Sgebngob Pbbsb
MKSOF
TNI NEW IRA PRINTINQ OOMMNY
LANCAtTtR« PA.
■H'
CONTENTS
PAOX
Henri Poincard zi
Author 'b Preface to the Translation 3
SCIENCE AND HYPOTHESIS
Introduction hj Bojee 9
Introduction 27
Past I. Number and Magnitude
Chapter I. — On the Nature of Mathematical Beasoning 31
Sjllogistic Deduction 31
Verification and Proof 32
Elements of Arithmetic 33
Reasoning hj Becurrence 37
Induction ... * 40
Mathematical Construction 41
Chapter II. — ^Mathematical Magnitude and Experience 43
Definition of Incommensurables 44
The Physical Continuum 46
Creation of the Mathematical Continuum 46
Measurable Magnitude 49
Various Bemarks (Curves without Tangents) 50
The Physical Continuum of Several Dimensions 52
The Mathematical Continuum of Several Dimensions 53
Part II. Space
Chapter HE. — The Non-Euclidean Geometries 55
The Bolyai-Lobachevski Geometry 56
Riemann 's Geometry 57
The Surfaces of Constant Curvature 58
Interpretation of Non-Euclidean Geometries 59
The Implicit Axioms 60
The Fourth Geometry 62
Lie's Theorem 62
Biemann 's Geometries 63
On the Nature of Axioms 63
Chapter IV. — Space and Geometry 66
Geometric Space and Perceptual Space 66
Visual Space 67
Tactile Space and Motor Space 68
Characteristics of Perceptual Space 69
Change of State and Change of Position 70
Conditions of Compensation 72
V
vi CONTENTS
Solid Bodies and (Geometry 72
Law of Homogeneity 74
The Non-Euclidean World 75
The World of Pour Dimenaiona 78
Conclusions 79
Chaptbb V. — ^Experience and Geometry 81
Geometry and Aatronomy 81
The Law of Belativity 83
Bearing of Experiments 86
Supplement (What is a Pointf ) 89
Ancestral Experience 91
Pabt m. Force
CHAPm VI. — The Classic Mechanics 92
The Principle of Inertia 93
The Law of Acceleration 97
Anthropomorphic Mechanics 103
The School of the Thread 104
Ohaptbr YII. — ^Belatiye Motion and Absolute Motion 107
The Principle of Belative Motion 107
Newton 's Argument 108
Chapter VIII. — ^Energy and Thermodynamics 115
Energetics 115
Thermodynamics 119
General Conclusions on Part HI 123
Past IV. Natwre
Chapthi IX.— Hypotheses in Physics 127
The Bdle of Experiment and (Generalization 127
The Unity of Nature 130
The Bdle of Hypothesis 133
Origin of Mathematical Physics 136
Chapter X. — The Theories of Modem Physics 140
Meaning of Physical Theories 140
Physics and Mechanism 144
Present State of the Science 148
Chapter XI. — The Calculus of Probabilities 155
Classification of the Problems of Probability 158
Probability in Mathematics 161
Probability in the Physical Sciences 164
Bouge et noir 167
The Probability of Causes 169
The Theory of Errors 170
Conclusions 172
Chapter XII. — Optics and Electricity 174
Fresnel 's Theory 174
Maxwell's Theory 175
The Mechanical Explanation of Physical Phenomena 177
CONTENTS vii
Xm.— Electrodynamics 184
Ampere's Theory 184
Closed Currents 185
Action of a Closed Current on a Portion of Current 186
Continuous Botations 187
Mutual Action of Two Open Currents 189
Induction 190
Theory of Helmholtz 191
Difficulties Baised by these Theories 193
Maxwell's Theory 193
Bowland 's Experiment 194
The Theory of Lorentz 196
THE VALUE OP SCIENCE
Translator 's Introduction 201
Does the Scientist Create Sciencef 201
The Mind Dispelling Optical Illusions 202
Euclid not Necessary 202
Without Hypotheses, no Science 203
What Outcomef 203
Introduction 205
Past I. The Mathematical Sciences
Chaptkb I. — ^Intuition and Logic in Mathematics 210
Crafteb, II.— -The Measure of Time 223
Chapter III.— The Notion of Space 235
Qualitative Geometry 238
The Physical Continuum of Several Dimensions 240
The Notion of Point 244
The Notion of Displacement 247
Visual Space 252
Chaptib IV. — Space and its Three Dimensions 256
The Group of Displacements 256
Identity of Two Points 259
Tactile Space 264
Identity of the Different Spaces 268
Space and Empiricism 271
B6le of the Semicircular Canals 276
Paet II. TTw Physical Sciences
Chaptee. V.^Analysis and Physics 279
Chapter VI. — ^Astronomy 289
Chapter VII. — The History of Mathematical Physics 297
The Physics of Central Forces 297
The Physics of the Principles 299
Chapter Vin. — ^The Present Crisis in Physics 303
The New Crisis 303
Camot's Principle 303
viii CONTENTS
The Principle of Eelativity 305
Newton's Principle 308
Lavoisier 's Principle 310
Majer 's Principle 312
Chapter IX. — The Future of Mathematical Physics 314
The Principles and Experiment 314
The BMe of the Analyst 314
Aberration and Astronomy 315
Electrons and Spectra 316
Conventions preceding Experiment 317
Futare Mathematical Physics 319
Part III. The Objective Value of Science
Chapter X. — Is Science Artificialf 321
The Philosophy of LeBoy 321
Science, Bule of Action 323
The Crude Fact and the Scientific Fact 325
Nominalism and the Universal Invariant 333
Chapter XI. — Science and Reality 340
Contingence and Determinism 340
Objectivity of Science 347
The notation of the Earth 353
Science for Its Own Sake 354
SCIENCE AND METHOD
Introduction 359
Book I. Science and the Scientist
Chapter I. — The Choice of Facts 362
Chapter II. — The Future of Mathematics 369
Chapter III. — ^Mathematical Creation 383
Chapter IV. — Chance 395
Book n. Maihematicdl Seasoning
Chapter I.— The Belativity of Space 413
Chapter II. — ^Mathematical Definitions and Teaching 430
Chapter III. — ^Mathematics and Logic 448
Chapter IV. — The New Logics 460
Chapter V. — The Latest Efforts of the Logisticians 472
Book III. The New Mechanics
Chapter I. — Mechanics and Badium 486
Chapter II. — ^Mechanics and Optics 496
Chapter HJ. — The New Mechanics and Astronomy 515
Book IV. Astronomic Science
Chapter I. — The Milky Way and the Theory of Gases 522
Chapter I. — ^French Geodesy 535
General Conclusions 544
Index 547
HENRI POINCARE
Sm George Darwin, worthy son of an immortal father, said,
referring to what Poincar^ was to him and to his work: **He
must be regarded as the presiding genius — or, shall I say, my
patron saint t"
Henri Poincar6 was born April 29, 1854, at Nancy, where his
father was a physician highly respected. His schooling was
broken into by the war of 1870-71, to get news of which he
learned to read the German newspapers. He outclassed the
other boys of his age in all subjects and in 1873 passed highest
into the Ecole Polytechnique, where, like John Bolyai at Maros
Y&s4rhely, he followed the courses in mathematics without taking
a note and without the syllabus. He proceeded in 1875 to the
School of Mines, and was Nomme, March 26, 1879. But he won
his doctorate in the University of Paris, August 1, 1879, and
was appointed to teach in the Faculty des Sciences de Caen,
December 1, 1879, whence he was quickly called to the Uni-
versity of Paris, teaching there from October 21, 1881, until his
death, July 17, 1912. So it is an error to say he started as an
engineer. At the early age of thirty-two he became a member
of TAcad^mie des Sciences, and, March 5, 1908, was chosen
Membre de TAcademie Frangaise. July 1, 1909, the number of
his writings was 436.
His earliest publication was in 1878, and was not important.
Afterward came an essay submitted in competition for the
Grand Prix offered in 1880, but it did not win. Suddenly there
came a change, a striking fire, a bursting forth, in February,
1881, and Poincare tells us the very minute it happened. Mount-
ing an omnibus, **at the moment when I put my foot upon the
step, the idea came to me, without anything in my previous
thoughts seeming to foreshadow it, that the transformations I had
used to define the Fuchsian functions were identical with those
of non-Euclidean geometry.'' Thereby was opened a perspec-
tive new and immense. Moreover, the magic wand of his whole
ix
X THE FOUNDATIONS OF SCIENCE
life-work had been grasped, the Aladdin's lamp had been rubbed,
non-Euclidean geometry, whose necromancy was to open up a
new theory of our universe, whose brilliant exposition was com-
menced in his book Science and Hypothesis, which has been
translated into six languages and has already had a circulation
of over 20,000. The non-Euclidean notion is that of the possi-
bility of alternative laws of nature, which in the Introduction
to the Electridte et Optique, 1901, is thus put: ''If therefore a
phenomenon admits of a complete mechanical explanation, it
will admit of an infinity of others which will account equally
well for all the peculiarities disclosed by experiment."
The scheme of laws of nature so largely due to Newton is
merely one of an infinite number of conceivable rational schemes
for helping us master and make experience; it is commode, con-
venient; but perhaps another may be vastly more advantageous.
The old conception of true has been revised. The first expres-
sion of the new idea occurs on the title page of John Bolyai's
marvelous Science Absolute of Space, in the phrase **haud un-
quam a priori decidenda."
With bearing on the history of the earth and moon system and
the origin of double stars, in formulating the geometric criterion
of stability, Poincar^ proved the existence of a previously un-
known pear-shaped figure, with the possibility that the progres-
sive deformation of this figure with increasing angular velocity
might result in the breaking up of the rotating body into two
detached masses. Of his treatise Les Methodes nouvelles de la
Mechanique celeste. Sir George Darwin says: **It is probable that
for half a century to come it will be the mine from wh^ch humbler
investigators will excavate their materials." Brilliant was his
appreciation of Poincar6 in presenting the gold medal of the
Royal Astronomical Society. The three others most akin in
genius are linked with him by the Sylvester medal of the Royal
Society, the Lobachevski medal of the Physico-Mathematical
Society of Kazan, and the Bolyai prize of the Hungarian Acad-
emy of Sciences. His work must be reckoned with the greatest
mathematical achievements of mankind.
The kernel of Poincar6's power lies in an oracle Sylvester often
quoted to me as from Hesiod : The whole is less than its part.
HENBI POINCABE xi
He penetrates at once the divine simplicity of the perfectly
general case, and thence descends, as from Olympus, to the
special concrete earthly particulars.
A combination of seemingly extremely simple analytic and
geometric concepts gave necessary general conclusions of im-
mense scope from which sprang a disconcerting wilderness of
possible deductions. And so he leaves a noble, fruitful heritage.
Says Love: ''His right is recognized now, and it is not likely
that future generations will revise the judgment, to rank among
the greatest mathematicians of all time."
Geobgb Bruce Halsted.
SCIENCE AND HYPOTHESIS
I
AUTHOR'S PREFACE TO THE
TRANSLATION
I AM exceedingly grateful to Dr. Halsted, who has been so
good as to present my book to American readers in a translation,
clear and faithful.
Every one knows that this savant has already taken the trouble
to translate many European treatises and thus has powerfully
contributed to make the new continent understand the thought
of the old.
Some people love to repeat that Anglo-Saxons have not the
same way of thinking as the Latins or as the Germans ; that they
have quite another way of understanding mathematics or of un-
derstanding physics ; that this way seems to them superior to all
others ; that they feel no need of changing it, nor even of know-
ing the ways of other peoples.
In that they would beyond question be wrong, but I do not
believe that is true, or, at least, that is true no longer. For some
time the English and Americans have been devoting themselves
much more than formerly to the better understanding of what is
thought and said on the continent of Europe.
To be sure, each people will preserve its characteristic genius,
and it would be a pity if it were otherwise, supposing such a
thing possible. If the Anglo-Saxons wished to become Latins,
they would never be more than bad Latins; just as the French,
in seeking to imitate them, could turn out only pretty poor
Anglo-Saxons.
And then the English and Americans have made scientific
eonquests they alone could have made ; they will make still more
of which others would be incapable. It would therefore be de-
plorable if there were no longer Anglo-Saxons.
But continentals have on their part done things an English-
man could not have done, so that there is no need either for
wishing all the world Anglo-Saxon.
Each has his characteristic aptitudes, and these aptitudes
3
4 SCIENCE AND HTP0THESI8
should be diverse, else would the scientific concert resemble a
quartet where every one wanted to play the violin.
And yet it is not bad for the violin to know what the violon-
cello is playing, and vice versa.
This it is that the English and Americans are comprehending
more and more; and from this point of view the translations
undertaken by Dr. Halsted are most opportune and timely.
Consider first what concerns the mathematical sciences. It
is frequently said the English cultivate them only in view of
their applications and even that they despise those who have
other aims; that speculations too abstract repel them as savor-
ing of metaphysic.
The English, even in mathematics, are to proceed always
from the particular to the general, so that they would never have
an idea of entering mathematics, as do many Germans, by the
gate of the theory of aggregates. They are always to hold, so to
speak, one foot in the world of the senses, and never burn the
bridges keeping them in communication with reality. They thus
are to be incapable of comprehending or at least of appreciat-
ing certain theories more interesting than utilitarian, such as the
non-Euclidean geometries. According to that, the first twK)
parts of this book, on number and space, should seem to them
void of all substance and would only baflBe them.
But that is not true. And first of all, are they such uncom-
promising realists as has been said? Are they absolutely refrac-
tory, I do not say to metaphysic, but at least to everything
metaphysical ?
Recall the name of Berkeley, bom in Ireland doubtless, but
immediately adopted by the English, who marked a natural and
necessary stage in the development of English philosophy.
Is this not enough to show they are capable of making ascen-
sions otherwise than in a captive balloon?
And to return to America, is not the Monist published at
Chicago, that review which even to us seems bold and yet which
finds readers?
And in mathematics? Do you think American geometers
are concerned only about applications? Far from it. The part
of the science they cultivate most devotedly is the theory of
AUTHOE'3 PREFACE TO TRANSLATION 6
groups of snbstitations, and under its most abstract form, the
farthest removed from the practical.
Moreover, Dr. Halsted gives regularly each year a review of
all productioDS relative to the non-Euclidean geometry, and he
has about him a public deeply interested in his work. He has
initiated this public into the ideaa of Hilbert, and he has even
written an elementary treatise on 'Rational Geometry,' based
on the principles of the renowned German savant.
To introduce this principle into teaching is surely this time
to bum all bridges of reliance upon sensory intuition, and this is,
I confess, a boldness which seems to me almost rash&ess.
The American public is therefore much better prepared than
has been thought for investigating the origin of the notion of
space.
Moreover, to analyze this concept is not to sactifiee reality to
I know not what phantom. The geometric language is after all
only a language. Space is only a word that we have believed
a thing. "What is the origin of this word and of other words
alsol What things do they hidet To ask this is permissible;
to forbid it would be, on the contrary, to be a dupe of words ;
it would be to adore a metaphysical idol, like savage peoples who
prostrate themselves before a statue of wood without daring to
take a look at what is within.
Iq the study of nature, the contrast between the Anglo-Saxon
spirit and the Latin spirit is still greater.
The Latins seek in general to put their thought in mathe-
matical form; the English prefer to express it by a material
representation.
Both doubtless rely only on experience for knowing the world;
when they happen to go beyond this, they consider their fore-
knowledge as only provisional, and they hasten to ask its defini-
tive confirmation from nature herself.
But experience is not all, and the savant is not passive; he
does not wait for the truth to come and find him, or for a
chance meeting to bring him face to face with it. He must go
to meet it, and it is for his thinking to reveal to him the way
leading thither. For that there is need of an instrument ; well,
just there begins the difference — the instrument the Latins ordi-
narily choose is not that preferred by the Anglo-Saxons.
6 SCIENCE AND HYPOTHESIS
For a Latin, truth can be expressed only by equations; it
must obey laws simple, logical, symmetric and fitted to satisfy
minds in love with mathematical elegance.
The Anglo-Saxon to depict a phenomenon will first be en-
grossed in making a model, and he will make it with common
materials, such as our crude, unaided senses show us them. He
also makes a hypothesis, he assumes implicitly that nature, in her
finest elements, is the same as in the complicated aggregates
which alone are within the reach of our senses. He concludes
from the body to the atom.
Both thelrefore make hypotheses, and this indeed is necessary,
since no scientist has ever been able to get on without them. The
essential thing is never to make them unconsciously.
From this point of view again, it would be well for these two
sorts of physicists to know something of each other; in study-
ing the work of minds so unlike their own, they will immedi-
ately recognize that in this work there has been an accumulation
of hypotheses.
Doubtless this will not suffice to make them comprehend that
they on their part have made just as many; each sees the mote
without seeing the beam ; but by their criticisms they will warn
their rivals, and it may be supposed these will not fail to render
them the same service.
The English procedure often seems to us crude, the analogies
they think they discover to us seem at times superficial ; they are
not sufficiently interlocked, not precise enough; they sometimes
permit incoherences, contradictions in terms, which shock a geo-
metric spirit and which the employment of the mathematical
method would immediately have put in evidence. But most often
it is, on the other hand, very fortunate that they have not per-
ceived these contradictions; else would they have rejected their
model and could not have deduced from it the brilliant results
they have often made to come out of it.
And then these very contradictions, when they end by per-
ceiving them, have the advantage of showing them the hypothet-
ical character of their conceptions, whereas the mathematical
method, by its apparent rigor and inflexible course, often inspires
in us a confidence nothing warrants, and prevents our looking
about us.
AUTHOR'S PREFACE TO TRANSLATION 7
From another point of view, however, the two conceptions are
very unlike, and if all must be said, they are very unlike because
of a common fault.
The English wish to make the world out of what we see. I
mean what we see with the unaided eye, not the microscope, nor
that still more subtile microscope, the human head guided by
scientific induction.
The Latin wants to make it out of formulas, but these for-
mulas are still the quintessenced expression of what we see. In
a word, both would make the unknown out of the known, and
their excuse is that there is no way of doing otherwise.
And yet is this legitimate, if the unknown be the simple and
the known the complex?
Shall we not get of the simple a false idea, if we think it like
the complex, or worse yet if we strive to make it out of elements
which are themselves compounds?
Is not each great advance accomplished precisely the day some
one has discovered under the complex aggregate shown by our
senses something far more simple, not even resembling it — as
when Newton replaced Kepler's three laws by the single law of
gravitation, which was something simpler, equivalent, yet unlike ?
One is justified in asking if we are not on the eve of just such
a revolution or one even more important. Matter seems on
the point of losing its mass, its solidest attribute, and resolving
itself into electrons. Mechanics must then give place to a
broader conception which will explain it, but which it will not
explain.
So it was in vain the attempt was made in England to con-
struct the ether by material models, or in Prance to apply to
it the laws of dynamic.
The ether it is, the unknown, which explains matter, the
known; matter is incapable of explaining the ether.
POINCARfi.
INTRODUCTION
BY PEOFE880B JOSIAH EOYCE
Habvasd University
The treatise of a master needs no commendation through the
words of a mere learner. But, since my friend and former fellow
student, the translator of this volume, has joined with another
of my colleagues. Professor Cattell, in asking me to undertake
the task of calling the attention of my fellow students to the
importance and to the scope of M. Poincare's volume, I accept
the office, not as one competent to pass judgment upon the book,
but simply as a learner, desirous to increase the number of those
amongst us who are already interested in the type of researches
to which M. Poincare has so notably contributed.
The branches of inquiry collectively known as the Philosophy
of Science have undergone great changes since the appearance of
Herbert Spencer's First Principles, that volume which a large
part of the general public in this country used to regard as the
representative compend of all modern wisdom relating to the
foundations of scientific knowledge. The summary which M.
Poincare gives, at the outset of his own introduction to the
present work, where he states the view which the 'superficial
observer' takes of scientific truth, suggests, not indeed Spencer's
own most characteristic theories, but something of the spirit in
which many disciples of Spencer interpreting their master's
formulas used to conceive the position which science occupies in
dealing with experience. It was well known to them, indeed,
that experience is a constant guide, and an inexhaustible source
both of novel scientific results and of unsolved problems; but
the fundamental Spencerian principles of science, such as *the
persistence of force,' the 'rhythm of motion' and the rest, were
treated by Spencer himself as demonstrably objective, although
9
10 SCIENCE AND HYPOTHESIS
indeed 'relative' truths, capable of being tested once for all by
the 'inconceivability of the opposite,' and certain to hold true for
the whole 'knowable' universe. Thus, whether one dwelt upon
the results of such a mathematical procedure as that to which M.
Poincar6 refers in his opening paragraphs, or whether, like Spen-
cer himself, one applied the 'first principles' to regions of less
exact science, this confidence that a certain orthodoxy regarding
the principles of science was established forever was characteristic
of the followers of the movement in question. Experience,
lighted up by reason, seemed to them to have predetermined for
all future time certain great theoretical results regarding the real
constitution of the 'knowable' cosmos. Whoever doubted this
doubted 'the verdict of science.'
Some of us well remember how, when Stallo's 'Principles and
Theories of Modem Physics' first appeared, this sense of scien-
tific orthodoxy was shocked amongst many of our American read-
ers and teachers of science. I myself can recall to mind some
highly authoritative reviews of that work in which the author
was more or less sharply taken to task for his ignorant presump-
tion in speaking with the freedom that he there used regarding
such sacred possessions of humanity as the fundamental concepts
of physics. That very book, however, has quite lately been
translated into German as a valuable contribution to some of the
most recent efforts to reconstitute a modem 'philosophy of
nature.' And whatever may be otherwise thought of Stallo's
critical methods, or of his results, there can be no doubt that, at
the present moment, if his book were to appear for the first
time, nobody would attempt to discredit the work merely on
account of its disposition to be agnostic regarding the objective
reality of the concepts of the kinetic theory of gases, or on
account of its call for a logical rearrangement of the fundamental
concepts of the theory of energy. We are no longer able so easily
to know heretics at first sight.
For we now appear to stand in this position: The control
of natural phenomena, which through the sciences men have
attained, grows daily vaster and more detailed, and in its de-
tails more assured. Phenomena men know and predict better
than ever. But regarding the most general theories, and the
INTRODUCTION 11
most fundamental, of science, there ia no longer an? notablt
Kieatific orthodoxy. Thus, as knowledge grows firmer and wider,
conceptual construction becomes less rigid. The field of the
theoretical philosophy of nature — ^yes, the field of the logic of
science — this whole region is to-day an open one. "Whoever will
work there must indeed accept the verdict of experience regard-
ing what happens in the natural world. So far he is indeed
bound. But he may undertake without hindrance from mere
tradition the task of trying afresh to reduce what happens
to conceptual unity. The cdrele-squares and the inventors of
devices for perpetual motion are indeed still as unwelcome in
scientific company as they were in the days when scientific
orthodoxy was more rigidly defined ; but that is not because the
foundations of geometry are now viewed as completely settled,
beyond controversy, nor yet because the 'persistence of force'
has been finally so defined>as to make the 'opposite ineonceiT-
able ' and the doctrine of energy beyond the reach of novel formu-
lations. No, the circle-squarers and the inventors of devices for
perpetual motion are to-day discredited, not because of any
unorthodoxy of their general philosophy of nature, but because
their views regarding special facts and processes stand in
conflict with certain equally special results of science which
themselves admit of very various general theoretical interpre-
tations. Certain properties of the irrational number ir are
known, in suificient multitude to justify the mathematician in
declining to listen to the arguments of the circle-squarer ; but,
despite great advances, and despite the assured results of Dede-
kind, of Cantor, of "Weierstrass and of various others, the gen-
eral theory of the logic of the numbers, rational and irrational,
still presents several important features of great obscurity ; and
the philosophy of the concepts of geometry yet remains, in sev-
eral very notable respects, unconquered territory, despite the
work of Hilbert and of Fieri, and of our author himself. The
ordinary inventors of the perpetual motion machines still stand
in conflict with accepted generalizations; but nobody knows as
yet what the final form of the theory of energy will be, nor can
any one say precisely what place the phenomena of the radioac-
tive bodies will occupy in that theory. The alchemists would not
12 SCIENCE AND BYP0TBESI8
be welcome workers in modem laboratories; yet some sorts of
transformation and of evolution of the elements are to-day
matters which theory can find it convenient, upon occasion, to
treat as more or less exactly definable possibilities; while some
newly observed phenomena tend to indicate, not indeed that the
ancient hopes of the alchemists were well founded, but that the
ultimate constitution of matter is something more fluent, less in-
variant, than the theoretical orthodoxy of a recent period sap-
posed. Again, regarding the foundations of biology, a theoret-
ical orthodoxy grows less possible, less definable, less conceiv-
able (even as a hope) the more knowledge advances. Once
'mechanism' and 'vitalism' were mutually contradictory theories
regarding the ultimate constitution of living bodies. Now they
are obviously becoming more and more 'points of view,' diverse
but not necessarily conflicting. So far as you find it convenient
to limit your study of vital processes to those phenomena which
distinguish living matter from all other natural obects, you may
assume, in the modern 'pragmatic' sense, the attitude of a 'neo-
vitalist. ' So far, however, as you are able to lay stress, with good
results, upon the many ways in which the life processes can be
assimilated to those studied in physics and in chemistry, yon
work as if you were a partisan of 'mechanics.' In any case,
your special science prospers by reason of the empirical discov-
eries that jou make. And your theories, whatever they are,
must not run counter to any positive empirical results. But
otherwise, scientific orthodoxy no longer predetermines what
alone it is respectable for you to think about the nature of living
substance.
This gain in the freedom of theory, coming, as it does, side by
side with a constant increase of a positive knowledge of nature,
lends itself to various interpretations, and raises various obvious
questions.
II
One of the most natural of these interpretations, one of the
most obvious of these questions, may be readily stated. Is not
the lesson of all these recent discussions simply this, that general
theories are simply vain, that a philosophy of nature is an idle
INTRODUCTION 13
dream, and that the results of science are coextensive with the
range of actual empirical observation and of successful predic-
tion? If this is indeed the lesson, then the decline of theoretical
orthodoxy in science is — ^like the eclipse of dogma in religion —
merely a further lesson in pure positivism, another proof that
nttn does best when he limits himself to thinking about what can
be found in human experience, and in trying to plan what can
be done to make human life more controllable and more reason-
able. What we are free to do as we please — ^is it any longer a
serious business? What we are free to think as we please — ^is it
of any further interest to one who is in search of truth? If
certain general theories are mere conceptual constructions, which
to-day are, and to-morrow are cast into the oven, why dignify
them by the name of philosophy? Has science any place for
such theories? Why be a *neo-vitalist,' or an 'evolutionist,' or
an * atomist, ' or an ' Energetiker ' ? Why not say, plainly : * * Such
and such phenomena, thus and thus described, have been ob-
served; such and such experiences are to be expected, since the
hypotheses by the terms of which we are required to expect
them have been verified too often to let us regard the agreement
with experience as due merely to chance; so much then with
reasonable assurance we know; all else is silence— or else is
some matter to be tested by another experiment?" Why not
limit our philosophy of science strictly to such a counsel of resig-
nation? Why not substitute, for the old scientific orthodoxy,
simply a confession of ignorance, and a resolution to devote our-
selves to the business of enlarging the bounds of actual em-
pirical knowledge?
Such comments upon the situation just characterized are fre-
quently made. Unfortunately, they seem not to content the
very age whose revolt from the orthodoxy of traditional theory,
whose uncertainty about all theoretical formulations, and whose
vast wealth of empirical discoveries and of rapidly advancing
special researches, would seem most to justify tliese very com-
ments. Never has there been better reason than there is to-day
to be content, if rational man could be content, with a pure pos-
itivism. The splendid triumphs of special research in the most
various fields, the constant increase in our practical control over
14 SCIENCE AND ETPOTHESIS
nature — ^these, our positive and growing possessions, stand in
glaring contrast to the failure of the scientific orthodoxy of a
former period to fix the outlines of an ultimate creed about the
nature of the knowable universe. Why not 'take the cash and
let the credit go'f Why pursue the elusive theoretical 'unifica-
tion' any further, when what we daily get from our sciences is
an increasing wealth of detailed information and of practical
guidance T
As a fact, however, the known answer of our own age to these
very obvious comments is a constant multiplication of new
efforts towards large and unifying theories. If theoretical ortho-
doxy is no longer clearly definable, theoretical construction was
never more rife. The history of the doctrine of evolution, even
in its most recent phases, when the theoretical uncertainties re-
garding the 'factors of evolution' are most insisted upon, is full
of illustrations of this remarkable union of scepticism in critical
work with courage regarding the use of the scientific imagination.
The history of those controversies regarding theoretical physics,
some of whose principal phases M. Poincare, in his book, sketches
with the hand of the master, is another illustration of the con-
sciousness of the time. Men have their freedom of thought in
these regions; and they feel the need of making constant and
constructive use of this freedom. And the men who most feel
this need are by no means in the majority of cases professional
metaphysicians — or students who, like myself, have to view all
these controversies amongst the scientific theoreticians from
without as learners. These large theoretical constructions are
due, on the contrary, in a great many cases to special workers,
who have been driven to the freedom of philosophy by the oppres-
sion of experience, and who have learned in the conflict with
special problems the lesson that they now teach in the form of
general ideas regarding the philosophical aspects of science.
Why, then, does science actually need general theories, despite
the fact that these theories inevitably alter and pass awayf
What is the service of a philosophy of science, when it is certain
that the philosophy of science which is best suited to the needs
of one generation must be superseded by the advancing insight
of the next generation? Why must that which endlessly grows^
INTRODUCTION 15
namdy, man's knowledge of the phenomenal order of natnre^
be constantly united in men's minds with that which is certain
to decay, namely, the theoretical formulation of special knowl-
edge in more or less completely unified systems of doctrine T
I understand our author's volume to be in the main an
answer to this question. To be sure, the compact and manifold
teachings which this text contains relate to a great many dif-
ferent special issues. A student interested in the problems of
the philosophy of mathematics, or in the theory of probabilities,
or in the nature and office of mathematical physics, or in still
other problems belonging to the wide field here discussed, may
find what he wants here and there in the text, even in case the
general issues which give the volume its unity mean little to
him, or even if he differs from the author's views regarding the
principal issues of the book. But in the main, this volume must
be regarded as what its title indicates — a critique of the nature
and place of hypothesis in the work of science and a study of the
logical relations of theory and fact. The result of the book is a
substantial justification of the scientific utility of theoretical con-
struction — an abandonment of dogma, but a vindication of the
rights of the constructive reason.
Ill
The most notable of the results of our author's investigation
of the logic of scientific theories relates, as I understand his work,
to a topic which the present state of logical investigation, just
summarized, makes especially important, but which has thus far
been very inadequately treated in the text-books of inductive
logic. The useful hypotheses of science are of two kinds :
1. The hypotheses which are valuable precisely because they
are either verifiable or else refutable through a definite appeal
to the tests furnished by experience ; and
2. The hypotheses which, despite the fact that experience sug-
gests them, are valuable despite, or even because, of the fact that
experience can neither confirm nor refute them. The contrast
between these two kinds of hypotheses is a prominent topic of
our author's discussion.
Hypotheses of the general type which I have here placed first
16 SCIENCE AND HYPOTHESIS
in order are the ones which the text-books of inductive logic and
those summaries of scientific method which are customary in the
course of the elementary treatises upon physical science are
already accustomed to recognize and to characterize. The value
of such hypotheses is indeed undoubted. But hypotheses of the
type which I have here named in the second place are far less
frequentiy recognized in a perfectly explicit way as useful aids
in the work of special science. One usually either fails to admit
their presence in scientific work, or else remains silent as to the
reasons of their usefulness. Our author's treatment of the work
of science is therefore especially marked by the fact that he ex-
plicitiy makes prominent both the existence and the scientific
importance of hypotheses of this second type. They occupy in
his discussion a place somewhat analogous to each of the two dis-
tinct positions occupied by the 'categories' and the 'forms of
sensibility/ on the one hand, and by the 'regulative principles of
the reason,' on the other hand, in the Kantian theory of our
knowledge of nature. That is, these hypotheses which can
neither be confirmed nor refuted by experience appear, in M.
Poincar6's account, partly (like the conception of * continuous
quantity') as devices of the understanding whereby we give
conceptual unity and an invisible connectedness to certain types
of phenomenal facts which come to us in a discrete form and in
a confused variety; and partly (like the larger organizing con-
cepts of science) as principles regarding the structure of the
world in its wholeness ; i. e., as principles in the light of which we
try to interpret our experience, so as to give to it a totality and
an inclusive unity such as Euclidean space, or such as the world
of the theory of energy is conceived to possess. Thus viewed, M.
Poincare's logical theory of this second class of hypotheses under-
takes to accomplish, with modem means and in the light of
to-day's issues, a part of what Kant endeavored to accomplish
in his theory of scientific knowledge with the limited means
which were at his disposal. Those aspects of science which are
determined by the use of the hypotheses of this second kind
appear in our author's account as constituting an essential
human way of viewing nature, an interpretation rather than
a portrayal or a prediction of the objective facts of nature, an
INTRODUCTION 17
adjustment of our conceptions of things to the internal needs
of our intelligence, rather than a grasping of things as they are
in themselves.
To be sure, M. Poincare's view, in this portion of his work,
obviously differs, meanwhile, from that of Kant, as well as this
agrees, in a measure, with the spirit of the Kantian epistemology.
I do not mean therefore to class our author as a Kantian. For
Kant, the interpretations imposed by the * forms of sensibility,'
and by the 'categories of the understanding,' upon our doctrine
of nature are rigidly predetermined by the unalterable 'form'
of our intellectual powers. We 'must' thus view facts, whatever
tiie data of sense must be. This, of course, is not M. Poincar^'s
view. A similarly rigid predetermination also limits the Kantian
'ideas of the reason' to a certain set of principles whose guidance
of the course of our theoretical investigations is indeed only
'regulative,' but is 'a priori,' and so unchangeable. For M.
Poincar^, on the contrary, all this adjustment of our interpre-
tations of experience to the needs of our intellect is something
far less rigid and unalterable, and is constantly subject to the
suggestions of experience. We must indeed interpret in our own
way; but our way is itself only relatively determinate; it is
essentially more or less plastic ; other interpretations of experience
are conceivable. Those that we use are merely the ones found to
be most convenient. But this convenience is not absolute neces-
sity. Unverifiable and irrefutable hypotheses in science are in-
deed, in general, indispensable aids to the organization and to the
guidance of our interpretation of experience. But it is expe-
rience itself which points out to us what lines of interpretation
will prove most convenient. Instead of Kant's rigid list of
a priori 'forms,' we consequently have in M. Poincare's account
a set of conventions, neither wholly subjective and arbitrary, nor
yet imposed upon us unambiguously by the external compulsion
of experience. The organization of science, so far as this organ-
ization is due to hypotheses of the kind here in question, thus
resembles that of a constitutional government — neither abso-
lutely necessary, nor yet determined apart from the will of the
subjects, nor yet accidental — a free, yet not a capricious estab-
lishment of good order, in conformity with empirical needs.
3
18 SCIENCE AND HYPOTHESIS
Characteristic remains, however, for our author, as, in his
decidedly contrasting way, for Kant, the thought that without
principles which at every stage transcend precise confirmation
through such experience as is then accessible the organization of
experience is impossible. Whether one views these principles as
conventions or as a priori 'forms,' they may therefore be de-
scribed as hypotheses, but as hypotheses that, while lying at the
basis of our actual physical sciences, at once refer to experience
and help us in dealing with experience, and are yet neither con-
firmed nor refuted by the experiences which we possess or which
we can hope to attain.
Three special instances or classes of instances, according to
our author's account, may be used as illustrations of this general
type of hypotheses. They are: (1) The hypothesis of the exist-
ence of continuous extensive quanta in nature; (2) The prin-
ciples of geometry; (3) The principles of mechanics and of the
general theory of energy. In case of each of these special types
of hypotheses we are at first disposed, apart from reflection, to
say that we find the world to be thus or thus, so that, for instance,
we can confirm the thesis according to which nature contains
continuous magnitudes; or can prove or disprove the physical
truth of the postulates of Euclidean geometry ; or can confirm by
definite experience the objective validity of the principles of
mechanics. A closer examination reveals, according to our
author, the incorrectness of all such opinions. H3rpotheses of
these various special types are needed ; and their usefulness can
be empirically shown. They are in touch with experience; and
that they are not merely arbitrary conventions is also verifiable.
They are not a priori necessities ; and we can easily conceive in-
telligent beings whose experience could be best interpreted with-
out using these hypotheses. Yet these hypotheses are not sub-
ject to direct confirmation or refutation by experience. They
stand then in sharp contrast to the scientific hypotheses of the
other, and more frequently recognized, type, i. e., to the hy-
potheses which can be tested by a definite appeal to experience.
To these other hypotheses our author attaches, of course, great
importance. His treatment of them is full of a living apprecia-
tion of the significance of empirical investigation. But the cen-
INTRODUCTION 19
tral problem of the logic of science thus becomes the problem of
the relation between the two fundamentally distinct types of
hypotheses, ♦. e., between those which can not be verified or re-
futed through experience, and those which can be empirically
tested.
IV
The detailed treatment which M. Poincar6 gives to the problem
thus defined must be learned from his text. It is no part of my
purpose to expound, to defend or to traverse any of his special
conclusions regarding this matter. Yet I can not avoid observ-
ing that, while M. Poincar^ strictly confines his illustrations and
his expressions of opinion to those regions of science wherein, as
special investigator, he is himself most at home, the issues which
he thus raises regarding the logic of science are of even more
critical importance and of more impressive interest when one
applies M. Poincare's methods to the study of the concepts and
presuppositions of the organic and of the historical and social
sciences, than when one confines one's attention, as our author
here does, to the physical sciences. It belongs to the province of
an introduction like the present to point out, however briefiy and
inadequately, that the significance of our author's ideas extends
far beyond the scope to which he chooses to confine their discussion.
The historical sciences, and in fact all those sciences such as
geology, and such as the evolutionary sciences in general, un-
dertake theoretical constructions which relate to past time. Hy-
potheses relating to the more or less remote past stand, however,
in a position which is very interesting from the point of view of
the logic of science. Directly speaking, no such hypothesis is
capable of confirmation or of refutation, because we can not
return into the past to verify by our own experience what then
happened. Yet indirectly, such hypotheses may lead to predic-
tions of coming experience. These latter will be subject to con-
troL Thus, Schliemann's confidence that the legend of Troy had
a definite historical foundation led to predictions regarding what
certain excavations would reveal. In a sense somewhat different
from that which filled Schliemann's enthusiastic mind, these pre-
dictions proved verifiable. The result has been a considerable
20 SCIENCE AND HYPOTHESIS
change in the attitude of historians toward the legend of Troy.
Geological investigation leads to predictions regarding the order
of the strata or the course of mineral veins in a district, regard-
ing the fossils which may be discovered in given formations, and
so on. These hypotheses are subject to the control of experience.
The various theories of evolutionary doctrine include many hy-
potheses capable of confirmation and of refutation by empirical
tests. Yet, despite all such empirical control, it still remains
true that whenever a science is mainly concerned with the remote
past, whether this science be archeology, or geology, or anthro-
pology, or Old Testament history, the principal theoretical con-
structions always include features which no appeal to present
or to accessible future experience can ever definitely test. Hence
the suspicion with which students of experimental science often
regard the theoretical constructions of their confreres of the sci-
ences that deal with the past. The origin of the races of men,
of man himself, of life, of species, of the planet ; the hypotheses
of anthropologists, of archeologists, of students of 'higher criti-
cism' — ^all these are matters which the men of the laboratory
often regard with a general incredulity as belonging not at all
to the domain of true science. Yet no one can doubt the im-
portance and the inevitableness of endeavoring to apply scientific
method to these regions also. Science needs theories regarding
the past history of the world. And no one who looks closer into
the methods of these sciences of past time can doubt that verifi-
able and unverifiable hypotheses are in all these regions inevitably
interwoven; so that, while experience is always the guide, the
attitude of the investigator towards experience is determined by
interests which have to be partially due to what I should call
that 'internal meaning,' that human interest in rational theoret-
ical construction which inspires the scientific inquiry; and the
theoretical constructions which prevail in such sciences are
neither unbiased reports of the actual constitution of an external
reality, nor yet arbitrary constructions of fancy. These con-
structions in fact resemble in a measure those which M. Poincarfi
in this book has analyzed in the case of geometry. They are
constructions molded, but not predetermined in their details, by
experience. We report facts ; we let the facts speak ; but we, as
INTRODUCTION 21
we inyestigate, in the popular phrase, Halk back' to the facts.
We interpret as well as report Man is not merely made for
science, but science is made for man. It expresses his deepest
intellectual needs, as well as his careful observations. It is an
effort to bring internal meanings into harmony with external
verifications. It attempts therefore to control, as well as to
submit, to conceive with rational unity, as well as to accept data.
Its arts are those directed towards self-possession as well as
towards an imitation of the outer reality which we find. It
seeks therefore a disciplined freedom of thought. The discipline
is as essential as the freedom; but the latter has also its place.
The theories of science are human, as well as objective, inter-
nally rational, as well as (when that is possible) subject to ex-
ternal tests.
In a field very different from that of the historical sciences,
namely, in a science of observation and of experiment, which is
at the same time an organic science, I have been led in the course
of some study of the history of certain researches to notice the
existence of a theoretical conception which has proved extremely
fruitful in guiding research, but which apparently resembles in
a measure the type of hypotheses of which M. Poincar4 speaks
when he characterizes the principles of mechanics and of the
theory of energy. I venture to call attention here to this con-
ception, which seems to me to illustrate M. Poincare's view of the
functions of hypothesis in scientific work.
The modem science of pathology is usually regarded as dating
from the earlier researches of Virchow, whose * Cellular Path-
ology' was the outcome of a very careful and elaborate induc-
tion. Virchow, himself, felt a strong aversion to mere specula-
tion. He endeavored to keep close to observation, and to relieve
medical science from the control of fantastic theories, such as
those of the Naturphilosophen had been. Yet Virchow 's re-
searches were, as early as 1847, or still earlier, already under the
guidance of a theoretical presupposition which he himself states
as follows: **We have learned to recognize," he says, **that dis-
eases are not autonomous organisms, that they are no entities
that have entered into the body, that they are no parasites which
take root in the body, but that they merely show tis the course of
22 8CIESCE AXD HYPOTHESIS
the vital proeeM$es under mltered ccmdUions" Cdaas sie nnr
AMauf der Lebensendieiiiiiiigeii anter Teiudcrten Bedingnn-
gen dAntdkn')-
The enoTiDoiis importmiiee of this theoredcal presupposition
for all the earljr so ccc ss c s of modem pmtiiologieal inresligation
k generalljr recognized by the experts. I do not doubt this
opinion. It spi>ear8 to be a eommonplsee of tiie history of this
aeienee. Bnt in Yirchow's later jrears this Tery presupposition
seemed to some of his contemporaries to be ealled in qaestion by
the soccesses of recent bacteriology. The qaestion arose whether
the theoretical foundations of Virchow's pathology had not been
set aside. And in fact the theoiy of the parasitical origin of
a vast number of diseased conditions has indeed come upon an
empirical basis to be generally recognized. Yet to the end of his
own career Virchow stoutly maintained that in all its essential
significance his own fundamental principle remained quite un-
touched by the newer discoreries. And, as a fact, this view
could indeed be maintained. For if diseases proved to be the
consequences of the presence of parasites, the diseases them-
selves, so far as they belonged to the diseased organism, were
still not the parasites, but were, as before, the reaction of the
organism to the verdnderie Bedingungen which the presence of
the parasites entailed. So Virchow could well insist And if
the famous principle in question is only stated with sufficient
generality, it amounts simply to saying that if a disease in-
volves a change in an organism, and if this change is subject to
law at all, then the nature of the organism and the reaction of
the organism to whatever it is which causes the disease must be
underHtood in case the disease is to be understood.
For this very reason, however, Virchow's theoretical principle
in its most general form could be neither confirmed nor refuted
by experience. It would remain empirically irrefutable, so far
as I can see, even if we should learn that the devil was the
true cause of all diseases. For the devil himself would then
simply predetermine the verdnderte Bedingungen to which the
diseased organism would be reacting. Let bullets or bacteria,
poisons or compressed air, or the devil be the Bedingungen to
which a diseased organism reacts, the postulate that Virchow
INTRODUCTION 23
states in the passage just quoted will remain irrefutable, if only
this postulate be interpreted to meet the case. For the principle
in question merely says that whatever entity it may be, bullet, or
poison, or devil, that affects the organism, the disease is not that
entity, but is the resulting alteration in the process of the
organism.
I insist, then, that this principle of Virchow's is no trial sup-
position, no scientific hypothesis in the narrower sense — capable
of being submitted to precise empirical tests. It is, on the
contrary, a very precious leading idea, a theoretical interpre-
tation of phenomena, in the light of which observations are to be
made — *a regulative principle' of research. It is equivalent to
a resolution to search for those detailed connections which link
the processes of disease to the normal process of the organism.
Such a search undertakes to find the true unity, whatever that
may prove to be, wherein the pathological and the normal proc-
esses are linked. Now without some such leading idea, the cellu-
lar pathology itself could never have been reached ; because the
empirical facts in question would never have been observed.
Hence this principle of Virchow's was indispensable to the
growth of his science. Yet it was not a verifiable and not a re-
futable hypothesis. One value of unverifiable and irrefutable
hyx)otheses of this type lies, then, in the sort of empirical
inquiries which they initiate, inspire, organize and guide. In
these inquiries hypotheses in the narrower sense, that is, trial
propositions which are to be submitted to definite empirical con-
trol, are indeed everywhere present. And the use of the other
sort of principles lies wholly in their application to experience.
Yet without what I have just proposed to call the 'leading ideas'
of a science, that is, its principles of an unverifiable and irre-
futable character, suggested, but not to be finally tested, by
experience, the hypotheses in the narrower sense would lack that
guidance which, as M. Poincare has shown, the larger ideas of
science give to empirical investigation.
V
I have dwelt, no doubt, at too great length upon one aspect
only of our author's varied and well-balanced discussion of the
24 SCIENCE AND HYPOTHESIS
problems and concepts of scientific theory. Of the hypotheses
in the narrower sense and of the value of direct empirical control,
he has also spoken with the authority and the originality which
belong to his position. And in dealing with the foundations of
mathematics he has raised one or two questions of great philo-
sophical import into which I have no time, even if I had the
right, to enter here. In particular, in speaking of the essence
of mathematical reasoning, and of the difficult problem of what
makes possible novel results in the field of pure mathematics, M.
Poincar6 defends a thesis regarding the office of 'demonstration
by recurrence' — ^a thesis which is indeed disputable, which has
been disputed and which I myself should be disposed, so far as
I at present understand the matter, to modify in some respects,
even in accepting the spirit of our author's assertion. Yet there
can be no doubt of the importance of this thesis, and of the fact
that it defines a characteristic that is indeed fundamental in a
wide range of mathematical research. The philosophical prob-
lems that lie at the basis of recurrent proofs and processes are,
as I have elsewhere argued, of the most fundamental importance.
These, then, are a few hints relating to the significance of
our author's discussion, and a few reasons for hoping that our
own students will profit by the reading of the book as those of
other nations have already done.
Of the person and of the life-work of our author a few words
are here, in conclusion, still in place, addressed, not to the stu-
dents of his own science, to whom his position is well known, but
to the general reader who may seek guidance in these pages.
Jules Henri Poincar6 was born at Nancy, in 1854, the son
of a professor in the Faculty of Medicine at Nancy. He
studied at the i^cole Polytechnique and at the i^cole des Mines,
and later received his doctorate in mathematics in 1879. In
1883 he began courses of instruction in mathematics at the
£cole Polytechnique ; in 1886 received a professorship of mathe-
matical physics in the Faculty of Sciences at Paris; then
became member of the Academy of Sciences at Paris, in 1887,
and devoted his life to instruction and investigation in the
regions of pure mathematics, of mathematical physics and of
celestial mechanics. His list of published treatises relating to
INTRODUCTION 25
yarious branches of his chosen sciences is long; and his ori-
ginal memoirs have included several momentous investigations,
which have gone far to transform more than one branch of
research. His presence at the International Congress of Arts
and Science in St. Louis was one of the most noticeable features
of that remarkable gathering of distinguished foreign guests.
In Poincar6 the reader meets, then, not one who is primarily a
speculative student of general problems for their own sake, but
an original investigator of the highest rank in several distinct,
although interrelated, branches of modem research. The theory
of functions — ^a highly recondite region of pure mathematics —
owes to him advances of the first importance, for instance, the
definition of a new type of functions. The 'problem of the three
bodies, ' a famous and fundamental problem of celestial mechanics,
has received from his studies a treatment whose significance has
been recognized by the highest authorities. His international
reputation has been confirmed by the conferring of more than one
important prize for his researches. His membership in the most
eminent learned societies of various nations is widely extended;
his volumes bearing upon various branches of mathematics and
of mathematical physics are used by special students in all parts
of the learned world ; in brief, he is, as geometer, as analyst and
as a theoretical physicist, a leader of his age.
Meanwhile, as contributor to the philosophical discussion of
the bases and methods of science, M. Poincar^ has long been
active. When, in 1893, the admirable Revue de Meiaphysique et
de Morale began to appear, M. Poincar^ was soon found amongst
the most satisfactory of the contributors to the work of that
journal, whose office it has especially been to bring philosophy
and the various special sciences (both natural and moral) into
a closer mutual understanding. The discussions brought to-
gether in the present volume are in large part the outcome of
M. Poincar^'s contributions to the Revue de Meiaphysique et de
Morale. The reader of M. Poincar^'s book is in presence, then,
of a great special investigator who is also a philosopher.
SCIENCE AND HYPOTHESIS
INTRODUCTION
Fob a superficial observer, scientific truth is beyond the possi-
bility of doubt ; the logic of science is infallible, and if the scien-
tists are sometimes mistaken, this is only from their mistaking
its rules.
''The mathematical verities flow from a small number of self-
evident propositions by a chain of impeccable reasonings; they
impose themselves not only on us, but on nature itself. They
fetter, so to speak, the Creator and only permit him to choose
between some relatively few solutions. A few experiments then
will suffice to let us know what choice he has made. From each
experiment a crowd of consequences will follow by a series of
mathematical deductions, and thus each experiment will make
known to us a comer of the universe."
Behold what is for many people in the world, for scholars get-
ting their first notions of physics, the origin of scientific certi-
tude. This is what they suppose to be the role of experimenta-
tion and mathematics. This same conception, a hundred years
ago, was held by many savants who dreamed of constructing the
world with as little as possible taken from experiment.
On a little more reflection it was perceived how great a place
hypothesis occupies; that the mathematician can not do without
it, still less the experimenter. And then it was doubted if all
these constructions were really solid, and believed that a breath
would overthrow them. To be skeptical in this fashion is still to
be superficial. To doubt everything and to believe everything
are two equally convenient solutions; each saves us from
thinking.
Instead of pronouncing a summary condemnation, we ought
therefore to examine with care the role of hypothesis; we shall
then recognize, not only that it is necessary, but that usually it is
27
28 SCIENCE AND HYPOTHESIS
le^timate. We shall also see that there are several sorts of hy-
potheses ; that some are verifiabley and once confirmed by experi-
ment become fruitful truths; that others, powerless to lead us
astray, may be useful to us in fixing our ideas; that others,
finally, are hypotheses only in appearance and are reducible to
disguised definitions or conventions.
These last are met with above all in mathematics and the
related sciences. Thence precisely it is that these sciences get
their rigor; these conventions are the work of the free activity
of our mind, which, in this domain, recognizes no obstacle. Here
our mind can affirm, since it decrees ; but let us understctnd that
while these decrees are imposed upon our science, which, without
them, would be impossible, they are not imposed upon nature.
Are they then arbitrary! No, else were they sterile. Experi-
ment leaves us our freedom of choice, but it guides us by aiding
us to discern the easiest way. Our decrees are therefore like
those of a prince, absolute but wise, who consults his council of
state.
Some people have been struck by this character of free conven-
tion recognizable in certain fundamental principles of the
sciences. They have wished to generalize beyond measure, and,
at the same time, they have forgotten that liberty is not license.
Thus they have reached what is called nominalism, and have
asked themselves if the savant is not the dupe of his own defi-
nitions and if the world he thinks he discovers is not simply
created by his own caprice.^ Under these conditions science
would be certain, but deprived of significance.
If this were so, science would be powerless. Now every day
we see it work under our very eyes. That could not be if it
taught us nothing of reality. Still, the things themselves are
not what it can reach, as the naive dogmatists think, but only
the relations between things. Outside of these relations there
is no knowable reality.
Such is the conclusion to which we shall come, but for that we
must review the series of sciences from arithmetic and geometry
to mechanics and experimental physics.
i-See Le B07, 'Science et Philosophie, ' Bevue de M^aphysique et de
Morale, 1901.
INTRODUCTION 29
What is the nature of mathematical reasoning f Is is really
deductivey as is commonly supposed? A deeper analysis shows
us that it is not, that it partakes in a certain measure of the
nature of inductive reasoning, and just because of this is it so
fruitful. None the less does it retain its character of rigor
absolute; this is the first thing that had to be shown.
Knowing better now one of the instruments which mathemat-
ics puts into the hands of the investigator, we had to analyze an-
other fundamental notion, that of mathematical magnitude. Do
we find it in nature, or do we ourselves introduce it there f And,
in this latter case, do we not risk marring everything! Com-
paring the rough data of our senses with that extremely complex
and subtile concept which mathematicians call magnitude, we are
forced to recognize a difference ; this frame into which we wish to
force everything is of our own construction; but we have not
made it at random. We have made it, so to speak, by measure
and therefore we can make the facts fit into it without changing
what is essential in them.
Another frame which we impose on the world is space.
Whence come the first principles of geometry! Are they im-
posed on us by logic ! Lobachevski has proved not, by creating
non-Euclidean geometry. Is space revealed to us by our senses !
Still no, for the space our senses could show us differs absolutely
from that of the geometer. Is experience the source of geom-
etry ? A deeper discussion will show us it is not. We therefore
conclude that the first principles of geometry are only conven-
tions ; but these conventions are not arbitrary and if transported
into another world (that I call the non-Euclidean world and seek
to imagine), then we should have been led to adopt others.
In mechanics we should be led to analogous conclusions, and
should see that the principles of this science, though more di-
rectly based on experiment, still partake of the conventional
character of the geometric postulates. Thus far nominalism
triumphs ; but now we arrive at the physical sciences, properly so
called. Here the scene changes; we meet another sort of hy-
potheses and we see their fertility. Without doubt, at first blush,
the theories seem to us fragile, and the history of science proves
to us how ephemeral they are; yet they do not entirely perish,
30 SCIENCE AND HYPOTHESIS
and of each of them something remains. It is this something
we most seek to disentangle, since there and there alone is the
veritable reality.
The method of the physical sciences rests on the induction
which makes ns expect the repetition of a phenomenon when the
circumstances under which it first happened are reproduced* U
all these circumstances could be reproduced at once, this prin-
ciple could be applied without fear; but that will never happen;
some of these circumstances will always be lacking. Are we
absolutely sure they are unimportant! Evidently not. That
may be probable, it can not be rigorously certain. Hence the
important role the notion of probability plays in the physical
sciences. The calculus of probabilities is therefore not merely
a recreation or a guide to players of baccarat, and we must seek
to go deeper with its foundations. Under this head I have been
able to give only very incomplete results, so strongly does this
vague instinct which lets us discern probability defy analysis.
After a study of the conditions under which the physicist
works, I have thought proper to show him at work. For that I
have taken instances from the history of optics and of electricity.
We shall see whence have sprung the ideas of Fresnel, of Max-
well, and what unconscious hypotheses were made by Ampere
and the other founders of electrodynamics.
PARTI
NUMBER AND MAGNITUDE
CHAPTER I
On the Nature of Mathematical BsASONiNa
The very possibility of the science of mathematics seems
an insoluble contradiction. If this science is deductive only in
appearance, whence does it derive that perfect rigor no one
dreams of doubting? If, on the contrary, all the propositions it
enunciates can be deduced one from another by the rules of
formal logic, why is not mathematics reduced to an immense
tautology? The syllogism can teach us nothing essentially new,
and, if everything is to spring from the principle of identity,
everything should be capable of being reduced to it. Shall we
then admit that the enunciations of all those theorems which fill
80 many volumes are nothing but devious ways of saying A is A ?
Without doubt, we can go back to the axioms, which are at
the source of all these reasonings. If we decide that these can
not be reduced to the principle of contradiction, if still less we
see in them experimental facts which could not partake of mathe-
matical necessity, we have yet the resource of classing them
among synthetic a priori judgments. This is not to solve the diflS-
culty, but only to baptize it ; and even if the nature of synthetic
judgments were for us no mystery, the contradiction would not
have disappeared, it would only have moved back ; syllogistic rea-
soning remains incapable of adding anything to the data given
it ; these data reduce themselves to a few axioms, and we should
find nothing else in the conclusions.
No theorem could be new if no new axiom intervened in its
demonstration; reasoning could give us only the immediately
31
32 SCIENCE AND HYPOTHESIS
evident verities borrowed from direct intuition ; it would be only
an intermediary parasite, and therefore should we not have good
reason to ask whether the whole syllogistic apparatus did not
serve solely to disguise our borrowing?
The contradiction will strike us the more if we open any book
on mathematics ; on every page the author will announce his in-
tention of generalizing some proposition already known. Does
the mathematical method proceed from the particular to the gen-
eral, and, if so, how then can it be called deductive f
If finally the science of number were purely analytic, or
could be analytically derived from a small number of Gfynthetic
judgments, it seems that a mind sufficiently powerful could at
a glance perceive all its truths; nay more, we might even hope
that some day one would invent to express them a language suffi-
ciently simple to have them appear self-evident to an ordinary
intelligence.
If we refuse to admit these consequences, it must be conceded
that mathematical reasoning has of itself a sort of creative virtue
and consequently differs from the syllogism.
The difference must even be profound. We shall not, for
example, find the key to the mystery in the frequent use of that
rule according to which one and the same uniform operation
applied to two equal numbers will give identical results.
All these modes of reasoning, whether or not they be reducible
to the syllogism properly so called, retain the analytic character,
and just because of that are powerless.
II
The discussion is old; Leibnitz tried to prove 2 and 2 make 4;
let us look a moment at his demonstration.
I will suppose the number 1 defined and also the operation
a? + 1 which consists in adding unity to a given number x.
These definitions, whatever they be, do not enter into the
course of the reasoning.
I define then the numbers 2, 3 and 4 by the equalities
(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4.
In the same way, I define the operation x + 2 by the relation:
MATHEMATICAL REASONING 33
(4) a? + 2= (« 4-1)4-1.
That presupposed, we have
2 4-1 4-1 = 3 4- 1 (Definition 2),
3 4-1 = 4 (Definition 3),
24-2= (2 4- 1)4-1 (Definition 4),
whence
24-2 = 4 Q.E.D.
It can not be denied that this reasoning is purely analytic.
But ask any mathematician: 'That is not a demonstration prop-
erly so called,' he will say to you: 'that is a verification.' We
have confined ourselves to comparing two purely conventional
definitions and have ascertained their identity ; we have learned
nothing new. Verification differs from true demonstration pre-
cisely because it is purely analytic and because it is sterile. It is
sterile because the conclusion is nothing but the premises trans-
lated into another language. On the contrary, true demonstration
is fruitful because the conclusion here is in a sense more general
than the premises.
The equality 2 + 2 = 4 is thus susceptible of a verification
only because it is particular. Every particular enunciation in
mathematics can always be verified in this same way. But if
mathematics could be reduced to a series of such verifications, it
would not be a science. So a chess-player, for example, does not
create a science in winning a game. There is no science apart
from the general.
It may even be said the very object of the exact sciences is to
spare us these direct verifications.
Ill
Let us, therefore, see the geometer at work and seek to catch
his 'process.
The task is not without diflSculty; it does not suflSce to open
a work at random and analyze any demonstration in it.
We must first exclude geometry, where the question is com-
plicated by arduous problems relative to the role of the postu-
lates, to the nature and the origin of the notion of space. For
analogous reasons we can not turn to the infinitesimal analysis.
34 SCIENCE AND HYPOTHESIS
We must seek mathematical thought where it has remained pure^
that is, in arithmetic.
A choice still is necessary; in the higher parts of the theory
of numbers, the primitive mathematical notions have already un-
dergone an elaboration so profound that it becomes difficult to
analyze them.
It is, therefore, at the beginning of arithmetic that we must
expect to find the explanation we seek, but it happens that pre-
cisely in the demonstration of the most elementary theorems the
authors of the classic treatises have shown the least precision and
rigor. We must not impute this to them as a crime; they have
yielded to a necessity ; beginners are not prepared for real mathe-
matical rigor ; they would see in it only useless and irksome sub-
tleties; it would be a waste of time to try prematurely to make
them more exacting; they must pass over rapidly, but without
skipping stations, the road traversed slowly by the founders of
the science.
Why is so long a preparation necessary to become habituated
to this perfect rigor, which, it would seem, should naturally im-
press itself upon all good minds? This is a logical and psy-
chological problem well worthy of study.
But we shall not take it up; it is foreign to our purpose; all
I wish to insist on is that, not to fail of our purpose, we must
recast the demonstrations of the most elementary theorems and
give them, not the crude form in which they are left, so as not to
harass beginners, but the form that will satisfy a skilled
geometer.
Definition op Addition. — I suppose already defined the
operation a; + 1, which consists in adding the number 1 to a
given number x.
This definition, whatever it be, does not enter into our sub-
sequent reasoning.
We now have to define the operation « -f a, which consists in
adding the number a to a given number x.
Supposing we have defined the operation
a?+(a — 1),
the operation a; + a will be defined by the equality
(1) x + a=lx-\' (a — 1)]+1.
MATHEMATICAL EEASONINQ 86
We shall know then what x-\-a \a when we know what
«-|- (<* — 1) is, and as I have supposed that to start with we
knew what a?-|-l ^ w® <^*^ define successively and *by recur-
rence ' the operations a? + 8, a; + 3, etc.
This definition deserves a moment's attention; it is of a par-
ticular nature which already distinguishes it from the purely
logical definition; the equality (1) contains an infinity of dis-
tinct definitions, each having a meaning only when one knows the
preceding.
Pbopebtibs op ADDrnoN. — Assodaiivity. — I say that
a+(& + c) = (a + &)+c.
In fact the theorem is true for c = l; it is then written
o+(& + l) = (o+b)+l,
which, apart from the difference of notation, is nothing but the
equality (1), by which I have just defined addition.
Supposing the theorem true for c=y, I say it will be true for
C=3y4-1.
In fact, supposing
(a + &)+7 = a+(& + 7),
it follows that
[(a + b)4-7]+l = [a+(& + 7)]+l
or by definition (1)
(a+ &) + (7 + 1) =a + (& + 7 4- 1) =a + [6 4- (7 + 1)],
which shows, by a series of purely analytic deductions, that the
theorem is true for y + 1.
Being true for c = 1, we thus see successively that so it is for
c=2, for c = 3, etc.
Commutaiivity. — 1° I say that
a + 1 = 1 + a.
The theorem is evidently true for a=il; we can verify by
purely analytic reasoning that if it is true for a=y it will be
true for a =y + 1 ; for then
(7 + 1)4-1= (1 + 7) +1 = 1 + (7 + 1);
now it is true for a = l, therefore it will be true for a = 2, for
a =3, etc., which is expressed by saying that the enunciated
proposition is demonstrated by recurrence.
36 SCIENCE AND HYPOTHESIS
2** I say that
The theorem has just been demonstrated for & =: 1 ; it can be
verified analytically that if it is true for b=fi,it will be true for
The proposition is therefore established by recurrence.
Definition op Multiplication. — ^We shall define multiplica-
tion by the equalities.
(1) axi = a.
(2) aXh = [aX (6 — l)] + o.
Like equality (1), equality (2) contains an infinity of defini-
tions ; having defined a X !> it enables us to define successively :
a X 2, a X 3, etc.
Properties op Multiplication. — Distributivity. — ^I say that
(a + 6) Xc=(oXc) + (bxc).
We verify analytically that the equality is true for c = l ; then
that if the theorem is true for c = y, it will be true for c =y + 1.
The proposition is, therefore, demonstrated by recurrence.
Commutativity, — 1° I say that
a X 1 = 1 X a.
The theorem is evident for a=l.
We verify analytically that if it is true for o = o, it will be
true for o =s o + 1.
2M say that
a X ft = & X o.
The theorem has just been proven for 6 = 1. We could verify
analytically that if it is true for b=py it will be true for
b = p + l.
IV
Here I stop this monotonous series of reasonings. But this
very monotony has the better brought out the procedure which is
uniform and is met again at each step.
This procedure is the demonstration by recurrence. We first
establish a theorem for n = 1 ; then we show that if it is true of
w — 1, it is true of n, and thence conclude that it is true for all
the whole numbers.
MATHEMATICAL REASONING 37
We have just seen how it may be used to demonstrate the rules
of addition and multiplication, that is to say, the rules of the
algebraic calculus ; this calculus is an instrument of transforma-
tion, which lends itself to many more differing combinations than
joes the simple syllogism; but it is still an instrument purely
analytic, and incapable of teaching us anything new. If mathe-
matics had no other instrument, it would therefore be forth-
with arrested in its development; but it has recourse anew to
the same procedure, that is, to reasoning by recurrence, and it is
able to continue its forward march.
If we look closely, at every step we meet again this mode of
reasoning, either in the simple form we have just given it, or
under a form more or less modified.
Here then we have the mathematical reasoning par excellence,
and we must examine it more closely.
The essential characteristic of reasoning by recurrence is that
it contains, condensed, so to speak, in a single formula, an
infinity of syllogisms.
That this may the better be seen, I will state one after another
these syllogisms which are, if you will allow me the expression,
arranged in 'cascade.'
These are of course hypothetical syllogisms.
The theorem is true of the number 1.
Now, if it is true of 1, it is true of 2.
Therefore it is true of 2.
Now, if it is true of 2, it is true of 3.
Therefore it is true of 3, and so on.
We see that the conclusion of each syllogism serves as minor to
the following.
Furthermore the majors of all our syllogisms can be reduced
to a single formula.
If the theorem is true of n — 1, so it is of n.
We see, then, that in reasoning by recurrence we confine our-
selves to stating the minor of the first syllogism, and the general
formula which contains as particular cases all the majors.
This never-ending series of syllogisms is thus reduced to a
phrase of a few lines.
38 SCIENCE AND HTP0THE8I8
It is now easy to comprehend why every particular conse-
quence of a theorem can, as I have explained above, be verified
by purely analytic procedures.
If instead of showing that our theorem is true of all num-
bers, we only wish to show it true of the number 6, for example,
it will sufSce for us to establish the first 5 syllogisms of our cas-
cade ; 9 would be necessary if we wished to prove the theorem for
the number 10; more would be needed for a larger number; but,
however great this number might be, we should always end
by reaching it, and the analytic verification would be possible.
And yet, however far we thus might go, we could never rise
to the general theorem, applicable to all numbers, which alone
can be the object of science. To reach this, an infinity of syl-
logisms would be necessary ; it would be necessary to overleap an
abyss that the patience of the analyst, restricted to the resources
of formal logic alone, never could fill up.
I asked at the outset why one could not conceive of a mind
sufSciently powerful to perceive at a glance the whole body of
mathematical truths.
The answer is now easy; a chess-player is able to combine
four moves, five moves, in advance, but, however extraordinary
he may be, he will never prepare more than a finite number of
them; if he applies his faculties to arithmetic, he will not be
able to perceive its general truths by a single direct intuition ; to
arrive at the smallest theorem he can not dispense with the aid
of reasoning by recurrence, for this is an instrument which
enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap
at a bound as many stages as we wish, it spares us verifications,
long, irksome and monotonous, which would quickly become im-
practicable. But it becomes indispensable as soon as we aim at
the general theorem, to which analytic verification would bring
us continually nearer without ever enabling us to reach it.
In this domain of arithmetic, we may think ourselves very far
from the infinitesimal analysis, and yet, as we have just seen,
the idea of the mathematical infinite already plays a preponder-
ant role, and without it there would be no science, because there
would be nothing general.
MATHEMATICAL BEA80NIN0 39
VI
The judgment on which reasoning by recurrence rests can be
put under other forms; we may say, for example, that in an
infinite collection of different whole numbers there is always one
which is less than all the others.
We can easily pass from one enunciation to the other and thus
get the illusion of having demonstrated the legitimacy of reason-
ing by recurrence. But we shall always be arrested, we shall
always arrive at an undemonstrable axiom which will be in
reality only the proposition to be proved translated into another
language*
We can not therefore escape the conclusion that the rule of
reasoning by recurrence is irreducible to the principle of con-
tradiction.
Neither can this rule come to us from experience; experience
could teach us that the rule is true for the first ten or hundred
numbers; for example, it can not attain to the indefinite series
of numbers, but only to a portion of this series, more or less long
but always limited.
Now if it were only a question of that, the principle of con-
tradiction would sufiSce ; it would always allow of our developing
as many i^llogisms as we wished ; it is only when it is a question
of including an infinity of them in a single formula, it is only
before the infinite that this principle fails, and there too, experi-
ence becomes powerless. This rule, inaccessible to analytic
demonstration and to experience, is the veritable type of the
S3mthetic a priori judgment. On the other hand, we can not
think of seeing in it a convention, as in some of the postulates of
geometry.
Why then does this judgment force itself upon us with an
irresistible evidence? It is because it is only the affirmation of
the power of the mind which knows itself capable of conceiving
the indefinite repetition of the same act when once this act is
possible. The mind has a direct intuition of this power, and
experience can only give occasion for using it and thereby
becoming conscious of it.
But, one will say, if raw experience can not legitimatize
reasoning by recurrence, is it so of experiment aided by indue-
40 SCIENCE AND HYPOTHESIS
tion f We see successively that a theorem is true of the number
1, of the number 2, of the number 3 and so on ; the law is evident,
we say, and it has the same warranty as every physical law based
on observations, whose number is very great but limited.
Here is, it must be admitted, a striking analogy with the usual
procedures of induction. But there is an essential difference.
Induction applied to the physical sciences is always uncertain,
because it rests on the belief in a general order of the universe,
an order outside of us. Mathematical induction, that is, demon-
stration by recurrence, on the contrary, imposes itself necessarily
because it is only the affirmation of a property of the mind itself.
VII
Mathematicians, as I have said before, always endeavor to
generalize the propositions they have obtained, and, to seek no
other example, we have just proved the equality :
a + l = l + a
and afterwards used it to establish the equality
which is manifestly more general.
Mathematics can, therefore, like the other sciences, proceed
from the particular to the general.
This is a fact which would have appeared incomprehensible
to us at the outset of this study, but which is no longer mys^
terious to us, since we have ascertained the analogies between
demonstration by recurrence and ordinary induction.
Without doubt recurrent reasoning in mathematics and in-
ductive reasoning in physics rest on different foundations, but
their march is parallel, they advance in the same sense, that is
to say, from the particular to the general.
Let us examine the case a little more closely.
To demonstrate the equality
it suffices to twice apply the rule
(1) a+l = l + a
and write
(2) a-h 2 = a + 1 -h 1 = 1 -ha + 1 = 1 + 1+0 = 2 + a.
MATHEMATICAL BEASONING 41
The equality (2) thus deduced in purely analytic way from
the equality (1) is, however, not simply a particular case of it;
it is something quite different.
We can not therefore even say that in the really analytic
and deductive part of mathematical reasoning we proceed from
the general to the particular in the ordinary sense of the word.
The two members of the equality (2) are simply combinations
more complicated than the two members of the equality (1), and
analysis only serves to separate the elements which enter into
these combinations and to study their relations.
Mathematicians proceed therefore *by construction,' they 'con-
struct' combinations more and more complicated. Coming back
then by the analysis of these combinations, of these aggregates,
80 to speak, to their primitive elements, they perceive the rela-
tions of these elements and from them deduce the relations of
the aggregates themselves.
This is a purely analytical proceeding, but it is not, however,
a proceeding from the general to the particular, because evi-
dently the aggregates can not be regarded as more particular
than their elements.
Oreat importance, and justly, has been attached to this pro-
cedure of 'construction,' and some have tried to see in it the
necessary and sufficient condition for the progress of the exact
sciences.
Necessary, without doubt ; but sufficient, no.
For a construction to be useful and not a vain toil for the
mind, that it may serve as stepping-stone to one wishing to
mount, it must first of all possess a sort of unity enabling us to
see in it something besides the juxtaposition of its elements.
Or, more exactly, there must be some advantage in considering
the construction rather than its elements themselves.
What can this advantage be t
Why reason on a polygon, for instance, which is always de-
composable into triangles, and not on the elementary triangles?
It is because there are properties appertaining to polygons
of any number of sides and that may be immediately applied to
any particular polygon.
Usually, on the contrary, it is only at the cost of the most
f
42 SCIENCE AND HYPOTHESIS
prolonged exertions that they could be fonnd by studying
directly the relations of the elementary triangles. The knowl-
edge of the general theorem spares us these efforts.
A construction, therefore, becomes interesting only when it
can be ranged beside other analogous constructions, forming spe-
cies of the same genus.
If the quadrilateral is something besides the juxtaposition of
two triangles, this is because it belongs to the genus polygon.
Moreover, one must be able to demonstrate the properties of
the genus without being forced to establish them successively for
each of the species.
To attain that, we must necessarily mount from the particular
to the general, ascending one or more steps.
The analytic procedure 'by construction' does not oblige us
to descend, but it leaves us at the same level.
We can ascend only by mathematical induction, which alone
can teach us something new. Without the aid of this induction,
different in certain respects from physical induction, but quite
as fertile, construction would be powerless to create science.
Observe finally that this induction is possible only if the same
operation can be repeated indefinitely. That is why the theory
of chess can never become a science, for the different moves of
the same game do not resemble one another.
CHAPTBE II
Mathematical Maqnttude and Expebienge
To learn what mathematicians understand by a continunmy
one should not inquire of geometry. The geometer always seeks
to represent to himself more or less the figures he studies, but
his representations are for him only instruments; in making
geometry he uses space just as he does chalk; so too much weight
should not be attached to non-essentials, often of no more im-
portance than the whiteness of the chalk.
The pure analyst has not this rock to fear. He has disen-
gaged the science of mathematics from all foreign elements, and
can answer our question: 'What exactly is this continuum about
which mathematicians reason T Many analysts who reflect on
their art have answered already; Monsieur Tannery, for example,
in his Introduction d la thSorie des fonctions d^une variable.
Let us start from the scale of whole numbers; between two
consecutive steps, intercalate one or more intermediary steps,
then between these new steps still others, and so on indefinitely.
Thus we shall have an unlimited number of terms; these will
be the numbers called fractional, rational or commensurable.
But this is not yet enough ; between these terms, which, however,
are already infinite in number, it is still necessary to intercalate
others called irrational or incommensurable. A remark before
going further. The continuum so conceived is only a collection
of individuals ranged in a certain order, infinite in number, it is
true, but exterior to one another. This is not the ordinary con-
ception, wherein is supposed between the elements of the con-
tinuum a sort of intimate bond which makes of them a whole,
where the point does not exist before the line, but the line before
the point Of the celebrated formula, *the continuum is unity
in multiplicity,' only the multiplicity remains, the unity has
disappeared. The analysts are none the less right in defining
their continuum as they do, for they always reason on just this
as soon as they pique themselves on their rigor. But this is
43
44 SCIENCE AND HYPOTHESIS
enough to apprise us that the veritable mathematical continuum
is a very different thing from that of the physicists and that of
the metaphysicians.
It may also be said perhaps that the mathematicians who are
content with this definition are dupes of words, that it is neces-
sary to say precisely what each of these intermediary steps is, to
explain how they are to be intercalated and to demonstrate that
it is possible to do it. But that would be wrong ; the only prop-
erty of these steps which is used in their reasonings^ is that of
being before or after such and such steps; therefore also this
alone should occur in the definition.
So how the intermediary terms should be intercalated need
not concern us ; on the other hand, no one will doubt the possi-
bility of this operation, unless from forgetting that possible, in
the language of geometers, simply means free from contradiction.
Our definition, however, is not yet complete, and I return to
it after this over-long digression.
Definition of Incommensurables. — The mathematicians of
the Berlin school, Kronecker in particular, have devoted them-
selves to constructing this continuous scale of fractional and irra-
tional numbers without using any material other than the whole
number. The mathematical continuum would be, in this view,
a pure creation of the mind, where experience would have no
part.
The notion of the rational number seeming to them to present
no diflSculty, they have chiefly striven to define the incommen-
surable number. But before producing here their definition, I
must make a remark to forestall the astonishment it is sure to
arouse in readers unfamiliar with the customs of geometers.
Mathematicians study not objects, but relations between ob-
jects; the replacement of these objects by others is therefore
indifferent to them, provided the relations do not change. The
matter is for them unimportant, the form alone interests them.
Without recalling this, it would scarcely be comprehensible
that Dedekind should designate by the name incommensurable
number a mere symbol, that is to say, something very different
iWith those contained in the special conventions which serve to define
addition and of which we shaU speak later.
MATHEMATICAL MAGNITUDE AND EXPERIENCE 45
from the ordinary idea of a quantity, which should be measurable
and almost tangible.
Let us see now what Dedekind's definition is:
The commensurable numbers can in an infinity of ways be
partitioned into two classes, such that any number of the first
dass is greater than any number of the second class.
It may happen that among the numbers of the first class
there is one smaller than all the others ; if, for example, we range
in the first class all numbers greater than 2, and 2 itself, and in
the second class all numbers less than 2, it is clear that 2 will be
the least of all numbers of the first class. The number 2 may be
chosen as symbol of this partition.
It may happen, on the contrary, that among the numbers of
the second class is one greater than all the others; this is the
ease, for example, if the first class comprehends all numbers
greater than 2, and the second all numbers less than 2, and 2
itself. Here again the number 2 may be chosen as symbol of this
partition.
But it may equally well happen that neither is there in the
first class a number less than aU the others, nor in the second
class a number greater than all the others. Suppose, for ex-
ample, we put in the first class all commensurable numbers whose
squares are greater than 2 and in the second all whose squares
are less than 2. There is none whose square is precisely 2. Evi-
dently there is not in the first class a number less than all the
others, for, however near the square of a number may be to 2,
we can always find a commensurable number whose square is
still closer to 2.
In Dedekind's view, the incommensurable number
V2 or (2)*
is nothing but the symbol of this particular mode of partition
of commensurable numbers; and to each mode of partition cor-
responds thus a number, commensurable or not, which serves as
its symbol.
But to be content with this would be to forget too far the
origin of these symbols ; it remains to explain how we have been
led to attribute to them a sort of concrete existence, and, besides,
46 SCIENCE AND HYPOTHESIS
does not the difiScolty begin even for the fractional numbers
themselves f Should we have the notion of these numbers if we
had not previously known a matter that we conceive as infinitely
divisible, that is to say, a continuum?
The Physical Continuum. — ^We ask ourselves then if the
notion of the mathematical continuum is not simply drawn from
experience. If it were, the raw data of experience, which are
our sensations, would be susceptible of measurement. We might
be tempted to believe they really are so, since in these latter days
the attempt has been made to measure them and a law has even
been formulated, known as Fechner's law, according to which
sensation is proportional to the logarithm of the stimulus.
But if we examine more closely the experiments by which
it has been sought to establish this law, we shall be led to a
diametrically opposite conclusion. It has been observed, for ex-
ample, that a weight A of 10 grams and a weight B of 11 grams
produce identical sensations, that the weight B is just as indis-
tinguishable from a weight C of 12 grams, but that the weight A
is easily distinguished from the weight C. Thus the raw results
of experience may be expressed by the following relations :
A = B, B=zC, A<C,
which may be regarded as the formula of the physical continuum.
But here is an intolerable discord with the principle of con-
tradiction, and the need of stopping this has compelled us to
invent the mathematical continuum.
We are, therefore, forced to conclude that this notion has
been created entirely by the mind, but that experience has given
the occasion.
We can not believe that two quantities equal to a third are
not equal to one another, and so we are led to suppose that A is
different from B and B from C, but that the imperfection of our
senses has not permitted of our distinguishing them.
Creation op the Mathematical Continuum. — First Stage.
So far it would sufSce, in accounting for the facts, to intercalate
between A and B a few terms, which would remain discrete.
What happens now if we have recourse to some instrument to
MATHEMATICAL MAGNITUDE AND EXPERIENCE 47
sapplement the feebleness of our senses, if, for example, we
make use of a microscope f Terms such as A and B, before indis-
tingnishable, appear now distinct ; but between A and B, now be-
come distinct, will be intercalated a new term, D, that we can
distingmsh neither from A nor from B. Despite the employ-^
ment of the most highly perfected methods, the raw results of our
experience will always present the characteristics of the physical
continuum with the contradiction which is inherent in it.
We shall escape it only by incessantly intercalating new terms
between the terms already distinguished, and this operation must
be continued indefinitely. We might conceive the stopping of
this operation if we could imagine some instrument sufSciently
powerful to decompose the physical continuum into discrete ele-
ments, as the telescope resolves the milky way into stars. But
this we can not imagine ; in fact, it is with the eye we observe the
image magnified by the microscope, and consequently this image
must always retain the characteristics of visual sensation and
consequently those of the physical continuum.
Nothing distinguishes a length observed directly from the
half of this length doubled by the microscope. The whole is
homogeneous with the part; this is a new contradiction, or
rather it would be if the number of terms were supposed finite ;
in fact, it is clear that the part containing fewer terms than the
whole could not be similar to the whole.
The contradiction ceases when the number of terms is regarded
as infinite ; nothing hinders, for example, considering the aggre-
gate of whole numbers as similar to the aggregate of even num-
bers, which, however, is only a part of it ; and, in fact, to each
whole number corresponds an even number, its double.
But it is not only to escape this contradiction contained in the
empirical data that the mind is led to create the concept of a
continuum, formed of an indefinite number of terms.
All happens as in the sequence of whole numbers. We have
the faculty of conceiving that a unit can be added to a collection
of units ; thanks to experience, we have occasion to exercise this
faculty and we become conscious of it; but from this moment
we feel that our power has no limit and that we can count in-
definitely, though we have never had to count more than a finite
number of objects.
48 SCIENCE AND HTPOTHESIS
Just so, as soon as we have been led to intercalate means
between two consecutive terms of a series, we feel that this opera-
tion can be continued beyond all limit, and that there is, so to
speak, no intrinsic reason for stopping.
As an abbreviation, let me call a mathematical continuum
of the first order every aggregate of terms formed according to
the same law as the scale of commensurable numbers. If we
afterwards intercalate new steps according to the law of for-
mation of incommensurable numbers, we shall obtain what we
will call a continuum of the second order.
Second Stage. — ^We have made hitherto only the first stride;
we have explained the origin of continua of the first order ; but it
is necessary to see why even they are not sufficient and why the
incommensurable numbers had to be invented.
If we try to imagine a line, it must have the characteristics
of the physical continuum, that is to say, we shall not be able
to represent it except with a certain breadth. Two lines then
will appear to us under the form of two narrow bands, and, if
we are content with this rough image, it is evident that if the
two lines cross they will have a common part.
But the pure geometer makes a further effort ; without entirely
renouncing the aid of the senses, he tries to reach the concept of
the line without breadth, of the point without extension. This
he can only attain to by regarding the line as the limit toward
which tends an ever narrowing band, and the point as the limit
toward which tends an ever lessening area. And then, our two
bands, however narrow they may be, will always have a common
area, the smaller as they are the narrower, and whose limit will
be what the pure geometer calls a point.
This is why it is said two lines which cross have a point in
common, and this truth seems intuitive.
But it would imply contradiction if lines were conceived as
continua of the first order, that is to say, if on the lines traced
by the geometer should be found only points having for coordi-
nates rational numbers. The contradiction would be manifest
as soon as one affirmed, for example, the existence of straights
and circles.
It is clear, in fact, that if the points whose coordinates are
MATHEMATICAL MAGNITUDE AND EXPERIENCE 49
commensurable were alone regarded as real, the circle inscribed
in a square and the diagonal of this square would not intersect,
since the coordinates of the point of intersection are incom-
mensurable.
That would not yet be sufiScient, because we should get in this
way only certain incommensurable numbers and not all those
numbers.
But conceive of a straight line divided into two rays. Each
of these rays will appear to our imagination as a band of a cer-
tain breadth; these bands moreover will encroach one on the
other, since there must be no interval between them. The com-
mon part will appear to us as a point which will always remain
when we try to imagine our bands narrower and narrower, so
that we admit as an intuitive truth that if a straight is cut into
two raya their common frontier is a point ; we recognize here the
conception of Dedekind, in which an incommensurable number
was regarded as the common frontier of two classes of rational
numbers.
Such is the origin of the continuum of the second order, which
is the mathematical continuum properly so called.
Resume. — ^In recapitulation, the mind has the faculty of cre-
ating symbols, and it is thus that it has constructed the mathe-
matical continuum, which is only a particular system of symbols.
Its power is limited only by the necessity of avoiding all contra-
diction ; but the mind only makes use of this faculty if experience
furnishes it a stimulus thereto.
In the case considered, this stimulus was the notion of the
physical continuum, drawn from the rough data of the senses.
But this notion leads to a series of contradictions from which it
is necessary successively to free ourselves. So we are forced to
imagine a more and more complicated system of symbols. That
at which we stop is not only exempt from internal contradiction
(it was so already at all the stages we have traversed), but
neither is it in contradiction with various propositions called in-
tuitive, which are derived from empirical notions more or less
elaborated.
Measubable Maonttude. — The magnitudes we have studied
hitherto are not measurable; we can indeed say whether a given
60 SCIENCE AND HTP0THESI8
one of these magnitudes is greater than another, but not whether
it is twice or thrice as great.
So far, I have only considered the order in which our temu
are ranged. But for most applications that does not suffice. We
must learn to compare the interval which separates any two
terms. Only on this condition does the continuum become a
measurable magnitude and the operations of arithmetic ap-
plicable.
This can only be done by the aid of a new and special con-
vention. We will agree that in such and such a case the interval
comprised between the terms A and B is equal to the interval
which separates C and D. For example, at the beginning of oiu
work we have set out from the scale of the whole numbers and we
have supposed intercalated between two consecutive steps n
intermediary steps ; well, these new steps will be by conventios
regarded as equidistant.
This is a way of defining the addition of two magnitudes, be-
cause if the interval AB is by definition equal to the interval CD^
the interval AD will be by definition the sum of the intervals
AB and AC.
This definition is arbitrary in a very large measure. It is not
completely so, however. It is subjected to certain conditions
and, for example, to the rules of commutativity and associativity
of addition. But provided the definition chosen satisfies these
rules, the choice is indifferent, and it is useless to particularize it.
Various Remarks. — ^We can now discuss several important
questions :
1** Is the creative power of the mind exhausted by the creation
of the mathematical continuiunf
No : the works of Du Bois-Beymond demonstrate it in a striking
way.
We know that mathematicians distinguish between infinitesi-
mals of different orders and that those of the second order are
infinitesimal, not only in an absolute way, but also in relatioi
to those of the first order. It is not difficult to imagine infinites-
imals of fractional or even of irrational order, and thus we find
again that scale of the mathematical continuum which has beei
dealt with in the preceding pages.
MATHEMATICAL MAGNITUDE AND EXPERIENCE 51
Farther, there are infinitesimals which are infinitely small in
relation to those of the first order, and, on the contrary, infinitely
great in relation to those of order 1 + c, and that however small
c may be. Here, then, are new terms intercalated in our series,
and if I may be permitted to revert to the phraseology lately em-
ployed which is very convenient though not consecrated by usage,
I shall say that thus has been created a sort of continuum of the
third order.
It would be easy to go further, but that would be idle; one
would only be imagining symbols without possible application,
and no one will think of doing that. The continuum of the third
order, to which the consideration of the different orders of infini-
tesimals leads, is itself not useful enough to have won citizenship,
and geometers regard it only as a mere curiosity. The mind uses
its creative faculty only when experience requires it.
2^ Once in possession of the concept of the mathematical con-
tinuum, is one safe from contradictions analogous to those which
gave birth to it?
No, and I will give an example.
One must be very wise not to regard it as evident that every
curve has a tangent ; and in fact if we picture this curve and a
straight as two narrow bands we can always so dispose them that
they have a part in common without crossing. If we imagine
then the breadth of these two bands to diminish indefinitely, this
common part will always subsist and, at the limit, so to speak, the
two lines will have a point in common without crossing, that is to
say, they will be tangent.
The geometer who reasons in this way, consciously or not, is
only doing what we have done above to prove two lines which
cut have a point in common, and his intuition might seem just as
legitimate.
It would deceive him however. We can demonstrate that
there are curves which have no tangent, if such a curve is de-
fined as an analytic continuum of the second order.
Without doubt some artifice analogous to those we have dis-
enssed above would have suflBced to remove the contradiction;
but, as this is met with only in very exceptional cases, it has
received no further attention.
52 SCIENCE AND HYPOTHESIS
Instead of seeking to reconcile intuition with analysis, we have
been content to sacrifice one of the two, and as analysis must
remain impeccable, we have decided against intuition.
The Physical Continuum op Several Dimensions. — ^We
have discussed above the physical continuum as derived from the
immediate data of our senses, or, if you wish, from the rough re-
sults of Fechner's experiments; I have shown that these results
are summed up in the contradictory formulas
A=zB, B = C, A<C.
Let us now see how this notion has been generalized and how
from it has come the concept of many-dimensional continua.
Consider any two aggregates of sensations. Either we can
discriminate them one from another, or we can not, just as in
Fechner's experiments a weight of 10 grams can be distinguished
from a weight of 12 grams, but not from a weight of 11 grams.
This is all that is required to construct the continuum of several
dimensions.
Let us call one of these aggregates of sensations an element.
That will be something analogous to the point of the mathe-
maticians; it will not be altogether the same thing however.
We can not say our element is without extension, since we can
not distinguish it from neighboring elements and it is thus
surrounded by a sort of haze. If the astronomical comparison
may be allowed, our * elements' would be like nebulae, whereas
the mathematical points would be like stars.
^p That being granted, a system of elements will form a con-
Hinuum if we can pass from any one of them to any other, by a
V,*- if series of consecutive elements such that each is indistinguish-
j able from the preceding. This linear series is to the line of the
^i" ■ mathematician what an isolated element was to the point.
/ Before going farther, I must explain what is meant by a
I cut. Consider a continuum C and remove from it certain of its
1 elements which for an instant we shall regard as no longer be-
\ longing to this continuum. The aggregate of the elements so
removed will be called a cut. It may happen that, thanks to this
cut, C may be subdivided into several distinct continua, the ag-
t gregate of the remaining elements ceasing to form a unique con-
/ tinuum.
V
V
MATHEMATICAL MAGNITUDE AND EXPERIENCE 63
There will then be on C two elements, A and B, that must be \
regarded as belonging to two distinct continua, and this will be J v ^
recognized because it will be impossible to find a linear series j ^
of consecutive elements of C, each of these elements indistin- \^ i^ '^
guishable from the preceding, the first being A and the last B, I
without one of the elements of this series being indistinguishable \ . /^
from one of the elements of the cut. j ^ a-'
On the contrary, it may happen that the cut made is insuffi-j
cient to subdivide the continuum C. To classify the physical!
continua, we will examine precisely what are the cuts which must \
be made to subdivide them.
If a physical continuum C can be subdivided by a cut reduc-
ing to a finite number of elements all distinguishable from one
another (and consequently forming neither a continuum, nor
several continua), we shall say C is a one-dimensional continuum.
If, on the contrary, C can be subdivided only by cuts which
are themselves continua, we shall say C has several dimen-
sions. If cuts which are continua of one dimension sufiSce, we
shall say C has two dimensions ; if cuts of two dimensions sufSce,
we shall say C has three dimensions, and so on.
Thus is defined the notion of the physical continuum of several
dimensions, thanks to this very simple fact that two aggregates
of sensations are distinguishable or indistinguishable.
The Mathematical Continuum op Several Dimensions. —
Thence the notion of the mathematical continuum of n dimen-
sions has sprung quite naturally by a process very like that we
discussed at the beginning of this chapter. A point of such a
continuum, you know, appears to us as defined by a system of
n distinct magnitudes called its coordinates.
These magnitudes need not always be measurable; there is,
for instance, a branch of geometry independent of the measure-
ment of these magnitudes, in which it is only a question of know-
ing, for example, whether on a curve ABC, the point B is be-
tween the points A and C, and not of knowing whether the arc
AB is equal to the arc BC or twice as great. This is what is
called Analysis Situs.
This is a whole body of doctrine which has attracted the
64 SCIENCE AND HYPOTHESIS
attention of the greatest geometers and where we see flow one
fram another a series of remarkable theorems. What distin-
guishes these theorems from those of ordinary geometry is that
they are purely qualitative and that they would remain true if
the figures were copied by a draughtsman so awkward as to
grossly distort the proportions and replace straights by strokes
more or less curved.
Through the wish to introduce measure next into the contin-
uum just defined this continuum becomes space, and geometry is
born. But the discussion of this is reserved for Part Second.
PART II
SPACE
CHAPTER III
The Non-Euclidean Geometries
Evert conclusion supposes premises ; these premises themselves
either are self-evident and need no demonstration, or can be
established only by relying upon other propositions, and since
we can not go back thus to infinity, every deductive science, and
in particular geometry, must rest on a certain number of unde-
monstrable axioms. All treatises on geometry begin, therefore,
by the enunciation of these axioms. But among these there is a
distinction to be made: Some, for example, * Things which are
equal to the same thing are equal to one another, ' are not propo-
sitions of geometry, but propositions of analysis. I regard them
as analytic judgments a priori, and shall not concern myself with
them.
But I must lay stress upon other axioms which are peculiar to
geometry. Most treatises enunciate three of these explicitly :
1® Through two points can pass only one straight;
2** The straight line is the shortest path from one point to
another ;
3** Through a given point there is not more than one parallel
to a given straight.
Although generally a proof of the second of these axioms is
omitted, it would be possible to deduce it from the other two and
from those, much more numerous, which are implicitly admitted
without enunciating them, as I shall explain further on.
It was long sought in vain to demonstrate likewise the third
axiom, known as Euclid^ s Postulate. What vast effort has been
wasted in this chimeric hope is truly unimaginable. Finally, in
55
56 SCIENCE AND HYPOTHESIS
the first quarter of the nineteenth century, and almost at the
same time, a Hungarian and a Russian, Bolyai and Lobachevski,
established irrefutably that this demonstration is impossible ; they
have almost rid us of inventors of geometries *sans postulatum';
since then the Academic des Sciences receives only about one or
two new demonstrations a year.
The question was not exhausted; it soon made a great
stride by the publication of Riemann's celebrated memoir en-
titled: Ueber die Hypothesen welche der Oeometrie zu Orunde
liegen. This paper has inspired most of the recent works of which
I shall speak further on, and among which it is proper to cite
those of Beltrami and of Helmholtz.
The Bolyai-Lobachevski Geometby. — If it were possible to
deduce Euclid's postulate from the other axioms, it is evident
that in denying the postulate and admitting the other axioms, we
should be led to contradictory consequences; it would therefore
be impossible to base on such premises a coherent geometry.
Now this is precisely what Lobachevski did.
He assumes at the start that: Through a given point can he
drawn two parallels to a given straight.
And he retains besides all Euclid's other axioms. From these
hypotheses he deduces a series of theorems among which it is
impossible to find any contradiction, and he constructs a
geometry whose faultless logic is inferior in nothing to that of
the Euclidean geometry.
The theorems are, of course, very different from those to which
we are accustomed, and they can not fail to be at first a little
disconcerting.
Thus the sum of the angles of a triangle is always less than
two right angles, and the diflference between this sum and two
right angles is proportional to the surface of the triangle.
It is impossible to construct a figure similar to a given figure
but of different dimensions.
If we divide a circumference into n equal parts, and draw
tangents at the points of division, these n tangents will form a
polygon if the radius of the circle is small enough; but if this
radius is sufficiently great they will not meet.
It is useless to multiply these examples; Lobachevski 's propo-
THE NON'EUCLIDEAN GEOMETRIES 67
sitions have no relation to those of Euclid, but they are not less
logically bound one to another.
Biemann's Geometby. — Imagine a world uniquely peopled
by beings of no thickness (height) ; and suppose these infinitely
flat' animals are all in the same plane and can not get out. Ad-
mit besides that this world is sufficiently far from others to be
free from their influence. While we are making hypotheses, it
costs us no more to endow these beings with reason and believe
them capable of creating a geometry. In that case, they will cer-
tainly attribute to space only two dimensions.
But suppose now that these imaginary animals, while remain-
ing without thickness, have the form of a spherical, and not of a
plane, figure, and are all on the same sphere without power to get
off. What geometry will they construct? First it is clear they
will attribute to space only two dimensions; what will play for
them the role of the straight line will be the shortest path from
one point to another on the sphere, that is to say, an arc of a great
circle ; in a word, their geometry will be the spherical geometry.
What they will call space will be this sphere on which they
must stay, and on which happen all the phenomena they can
know. Their space will therefore be unbou^ided since on a
sphere one can always go forward without ever being stopped,
and yet it will be finite; one can never find the end of it, but one
can make a tour of it.
Well, Riemann's geometry is spherical geometry extended to
three dimensions. To construct it, the German mathematician
had to throw overboard, not only Euclid's postulate, but also the
first axiom : Only one straight can pass through two points.
On a sphere, through two given points we can draw in general
only one great circle (which, as we have just seen, would play the
role of the straight for our imaginary beings) ; but there is an
exception : if the two given points are diametrically opposite, an
infinity of great circles can be drawn through them.
In the same way, in Riemann's geometry (at least in one of
its forms) , through two points will pass in general only a single
straight; but there are exceptional cases where through two
points an infinity of straights can pass.
68 SCIENCE AND HYPOTHESIS
There is a sort of opposition between Riemann's geometry and
that of Lobaehevski.
Thus the sum of the angles of a triangle is :
Equal to two right angles in Euclid's geometry;
Less than two right angles in that of Lobaehevski ;
Greater than two right angles in that of Biemann.
The number of straights through a given point that can be
drawn coplanar to a given straight, but nowhere meeting it, is
equal :
To one in Euclid's geometry;
To zero in that of Riemann ;
To infinity in that of Lobaehevski.
Add that Biemann 's space is finite, although unbounded, in
the sense given above to these two words.
The Surfaces op Constant Cubvatube. — One objection still
remained possible. The theorems of Lobaehevski and of Bie-
mann present no contradiction ; but however numerous the con-
sequences these two geometers have drawn from their hypotheses,
they must have stopped before exhausting them, since their num-
ber would be infinite ; who can say then that if they had pushed
their deductions farther they would not have eventually reached
some contradiction f
This diflSculty does not exist for Biemann 's geometry, pro-
vided it is limited to two dimensions; in fact, as we have seen,
two-dimensional Riemannian geometry does not differ from spher-
ical geometry, which is only a branch of ordinary geometry, and
consequently is beyond all discussion.
Beltrami, in correlating likewise Lobaehevski 's two-dimen-
sional geometry with a branch of ordinary geometry, has equally
refuted the objection so far as it is concerned.
Here is how he accomplished it. Consider any figure on a
surface. Imagine this figure traced on a flexible and inextensible
canvas applied over this surface in such a way that when the
canvas is displaced and deformed, the various lines of this figure
can change their form without changing their length. In gen-
eral, this flexible and inextensible figure can not be displaced
without leaving the surface ; but there are certain particular sur-
TBE NON-EUCLIDEAN GEOMETRIES 69
faces for which such a movement would be possible ; these are the
surfaces of constant curvature.
If we resume the comparison made above and imagine beings
without thickness living on one of these surfaces, they will regard
as possible the motion of a figure all of whose lines remain con-
stant in length. On the contrary, such a movement would appear
absurd to animals without thickness living on a surface of vari-
able curvature.
These surfaces of constant curvature are of two sorts: Some
are of positive curvature, and can be deformed so as to be applied
over a sphere. The geometry of these surfaces reduces itself
therefore to the spherical geometry, which is that of Riemann.
The others are of negative curvature. Beltrami has shown
that the geometry of these surfaces is none other than that of
Lobachevski. The two-dimensional geometries of Riemann and
Lobachevski are thus correlated to the Euclidean geometry.
Interpretation op Non-Euclidean Geometries. — So van-
ishes the objection so far as two-dimensional geometries are con-
cerned.
It would be easy to extend Beltrami's reasoning to three-
dimensional geometries. The minds that space of four dimen-
sions does not repel will see no diflSculty in it, but they are few.
I prefer therefore to proceed otherwise.
Consider a certain plane, which I shall call the fundamental
plane, and construct a sort of dictionary, by making correspond
each to each a double series of terms written in two columns, just
as correspond in the ordinary dictionaries the words of two lan-
guages whose significance is the same :
Space: Portion of space situated above the fundamental plane.
Plane: Sphere cutting the fundamental plane orthogonally.
Straight: Circle cutting the fundamental plane orthogonally.
Sphere: Sphere.
Circle: Circle.
Angle: Angle.
Distance between two points: Logarithm of the cross ratio of
these two points and the intersections of the fundamental plane
with a circle passing through these two points and cutting it
orthogonally. Etc., Etc.
60 SCIENCE AND HYPOTHESIS
Now take Lobachevski's theorems and translate them with
the aid of this dictionary as we transate a German text with the
aid of a German-English dictionary. We shall thus, obtain the-
orems of the ordinary geometry. For example, that theorem of
Lobachevski : ^the sum of the angles of a triangle is less than two
right angles' is translated thus: ^'If a curvilinear triangle has
for sides circle-arcs which prolonged would cut orthogonally the
fundamental plane, the sum of the angles of this curvilinear tri-
angle will be less than two right angles." Thus, however far the
consequences of Lobachevski's hypotheses are pushed, they will
never lead to «a contradiction. In fact, if two of Lobachevski's
theorems were contradictory, it would be the same with the trans-
lations of these two theorems, made by the aid of our dictionary,
but these translations are theorems of ordinary geometry and no
one doubts that the ordinary geometry is free from contradiction.
Whence comes this certainty and is it justified? That is a ques-
tion I can not treat here because it would require to be enlarged
upon, but which is very interesting and I think not insoluble.
Nothing remains then of the objection above formulated.
This is not all. Lobachevski's geometry, susceptible of a concrete
interpretation, ceases to be a vain logical exercise and is capal)le
of applications ; I have not the time to speak here of these appli-
cations, nor of the aid that Klein and I have gotten from them
for the integration of linear differential equations.
This interpretation moreover is not unique, and several dic-
tionaries analogous to the preceding could be constructed, which
would enable us by a simple 'translation' to transform Loba-
chevski's theorems into theorems of ordinary geometry.
The Implicit Axioms. — Are the axioms explicitly enunciated
in our treatises the sole foundations of geometry? We may be
assured of the contrary by noticing that after they are succes-
sively abandoned there are still left over some propositions com-
mon to the theories of Euclid, Lobachevski and Riemann. These
propositions must rest on premises the geometers admit without
enunciation. It is interesting to try to disentangle them from
the classic demonstrations.
Stuart Mill has claimed that every definition contains an
THE NON'EUCLIDEAN GEOMETRIES 61
axiom, because in defining one affirms implicitly the existence
of the object defined. This is going much too far ; it is rare that
in mathematics a definition is given without its being followed by
the demonstration of the existence of the object defined, and
when this is dispensed with it is generally because the reader
can easily supply it. It must not be forgotten that the word
existence has not the same sense when it refers to a mathematical
entity and when it is a question of a material object. A mathe-
matical entity exists, provided its definition implies no contradic-
tioUy either in itself, or with the propositions already admitted.
But if Stuart Mill's observation can not be applied to all
definitions, it is none the less just for some of them. The plane
is sometimes defined as follows :
The plane is a surface such that the straight which joins any
two of its points is wholly on this surface.
This definition manifestly hides a new axiom; it is true we
might change it, and that would be preferable, but then we
should have to enunciate the axiom explicitly.
Other definitions would suggest reflections not less important.
Such, for example, is that of the equality of two figures ; two
figures are equal when they can be superposed; to superpose
them one must be displaced until it coincides with the other ; but
how shall it be displaced? If we should ask this, no doubt we
should be told that it must be done without altering the shape
and as a rigid solid. The vicious circle would then be evident.
In fact this definition defines nothing; it would have no mean-
ing for a being living in a world where there were only fluids.
If it seems clear to us, that is because we are used to the proper-
ties of natural solids which do not differ much from those of the
ideal solids, all of whose dimensions are invariable.
Yet, imperfect as it may be, this definition implies an axiom.
The possibility of the motion of a rigid figure is not a self-
evident truth, or at least it is so only in the fashion of Euclid's
postulate and not as an analytic judgment a priori would be.
Moreover, in studying the definitions and the demonstrations
of geometry, we see that one is obliged to admit without proof
not only the possibility of this motion, but some of its properties
besides.
62 SCIENCE AND HYPOTHESIS
This is at once seen from the definition of the straight line.
Many defective definitions have been given, but the true one is
that which is implied in all the demonstrations where the straight
line enters:
''It may happen that the motion of a rigid figure is such that
all the points of a line belonging to this figure remain motionless
while all the points situated outside of this line move. Such a
line will be called a straight line." We have designedly, in this
enunciation, separated the definition from the axiom it implies.
Many demonstrations, such as those of the cases of the equality
of triangles, of the possibility of dropping a perpendicular from
a point to a straight, presume propositions which are not enun-
ciated, for they require the admission that it is possible to trans-
port a figure in a certain way in space.
The Fourth Geometry. — Among these implicit axioms, there
is one which seems to me to merit some attention, because when
it is abandoned a fourth geometry can be constructed as coherent
as those of Euclid, Lobachevski and Biemann.
To prove that a perpendicular may always be erected at a
point A to a straight AB, we consider a straight AC movable
around the point A and initially coincident with the fixed
straight AB; and we make it turn about the point A until it
comes into the prolongation of AB.
Thus two propositions are presupposed : First, that such a ro-
tation is possible, and next that it may be continued until the
two straights come into the prolongation one of the other.
If the first point is admitted and the second rejected, we are
led to a series of theorems even stranger than those of Loba-
chevski and Riemann, but equally exempt from contradiction.
I shall cite only one of these theorems and that not the most
singular: A real straight may be perpendicular to itself.
LiE^s Theorem. — The number of axioms implicitly intro-
duced in the classic demonstrations is greater than necessary, and
it would be interesting to reduce it to a minimum. It may first
be asked whether this reduction is possible, whether the number
of necessary axioms and that of imaginable geometries are not
infinite.
THE NON^-EUCLIDEAN GEOMETBIES 63
A theorem of Sophus Lie dominates this whole discussion. It
may be thus enunciated:
Suppose the following premises are admitted:
1^ Space has n dimensions;
2^ The motion of a rigid figure is possible;
3^ It requires p conditions to determine the position of this
figure in space.
The number of geometries compatible with these premises luill
he limited.
I may even add that if n is given, a superior limit can be
assigned to p.
If therefore the possibility of motion is admitted, there can
be invented only a finite (and even a rather small) number of
three-dimensional geometries.
Biemann's Geometries. — ^Tet this result seems contradicted
by Biemann, for this savant constructs an infinity of different
geometries, and that to which his name is ordinarily given is only
a particular case.
All depends, he says, on how the length of a curve is defined.
NW, there is an infinity of ways of defining this length, and each
of them may be the starting point of a new geometry.
That is perfectly true, but most of these definitions are incom-
patible with the motion of a rigid figure, which in the theorem
of Lie is supposed possible. These geometries of Rieraann, in
many ways so interesting, could never therefore be other than
purely analytic and would not lend themselves to demonstrations
analogous to those of Euclid.
On the Nature op Axioms. — Most mathematicians regard
Lobachevski's geometry only as a mere logical curiosity; some of
them, however, have gone farther. Since several geometries are
possible, is it certain ours is the true one ? Experience no doubt
teaches us that the sum of the angles of a triangle is equal to two
right angles ; but this is because the triangles we deal with are
too little; the difference, according to Lobachevski, is propor-
tiozud to the surface of the triangle ; will it not perhaps become
KQgible when we shall operate on larger triangles or when our
nteasurements shall become more precise ? The Euclidean geom-
etry would thus be only a provisional geometry.
64 SCIENCE AND HYPOTHESIS
To discuss this opinion, we should first ask ourselves what
is the nature of the geometric axioms.
Are they synthetic a priori judgments, as Kant said!
They would then impose themselves upon us with such force
that we could not conceive the contrary proposition, nor build
upon it a theoretic edifice. There would be no non-Euclidean
geometry.
To be convinced of it take a veritable synthetic a priori
judgment, the following, for instance, of which we have seen
the preponderant role in the first chapter :
// a theorem is true for the number 1, and if it has been proved
that it is true of n-\-l provided it is true of n, it u)iU be true of
all the positive whole numbers.
Then try to escape from that and, denying this proposition,
try to found a false arithmetic analogous to non-Euclidean
geometry — ^it can not be done ; one would even be tempted at first
blush to regard these judgments as analytic.
Moreover, resuming our fiction of animals without thickness,
we can hardly admit that these beings, if their minds are like
ours, would adopt the Euclidean geometry which would be con-
tradicted by all their experience.
Should we therefore conclude that the axioms of geometry are
experimental verities? But we do not experiment on ideal
straights or circles; it can only be done on material objects. On
what then could be based experiments which should serve as
foundation for geometry? The answer is easy.
We have seen above that we constantly reason as if the geo-
metric figures behaved like solids. What geometry would bor-
row from experience would therefore be the properties of these
bodies. The properties of light and its rectilinear propagation
have also given rise to some of the propositions of geometry,
and in particular those of projective geometry, so that from this
point of view one would be tempted to say that metric geometry
is the study of solids, and projective, that of light.
But a difficulty remains, and it is insurmountable. If geom-
etry were an experimental science, it would not be an exact
science, it would be subject to a continual revision. Nay, it
would from this very day be convicted of error, since we know
that there is no rigorously rigid solid.
THE NON-^EUCLIDEAN GEOMETRIES 66
The axioms of geometry therefore are neither synthetic a
priori judgments nor experimental facts.
They are conventions; our choice amon^ all possible conven-
tions is guided by experimental facts ; but it remains free and is
limited only by the necessity of avoiding all contradiction. Thus
it is that the postulates can remain rigorously true even though
the experimental laws which have determined their adoption are
only approximative.
In other words, the ctxioms of geometry (I do not speak of
those of arithmetic) are merely disguised definitions.
Then what are we to think of that question : Is the Euclidean
geometry truet
It has no meaning.
As well ask whether the metric system is true and the old
measures false ; whether Cartesian coordinates are true and polar
coordinates false. One geometry can not be more true than an-
other; it can only be more convenient.
Now, Euclidean geometry is, and will remain, the most con-
venient :
1^ Because it is the simplest ; and it is so not only in conse-
quence of our mental habits, or of I know not what direct in-
tuition that we may have of Euclidean space ; it is the simplest in
itself, just as a polynomial of the first degree is simpler than one
of the second; the formulas of spherical trigonometry are more
complicated than those of plane trigonometry, and they would
still appear so to an analyst ignorant of their geometric signifi-
cation.
2® Because it accords sufficiently well with the properties of
natural solids, those bodies which our hands and our eyes com-
pare and with which we make our instruments of measure.
6
CHAPTER IV
Space and Geometry
Let us begin by a little paradox.
Beings with minds like ours, and having the same senses as
we, but without previous education, would receive from a suitably
chosen external world impressions such that they would be led
to construct a geometry other than that of Euclid and to localize
the phenomena of that external world in a non-Euclidean space,
or even in a space of four dimensions.
As for us, whose education has been accomplished by our
actual world, if we were suddenly transported into this new
world, we should have no difficulty in referring its phenomena to
our Euclidean space. Conversely, if these beings were trans-
ported into our environment, they would be led to relate our
phenomena to non-Euclidean space.
Nay more; with a little effort we likewise could do it. A
person who should devote his existence to it might perhaps attain
to a realization of the fourth dimension.
Geometric Space and Perceptual Space. — It is often said
the images of external objects are localized in space, even that
they can not be formed except on this condition. It is also said
that this space, which serves thus as a ready prepared frame for
our sensations and our representations, is identical with that of
the geometers, of which it possesses all the properties.
To all the good minds who think thus, the preceding state-
ment must have appeared quite extraordinary. But let us see
whether they are not subject to an illusion that a more profound
analysis would dissipate.
What, first of all, are the properties of space, properly so
called! I mean of that space which is the object of geometry
and which I shall call geometric space.
The following are some of the most essential:
1** It is continuous;
66
SPACE AND GEOMETBT 67
2° It is infinite;
3° It has three dimensions;
4° It is homogeneous, that is to say, all its i>oints are identical
one with another;
5^ It is isotropic, that is to say, all the straights which pass
through the same point are identical one with another.
Compare it now to the frame of our representations and our
sensations, which I may call perceptual space.
Visual Space. — Consider first a purely visual impression, due
to an image formed on the bottom of the retina.
A cursory analysis shows us this image as continuous, but as
possessing only two dimensions; this already distinguishes from
geometric space what we may call pure visual space.
Besides, this image is enclosed in a limited frame.
Finally, there is another difference not less important: this
pure visvM space is not homogeneous. All the points of the
retina, aside from the images which may there be formed, do not
play the same role. The yellow spot can in no way be regarded
as identical with a point on the border of the retina. In fact, not
only does the same object produce there much more vivid im-
pressions, but in every limited frame the point occupying the
center of the frame will never appear as equivalent to a point
near one of the borders.
No doubt a more profound analysis would show us that this
continuity of visual space and its two dimensions are only an
illusion ; it would separate it therefore still more from geometric
space, but we shall not dwell on this remark.
Sight, however, enables us to judge of distances and conse-
quently to perceive a third dimension. But every one knows
that this perception of the third dimension reduces itself to the
sensation of the effort at accommodation it is necessary to make,
and to that of the convergence which must be given to the two
eyes, to perceive an object distinctly.
These are muscular sensations altogether different from the
visual sensations which have given us the notion of the first two
dimensions. The third dimension therefore will not appear to
us as playing the same role as the other two. What may be
called complete visual space is therefore not an isotropic space.
68 SCIENCE AND HYPOTHESIS
It has, it is true, precisely three dimensions, which means that
the elements of our visual sensations (those at least which com-
bine to form the notion of extension) will be completely de-
fined when three of them are known; to use the language of
mathematics, they will be functions of three independent
variables.
But examine the matter a little more closely. The third
dimension is revealed to us in two different ways: by the effort
of accommodation and by the convergence of the eyes.
No doubt these two indications are always concordant, there
is a constant relation between them, or, in mathematical terms,
the two variables which measure these two muscular sensations
do not appear to us as independent ; or again, to avoid an appeal
to mathematical notions already rather refined, we may go back
to the language of the preceding chapter and enunciate the same
fact as follows : If two sensations of convergence, A and B, are
indistinguishable, the two sensations of accommodation, A' and
B'f which respectively accompany them, will be equally indistin-
guishable.
But here we have, so to speak, an experimental fact; a priori
nothing prevents our supposing the contrary, and if the contrary
takes place, if these two muscular sensations vary independently
of one another, we shall have to take account of one more inde-
pendent variable, and 'complete visual space' will appear to us
as a physical continuum of four dimensions.
We have here even, I will add, a fact of external experience.
Nothing prevents our supposing that a being with a mind like
ours, having the same sense organs that we have, may be placed
in a world where light would only reach him after having
traversed reflecting media of complicated form. The two indi-
cations which serve us in judging distances would cease to be
connected by a constant relation. A being who should achieve
in such a world the education of his senses would no doubt
attribute four dimensions to complete visual space.
Tactile Space and Motor Space. — ^'Tactile space' is still
more complicated than visual space and farther removed from
geometric space. It is superfluous to repeat for touch the discus-
sion I have given for sight.
SPACE AND GEOMETBT 69
But apart from the data of sight and touch, there are other
sensations which contribute as much and more than they to the
genesis of the notion of space. These are known to every one;
they accompany all our movements, and are usually called mus-
cular sensations.
The corresponding frame constitutes what may be called motor
space.
Each muscle gives rise to a special sensation capable of aug-
menting or of diminishing, so that the totality of our muscular
sensations will depend upon as many variables as we have
muscles. From this point of view, motor space would have as
many dimensions as we have mtiscles.
I know it will be said that if the muscular sensations con-
tribute to form the notion of space, it is because we have the
sense of the direction of each movement and that it makes an
integrant part of the sensation. If this were so, if a muscular
sensation could not arise except accompanied by this geometric
9ense of direction, geometric space would indeed be a form im-
posed upon our sensibility.
But I perceive nothing at all of this when I analyze my sen-
sations.
What I do see is that the sensations which correspond to move-
ments in the same direction are connected in my mind by a mere
association of ideas. It is to this association that what we call
'the sense of direction' is reducible. This feeling therefore can
not be found in a single sensation.
This association is extremely complex, for the contraction of
the same muscle may correspond, according to the position of the
limbs, to movements of very different direction.
Besides, it is evidently acquired; it is, like all associations of
ideas, the result of a habit; this habit itself results from very
numerous experiences; without any doubt, if the education of our
senses had been accomplished in a different environment, where
we should have been subjected to different impressions, con-
trary habits would have arisen and our muscular sensations
would have been associated according to other laws.
Chabacteristics op Perceptual Space. — Thus perceptual
space, under its triple form, visual, tactile and motor, is essen-
tially different from geometric space.
70 SCIENCE AND HYPOTHESIS
It is neither homogeneous, nor isotropic ; one can not even say
that it has three dimensions.
It is often said that we 'project' into geometric space the
objects of our external perception; that we 'localize' them.
Has this a meaning, and if so whatf
Does it mean that we represent to ourselves external objects in
geometric space f
Our representations are only the reproduction of our sensa-
tions; they can therefore be ranged only in the same frame as
these, that is to say, in perceptual space.
It is as impossible for us to represent to ourselves external
bodies in geometric space, as it is for a painter to paint on a
plane canvas objects with their three dimensions.
Perceptual space is only an image of geometric space, an
image altered in shape by a sort of perspe<Hive, and we can repre-
sent to ourselves objects only by bringing them under the laws of
this perspective.
Therefore we do not represent to ourselves external bodies in
geometric space, but we reason on these bodies as if they were
situated in geometric space.
When it is said then that we 'localize' such and such an object
at such and such a point of space, what does it meant
It simply means that we represent to ourselves the movements
it would be necessary to m>ake to reach that object; and one may
not say that to represent to oneself these movements, it is neces-
sary to project the movements themselves in space and that the
notion of space must, consequently, pre-exist.
When I say that we represent to ourselves these movements,
I mean only that we represent to ourselves the muscular sensa-
tions which accompany them and which have no geometric char-
acter whatever, which consequently do not at all imply the pre-
existence of the notion of space.
Change op State and Change op Position. — ^But, it will
be said, if the idea of geometric space is not imposed upon our
mind, and if, on the other hand, none of our sensations can
furnish it, how could it have come into existence?
This is what we have now to examine, and it will take some
time, but I can summarize in a few words the attempt at explana-
tion that I am about to develop.
SPACE AND GEOMETRY 71
None of our sensations, isolated, could have conducted us to
ike idea of space; we are led to it only in studying the laws,
according to which these sensations succeed each other.
We see first that our impressions are subject to change; but
among the changes we ascertain we are soon led to make a dis-
tinction.
At one time we say that the objects which cause these im-
pressions have changed state, at another time that they have
changed position, that they have only been displaced.
Whether an object changes its state or merely its position,
this is always translated for us in the same manner: by a modifi-
cation in an aggregate of impressions.
How then could we have been led to distinguish between the
twof It is easy to account for. If there has only been a
change of position, we can restore the primitive aggregate of
impressions by making movements which replace us opposite the
mobile object in the same relative situation. We thus correct
the modification that happened and we reestablish the initial
state by an inverse modification.
If it is a question of sight, for example, and if an object
changes its place before our eye, we can * follow it with the
eye' and maintain its image on the same point of the retina by
appropriate movements of the eyeball.
These movements we are conscious of because they are volun-
tary and because they are accompanied by muscular sensations,
but that does not mean that we represent them to ourselves in
geometric space.
So what characterizes change of position, what distinguishes
it from change of state, is that it can always be corrected in this
way.
It may therefore happen that we pass from the totality of
impressions A to the totality B in two different ways :
1** Involuntarily and without experiencing muscular sensa-
tions ; this happens when it is the object which changes place ;
2*" Voluntarily and with muscular sensations; this happens
when the object is motionless, but we move so that the object has
relative motion with reference to us.
If this be so, the passage from the totality A to the totality B
is only a change of position.
72 SCIENCE AND STPOTHESIS
It follows from this that sight and toach could not have
given UB the notion of space without the aid of the 'muscular
sense.'
Not only could this notion not be derived from a single sen'
sation or even from a series of sensations, but what is more, an
immobile being could never have acquired it, since, not being
able to correct by his movements the effects of the changes of
position of exterior objects, he would have had no reason what-
ever to distinguish them from changes of state. Just as little
could he have acquired it if his motions had not been voluntary
or were unaccompanied by any sensations.
Conditions op Compensation. — How is a like compensation
possible, of such sort that two changes, otherwise independent of
each other, reciprocally correct each othert
A mind already familiar with geometry would reason as fol-
lows: Evidently, if there is to be compensation, the various
parts of the external object, on the one hand, and the various
sense organs, on the other hand, must be in the same relative
poffltioD after the double change. And, for that to be the case,
the various parts of the external object must likewise have
retained in reference to each other the same relative pontion,
and the same must be true of the various parts of our body in
regard to each other.
In other words, the external object, in the first change, must
be displaced as is a rigid solid, nnd so must it be with the whole
of our body in the second change which corrects the first.
Under these conditions, oompunsation may take place.
But we who as yet know nothing of geometry, since for ia"fl|
notion of space is not yet formed, we can not reason i
can not foresee a priori whether compensatiou is pOssibl
experience teaches us that it sometimes happens, and it j
this experimental fact that we star: to distingoiili t
state from changes of position.
Solid Bodies and Gbouets
there are some which ^tqv
ceptible of being thus eon
our own body; these are
SPACE AND GEOMETRY
V3
whose form is variable, only exceptionally undergo like displace-
ments (change of position without change of form). When «
body changes ita place and its shape, we can no longer, by appro-
priate movements, bring back our sense-organs into the same
relative situation with regard to this body; consequently we can
DO longer reestablish the primitive totality of impressions.
It is only later, and as a consequence of new experiences, that
we learn how to decompose the bodies of variable form into
smaller elements, such that each ia displaced almost in accord-
ance with the same laws as solid bodies. Thus we distinguish
'deformations" from other changes of state; in these deforma-
tions, each element undergoes a mere change of position, which
can he corrected, but the modification undergone by the aggre-
gate is more profound and is no longer susceptible of correction
by a correlative movement.
8uch a notion is already very complex and must have been
relatively late in appearing ; moreover it could not have arisen if
the observation of solid bodies f d not already taught us to dis-
tiDguish changes of portion.
Therefore, if there were no solid bodies in nature, there wcndd
ht no geometry.
Another remark also deserves a moment's attention. Suppose
a solid body to occupy successively the positions a. and p; in its
first position, it wrill produce on us the totality of impressions A,
lity of impressions B. Let
viag qualities entirely djffer-
ilerent color. Suppose it to
us the totality of im-
the totality of irn-
eommon with
;y B'. The trau-
that from the
:e8 which in
74 SCIENCE AND HYPOTHESIS
It is simply because they can both be corrected by the same
correlative movement of our body.
'Correlative movement' therefore constitutes the sole connec-
tion between two phenomena which otherwise we never should
have dreamt of likening.
On the other hand, our body, thanks to the number of its
articulations and muscles, may make a multitude of different
movements; but all are not capable of 'correcting' a modification
of external objects ; only those will be capable of it in which our
whole body, or at least all those of our sense-organs which come
into play, are displaced as a whole, that is, without their relative
positions varying, or in the fashion of a solid body.
To summarize:
1^ We are led at first to distinguish two categories of phe-
nomena :
Some, involuntary, unaccompanied by muscular sensations, are
attributed by us to external objects ; these are external changes ;
Others, opposite in character and attributed by us to the
movements of our own body, are internal changes ;
2** We notice that certain changes of each of these categories
may be corrected by a correlative change of the other category;
3** We distinguish among external changes those which have
thus a correlative in the other category; these we call displace-
ments; and just so among the internal changes, we distinguish
those which have a correlative in the first category.
Thus are defined, thanks to this reciprocity, a particular class
of phenomena which we call displacements.
The laws of these phenomena constitute the object of geometry.
Law op Homogeneity. — The first of these laws is the law of
homogeneity.
Suppose that, by an external change a, we pass from the total-
ity of impressions A to the totality B, then that this change
a is corrected by a correlative voluntary movement j8, so that we
are brought back to the totality A.
Suppose now that another external change a makes us pass
anew from the totality A to the totality B.
Experience teaches us that this change a is, like a, sus-
ceptible of being corrected by a correlative voluntary movement
SPACE AND GEOMETRY 75
fif and that this moyement p' corresponds to the same mnscnlar
sensations as the movement p which corrected a.
This fact is usually enunciated by saying that space is homo-
geneous and isotropic.
It may also be said that a movement which has once been pro-
duced may be repeated a second and a third time, and so on,
without its properties varying.
In the first chapter, where we discussed the nature of mathe-
matical reasoning, we saw the importance which must be
attributed to the possibility of repeating indefinitely the same
operation.
It is from this repetition that mathematical reasoning gets its
I>ower; it is, therefore, thanks to the law of homogeneity, that it
has a hold on the geometric facts.
For completeness, to the law of homogeneity should be added
a multitude of other analogous laws, into the details of which I
do not wish to enter, but which mathematicians sum up in a word
by saying that displacements form 'a group.'
The Non-Eucudean World. — If geometric space were a
frame imposed on each of our representations, considered indi-
vidually, it would be impossible to represent to ourselves an
image stripped of this frame, and we could change nothing of
our geometry.
But this is not the ease ; geometry is only the resume of the
laws according to which these images succeed each other. Noth-
ing then prevents us from imagining a series of representations,
similar in all points to our ordinary representations, but suc-
ceeding one another according to laws different from those to
which we are accustomed.
We can conceive then that beings who received their educa-
tion in an environment where these laws were thus upset might
have a geometry very different from ours.
Suppose, for example, a world enclosed in a great sphere and
subject to the following laws:
The temperature is not uniform; it is greatest at the center,
and diminishes in proportion to the distance from the center, to
sink to absolute zero when the sphere is reached in which this
world is enclosed.
76 SCIENCE AND HYPOTHESIS
To specify still more precisely the law in accordance with
which this temperature varies: Let B be the radius of the lim-
iting sphere; let r be the distance of the point considered from
the center of this sphere. The absolute temperature shall be
proportional to J2* — r*.
I shall further suppose that, in this world, all bodies have
the same coefScient of dilatation, so that the length of any rule
is proportional to its absolute temperature.
Finally, I shall suppose that a body transported from one
point to another of different temperature is put immediately into
thermal equilibrium with its new environment.
Nothing in these hypotheses is contradictory or unimaginable.
A movable object will then become smaller and smaller in pro-
portion as it approaches the limit-sphere.
Note first that, though this world is limited from the point
of view of our ordinary geometry, it will appear infinite to its
inhabitants.
In fact, when these try to approach the limit-sphere, they cool
off and become smaller and smaller. Therefore the steps they
take are also smaller and smaller, so that they can never reach the
limiting sphere.
If, for us, geometry is only the study of the laws according
to which rigid solids move, for these imaginary beings it will be
the study of the laws of motion of solids distorted hy the differ-
ences of temperature just spoken of.
No doubt, in our world, natural solids likewise undergo varia-
tions of form and volume due to warming or cooling. But we
neglect these variations in laying the foundations of geometry,
because, besides their being very slight, they are irregular and
consequently seem to us accidental.
In our hypothetical world, this would no longer be the case,
and these variations would follow regular and very simple laws.
Moreover, the various solid pieces of which the bodies of its
inhabitants would be composed would undergo the same, varia-
tions of form and volume.
I will make still another hypothesis; I will suppose light
traverses media diversely refractive and such that the index of
refraction is inversely proportional to J2* — r*. It is easy to
SPACE AND GEOMETRY 77
see that, under these conditions, the rays of light would not be
rectilinear, but circular.
To justify what precedes, it remains for me to show that
certain changes in the position of external objects can be cor-
reded by correlative movements of the sentient beings inhabit-
ing this imaginary world, and that in such a way as to restore the
primitive aggregate of impressions experienced by these sentient
beings.
Suppose in fact that an object is displaced, undergoing de-
formation, not as a rigid solid, but as a solid subjected to unequal
dilatations in exact conformity to the law of temperature above
supposed. Permit me for brevity to call such a movement a
nan-Euclidean displacement.
If a sentient being happens to be in the neighborhood, his
impressions will be modified by the displacement of the object,
but he can reestablish them by moving in a suitable manner. It
suffices if finally the aggregate of the object and the sentient
being, considered as forming a single body, has undergone one of
those particular displacements I have just called non-Euclidean.
This is possible if it be supposed that the limbs of these beings
dilate according to the same law as the other bodies of the world
they inhabit.
Although from the point of view of our ordinary geometry
there is a deformation of the bodies in this displacement and
their various parts are no longer in the same relative position,
nevertheless we shall see that the impressions of the sentient
being have once more become the same.
In fact, though the mutual distances of the various parts may
have varied, yet the parts originally in contact are again in
contact. Therefore the tactile impressions have not changed.
On the other hand, taking into account the hypothesis made
above in regard to the refraction and the curvature of the rays
of light, the visual impressions will also have remained the same.
These imaginary beings will therefore like ourselves be led
to classify the phenomena they witness and to distinguish among
them the * changes of position' susceptible of correction by a cor-
relative voluntary movement.
If they construct a geometry, it will not be, as ours is, the
78 SCIENCE AND HYPOTHESIS
study of the moyements of our rigid solids ; it will be the study
of the changes of position which they will thus have distin-
guished and which are none other than the 'non-Euclidean dis-
placements'; t^ tvUl he non-EucUdean geometry.
Thus beings like ourselves, educated in such a world, would
not have the same geometry as ours.
The World op Four Dimensions. — ^We can represent to our-
selves a four-dimensional world just as well as a non-Euclidean.
The sense of sight, even with a single eye, together with the
muscular sensations relative to the movements of the eyeball,
would sufSce to teach us space of three dimensions.
The images of external objects are painted on the retina, which
is a two-dimensional canvas; they are perspectives.
But, as eye and objects are movable, we see in succession vari-
ous perspectives of the same body, taken from different points
of view.
At the same time, we find that the transition from one per-
spective to another is often accompanied by muscular sensations.
If the transition from the perspective A to the perspective
B, and that from the perspective A' to the perspective B' are
accompanied by the same muscular sensations, we liken them one
to the other as operations of the same nature.
Studying then the laws according to which these operations
combine, we recognize that they form a group, which has the
same structure as that of the movements of rigid solids.
Now, we have seen that it is from the properties of this group
we have derived the notion of geometric space and that of three
dimensions.
We understand thus how the idea of a space of three dimen-
sions could take birth from the pageant of these perspectives,
though each of them is of only two dimensions, since they follow
one another according to certain laws.
Well, just as the perspective of a three-dimensional figure
can be made on a plane, we can make that of a four-dimensional
figure on a picture of three (or of two) dimensions. To a
geometer this is only child's play.
We can even take of the same figure several perspectives from
several different points of view.
SPACE AND GEOMETRY 79
We can easily represent to ourselves these perspectives, since
thej are of only three dimensions.
Imagine that the various perspectives of the same object suc-
ceed one another, and that the transition from one to the other
is accompanied by muscular sensations.
We shall of course consider two of these transitions as two
operations of the same nature when they are associated with the
same muscular sensations.
Nothing then prevents us from imagining that these opera-
tions combine according to any law we choose, for example, so as
to form a group with the same structure as that of the move-
ments of a rigid solid of four dimensions.
Here there is nothing unpicturable, and yet these sensations
are precisely those which would be felt by a being possessed of
a two-dimensional retina who could move in space of four dimen-
sions. In this sense we may say the fourth dimension is
imaginable.
CONCLUSIONS. — ^We see that experience plays an indispensable
role in the genesis of geometry ; but it would be an error thence
to conclude that geometry is, even in part, an experimental
science.
If it were experimental, it would be only approximative and
provisional. And what rough approximation!
Geometry would be only the study of the movements of solids ;
but in reality it is not occupied with natural solids, it has for
object certain ideal solids, absolutely rigid, which are only a
simplified and very remote image of natural solids.
The notion of these ideal solids is drawn from all parts of our
mind, and experience is only an occasion which induces us to
bring it forth from them.
The object of geometry is the study of a particular * group';
but the general group concept pre-exists, at least potentially, in
our minds. It is imposed on us, not as form of our sense, but as
form of our understanding.
Only, from among all the possible groups, that must be chosen
which will be, so to speak, the standard to which we shall refer
natural phenomena.
Experience guides us in this choice without forcing it upon
80 SCIENCE AND HYPOTHESIS
us; it tells us not which is the truest geometry, but which is the
most convenient.
Notice that I have been able to describe the fantastic worlds
above imagined without ceasing to employ the language of ordi-
nary geometry.
And, in fact, we should not have to change it if transported
thither.
Beings educated there would doubtless find it more convenient
to create a geometry different from ours, and better adapted to
their impressions. As for us, in face of the same impressions, it
is certain we should find it more convenient not to change our
habits.
CHAPTER V
Experience and Qeometby
1. Already in the preceding pages I have several times tried
to show that the principles of geometry are not experimental
facts and that in particular Euclid's postulate can not be proven
experimentally.
However decisive appear to me the reasons already given, I
believe I should emphasize this point because here a false idea
is profoundly rooted in many minds.
2. If we construct a material circle, measure its radius and
circumference, and see if the ratio of these two lengths is equal
to ir, what shall we have done T We shall have made an experi-
ment on the properties of the matter with which we constructed
this round thing, and of that of which the measure used was made.
3. Qeometry and Astronomy. — The question has also been
put in another way. If Lobachevski's geometry is true, the paral-
lax of a very distant star will be finite; if Riemann's is true, it
will be negative. These are results which seem within the reach
of experiment, and there have been hopes that astronomical obser-
vations might enable us to decide between the three geometries.
But in astronomy * straight line' means simply *path of a ray
of light. '
If therefore negative parallaxes were found, or if it were
demonstrated that all parallaxes are superior to a certain limit,
two courses would be open to us; we might either renounce
Euclidean geometry, or else modify the laws of optics and sup-
pose that light does not travel rigorously in a straight line.
It is needless to add that all the world would regard the latter
solution as the more advantageous.
The Euclidean geometry has, therefore, nothing to fear from
fresh experiments.
4. Is the position tenable, that certain phenomena, possible
in Euclidean space, would be impossible in non-Euclidean space,
7 81
82 SCIENCE AND ETPOTBESIS
SO that ezperienee, in eetsbliBhing these phenomena, woald di-
rectly contradict the non-Euclidean hypothesis t For my part I
think no such question can be put. To my mind it is precisely
equivalent to the following, whose absurdity is patent to all eyes:
are there lengths expressible in meters and centimeters, but which
can not be measured in fathoms, feet and inches, so that experi-
ence, in ascertaining the existence of these lengths, would directly
contradict the hypothesis that there are fathoms divided into
six feet I
Examine the question more closely. I suppose that the straight
line possesses in Euclidean space any two properties which I
shall call A and B ; that in non-Euclidean space it still possesses
the property A, but no longer has the property B ; finally I sup*
pose that in both Euclidean and non-Euclidean space the straight
line is the only line having the property A.
If this were so, experience would be capable of deciding between
the hypothesis of Euclid and that of Lobacbevski. It woold be
ascertained that a definite concrete object, accessible to experi-
ment, for example, a pencil of rays of light, possesBea the proper^
A ; we should conclude that it is rectilinear, and then inrectitgatt
whether or not it has the property B. W^M
But this is not so; no property exists which, like this propeHI^^
A, can be an absolute criterion enabling us to recognize the
straight line and to distinguish it from every other line.
Shall we say, for instance: "the following is such a pre
the straight line is a line such that a figure of which 1
forma a part can be moved without the mutual d
points varying and so that all points oF this line remain t
This, in fact, is a property which, in Euclidean or D
can space, belongs to the straight and belongs only t
how shall we ascertain experimentally whether it h
or that concrete object? It will be necessary i
tances, and how slmll one know i
which I have measured 'with my m
sents the abstract distance T
We have only pushed twfl^
In reality the prop«t7 }
the straight line al<ni^ i*
EXFESIENCB AND GEOMETRY
distance. For it to serve as absolute criterion, we should have
to be able to establish not only that it doea not also belong to a
line other than the straight and to distance, but in addition that
it does not belong to a line other than the straight and to a
ma^tude other than distance. Now thia is not true.
It is therefore impossible to imagine a concrete experiment
which can be interpreted in the Euclidean system and not in the
Lobachevskian system, so that I may conclude :
No experience will ever be in contradiction to Euclid's pos-
tulate; nor, on the other hand, will any experience ever contra-
dict the postulate of Lobachevski.
5. But it is not enough that the Euclidean (or non-Euclidean)
geometry can never be directly contradicted by experience. Might
it not happen that it can accord with experience only by violating
the principle of sufficient reason or that of the relativity of space T
I will explain myself: consider any material system; we shall
have to regard, on the one hand, 'the state' of the various bodies
of this system (for instance, their temperature, their electric
potential, etc.), and, on the other hand, their position in space;
and among the data which enable us to define this position we
aball, moreover, distinguish the mutual distances of these bodies,
which define their relative positions, from the conditions which
define the absolute position of the system and its absolute orien-
tation in sjwce.
Til.' ■ i ■ I'i i ■; ■:; i :! i '. 'ippea in this system
will ' : their mutual dis-
lani!' - ' . ■ •■■. ity of space, they
H-ill not Jtj- ' tionof the
In ..i»>-r- '• . iiitiml dis-
tail''-' .r iiL'se
84 SCIENCE AND HYPOTHESIS
the non-Euclidean hypothesis. Well, we have made a series of
experiments ; we have interpreted them on the Euclidean hyjxoth-
esis, and we have recognized that these experiments thus inter-
preted do not violate this 'law of relativity.'
We now interpret them on the non-Euclidean hypothesis:
this is always possible ; only the non-Euclidean distances of our
different bodies in this new interpretation will not generally be the
same as the Euclidean distances in the primitive interpretation.
Will our experiments, interpreted in this new manner, still
be in accord with our 'law of relativity'! And if there were
not this accord, should we not have also the right to say experi-
ence bad proven the falsity of the non-Euclidean geometry?
It is easy to see that this is an idle fear; in fact, to apply
the law of relativity in all rigor, it must be applied to the entire
universe. For if only a part of this universe were considered,
and if the absolute position of this part happened to vary, the
distances to the other bodies of the universe would likewise vary,
their influence on the part of the universe considered would con-
sequently augment or diminish, which might modify the laws
of the phenomena happening there.
But if our system is the entire universe, experience is power-
less to give information about its absolute position and orienta-
tion in space. All that our instruments, however perfected they
may be, can tell us will be the state of the various parts of the
tmiverse and their mutual distances.
So our law of relativity may be thus enunciated :
The readings we shall be able to make on our instruments at
any instant will depend only on the readings we could have made
on these same instruments at the initial instant.
Now such an enunciation is independent of every interpreta-
tion of experimental facts. If the law is true in the Euclidean
interpretation, it will also be true in the non-Euclidean interpre-
tation.
Allow me here a short digression. I have sx)oken above of
the data which define the position of the various bodies of the
system ; I should likewise have spoken of those which define their
velocities; I should then have had to distinguish the velocities
with which the mutual distances of the different bodies vary;
EXPERIENCE AND GEOMETRY 86
and, on the other hand, the velocities of translation and rotation
of the system, that is to say, the velocities (with which its absolute
position land orientation vary.
To fully satisfy the mind, the law of relativity should be
expressible thus :
The state of bodies and their mutual distances at any instant,
as well as the velocities with which these distances vary at this
same instant, will depend only on the state of those bodies and
their mutual distances at the initial instant, and the velocities
with which these distances vary at this initial instant, but they
will not depend either upon the absolute initial position of the
system, or upon its absolute orientation, or upon the velocities
with which this absolute position and orientation varied at the
initial instant.
Unhappily the law thus enunciated is not in accord with ex-
periments, at least as they are ordinarily interpreted.
Suppose a man be transported to a planet whose heavens were
always covered with a thick curtain of clouds, so that he could
never see the other stars ; on that planet he would live as if it
were isolated in space. Yet this man could become aware that it
turned, either by measuring its oblateness (done ordinarily by
the aid of astronomic observations, but capable of being done by
purely geodetic means) , or by repeating the experiment of Fou-
cault's pendulum. The absolute rotation of this plcmet could
therefore be made evident.
That is a fact which shocks the philosopher, but which the
physicist is compelled to accept.
We know that from this fact Newton inferred the existence
of absolute space ; I myself am quite unable to adopt this view.
I shall explain why in Part III. For the moment it is not my
intention to enter upon this diflSculty.
Therefore I must resign myself, in the enunciation of the law
of relativity, to including velocities of every kind among the data
which define the state of the bodies.
However that may be, this difficulty is the same for Euclid's
geometry as for Lobachevski's; I therefore need not trouble my-
self with it, and have only mentioned it incidentally.
86 SCIENCE AND HYPOTHESIS
What is important is the conclusion: experiment can not de-
cide between Euclid and Lobachevski.
To sum up, whichever way we look at it, it is impossible to
discover in geometric empiricism a rational meaning.
6. Experiments only teach us the relations of bodies to one
another; none of them bears or can bear on the relations of bodies
with space, or on the mutual relations of different parts of space.
"Yes," you reply, "a single experiment is insuflScient, be-
cause it gives me only a single equation with several unknowns ;
but when I shall have made enough experiments I shall have
equations enough to calculate all my unknowns."
To know the height of the mainmast does not sufSce for calcu-
lating the age of the captain. When you have measured every
bit of wood in the ship you will have many equations, but
you will know his age no better. All your measurements bear-
ing only on your bits of wood can reveal to you nothing except
concerning these bits of wood. Just so your experiments, how-
ever numerous they may be, bearing only on the relations of
bodies to one another, will reveal to us nothing about the mutual
relations of the various parts of space.
7. Will you say that if the experiments bear on the bodies,
they bear at least upon the geometric properties of the bodies?
But, first, what do you understand by geometric properties of
the bodies? I assume that it is a question of the relations of the
bodies with space ; these properties are therefore inaccessible to
experiments which bear only on the relations of the bodies to one
another. This alone would suffice to show that there can be no
question of these properties.
StiU let us begin by coming to an understanding about the
sense of the phrase: geometric properties of bodies. When I
say a body is composed of several parts, I assume that I do not
enunciate therein a geometric property, and this would remain
true even if I agreed to give the improper name of points to the
smallest parts I consider.
When I say that such a part of such a body is in contact
with such a part of such another body, I enunciate a proposition
which concerns the mutual relations of these two bodies and not
their relations with space.
EXPEBIENCE AND GEOMETRY 87
I suppose you will grant me these are not geometric properties;
at least I am sure you will grant me these properties are inde-
pendent of all knowledge of metric geometry.
This presupposed, I imagine that we have a solid body formed
of eight slender iron rods, OA, OB, OC, OD, OE, OF, 00, OH,
united at one of their extremities 0. Let us besides have a second
«olid body, for example a bit of wood, to be marked with three
little flecks of ink which I shall call a, p, y. I further suppose it
ascertained that apy may be brought into contact with AOO (I
mean a with A, and at the same time fi with O and y with 0),
then that we may bring successively into contact aPy with BOO,
COO, DOO, EOO, FOO, then with AHO, BEO, CEO, DEO,
EEO, FEO, then ay successively with AB, BC, CD, DE, EF, FA.
These are determinations we may make without having in
advance any notion about form or about the metric properties of
space. They in no wise bear on the 'geometric properties of
bodies.' And these determinations will not be possible if the
bodies experimented upon move in accordance with a group
having the same structure as the Lobachevskian group (I mean
according to the same laws as solid bodies in LobachevsM's geom-
etry). They suffice therefore to prove that these bodies move in
accordance with the Euclidean group, or at least that they do
not move according to the Lobachevskian group.
That they are compatible with the Euclidean group is easy
to see. For they could be made if the body apy was a rigid
solid of our ordinary geometry presenting the form of a right-
angled triangle, and if the points ABCDEFOE were the summits
of a polyhedron formed of two regular hexagonal pyramids of our
ordinary geometry, having for common base ABCDEF and for
apices the one and the other E.
Suppose now that in place of the preceding determination it
is observed that as above aPy can be successively applied to AOO,
BOO, COO, DOO, EOO, AEO, BEO, CEO, DEO, EEO, FEO,
then that ap (and no longer ay) can be successively applied to
AB, BC, CD, DE, EF and FA.
These are determinations which could be made if non-Euclid-
ean geometry were true, if the bodies aPy and OABCDEFOE
were rigid solids, and if the first were a right-angled triangle
88 SCIENCE AND HYPOTHESIS
and the second a double regular hexagonal pyramid of snitaible
dimensions.
Therefore these new determinations are not possible if the
bodies move according to the Euclidean group ; but they become
so if it be supposed that the bodies move according to the Loba-
chevskian group. They would suffice, therefore (if one made
them), to prove that the bodies in question do not move accord-
ing to the Euclidean group.
Thus, without making any hypothesis about form, about the
nature of space, about the relations of bodies to space, and with-
out attributing to bodies any geometric property, I have made
observations which have enabled me to show in one case that
the bodies experimented upon move according to a group whose
structure is Euclidean, in the other case that they move according
to a group whose structure is Lobachevskian.
And one may not say that the first aggregate of determinations
would constitute an experiment proving that space is Euclidean,
and the second an experiment proving that space is non-Euclidean.
In fact one could imagine (I say imagine) bodies moving so
as to render possible the second series of determinations. And
the proof is that the first mechanician met could construct such
bodies if he cared to take the pains and make the outlay. You
will not conclude from that, however, that space is non-Euclidean.
Nay, since the ordinary solid bodies would continue to exist
when the mechanician had constructed the strange bodies of which
I have just spoken, it would be necessary to conclude that space is
at the same time Euclidean and non-Euclidean.
Suppose, for example, that we have a great sphere of radius B
and that the temperature decreases from the center to the surface
of this sphere according to the law of which I have spoken in
describing the non-Euclidean world.
"We might have bodies whose expansion would be negligible
and whioh would act like ordinary rigid solids ; and, on the other
hand, bodies very dilatable and which would act like non-Euclidean
solids. We might have two double pyramids OABCDEFOH and
O'A'B'C'D'E'F'G'W and two triangles afiy and a'p^y'. The first
double pyramid might be rectilinear and the second curvilinear;.
EXPERIENCE AND GEOMETRY 89
the triangle aPy might be made of inexpansible matter and the
other of a very dilatable matter.
It would then be possible to make the first observations with
the double pyramid OAH and the triangle aPy, and the second
with the double pyramid O'A'H' and the triangle a'fify. And
then experiment would seem to prove first that the Euclidean
geometry is true and then that it is false.
Experiments therefore have a bearing, not on space, hut on
bodies.
Supplement
8. To complete the matter, I ought to speak of a very delicate
question, which would require long development; I shall confine
myself to summarizing here what I have expounded in the Revue
de MStaphysique et de Morale and in The Monist. When we
say space has three dimensions, what do we mean t
We have seen the importance of those 'internal changes'
revealed to us by our muscular sensations. They may serve to
characterize the various attitudes of our body. Take arbitrarily
as origin one of these attitudes A. When we pass from this
initial attitude to any other attitude B, we feel a series of mus-
cular sensations, and this series 8 will define B. Observe, how-
ever, that we shall often regard two series 8 and 8' as defining
the same attitude B (since the initial and final attitudes A and B
remaining the same, the intermediary attitudes and the corre-
sponding sensations may differ). How then shall we recognize
the equivalence of these two series ! Because they may serve to
compensate the same external change, or more generally because,
when it is a question of compensating an external change, one
of the series can be replaced by the other. Among these series,
we have distinguished those which of themselves alone can com-
pensate an external change, and which we have called 'displace-
ments.' As we can not discriminate between two displacements
which are too close together, the totality of these displacements
presents the characteristics of a physical continuum ; experience
teaches us that they are those of a physical continuum of six
dimensions; but we do not yet know how many dimensions
space itself has, we must first solve another question.
What is a point of space! Everybody thinks he knows, but
90 SCIENCE AND HYPOTHESIS
that is an illusion. What we see when we try to represent to our-
selves a point of space is a black speck on white paper, a speck of
chalk on a blackboard, always an object. The question should
therefore be understood as follows :
What do I mean when I say the object B is at the same point
that the object A occupied just now t Or further, what criterion
will enable me to apprehend thist
I mean that, although I have not budged (which my muscular
sense tells me) , my first finger which just now touched the object A
touches at present the object B. I could have used other criteria ,*
for instance another finger or the sense of sight. But the first
criterion is sufficient; I know that if it answers yes, all the other
criteria will give the same response. I know it by experience, I
can not know it a priori. For the same reason I say that touch
can not be exercised at a distance ; this is another way of enunci-
ating the same experimental fact. And if, on the contrary, I say
that sight acts at a distance, it means that the criterion furnished
by sight may respond yes while the others reply no.
And in fact, the object, although moved away, may form its
image at the same point of the retina. Sight responds yes, the
object has remained at the same point and touch answers no,
because my finger which just now touched the object, at present
touches it no longer. If experience had shown us that one finger
may respond no when the other says yes, we should likewise
say that touch acts at a distance.
In short, for each attitude of my body, my first finger deter-
mines a point, and this it is, and this alone, which defines a point
of space.
To each attitude corresponds thus a point ; but it often happens
that the same point corresponds to several different attitudes (in
this case we say our finger has not budged, but the rest of the
body has moved). We distinguish, therefore, among the changes
of attitude those where the finger does not budge. How are we
led thereto f It is because often we notice that in these changes
the object which is in contact with the finger remains in contact
with it.
Range, therefore, in the same class all the attitudes obtainable
from each other by one of the changes we have thus distinguished.
EXPERIENCE AND GEOMETRY 91
To all the attitudes of the class will correspond the same point
of space. Therefore to each class will correspond a point and to
each point a class. But one may say that what experience arrives
at is not the point, it is this class of changes or, better, the cor-
responding class of muscular sensations.
And when we say space has three dimensions, we simply mean
that the totality of these classes appears to us with the character-
istics of a physical continuum of three dimensions.
One might be tempted to conclude that it is experience which
has taught us how many dimensions space has. But in reality
here also our experiences have bearing, not on space, but on our
body and its relations with the neighboring objects. Moreover
they are excessively crude.
In our mind pre-existed the latent idea of a certain number
of groups — ^those whose theory Lie has developed. Which group
shall we choose, to make of it a sort of standard with which to com-
pare natural phenomena? And, this group chosen, which of its
sub-groups shall we take to characterize a point of space t Ex-
perience has guided us by showing us which choice best adapts
itself to the properties of our body. But its role is limited to that.
Ancestral Experience
It has often been said that if individual experience could
not create geometry the same is not true of ancestral experience.
But what does that meant Is it meant that we could not experi-
mentally demonstrate Euclid's postulate, but that our ancestors
have been able to do it f Not in the least. It is meant that by
natural selection our mind has adapted itself to the conditions of
the external world, that it has adopted the geometry most advan-
tageous to the species: or in other words the most convenient.
This is entirely in conformity with our conclusions ; geometry is
not true, it is advantageous.
PART III
FORCE
CHAPTER VI
The Classic Mechanics
The English teach mechanics as an experimental science; on
the continent it is always expounded as more or less a deductive
and a priori science. The English are right, that goes without
saying; but how could the other method have been persisted in
so longf Why have the continental savants who have sought to
get out of the ruts of their predecessors been usually unable to
free themselves completely !
On the other hand, if the principles of mechanics are only of
experimental origin, are they not therefore only approximate and
provisional! Might not new experiments some day lead us to
modify or even to abandon them ?
Such are the questions which naturally obtrude themselves,
and the diflSculty of solution comes principally from the fact
that the treatises on mechanics do not clearly distinguish between
what is experiment, what is mathematical reasoning, what is con-
vention, what is hypothesis.
That is not all :
1** There is no absolute space and we can conceive only of
relative motions ; yet usually the mechanical facts are enunciated
as if there were an absolute space to which to refer them.
2° There is no absolute time; to say two durations are equal
is an assertion which has by itself no meaning and which can
acquire one only by convention.
3° Not only have we no direct intuition of the equality of
two durations, but we have not even direct intuition of the
92
\
THE CLASSIC MECHANICS 98
fiimnltaneity of two events occurring in different places: this I
liave explained in an article entitled La mesure du iemps.^ '
4** Finally, our Euclidean geometry is itself only a sort of
convention of language; mechanical facts might be enunciated
with reference to a non-Euclidean space which would be a guide
less convenient than, but just as legitimate as, our ordinary space ;
the enunciation would thus become much more complicated, but it
would remain possible.
Thus absolute space, absolute time, geometry itself, are not
conditions which impose themselves on mechanics ; all these things
are no more antecedent to mechanics than the French language is
logically antecedent to the verities one expresses in French.
We might try to enunciate the fundamental laws of mechanics
in a language independent of all these conventions; we should
thus without doubt get a better idea of what these laws are in
themselves ; this is what M. Andrade has attempted to do, at least
in part, in his Leqons de mecanique physique.
The enunciation of these laws would become of course much
more complicated, because all these conventions have been devised
expressly to abridge and simplify this enunciation.
As for me, save in what concerns absolute space, I shall ignore
all these difficulties ; not that I fail to appreciate them, far from
that; but we have sufficiently examined them in the first two
parts of the book.
I shall therefore admit, provisionally, absolute time and Eu-
clidean geometry.
The Principle op Inertia. — ^A body acted on by no force can
only move uniformly in a straight line.
Is this a truth imposed a priori upon the mindf If it were
80, how should the Greeks have failed to recognize it? How could
they have believed that motion stops when the cause which gave
birth to it ceases ! Or again that every body if nothing prevents,
will move in a circle, the noblest of motions?
If it is said that the velocity of a body can not change if there
is no reason for it to change, could it not be maintained just as
well that the position of this body can not change, or that the
^Eevue de M4taphysique et de Morale, t. YI., pp. 1-13 (January, 1898).
94 SCIENCE AND HYPOTHESIS
curvature of its trajectory can not change, if no external canse
intervenes to modify themt
Is the principle of inertia, which is not an a priori truth,
therefore an experimental factt But has any one ever experi-
mented on bodies withdrawn from the action of every force t and,
if so, how was it known that these bodies were subjected to no
force t The example ordinarily cited is that of a baU rolling a
very long time on a marble table ; but why do we say it is sub-
jected to no force t Is this because it is too remote from all other
bodies to experience any appreciable action from themt Yet it
is not farther from the earth than if it were thrown freely into
the air ; and every one knows that in this case it would experience
the influence of gravity due to the attraction of the earth.
Teachers of mechanics usually pass rapidly over the example
of the ball ; but they add that the principle of inertia is verified
indirectly by its consequences. They express themselves badly;
they evidently mean it is possible to verify various consequences
of a more general principle, of which that of inertia is only a
particular case.
I shall propose for this general principle the following enun-
ciation :
The acceleration of a body depends only upon the position
of this body and of the neighboring bodies and upon their
velocities.
Mathematicians would say the movements of all the material
molecules of the universe depend on differential equations of the
second order.
To make it clear that this is really the natural generalization
of the law of inertia, I shall beg you to permit me a bit of fiction.
The law of inertia, as I have said above, is not imposed upon us
a priori; other laws would be quite as compatible with the prin-
ciple of suflScient reason. If a body is subjected to no force, in
lieu of supposing its velocity not to change, it might be supposed
that it is its position or else its acceleration which is not to change.
Well, imagine for an instant that one of these two hypothetical
laws is a law of nature and replaces our law of inertia. What
would be its natural generalization? A moment's thought will
show us.
THE CLASSIC MECHANICS 95
In the first case, we must suppose that the velocity of a body
depends only upon its position and upon that of the neighboring
bodies; in the second case that the change of acceleration of a
body depends only upon the position of this body and of the
neighboring 'bodies, upon their velocities and upon their acceler-
ations.
Or to speak the language of mathematics, the differential
equations of motion would be of the first order in the first case,
and of the third order in the second case.
Let us slightly modify our fiction. Suppose a world analogous
to our solar system, but where, by a strange chance, the orbits of
all the planets are without eccentricity and without inclination.
Suppose further that the masses of these planets are too slight
for their mutual perturbations to be sensible. Astronomers in-
habiting one of these planets could not fail to conclude that the
orbit of a star can only be circular and parallel to a certain plane ;
the position of a star at a given instant would then suffice to de-
termine its velocity and its whole path. The law of inertia which
they would adopt would be the first of the two hypothetical laws
I have mentioned.
Imagine now that this system is some day traversed with great
velocity by a body of vast mass, coming from distant constella-
tions. All the orbits would be profoundly disturbed. Still our
astronomers would not be too greatly astonished ; they would very
well divine that this new star was alone to blame for all the
mischief. *'But," they would say, *'when it is gone, order will
of itself be reestablished ; no doubt the distances of the planets
from the sun will not revert to what they were before the cata-
clysm, but when the perturbing star is gone, the orbits will again
become circular."
It would only be when the disturbing body was gone and when
nevertheless the orbits, in lieu of again becoming circular, became
elliptic, that these astronomers would become conscious of their
error and the necessity of remaking all their mechanics.
I have dwelt somewhat upon these hypotheses because it seems
to me one can clearly comprehend what our generalized law of
inertia really is only in contrasting it with a contrary hypothesis.
Well, now, has this generalized law of inertia been verified by
96 SCIENCE AND HYPOTHESIS
experiment, or can it bef When Newton wrote the Prindpia
he quite regarded this truth as experimentally acquired and dem-
onstrated. It was so in his eyes, not only through the anthropo-
morphism of which we shall speak further on, but through the
work of Galileo. It was so even from Kepler's laws themselves;
in accordance with these laws, in fact, the path of a planet is
completely determined by its initial position and initial velocity;
this is just what our generalized law of inertia requires.
For this principle to be only in appearance true, for one to
have cause to dread having some day to replace it by one of the
analogous principles I have just now contrasted with it, would be
necessary our having been misled by some amazing chance, like
that which, in the fiction above developed, led into error our
imaginary astronomers.
Such a hypothesis is too unlikely to delay over. No one will
believe that such coincidences can happen; no doubt the prob-
ability of two eccentricities being both precisely null, to within
errors of observation, is not less than the probability of one being
precisely equal to 0.1, for instance, and the other to 0.2, to within
errors of observation. The probability of a simple event is not
less than that of a complicated event ; and yet, if the first happens,
we shall not consent to attribute it to chance ; we should not believe
that nature had acted expressly to deceive us. The hypothesis of
an error of this sort being discarded, it may therefore be admitted
that in so far as astronomy is concerned, our law has been veri-
fied by experiment.
But astronomy is not the whole of physics.
May we not fear lest some day a new experiment should come
to falsify the law in some domain of physics t An experimental
law is always subject to revision; one should always expect to see
it replaced by a more precise law.
Yet no one seriously thinks that the law we are speaking of
will ever be abandoned or amended. Whyt Precisely because
it can never be subjected to a decisive test.
First of all, in order that this trial should be complete, it
would be necessary that after a certain time all the bodies in the
universe should revert to their initial positions with their initial
THU CLASSIC MECHANICS 97
velocities. It might then be seen whether, starting from this
moment, they would resume their original paths.
But this test is impossible, it can be only partially applied,
and, however well it is made, there will always be some bodies
which will not revert to their initial positions ; thus every deroga-
tion of the law will easily find its explanation.
This is not all ; in astronomy we see the bodies whose motions
we study and we usually assume that they are not subjected to the
action of other invisible bodies. Under these conditions our law
must indeed be either verified or not verified.
But it is not the same in physics ; if the physical phenomena
are due to motions, it is to the motions of molecules which we do
not see. If then the acceleration of one of the bodies we see
appears to us to depend on something else besides the positions
or velocities of other visible bodies or of invisible molecules whose
existence we have been previously led to admit, nothing prevents
our supposing that this something else is the position or the
velocity of other molecules whose presence we have not before
suspected. The law will find itself safeguarded.
Permit me to employ mathematical language a moment to
express the same thought under another form. Suppose we ob-
serve n molecules and ascertain that their 3n coordinates satisfy
a system of 3n differential equations of the fourth order (and
not of the second order as the law of inertia would require) . We
know that by introducing 3n auxiliary variables, a system of 3n
equations of the fourth order can be reduced to a system of 6n
equations of the second order. If then we suppose these 3n
auxiliary variables represent the coordinates of n invisible mole-
cules, the result is again in conformity with the law of inertia.
To sum up, this law, verified experimentally in some particular
cases, may unhesitatingly be extended to the most general cases,
since we know that in these general cases experiment no longer
is able either to confirm or to contradict it.
The Law op Acceleration. — The acceleration of a body is
equal to the force acting on it divided by its mass. Can this law
be verified by experiment! For that it would be necessary to
8
98 SCIENCE AND HYPOTHESIS
measure the three magnitudes which figure in the enunciation:
acceleration, force and mass.
I assume that acceleration can be measured, for I pass over
the difSculty arising from the measurement of time. But how
measure force, or mass t We do not even know what they are.
What is mass? According to Newton, it is the product of the
volume by the density. According to Thomson and Tait, it would
be better to say that density is the quotient of the mass by the
volume. What is force f It is, replies Lagrange, that which
moves or tends to move a body. It is, Kirchhoff will say, the
product of the mass by the acceleration. But then, why not say
the mass is the quotient of the force by the acceleration t
These di£Sculties are inextricable.
When we say force is the cause of motion, we talk metaphysics,
and this definition, if one were content with it, would be abso-
lutely sterile. For a definition to be of any use, it must teach us
to measure force ; moreover that sufSces ; it is not at all necessary
that it teach us what force is in itself, nor whether it is the cause
or the effect of motion.
We must therefore first define the equality of two forces.
When shall we say two forces are equal! It is, we are told,
when, applied to the same mass, they impress upon it the same
acceleration, or when, opposed directly one to the other, they pro-
duce equilibrium. This definition is only a sham. A force applied
to a body can not be uncoupled to hook it up to another body,
as one uncouples a locomotive to attach it to another train. It
is therefore impossible to know what acceleration such a force,
applied to such a body, would impress upon such another body,
if it were applied to it. It is impossible to know how two forces
which are not directly opposed would act, if they were directly
opposed.
It is this definition we try to materialize, so to speak, when
we measure a force with a dynamometer, or in balancing it with
a weight. Two forces F and F\ which for simplicity I will sup-
pose vertical and directed upward, are applied respectively to two
bodies C and C ; I suspend the same heavy body P first to the
body C, then to the body C ; if equilibrium is produced in both
cases, I shall conclude that the two forces F and F' are equal to
THE CLASSIC MECHANICS 99
one another, since they are each equal to the weight of the body P.
But am I sure the body P has retained the same weight when
I have transported it from the first body to the second t Far from
it; I am sure of the contrary; I know the intensity of gravity
varies from one point to another, and that it is stronger, for
instance, at the pole than at the equator. No doubt the difference
is very slight and, in practise, I shall take no account of it; but
a properly constructed definition should have mathematical
rigor; this rigor is lacking. What I say of weight would evi-
dently apply to the force of the resiliency of a dynamometer,,
which the temperature and a multitude of circumstances may
cause to vary. i
This is not aU; we can not say the weight of the body P
may be applied to the body C and directly balance the force P.
What is applied to the body C is the action A of the body P on
the body C ; the body P is submitted on its part, on the one hand,
to its weight; on the other hand, to the reaction B of the body C
on P. Finally, the force F is equal to the force A, since it balances
it; the force A is equal to 2?, in virtue of the principle of the
equality of action and reaction ; lastly, the force B is equal to the
weight of P, since it balances it It is from these three equalities
we deduce as consequence the equality of P and the weight of P.
We are therefore obliged in the definition of the equality of
the two forces to bring in the principle of the equality of action
and reaction; on this account, this principle must no longer be
regarded as an experimental law, but as a definition.
For recognizing the equality of two forces here, we are then
in possession of two rules : equality of two forces which balance ;
equality of action and reaction. But, as we have seen above,
these two rules are insuflScient ; we are obliged to have recourse to
a third rule and to assume that certain forces, as, for instance, the
weight of a body, are constant in magnitude and direction. But
this third rule, as I have said, is an experimental law ; it is only
approximately true ; it is a bad defimtion.
We are therefore reduced to Kirchhoff's definition; force is
equal to the m^iss multipled by the acceleration. This 'law of
Newton' in its turn ceases to be regarded as an experimental law,
it is now only a definition. But this definition is still insufficient,
100 SCIENCE AND BTP0THESI8
for we do not know what mass is. It enables us donbUess to cal-
culate the relation of two forces applied to the same body at dif-
ferent instants; it teaches us nothing about the relation of two
forces applied to two different bodies.
To complete it, it is necessary to go back anew to Newton's
third law (equality of action and reaction), regarded again, not
as an experimental law, but as a definition. Two bodies A and B
act one upon the other; the acceleration of A multiplied by the
mass of A is equal to the action of B upon A ; in the same way,
the product of the acceleration of B by its mass is equal to the
reaction of A upon B, As, by definition, action is equal to reac-
tion, the masses of A and B are in the inverse ratio of their
accelerations. Here we have the ratio of these two masses defined,
and it is for experiment to verify that this ratio is constant.
That would be all very well if the two bodies A and B alone
were present and removed from the action of the rest of the
world. This is not at all the case ; the acceleration of A is not due
merely to the action of B, but to that of a multitude of other
bodies C, D, . . . To apply the preceding rule, it is therefore
necessary to separate the acceleration of A into many components,
and discern which of these components is due to the action of B.
This separation would still be possible, if we should assume
that the action of C upon A is simply adjoined to that of B
upon A, without the presence of the body C modifying the action
of B upon A ; or the presence of B modifying the action of C
upon ^ ; if we should assume, consequently, that any two bodies
attract each other, that their mutual action is along their join
and depends only upon their distance apart; if, in a word, we
assume the hypothesis of central forces.
You know that to determine the masses of the celestial bodies
we use a wholly different principle. The law of gravitation
teaches us that the attraction of two bodies is proportional to
their masses ; if r is their distance apart m and m' their masses,
k a constant, their attraction will be kmm'/r^.
What we are measuring then is not mass, the ratio of force to
acceleration, but the attracting mass; it is not the inertia of the
body, but its attracting force.
This is an indirect procedure, whose employment is not theo-
THE CLASSIC MECHANICS 101
retically indispensable. It might very well have been that attrac-
tion was inversely proportional to the square of the distance with-
ont being proportional to the product of the masses, that it was
equal to //r^, but without our having / = kmm'.
If it were so, we could nevertheless, by observation of the
relative motions of the heavenly bodies, measure the masses of
these bodies.
But have we the right to admit the hypothesis of central
forces t Is this hypothesis rigorously exact t Is it certain it
will never be contradicted by experiment t Who would dare
affirm that t And if we must abandon this hypothesis, the whole
edifice so laboriously erected will crumble.
We have no longer the right to speak of the component of
the acceleration of A due to the action of B. We have no means
of distinguishing it from that due to the action of C or of another
body. The rule for the measurement of masses becomes in-
applicable.
What remains then of the principle of the equality of action
and reaction! If the hypothesis of central forces is rejected,
this principle should evidently be enunciated thus : the geometric
resultant of all the forces applied to the various bodies of a
fifystem isolated from all external action will be null. Or, in
other words, the motion of the center of gravity of this system
will he rectilinear and uniform.
There it seems we have a means of defining mass ; the position
of the center of gravity evidently depends on the values attrib-
uted to the masses ; it will be necessary to dispose of these values
in such a way that the motion of the center of gravity may be
rectilinear and uniform; this will always be possible if Newton's
third law is true, and possible in general only in a single way.
But there exists no system isolated from all external action;
all the parts of the universe are subject more or less to the action
of all the other parts. The law of the motion of the center of
gravity is rigorously true only if applied to the entire universe.
But then, to get from it the values of the masses, it would be
necessary to observe the motion of the center of gravity of the
universe. The absurdity of this consequence is manifest; we
know only relative motions ; the motion of the center of gravity
of the universe will remain for us eternally unknown.
102 SCIENCE AND HYPOTHESIS
Therefore nothing remains and our efforts have been fmiileas;
we are driven to the following definition, which is only an
avowal of powerlessness: masses a/re coefficients it is convenient
to introduce into calculations.
We could reconstruct all mechanics by attributing different
values to all the masses. This new mechanics would not be in
contradiction either with experience or with the general prin-
ciples of dynamics (principle of inertia, proportionality of
forces to masses and to accelerations, equality of action and
reaction, rectilinear and uniform motion of the center of gravis,
principle of areas).
Only the equations of this new mechanics would be less simple.
Let us understand clearly : it would only be the first terms which
would be less simple, that is those experience has already made us
acquainted with; perhaps one could alter the masses by small
quantities without the complete equations gaining or losing in
simplicity.
Hertz has raised the question whether the principles of me-
chanics are rigorously true. **In the opinion of many physi-
cists," he says, **it is inconceivable that the remotest experience
should ever change anything in the immovable principles of
mechanics; and yet, what comes from experience may always
be rectified by experience.'' After what we have just said, these
fears will appear groundless.
The principles of dynamics at first appeared to us as experi-
mental truths; but we have been obliged to use them as defini-
tions. It is hy definition that force is equal to the product of
mass by acceleration; here, then, is a principle which is hence-
forth beyond the reach of any further experiment. It is in the
same way by definition that action is equal to reaction.
But then, it will be said, these unverifiable principles are abso-
lutely devoid of any significance ; experiment can not contradict
them; but they can teach us nothing useful; then what is the
use of studying dynamics!
This over-hasty condemnation would be unjust. There is not
in nature any system perfectly isolated, perfectly removed from
all external action ; but there are systems almost isolated.
If such a system be observed, one may study not only the
THE CLASSIC MECHANICS 103
relative motion of its various parts one in reference to another,
but also the motion of its center of gravity in reference to the
other parts of the universe. We ascertain then that the motion
of this center of gravity is almost rectilinear and unif orm, in
conformity with Newton's third law.
That is an experimental truth, but it can not be invalidated
by experience; in fact, what would a more precise experiment
teach ust It would teach us that the law was only almost true;
but that we knew already.
We can now understand how experience has been able to serve
as basis for the principles of mechanics and yet wUl never be
able to contradict them.
Anthropomorphio Mechanics. — **Kirchhoff," it will be said,
''has only acted in obedience to the general tendency of mathe-
maticians toward nominalism ; from this his ability as a physicist
has not saved him. He wanted a definition of force, and he
took for it the first proposition that presented itself; but we
need no definition of force : the idea of force is primitive, irre-
ducible, indefinable; we know all that it is, we have a direct
intuition of it. This direct intuition comes from the notion of
effort, which is familiar to us from infancy."
But first, even though this direct intuition made known to
us the real nature of force in itself, it would be insufficient as a
foundation for mechanics; it would besides be wholly useless.
What is of importance is not to know what force is, but to know
how to measure it.
Whatever does not teach us to measure it is as useless to
mechanics as is, for instance, the subjective notion of warmth
and cold to the physicist who is studying heat. This subjective
notion can not be translated into numbers, therefore it is of no
use; a scientist whose skin was an absolutely bad conductor of
heat and who, consequently, would never have felt either sensa-
tions of cold or sensations of warmth, could read a thermometer
just as well as any one else, and that would suffice him for con-
structing the whole theory of heat.
Now this immediate notion of effort is of no use to us for
measuring force ; it is dear, for instance, that I should feel more
104 SCIENCE AND HYPOTHESIS
fatigue in lifting a weight of fifty kilos than a man accnstomed
to carry burdens.
But more than that: this notion of effort does not teach us
the real nature of force; it reduces itself finally to a remem-
brance of muscular sensations, and it will hardly be maintained
that the sun feels a muscular sensation when it draws the earth.
All that can there be sought is a symbol, less precise and less
convenient than the arrows the geometers use, but just as remote
from the reality.
Anthropomorphism has played a considerable historic role in
the genesis of mechanics; perhaps it will still at times furnish
a symbol which will appear convenient to some minds ; but it can
not serve as foundation for anything of a truly scientific or
philosophic character.
'The School op the Thread.' — ^M. Andrade, in his Legons
de mechanique physique, has rejuvenated anthropomorphic me-
chanics. To the school of mechanics to which Kirchhoff belongs,
he opposes that which he bizarrely calls the school of the thread.
This school tries to reduce everything to ''the consideration
of certain material systems of negligible mass, envisaged in the
state of tension and capable of transmitting considerable efforts
to distant bodies, systems of which the ideal type is the thread."
A thread which transmits any force is slightly elongated under
the action of this force; the direction of the thread tells us the
direction of the force, whose magnitude is measured by the
elongation of the thread.
One may then conceive an experiment such as this. A body
A is attached to a thread ; at the other extremity of the thread
any force acts which varies until the thread takes an elongation
a; the acceleration of the body A is noted; A is detached and
the body B attached to the same thread; the same force or
another force acts anew, and is made to vary until the thread
takes again the elongation a ; the acceleration of the body B is
noted. The experiment is then renewed with both A and B,
but so that the thread takes the elongation p. The four observed
accelerations should be proportional. We have thus an experi-
mental verification of the law of acceleration above enunciated.
Or still better, a body is submitted to the simultaneous action
THE CLASSIC MECHANICS 105
of several identical threads in equal tension, and by experiment
it is sought what must be the orientations of all these threads that
the body may remain in equilibrium. We have then an experi-
mental verification of the law of the composition of forces.
But, after all, what have we done! We have defined the
force to which the thread is subjected by the deformation under-
gone by this thread, which is reasonable enough ; we have further
assumed that if a body is attached to this thread, the effort trans-
mitted to it by the thread is equal to the action this body exercises
on this thread ,* after all, we have therefore used the principle of
the equality of action and reaction, in considering it, not as an
experimental truth, but as the very definition of force.
This definition is just as conventional as Kirchhoff's, but far
less generaL
All forces are not transmitted by threads (besides, to be able
to compare them, they would all have to be transmitted by iden-
tical threads). Even if it should be conceded that the earth is
attached to the sun by some invisible thread, at least it would be
admitted that we have no means of measuring its elongation.
Nine times out of ten, consequently, our definition would be at
fault ; no sort of sense could be attributed to it, and it would be
necessary to fall back on Kirchhoff 's.
Why then take this d6tourt You admit a certain definition
of force which has a meaning only in certain particular cases.
In these cases you verify by experiment that it leads to the law
of acceleration. On the strength of this experiment, you then
take the law of acceleration as a definition of force in all the
other cases.
Would it not be simpler to consider the law of acceleration as
a definition in all cases, and to regard the experiments in ques-
tion, not as verifications of this law, but as verifications of the
principle of reaction, or as demonstrating that the deformations
of an elastic body depend only on the forces to which this body is
subjected t
And this is without taking into account that the conditions
under which your definition could be accepted are never fulfilled
except imperfectly, that a thread is never without mass, that it
is never removed from every force except the reaction of the
bodies attached to its extremities.
106 SCIENCE AND HTP0THESI8
Andrade's ideas are nevertheless very interesting; if th^
do not satisfy our logical craving, they make us understand
better the historic genesis of the fundamental ideas of mechanics.
The reflections they suggest show us how the human mind has
raised itself from a naive anthropomorphism to the present con-
ceptions of science.
We see at the start a very particular and in sum rather crude
experiment ; at the finish, a law perfectly general, perfectly pre-
cise, the certainty of which we regard as absolute. This cer-
tainty we ourselves have bestowed upon it voluntarily, so to
speak, by looking upon it as a convention.
Are the law of acceleration, the rule of the composition of
forces then only arbitrary conventions t Conventions, yes; arbi-
trary, no ; they would be if we lost sight of the experiments which
led the creators of the science to adopt them, and which, imper-
fect as they may be, suffice to justify them. It is well that from
time to time our attention is carried back to the experimental
origin of these conventions.
CHAPTER VII
Belativb Motion and Absolute Motion
The Principle of Relative Motion. — The attempt has some-
times been made to attach the law of acceleration to a more
general principle. The motion of any system must obey the
same laws, whether it be referred to fixed axes, or to movable
axes carried along in a rectilinear and uniform motion. This is
the principle of relative motion, which forces itself upon us for
two reasons: first, the commonest experience confirms it, and
second, the contrary hypothesis is singularly repugnant to the
mind.
Assume it then, and consider a body subjected to a force;
the relative motion of this body, in reference to an observer
moving with a uniform velocity equal to the initial velocity of
the body, must be identical to what its absolute motion would be
if it started from rest. We conclude hence that its acceleration
can not depend upon its absolute velocity ; the attempt has even
been made to derive from this a demonstration of the law of
acceleration.
There long were traces of this demonstration in the regula-
tions for the degree B. ^s Sc. It is evident that this attempt is
idle. The obstacle which prevented our demonstrating the law
of acceleration is that we had no definition of force ; this obstacle
subsists in its entirety, since the principle invoked has not fur-
nished us the definition we lacked.
The principle of relative motion is none the less highly inter-
esting and deserves study for its own sake. Let us first try to
enunciate it in a precise manner.
We have said above that the accelerations of the different
bodies forming part of an isolated system depend only on their
relative velocities and positions, and not on their absolute veloc-
ities and positions, provided the movable axes to which the rela-
tive motion is referred move uniformly in a straight line. Or, if
107
108 SCIENCE AND HYPOTHESIS
we prefer, their acceleratioiis depend only on the differences o£
their velocities and the differences of their coordinates, and not
on the absolute values of these velocities and coordinates.
If this principle is true for relative accelerations, or rather
for differences of acceleration, in combining it with the law of
reaction we shall thence deduce that it is still true of absolute
accelerations.
It then remains to be seen how we may demonstrate that the
differences of the accelerations depend only on the differences
of the velocities and of the coordinates, or, to speak in math-
ematical language, that these differences of coordinates satisfy
differential equations of the second order.
Can this demonstration be deduced from experiments or from
a priori considerations t
Recalling what we have said above, the reader can answer for
himself.
Thus enunciated, in fact, the principle of relative motion
singularly resembles what I called above the generalized principle
of inertia ; it is not altogether the same thing, since it is a ques-
tion of the differences of coordinates and not of the coordinates
themselves. The new principle teaches us therefore something
more than the old, but the same discussion is applicable and
would lead to the same conclusions; it is unnecessary to return
to it.
Newton's Argument. — Here we encounter a very important
and even somewhat disconcerting question. I have said the prin-
ciple of relative motion was for us not solely a result of experi-
ment and that a priori every contrary hypothesis would be re-
pugnant to the mind.
But then, why is the principle true only if the motion of the
movable axes is rectilinear and uniform! It seems that it ought
to impose itself upon us with the same force, if this motion is
varied, or at any rate if it reduces to a uniform rotation. Now,
in these two cases, the principle is not true. I will not dwell
long on the case where the motion of the axes is rectilinear with-
out being uniform; the paradox does not bear a moment's exam-
ination. If I am on board, and if the train, striking any ob-
RELATIVE MOTION AND ABSOLUTE MOTION 109
stade, stops suddenly, I shall be thrown against the seat in front
of me, although I have not been directly subjected to any force.
There is nothing mysterious in that; if I have undergone the
action of no external force, the train itself has experienced an
external impact. There can be nothing paradoxical in the rela-
tive motion of two bodies being disturbed when the motion of
one or the other is modified by an external cause.
I will pause longer on the case of relative motions referred to
axes which rotate uniformly. If the heavens were always
covered with clouds, if we had no means of observing the stars,
we nevertheless might conclude that the earth turns round; we
could learn this from its flattening or again by the Foucault pen-
dulum experiment.
And yet, in this case, would it have any meaning, to say the
earth turns round! If there is no absolute space, can one turn
without turning in reference to something else! and, on the other
hand, how could we admit Newton's conclusion and believe in
absolute space t
But it does not su£5ce to ascertain that all possible solutions
are equally repugnant to us ; we must analyze, in each case, the
reasons for our repugnance, so as to make our choice intelli-
gently. The long discussion which follows will therefore be
excused.
Let us resume our fiction: thick clouds hide the stars from
men, who can not observe them and are ignorant even of their
existence; how shall these men know the earth turns round!
Even more than our ancestors, no doubt, they will regard the
ground which bears them as fixed and immovable; they will
await much longer the advent of a Copernicus. But in the end
the Copernicus would come — ^how!
The students of mechanics in this world would not at first be
confronted with an absolute contradiction. In the theory of
relative motion, besides real forces, two fictitious forces are met
which are called ordinary and compound centrifugal force. Our
imaginary scientists could therefore explain everything by re-
garding these two forces as real, and they would not see therein
any contradiction of the generalized principle of inertia, for
these forces would depend, the one on the relative positions of
110 SCIENCE AND HYPOTHESIS
the various parts of the system, as real attractions do, the other
on their relative velocities, as real frictions do.
Many difSculties, however, would soon awaken their attention;
if they succeeded in realizing an isolated system, the center of
gravity of this system would not have an almost rectilinear path.
They would invoke, to explain this fact, the centrifugal forces
which they would regard as real, and which they would attribute
no doubt to the mutual actions of the bodies. Only they would
not see these forces become null at great distances, that is to say
in proportion as the isolation was better realized; far from it;
centrifugal force increases indefinitely with the distance.
This difSculty would seem to them already sufSciently great;
and yet it would not stop them long ; they would soon imagine
some very subtile medium, analogous to our ether, in which all
bodies would be immersed and which would exert a repellent
action upon them.
But this is not all. Space is symmetric, and yet the laws of
motion would not show any synunetry; they would have to dis-
tinguish between right and left. It would be seen for instance
that cyclones turn always in the same sense, whereas by reason
of symmetry these winds should turn indifferently in one sense
and in the other. If our scientists by their labor had succeeded
in rendering their universe perfectly symmetric, this symmetry
would not remain, even though there was no apparent reason
why it should be disturbed in one sense rather than in the other.
They would get themselves out of the diflBculty doubtless, they
would invent something which would be no more extraordinary
than the glass spheres of Ptolemy, and so it would go on, com-
plications accumulating, until the long-expected Copernicus
sweeps them all away at a single stroke, saying: It is much
simpler to assume the earth turns round.
And just as our Copernicus said to us : It is more convenient
to suppose the earth turns round, since thus the laws of astron-
omy are expressible in a much simpler language ; this one would
say: It is more convenient to suppose the earth turns round,
since thus the laws of mechanics are expressible in a much
simpler language.
This does not preclude maintaining that absolute space, that
RELATIVE MOTION AND ABSOLUTE MOTION
is to say the mark to which it would be necessary to refer the
earth to know whether it really moves, has no objective existence.
Hence, this afSnration; 'the earth turns round' has no mean-
ing, since it can be verified by no experiment; since such an
experiment, not only could not be either realized or dreamed by
the boldest Jules Verne, but can not be conceived of without con-
tradiction; or rather these two propositions; 'the earth turns
round,' and, 'it is more convenient to suppose the earth turns
round' have the same meaning; there is nothing more in the one
than in the other.
Perhaps one will not be content even with that, and will find
it already shocking that among all the hypotheses, or rather
all the conventions we can make on this subject, there is one more
convenient than the others.
But if it has been admitted without difficulty when it was a
question of the laws of astronomy, why should it be shocking in
that which concerns mechanics ?
We have seen that the coordinates of bodies are determined
by differential equations of the second order, and that so are the
differences of tJiese coordinates. This is what we have called
the generalized principle of inertia and the principle of relative
motion. If the distances of these bodies were determined like-
wise by equations of the second order, it seems that the mind
ought to be entirely satisfied. In what ineasure does the mind
get this satisfaction and why is it not content with it!
To account for this, we had better take a simple example.
I suppose a sj'stem analogous to our solar systv^m, but where one
can not perceive fixed stars foreign to this system, so that astron-
omers can observe only the mutual distances of the planets and
the sun, and not the absolute longitudes of the planets. If we
deduce directly from Newton's law the differential equations
which define the variation o£ these distances, these equations will
not be of the second order. I mean that if. besides Newton's law,
one knew the initial values of these distances and of their de-
rivatives with respect to tlie time, that would not suffice to deter-
mine the valnes of these same distances at a subsequent instant.
There would still be lacking one datum, and this datum might be
ior instance what astronomers call the area-constant.
112 SCIENCE AND HYPOTHESIS
But here two different points of view may be taken ; we may
distinguish two sorts of constants. To the eyes of the physicist
the world reduces to a series of phenomena, depending, on the
one handy solely upon the initial phenomena ; on the other hand,
upon the laws which bind the consequents to the antecedents.
If then observation teaches us that a certain quantity is a con-
stant, we shall have the choice between two conceptions.
Either we shall assume that there is a law requiring this
quantity not to vary, but that by chance, at the beginning of
the ages, it had, rather than another, this value it has been
forced to keep ever since. This quantity might then be called
an accidental constant.
Or else we shall assume, on the contrary, that there is a law
of nature which imposes upon this quantity such a value and
not such another.
We shall then have what we may call an essential constant.
For example, in virtue of Newton's laws, the duration of the
revolution of the earth must be constant. But if it is 366
sidereal days and something over, and not 300 or 400, this is in
consequence of I know not what initial chance. This is an
accidental constant. If, on the contrary, the exponent of the
distance which figures in the expression of the attractive force is
equal to — 2 and not to — 3, this is not by chance, but because
Newton's law requires it. This is an essential constant.
I know not whether this way of giving chance its part is
legitimate in itself, and whether this distinction is not somewhat
artificial ; it is certain at least that, so long as nature shall have
secrets, this distinction will be in application extremely arbitrary
and always precarious.
As to the area-constant, we are accustomed to regard it as
accidental. Is it certain our imaginary astronomers would do
the same? If they could have compared two different solar
systems, they would have the idea that this constant may have
several different values; but my very supposition in the begin-
ning was that their system should appear as isolated, and that
they should observe no star foreign to it. Under these condi-
tions, they would see only one single constant which would have
a single value absolutely invariable; they would be led without
any doubt to regard it as an essential constant.
RELATIVE MOTION AND ABSOLUTE MOTION 113
A word in passing to forestall an objection: the inhabitants
of this imaginary world could neither observe nor define the
area-constant as we do, since the absolute longitudes escape them;
that would not preclude their being quickly led to notice a cer-
tain constant which would introduce itself naturally into their
equations and which would be nothing but what we call the area-
constant.
But then see what would happen. If the area-constant is
regarded as essential, as depending upon a law of nature, to cal-
culate the distances of the planets at any instant it will suffice
to know the initial values of these distances and those of their
first derivatives. From this new point of view, the distances will
be determined by differential equations of the second order.
Yet would the mind of these astronomers be completely satis-
fied T I do not believe so; first, they would soon perceive that
in differentiating their equations and thus raising their order,
these equations became much simpler. And above all they would
be struck by the difficulty which comes from symmetry. It
would be necessary to assume different laws, according as the
aggregate of the planets presented the figure of a certain polyhe-
dron or of the symmetric polyhedron, and one would escape from
this consequence only by regarding the area-constant as acci-
dental.
I have taken a very special example, since I have supposed
astronomers who did not at all consider terrestrial mechanics,
and whose view was limited to the solar system. Our universe is
more extended than theirs, as we have fixed stars, but still it too
is limited, and so we might reason on the totality of our universe
as the astronomers on their solar system.
Thus we see that finally we should be led to conclude that the
equations which define distances are of an order superior to the
second. Why should we be shocked at that, why do we find it
perfectly natural for the series of phenomena to depend upon
the initial values of the first derivatives of these distances, while
we hesitate to admit that they may depend on the initial values of
the second derivatives? This can only be because of the habits
of mind created in us by the constant study of the generalized
principle of inertia and its consequences.
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CHAPTER VIII
Energy and Thermodynamics
Energetics. — The difficulties inherent in the classic mechan-
ics have led certain minds to prefer a new system they call
energetics.
Energetics took its rise as an outcome of the discovery of the
principle of the conservation of energy. Helmholtz gave it its
final form.
It begins by defining two quantities which play the funda-
mental role in this theory. They are kinetic energy, or vis viva,
and potential energy.
All the changes which bodies in nature can undergo are regu-
lated by two experimental laws:
1^ The sum of kinetic energy and potential energy is con-
stant. This is the principle of the conservation of energy.
2° If a system of bodies is at A at the time ^o ^^^ &t B at
the time t^, it always goes from the first situation to the second
in such a way that the mean value of the difference between the
two sorts of energy, in the interval of time which separates the
two epochs ^0 <^d ^1) ^^7 ^^ ^ small as possible.
This is Hamilton's principle, which is one of the forms of the
principle of least action.
The energetic theory has the following advantages over the
classic theory:
1° It is less incomplete; that is to say, Hamilton's principle
and that of the conservation of energy teach us more than the
fundamental principles of the classic theory, and exclude certain
motions not realized in nature and which would be compatible
with the classic theory :
2° It saves us the hypothesis of atoms, which it was almost
impossible to avoid with the classic theory.
But it raises in its turn new difficulties :
The definitions of the two sorts of energy would raise diffi-
culties almost as great as those of force and mass in the first
115
116 SCIENCE AND HYPOTHESIS
STstem. Yet they may be gotten over more easily, at least in
the simplest cases.
Suppose an isolated system formed of a certain number of
material points; suppose these points subjected to forces depend-
ing only on their relative position and their mutual distances,
and independent of their velocities. In virtue of the principle
of the conservation of energy, a function of forces must exist.
In this simple case the enunciation of the principle of the
conservation of energy is of extreme simplicity. A certain quan-
tity, accessible to experiment, must remain constant. This quan-
tity is the sum of two terms ; the first depends only on the posi-
tion of the material points and is independent of their velocities;
the second is proportional to the square of these velocities. This
resolution can take place only in a single way.
The first of these terms, which I shall call U, will be the
potential energy; the second, which I shall call T, will be the
kinetic energy.
It is true that ii T+U ib a constant, so is any function of
T+U,
</>(T+ U),
But this function <I>{T^U) will not be the sum of two terms the
one independent of the velocities, the other proportional to the
square of these velocities. Among the functions which remain
constant there is only one which enjoys this property, that is
T+ 17 (or a linear function ot T-^-Uy which comes to the same
thing, since this linear function may always be reduced to T -j- 17
by change of unit and of origin). This then is what we shall
call energy ; the first term we shall call potential energy and the
second kinetic energy. The definition of the two sorts of energy
can therefore be carried through without any ambiguity.
It is the same with the definition of the masses. Kinetic
energy, or vis viva, is expressed very simply by the aid of the
masses and the relative velocities of all the material points with
reference to one of them. These relative velocities are accessible
to observation, and, when we know the expression of the kinetic
energy as function of these relative velocities, the coefScients of
this expression will give us the masses.
ENERGY AND THERMODYNAMICS 117
Thus, in this simple case, the fundamental ideas may be de-
fined without difSculty. But the difficulties reappear in the
more complicated cases and, for instance, if the forces, in lieu
of depending only on the distances, depend also on the velocities.
For example, Weber supposes the mutual action of two electric
molecules to depend not only on their distance, but on their veloc-
ity and their acceleration. If material points should attract each
other according to an analogous law, V would depend on the
velocity, and might contain a term proportional to the square of
the velocity.
Among the terms proportional to the squares of the velocities^
how distinguish those which come from T or from 17 T Conse-
quently, how distinguish the two parts of energy T
But still more; how define energy itself T We no longer have
any reason to take as definition T^TJ rather than any other
function of T + U, when the property which characterized T + Z7
has disappeared, that, namely, of being the sum of two terms of
a particular form.
But this is not all; it is necessary to take account, not only
of mechanical energy properly so called, but of the other forms
of energy, heat, chemical energy, electric energy, etc. The prin-
ciple of the conservation of energy should be written :
r-f i7 + o=coii8t.
where T would represent the sensible kinetic energy, TJ the poten-
tial energy of position, depending only on the position of the
bodies, Q the internal molecular energy, under the thermal,
chemic or electric form.
All would go well if these three terms were absolutely distinct,
if T were proportional to the square of the velocities, TJ inde-
pendent of these velocities and of the state of the bodies, Q inde-
pendent of the velocities and of the positions of the bodies and
dependent only on their internal state.
The expression for the energy could be resolved only in one
single way into three terms of this form.
But this is not the case ; consider electrified bodies ; the electro-
static energy due to their mutual action will evidently depend
upon their charge, that is to say, on their state ; but it will equally
118 SCIENCE AND HYPOTHESIS
depend upon their position. If these bodies are in motion, they
will act one upon another eleetrodynamically and the electro-
dynamic energy will depend not only upon their state and their
position, but upon their velocities.
We therefore no longer have any means of making the sepa-
ration of the terms which should make part of T, of U and of Q,
and of separating the three parts of energy.
If (T+U + Q) is constant so is any function <^(T + 17 -fQ).
If T + J7 + P were of the particular form I have above
considered, no ambiguity would result; among the functions
^(T + J7 + Q) which remain constant, there would only be one
of this particular form, and that I should convene to call energy.
But as I have said, this is not rigorously the case; among
the functions which remain constant, there is none which can
be put rigorously under this particular form ; hence, how choose
among them the one which should be called energy T We no
longer have anything to guide us in our choice.
There only remains for us one enunciation of the principle of
the conservation of enei^: There is something which remains
constant. Under this form it is in its turn out of the reach of
experiment and reduces to a sort of tautology. It is clear that if
the world is governed by laws, there will be quantities which will
remain constant. Like Newton's laws, and, for an analogous
reason, the principle of the conservation of energy, founded on
experiment, could no longer be invalidated by it.
This discussion shows that in passing from the classic to the
energetic system progress has been made ; but at the same time
it shows this progress is insufScient.
Another objection seems to me still more grave: the prin-
ciple of least action is applicable to reversible phenomena; but it
is not at all satisfactory in so far as irreversible phenomena are
concerned ; the attempt by Helmholtz to extend it to this kind of
phenomena did not succeed and could not succeed ; in this regard
everything remains to be done. The very statement of the prin-
ciple of least action has something about it repugnant to the mind.
To go from one point to another, a material molecule, acted upon
by no force, but required to move on a surface, will take the
geodesic line, that is to say, the shortest path.
ENERGY AND THERMODYNAMICS 119
This molecule seems to know the point whither it is to go, to
foresee the time it would take to reach it by such and such
a route, and then to choose the most suitable path. The state-
ment presents the molecule to us, so to speak, as a living and
free being. Clearly it would be better to replace it by an enun-
ciation less objectionable, and where, as the philosophers would
say, final causes would not seem to be substituted for efficient
causes.
Thermodynamics.^ — The role of the two fundamental prin-
ciples of thermodynamics in all branches of natural philosophy
becomes daily more important. Abandoning the ambitious the-
ories of forty years ago, which were encumbered by molecular
hypotheses, we are trying to-day to erect upon thermodynamics
alone the entire edifice of mathematical physics. Will the two
principles of Mayer and of Clausius assure to it foundations
solid enough for it to last some timet No one doubts it; but
whence comes this confidence T
An eminent physicist said to me one day d propos of the law
of errors: "All the world believes it firmly, because the mathe-
maticians imagine that it is a fact of observation, and the ob-
servers that it is a theorem of mathematics." It was long so for
the principle of the conservation of energy. It is no longer so
to-day ; no one is ignorant that this is an experimental fact.
But then what gives us the right to attribute to the principle
itself more generality and more precision than to the experiments
which have served to demonstrate it? This is to ask whether
it is legitimate, as is done every day, to generalize empirical
data, and I shall not have the presumption to discuss this ques-
tion, after so many philosophers have vainly striven to solve
it. One thing is certain; if this power were denied us, science
could not exist or, at least, reduced to a sort of inventory, to
the ascertaining of isolated facts, it would have no value for us,
since it could give no satisfaction to our craving for order and
harmony and since it would be at the same time incapable of
foreseeing. As the circumstances which have preceded any fact
will probably never be simultaneously reproduced, a first general-
1 The following lines are a partial reproduction of the preface of mj
book Thermodynamigue,
120 SCIENCE AND HYPOTHESIS
ization is already necessary to foresee whether this fact will be
reproduced again after the least of these circumstances shall
be changed.
But every proposition may be generalized in an infinity of
ways. Among all the generalizations possible, we must choose,
and we can only choose the simplest. We are therefore led to act
as if a simple law were, other things being equal, more probable
than a complicated law.
Half a century ago this was frankly confessed, and it was
proclaimed that nature loves simplicity; she has since too often
given us the lie. To-day we no longer confess this tendency,
and we retain only so much of it as is indispensable if science
is not to become impossible.
In formulating a general, simple and precise law on the basis
of experiments relatively few and presenting certain divergences,
we have therefore only obeyed a necessity from which the human
mind can not free itself.
But there is something more, and this is why I dwell upon
the point.
No one doubts that Mayer's principle is destined to survive
all the particular laws from which it was obtained, just as New-
ton's law has survived Kepler's laws, from which it sprang,
and which are only approximative if account be taken of
perturbations.
Why does this principle occupy thus a sort of privileged place
among all the physical lawsT There are many little reasons
for it.
First of all it is believed that we could not reject it or even
doubt its absolute rigor without admitting the possibility of per-
petual motion ; of course we are on our guard at such a prospect,
and we think ourselves less rash in afiSrming Mayer's principle
than in denying it.
That is perhaps not wholly accurate ; the impossibility of per-
petual motion implies the conservation of energy only for re-
versible phenomena.
The imposing simplicity of Mayer's principle likewise con-
tributes to strengthen our faith. In a law deduced immediately
from experiment, like Mariotte's, this simplicity would rather
ENERGY AND THEBM0DTNAMIC8 121
seem to us a reason for distrust; but here this is no longer the
case; we see elements, at first sight disparate, arrange them-
selves in an unexpected order and form a harmonious whole ; and
we refuse to believe that an unforeseen harmony may be a
simple effect of chance. It seems that our conquest is the dearer
to us the more effort it has cost us, or that we are the surer of
having wrested her true secret from nature the more jealously
she has hidden it from us.
But those are only little reasons; to establish Mayer's law as
an absolute principle, a more profound discussion is necessary.
But if this be attempted, it is seen that this absolute principle is
not even easy to state.
In each particular case it is clearly seen what energy is and at
least a provisional definition of it can be given; but it is im-
jKMSsible to find a general definition for it.
If we try to enunciate the principle in all its generality and
apply it to the universe, we see it vanish, so to speak, and nothing
is left but this : There is something which remains constant.
But has even this any meaning T In the determinist hypoth-
esis, the state of the universe is determined by an extremely great
number n of parameters which I shall call x^, X2, > . . Xn. As
soon as the values of these n parameters at any instant are
known, their derivatives with respect to the time are likewise
known and consequently the values of these same parameters at
a preceding or subsequent instant can be calculated. In other
words, these n parameters satisfy n differential equations of the
first order.
These equations admit of n — 1 integrals and consequently
there are n — 1 functions of x^, Xz, ... Xn, which remain
constant. If then we say there is something which remains
constant, we only utter a tautology. We should even be puzzled
to say which among all our integrals should retain the name of
energy.
Besides, Mayer's principle is not understood in this sense
when it is applied to a limited system. It is then assumed that
p of our parameters vary independently, so that we only have
n — p relations, generally linear, between our n parameters and
their derivatives.
122 SCIENCE AND HYPOTHESIS
To simplify the enunciation, suppose that the sum of the
work of the external forces is null, as well as that of the quan-
tities of heat given off to the outside. Then the signification
of our principle will be:
There is a combination of these n — p relations whose first
member is an exact differential; and then this differential vanish-
ing in virtue of our n — p relations, its integral is a constant
and this integral is called energy.
But how can it be possible that there are several parameters
whose variations are independent T That can only happen under
the influence of external forces (although we have supposed, for
simplicity, that the algebraic sum of the effects of these forces
is null). In fact, if the system were completely isolated from
all external action, the values of our n parameters at a given
instant would suffice to determine the state of the system at any
subsequent instant, provided always we retain the determinist
hypothesis; we come back therefore to the same difficulty as
above.
If the future state of the system is not entirely determined by
its present state, this is because it depends besides upon the
state of bodies external to the system. But then is it probable
that there exist between the parameters x, which define the state
of the system, equations independent of this state of the external
bodies ? and if in certain cases we believe we can find such, is this
not solely in consequence of our ignorance and because the influ-
ence of these bodies is too slight for our experimenting to
detect it?
If the system is not regarded as completely isolated, it is
probable that the rigorously exact expression of its internal
energy will depend on the state of the external bodies. Again,
I have above supposed the sum of the external work was null,
and if we try to free ourselves from this rather artificial restric-
tion, the enunciation becomes still more difficult.
To formulate Mayer's principle in an absolute sense, it is
therefore necessary to extend it to the whole universe, and then
we find ourselves face to face with the very difficulty we sought
to avoid.
In conclusion, using ordinary language, the law of the con-
ENERGY AND THERMODYNAMICS 128
senration of energy can have only one signification, which is
that there is a property conunon to all the possibilities; but on
the determinist hypothesis there is only a single possibility, and
then the law has no longer any meaning.
On the indeterminist hypothesis, on the contrary, it would
have a meaning, eveh if it were taken in an absolute sense; it
would appear as a limitation imposed upon freedom.
But this word reminds me that I am digressing and am on
the point of leaving the domain of mathematics and physics. I
check myself therefore and will stress of all this discussion only
one impression, that Mayer's law is a form flexible enough for
us to put into it almost whatever we wish. By that I do not mean
it corresponds to no objective reality, nor that it reduces itself
to a mere tautology, since, in each particular case, and provided
one does not try to push to the absolute, it has a perfectly clear
meaning.
This flexibility is a reason for believing in its permanence,
and as, on the other hand, it will disappear only to lose itself
in a higher harmony, we may work with confidence, supporting
ourselves upon it, certain beforehand that our labor will not be
lost.
Almost everything I have just said applies to the principle
of Clausius. What distinguishes it is that it is expressed by
an inequality. Perhaps it will be said it is the same with all
physical laws, since their precision is always limited by errors
of observation. But they at least claim to be first approxima-
tions, and it is hoped to replace them little by little by laws more
and more precise. If, on the other hand, the principle of Clau-
sius reduces to an inequality, this is not caused by the imper-
fection of our means of observation, but by the very nature of
the question.
General Conclusions on Pabt Thibd
The principles of mechanics, then, present themselves to us
under two different aspects. On the one hand, they are truths
founded on experiment and approximately verified so far as
eoncems almost isolated systems. On the other hand, they are
124 SCIENCE AND HYPOTHESIS
postulates applicable to the totality of the universe and reg
as rigorously true.
If these postulates possess a generality and a certainty
are lacking to the experimental verities whence they are d
this is because they reduce in the last analysis to a mer
vention which we have the right to make, because we are c
beforehand that no experiment can ever contradict it.
This convention, however, is not absolutely arbitrary; i
not spring from our caprice ; we adopt it because certain e
ments have shown us that it would be convenient.
Thus is explained how experiment could make the prio
of mechanics, and yet why it can not overturn them.
Compare with geometry: The fundamental propositi^
geometry, as for instance Euclid's postulate, are nothing
than conventions, and it is just as unreasonable to ii
whether they are true or false as to ask whether the metri
tem is true or false.
Only, these conventions are convenient, and it is certain e
ments which have taught us that.
At first blush, the analogy is complete; the role of e
ment seems the same. One will therefore be tempted tc
Either mechanics must be regarded as an experimental sc
and then the same must hold for geometry ; or else, on th
trary, geometry is a deductive science, and then one may \
much of mechanics.
Such a conclusion would be illegitimate. The experi
which have led us to adopt as more convenient the fundan
conventions of geometry bear on objects which have nothi
common with those geometry studies ; they bear on the prop
of solid bodies, on the rectilinear propagation of light,
are experiments of mechanics, experiments of optics; the
not in any way be regarded as experiments of geometry,
even the principal reason why our geometry seems conv<
to us is that the different parts of our body, our eye, our \
have the properties of solid bodies. On this account, our f
mental experiments are preeminently physiological experii
which bear, not on space which is the object the geometer
ENERGY AND THERMODTNAMICS 125
stady, but on his body, that is to say, on the instrument he must
use for this study.
On the contrary, the fundamental conventions of mechanics,
and the experiments which prove to us that they are convenient,
bear on exactly the same objects or on analogous objects. The
conventional and general principles are the natural and direct
generalization of the experimental and particular principles.
Let it not be said that thus I trace artificial frontiers between
the sciences ; that if I separate by a barrier geometry properly
so called from the study of solid bodies, I could just as well erect
one between experimental mechanics and the conventional me>
chanics of the general principles. In fact, who does not see that
in separating these two sciences I mutilate them both, and that
what will remain of conventional mechanics when it shall be
isolated will be only a very small thing and can in no way be com-
pared to that superb body of doctrine called geometry!
One sees now why the teaching of mechanics should remain
exi>erimental.
Only thus can it make us comprehend the genesis of the science,
and that is indispensable for the complete understanding of the
science itself.
Besides, if we study mechanics, it is to apply it; and we can
apply it only if it remains objective. Now, as we have seen, what
the principles gain in generality and certainty they lose in objec-
tivity. It is, therefore, above all with the objective side of the
principles that we must be familiarized early, and that can be
done only by going from the particular to the general, instead of
the inverse.
The principles are conventions and disguised definitions. Yet
they are drawn from experimental laws; these laws have, so
to speak, been exalted into principles to which our mind attri-
butes an absolute value.
Some philosophers have generalized too far; they believed the
principles were the whole science and consequently that the whole
adence was conventional.
This paradoxical doctrine, called nominalism, will not bear
examination.
126 SCIENCE AND HYPOTHESIS
How can a law become a principle T It expressed a relation
between two real terms A and B. But it was not rigorooaly tme,
it was only approximate. We introduce arbitrarily an inter-
mediary term C more or less fictitious, and C is by definition that
which has with A exactly the relation expressed by the law.
Then our law is separated into an absolute and rigorous prin-
ciple which expresses the relation of A to C and an experimental
law, approximate and subject to revision, which expresses the
relation ot C io B. It is clear that, however far this partition is
pushed, some laws will always be left remaining.
We go to enter now the domain of laws properly so called.
PART IV
NATURE
CHAPTER IX
Hypotheses in Physics
The RdiiE of Experiment and Qenebauzation. — Experiment
is the sole source of truth. It alone can teach us anything new ;
it alone can give us certainty. These are two points that can not
be questioned.
But then, if experiment is everything, what place will remain
for mathematical physics T What has experimental physics to do
with such an aid, one which seems useless and perhaps even
dangerous T
And yet mathematical physics exists, and has done unquestion-
able service. We have here a fact that must be explained.
The explanation is that merely to observe is not enough. We
must use our observations, and to do that we must generalize.
This is what men always have done ; only as the memory of past
errors has made them more and more careful, they have observed
more and more, and generalized less and less.
Every age has ridiculed the one before it, and accused it of
having generalized too quickly and too naively. Descartes pitied
the lonians; Descartes, in his turn, makes us smile. No doubt
our children will some day laugh at us.
But can we not then pass over immediately to the goal? Is
not this the means of escaping the ridicule that we foresee 1 Can
we not be content with just the bare experiment?
No, that is impossible; it would be to mistake utterly the
tme nature of science. The scientist must set in order. Science
is built up with facts, as a house is with stones. But a collection
of facts is no more a science than a heap of stones is a house.
127
128 SCIENCE AND HYPOTHESIS
And above all the scientist most foresee. Garlyle has some-
where said something like this: '^ Nothing but facts are of im-
portance. John Lackland passed by here. Here is something
that is admirable. Here is a reality for which I wonld give all
the theories in the world." Garlyle was a fellow countryman of
Bacon ; but Bacon would not have said that. That is the language
of the historian. The physicist would say rather: *'John Lack-
land passed by here; that makes no difference to me, for he
never will pass this way again."
We all know that there are good experiments and poor ones.
The latter will accumulate in vain ; though one may have made a
hundred or a thousand, a single piece of work by a true master,
by a Pasteur, for example, will sufSce to tumble them into oblivion.
Bacon would have well understood this ; it is he who invented the
phrase Experimentum crucis. But Garlyle would not have under-
stood it. A fact is a fact. A pupil has read a certain number on
his thermometer; he has taken no precaution; no matter, he has
read it, and if it is only the fact that counts, here is a reality of
the same rank as the peregrinations of King John Lackland. Why
is the fact that this pupil has made this reading of no interest,
while the fact that a skilled physicist had made another reading
might be on the contrary very important! It is because from the
first reading we could not infer anything. What then is a good
experiment? It is that which informs us of something besides
an isolated fact ; it is that which enables us to foresee, that is, that
which enables us to generalize.
For without generalization foreknowledge is impossible. The
circumstances under which one has worked will never reproduce
themselves all at once. The observed action then will never recur ;
the only thing that can be afSrmed is that under analogous cir-
cumstances an analogous action will be produced. In order to
foresee, then, it is necessary to invoke at least analogy, that is to
say, already then to generalize.
No matter how timid one may be, still it is necessary to inter-
polate. Experiment gives us only a certain number of isolated
I>oints. We must unite these by a continuous line. This is a
veritable generalization. But we do more ; the curve that we shall
trace will pass between the observed points and near these points ;
HYPOTHESES IN PHYSICS 129
it will not pass through these points themselves. Thus one does
ZK>t restrict himself to generalizing the experiments, but corrects
them ; and the physicist who should try to abstain from these cor-
rections and really be content with the bare experiment, would be
forced to enunciate some very strange laws.
The bare facts, then, would not be enough for us; and that is
why we must have science ordered, or rather organized.
It is often said experiments must be made without a pre-
conceived idea. That is impossible. Not only would it make
all experiment barren, but that would be attempted which could
not be done. Every one carries in his mind his own conception
of the world, of which he can not so easily rid himself. We must,
for instance, use language ; and our language is made up only of
preconceived ideas and can not be otherwise. Only these are
unconscious preconceived ideas, a thousand times more dangerous
than the others.
Shall we say that if we introduce others, of which we are
fully conscious, we shall only aggravate the evil? I think not.
I believe rather that they will serve as counterbalances to each
other — I was going to say as antidotes ; they will in general accord
ill with one another — they will come into conflict with one another,
and thereby force us to regard things under different aspects.
This is enough to emancipate us. He is no longer a slave who
can choose his master.
Thus, thanks to generalization, each fact observed enables us
to foresee a great many others ; only we must not forget that the
first alone is certain, that all others are merely probable. No
matter how solidly founded a prediction may appear to us, we are
never absolutely sure that experiment will not contradict it, if
we undertake to verify it. The probability, however, is often so
great that practically we may be content with it. It is far better
to foresee even without certainty than not to foresee at all.
One must, then, never disdain to make a verification when
opportunity offers. But all experiment is long and difficult ; the
workers are few ; and the number of facts that we need to foresee
is immense. Compared with this mass the number of direct verifi-
cations that we can make will never be anything but a negligible
quantity.
10
130 SCIENCE AND HTPOTHESIS
Of this few tliat we can directly attain, we mnst make the best
nse ; it is very necessary to get from every experiment the greatest
IKMSsible number of predictions, and with the highest possible
degree of probability. The problem is, so to sp^ik, to increase
the yield of the scientific machine.
Let XLS compare science to a library that ought to grow continu-
*
ally. The librarian has at his disposal for his purchases only
insufficient funds. He ought to make an effort not to waste them.
It is experimental physics that is entrusted with the purchases.
It alone, then, can enrich the library.
As for mathematical physics, its task will be to make out the
catalogue. If the catalogue is well made, the library will not be
any richer, but the reader will be helped to use its riches.
And even by showing the librarian the gaps in his collections,
it will enable him to make a judicious use of his funds ; which is all
the more important because these funds are entirely inadequate.
Such, then, is the role of mathematical physics. It must direct
generalization in such a manner as to increase what I just now
called the yield of science. By what means it can arrive at this,
and how it can do it without danger, is what remains for us to
investigate.
Thb Unity op Nature. — ^Let us notice, first of all, that every
generalization implies in some measure the belief in the unity
and simplicity of nature. As to the unity there can be no diffi-
culty. If the different parts of the universe were not like the
members of one body, they would not act on one another, they
would know nothing of one another ; and we in particular would
know only one of these parts. We do not ask, then, if nature is
one, but how it is one.
As for the second point, that is not such an easy matter. It is
not certain that nature is simple. Can we without danger act
as if it were !
There was a time when the simplicity of Mariotte's law was
an argument invoked in favor of its accuracy ; when Fresnel him-
self, after having said in a conversation with Laplace that nature
was not concerned about analytical difficulties, felt himself
obliged to make explanations, in order not to strike too hard
at prevailing opinion.
HYPOTHESES /JV PHYSICS
To-day ideas have greatly changed ; and yet, those who do not
believe that natural laws have to be simple, are still often obliged
to act as if they did. They could not entirely avoid this neceesity
without making impossible all generalization, and consequently
all science.
It is clear that any fact can be generalized in an infinity of
■ways, and it is a question of choice. The choipe can be guided
only by considerations of amplieity. Let us take the most com-
monplace case, that of interpolation. We pass a continuous line,
AS regular as possible, between the points given by observation.
Why do we avoid points making angles and too abrupt turns T
Why do we not make our curve describe the most capricious zig-
aagsl It is because we know beforehand, or believe we know, that
the law to be expressed can not be so complicated as all that.
We may calculate the mass of Jupiter from either the move-
ments of its satellites, or the perturbations of the major planets,
or those of the minor planets. If we take the averages of the
determinations obtained by these three methods, we find three
numbers very close together, but different. We might interpret
this result by supposing that the coefficient of gravitation is not
the same in the tliree eases. The observations would certainly be
maeh better represented. Why do we reject this interpretation t
Not because it is absurd, but because it is needlessly complicated.
We shall only accept it when we are forced to, and that is not yet.
To sum up, ordinarily every law is held to be simple till the
contrary is proved.
This custom is imposed upon physicists by the causes that I
have just explained. But how shall we justify it in the presence
of discoveries that show us every day new details that are richer
and more complex V How shall we even reconcile it with the
belief in the unity of nature! For if everything depends on
rveiything, relationships where flo many diverse factors enter can
DO longer be smple.
If we study the history of science, we see happen two invetse
phenomena, so to speak. Sometimes simplicity hides under com-
plex appearances ; sometimes it is the simplicity which is appar-
ent, and which disguises extremely complicated realities.
What is more complicated than the confused movements of
i
132 SCIENCE AND HYPOTHESIS
the planets T What simpler than Newton's lawT Here nature,
making sport, as Fresnel said, of analytical difficulties, employs
only simple means, and by combining them produces I know not
what inextricable tangle. Here it is the hidden simplicity which
must be discovered.
Examples of the opposite abound. In the kinetic theory of
gases, one deals with molecules moving with great velocities,
whose paths, altered by incessant collisions, have the most capri-
cious forms and traverse space in every direction. The observable
result is Mariotte's simple law. Every individual fact was com-
plicated. The law of great numbers has reestablished simplicity
in the average. Here the simplicity is merely apparent, and only
the coarseness of our senses prevents our perceiving the complexity.
Many phenomena obey a law of proportionality. But why!
Because in these phenomena there is something very small. The
simple law observed, then, is only a result of the general ana-
lytical rule that the infinitely small increment of a function is
proportional to the increment of the variable. As in reality our
increments are not infinitely small, but very small, the law of
proportionality is only approximate, and the simplicity is only
apparent. What I have just said applies to the rule of the super-
position of small motions, the use of which is so fruitful, and
which is the basis of optics.
And Newton's law itself? Its simplicity, so long undetected,
is perhaps only apparent. Who knows whether it is not due to
some complicated mechanism, to the impact of some subtile matter
animated by irregular movements, and whether it has not become
simple only through the action of averages and of great num-
bers? In any ease, it is difficult not to suppose that the true law
contains complementary terms, which would become sensible at
small distances. If in astronomy they are negligible as modify-
ing Newton's law, and if the law thus regains its simplicity, it
would be only because of the immensity of celestial distances.
No doubt, if our means of investigation should become more
and more penetrating, we should discover the simple under the
complex, then the complex under the simple, then again the simple
under the complex, and so on, without our being able to foresee
what wiU be the last term.
ETP0THE8E8 IN PHYSICS 133
We must stop somewhere, and that science may be possible, we
must stop when we have found simplicity. This is the only ground
on which we can rear the edifice of our generalizations. But
this simplicity being only apparent, will the ground be firm
enough? This is what must be investigated.
For that purpose, let us see what part is played in our gener-
alizations by the belief in simplicity. We have verified a simple
law in a good many particular cases ; we refuse to admit that this
agreement, so often repeated, is simply the result of chance, and
conclude that the law must be true in the general case.
Kepler notices that a planet's positions, as observed by Tycho,
are all on one ellipse. Never for a moment does he have the
thought that by a strange play of chance Tycho never observed
the heavens except at a moment when the real orbit of the planet
happened to cut this ellipse.
What does it matter then whether the simplicity be real, or
whether it covers a complex reality? Whether it is due to the
influence of great numbers, which levels down individual diflfer-
ences, or to the greatness or smallness of certain quantities, which
allows us to neglect certain terms, in no case is it due to chance.
This simplicity, real or apparent, always has a cause. We can
always follow, then, the same course of reasoning, and if a simple
law has been observed in several particular cases, we can legiti-
mately suppose that it will still be true in analogous cases. To
refuse to do this would be to attribute to chance an inadmis-
sible role.
There is, however, a diflFerence. If the simplicity were real
and essential, it would resist the increasing precision of our means
of measure. If then we believe nature to be essentially simple,
we must, from a simplicity that is approximate, infer a simplicity
that is rigorous. This is what was done formerly; and this is
what we no longer have a right to do.
The simplicity of Kepler's laws, for example, is only apparent.
That does not prevent their being applicable, very nearly, to all
systems analogous to the solar system ; but it does prevent their
being rigorously exact.
The RdLE op Hypothesis. — All generalization is a hypothesis.
Hypothesis, then, has a necessary role that no one has ever con-
134 SCIENCE AND HYPOTHESIS
tested. Only, it ought always, as soon as possible and as often
as possible, to be subjected to verification. And, of course, if it
does not stand this test, it ought to be abandoned without reserve.
This is what we generally do, but sometimes with rather an ill
humor.
Well, even this ill humor is not justified. The physicist who
has just renounced one of his hypotheses ought, on the contraryi
to be full of joy ; for he has found an unexpected opportunity '
for discovery. His hypothesis, I imagine, had not been adopted
without consideration ; it took account of all the known factors
that it seemed could enter into the phenomenon. If the test does
not support it, it is because there is something unexpected and
extraordinary ; and beoause there is going to be something found
that is unknown and new.
Has the discarded hypothesis, then, been barren T Far from
that, it may be said it has rendered more service than a true
hypothesis. Not only has it been the occasion of the decisive
experiment, but, without having made the hypothesis, the experi-
ment would have been made by chance, so that nothing would
have been derived from it. One would have seen nothing ex-
traordinary ; only one fact the more would have been catalogued
without deducing from it the least consequence.
Now on what condition is the use of hypothesis without danger T
The firm determination to submit to experiment is not enough ;
there are still dangerous hypotheses; first, and above all, those
which are tacit and unconscious. Since we make them without
knowing it, we are powerless to abandon them. Here again, then,
is a service that mathematical physics can render us. By the
precision that is characteristic of it, it compels us to formulate
all the hypotheses that we should make without it, but uncon-
sciously.
Let us notice besides that it is important not to multiply
hypotheses beyond measure, and to make them only one after the
other. If we construct a theory based on a number of hypotheses,
and if experiment condemns it, which of our premises is it neces-
sary to change 1 It will be impossible to know. And inversely,
if the experiment succeeds, shall we believe that we have demon-
HYPOTHESES IN PHYSICS 135
strated all the hypotheses at onceT Shall we believe that with
one single equation we have determined several unknowns T
We must equally take care to distinguish between the different
kinds of hypotheses. There are first those which are perfectly
natural and from which one can scarcely escape. It is difficult
not to suppose that the influence of bodies very remote is quite
negligible, that small movements follow a linear law, that the
effect is a continuous function of its cause. I will say as much
of the conditions imposed by symmetry. All these hypotheses
form, as it were, the common basis of all the theories of mathe-
matical physics. They are the last that ought to be abandoned.
There is a second class of hypotheses, that I shall term neutral.
In most questions the analyst assumes at the beginning of his
calculations either that matter is continuous or, on the contrary,
that it is formed of atoms. He might have made the opposite
assumption without changing his results. He would only have
had more trouble to obtain them ; that is all. If, then, experiment
confirms his conclusions, will he think that he has demonstrated,
for instance, the real existence of atoms T
In optical theories two vectors are introduced, of which one
is regarded as a velocity, the other as a vortex. Here again is
a neutral hypothesis, since the same conclusions would have been
reached by taking precisely the opposite. The success of the
experiment, then, can not prove that the first vector is indeed a
velocity ; it can only prove one thing, that it is a vector. This
is the only hypothesis that has really been introduced in the
premises. In order to give it that concrete appearance which the
weakness of our minds requires, it has been necessary to consider
it either as a velocity or as a vortex, in the same way that it has
been necessary to represent it by a letter, either z or y. The
result, however, whatever it may be, will not prove that it was
right or wrong to regard it as a velocity any more than it will
prove that it was right or wrong to call it x and not y.
These neutral hypotheses are never dangerous, if only their
character is not misunderstood. They may be useful, either as
devices for computation, or to aid our understanding by concrete
images, to fix our ideas as the saying is. There is, then, no occa-
sion to exclude them.
136 SCIENCE AND HTP0THE8I8
The hypotheses of the third class are the real generalizations.
They are the ones that experiment must confirm or invalidate.
Whether verified or condemned, they will always be froitfoL
But for the reasons that I have set forth, they will only be fruit-
ful if they are not too numerous.
Origin op Mathematical Physics. — ^Let us penetrate further,
and study more closely the conditions that have permitted the
development of mathematical physics. We observe at once that
the efforts of scientists have always aimed to resolve the complex
phenomenon directly given by experiment into a very large num-
ber of elementary phenomena.
This is done in three different ways : first, in time. Instead of
embracing in its entirety the progressive development of a
phenomenon, the aim is simply to connect each instant with the
instant immediately preceding it. It is admitted that the actual
state of the world depends only on the immediate past, without
being directly influenced, so to speak, by the memory of a distant
past. Thanks to this postulate, instead of studying directly the
whole succession of phenomena, it is possible to confine ourselves
to writing its * differential equation.' For Kepler's laws we sul>
stitute Newton's law.
Next we try to analyze the phenomenon in space. What ex-
periment gives us is a confused mass of facts presented on a
stage of considerable extent. We must try to discover the ele-
mentary phenomenon, which will be, on the contrary, localized in
a very small region of space.
Some examples will perhaps make my thought better under-
stood. If we wished to study in all its complexity the distribu-
tion of temperature in a cooling solid, we should never succeed.
Everything becomes simple if we reflect that one point of the
solid can not give up its heat directly to a distant point ; it will
give up its heat only to the points in the immediate neighbor-
hood, and it is by degrees that the flow of heat can reach other
parts of the solid. The elementary phenomenon is the exchange
of heat between two contiguous points. It is strictly localized,
and is relatively simple, if we admit, as is natural, that it is not
influenced by the temperature of molecules whose distance is
sensible.
HYPOTHESES IN PHYSICS 137
I bend a rod. It is going to take a very complicated form,
the direct study of which would be impossible. But I shall be
able, however, to attack it, if I observe that its flexure is a result
only of the deformation of the very small elements of the rod, and
that the deformation of each of these elements depends only on
the forces that are directly applied to it, and not at all on those
which may act on the other elements.
In all these examples, which I might easily multiply, we
admit that there is no action at a distance, or at least at a great
distance. This is a hypothesis. It is not always true, as the
law of gravitation shows us. It must, then, be submitted to veri-
fication. If it is confirmed, even approximately, it is precious,
for it will enable us to make mathematical physics, at least by
successive approximations.
If it does not stand the test, we must look for something else
analogous; for there are still other means of arriving at the
elementary phenomenon. If several bodies act simultaneously,
it may happen that their actions are independent and are simply
added to one another, either as vectors or as scalars. The ele-
mentary phenomenon is then the action of an isolated body. Or
again, we have to deal with small movements, or more generally
with small variations, which obey the well-known law of super-
position. The observed movement will then be decomposed into
simple movements, for example, sound into its harmonics, white
light into its monochromatic components.
When we have discovered in what direction it is advisable to
look for the elementary phenomenon, by what means can we
reach it?
First of all, it will often happen that in order to detect it,
or rather to detect the part of it useful to us, it will not be neces-
sary to penetrate the mechanism ; the law of great numbers will
suffice.
Let us take again the instance of the propagation of heat.
Every molecule emits rays toward every neighboring molecule.
According to what law, we do not need to know. If we should
make any supposition in regard to this, it would be a neutral
hypothesis and consequently useless and incapable of verification.
And, in fact, by the action of averages and thanks to the sym-
138 SCIENCE AND HYPOTHESIS
metry of the medium, all the differences are leveled down, and
whatever hypothesis may be made, the result is always the same.
The same circumstance is presented in the theory of electricity
and in that of capillarity. The neighboring molecules attract
and repel one another. We do not need to know according to
what law; it is enough for us that this attraction is sensible only
at small distances, that the molecules are very numerous, that
the medium is symmetrical, and we shall only have to let the law
of great numbers act.
Here again the simplicity of the elementary phenomenon
was hidden under the complexity of the resultant observable phe-
nomenon ; but, in its turn, this simplicity was only apparent, and
concealed a very complex mechanism.
The best means of arriving at the elementary phenomenon
would evidently be experiment. We ought by experimental con-
trivance to dissociate the complex sheaf that nature offers to our
researches, and to study with care the elements as much isolated
as possible. For example, natural white light would be decom-
posed into monochromatic lights by the aid of the prism, and
into polarized light by the aid of the polarizer.
Unfortunately that is neither always possible nor always suflS-
cient, and sometimes the mind must outstrip experiment. I shall
cite only one example, which has always struck me forcibly.
If I decompose white light, I shall be able to isolate a small part
of the spectrum, but however small it may be, it will retain a
certain breadth. Likewise the natural lights, called monochrO'
matic, give us a very narrow line, but not, however, infinitely
narrow. It might be supposed that by studying experimentally
the properties of these natural lights, by working with finer and
finer lines of the spectrum, and by passing at last to the limit, so
to speak, we should succeed in learning the properties of a light
strictly monochromatic.
That would not be accurate. Suppose that two rays emanate
from the same source, that we polarize them first in two perpen-
dicular planes, then bring them back to the same plane of polari-
zation, and try to make them interfere. If the light were strictly
monochromatic, they would interfere. With our lights, which
are nearly monochromatic, there will be no interference, and
HYPOTHESES IN PHYSICS 139
that no matter how narrow the line. In order to be otherwise
it would have to be several million times as narrow as the finest
known lines.
Here, then, the passage to the limit would have deceived us.
The mind must outstrip the experiment, and if it has done so
with success, it is because it has allowed itself to be guided by the
instinct of simplicity.
The knowledge of the elementary fact enables us to put the
problem in an equation. Nothing remains but to deduce from
this by combination the complex fact that can be observed and
verified. This is what is called integration, and is the business
of the mathematician.
It may be asked why, in physical sciences, generalization so
readily takes the mathematical form. The reason is now easy to
see. It is not only because we have numerical laws to express ; it
is because the observable phenomenon is due to the superposition
of a great number of elementary phenomena all alike. Thus
quite naturally are introduced differential equations.
It is not enough that each elementary phenomenon obeys sim-
ple laws ; all those to be combined must obey the same law. Then
only can the intervention of mathematics be of use ; mathematics
teaches us in fact to combine like with like. Its aim is to learn
the result of a combination without needing to go over the com-
bination piece by piece. If we have to repeat several times the
same operation, it enables us to avoid this repetition by telling us
in advance the result of it by a sort of induction. I have ex-
plained this above, in the chapter on mathematical reasoning.
But, for this, all the operations must be alike. In the opposite
case, it would evidently be necessary to resign ourselves to doing
them in reality one after another, and mathematics would become
useless.
It is then thanks to the approximate homogeneity of the
matter studied by physicists, that mathematical physics could be
bom.
In the natural sciences, we no longer find these conditions:
homogeneity, relative independence of remote parts, simplicity
of the elementary fact ; and this is why naturalists are obliged
to resort to other methods of generalization.
CHAPTER X
The Theories of Modebn Physics
Meaning op Physical Theories. — The laity are stmck to
see how ephemeral scientific theories are. After some years of
prosperity, they see them successively abandoned ; they see ruins
accumulate upon ruins ; they foresee that the theories fashionable
to-day will shortly succumb in their turn and hence they con-
clude that these are absolutely idle. This is what they call the
bankruptcy of science.
Their scepticism is superficial ; they give no account to them-
selves of the aim and the role of scientific theories; otherwise
they would comprehend that the ruins may still be good for
something.
No theory seemed more solid than that of Fresnel which
attributed light to motions of the ether. Yet now Maxwell's
is preferred. Does this mean the work of Fresnel was in vainT
No, because the aim of Fresnel was not to find out whether
there is really an ether, whether it is or is not formed of atoms,
whether these atoms really move in this or that sense ; his object
was to foresee optical phenomena.
Now, Fresnel's theory always permits of this, to-day as well
as before Maxwell. The diflFerential equations are always true;
they can always be integrated by the same procedures and the
results of this integration always retain their value.
And let no one say that thus we reduce physical theories to
the role of mere practical recipes; these equations express rela-
tions, and if the equations remain true it is because these rela-
tions preserve their reality. They teach us, now as then, that
there is such and such a relation between some thing and some
other thing; only this something formerly we called motion; we
now call it electric current. But these appellations were only
images substituted for the real objects which nature will eternally
hide from us. The true relations between these real objects are
the only reality we can attain to, and the only condition is that
140
THE THEORIES OF MODERN PHYSICS 141
the same relations exist between these objects as between the
images by which we are forced to replace them. If these rela-
tions are known to us, what matter if we deem it convenient
to replace one image by another.
That some periodic phenomenon (an electric oscillation, for
instance) is really due to the vibration of some atom which, act-
ing like a pendulum, really moves in this or that sense, is neither
certain nor interesting. But that between electric oscillation,
the motion of the pendulum and all periodic phenomena there
exists a close relationship which corresponds to a profound real-
ity ; that this relationship, this similitude, or rather this parallel-
ism extends into details ; that it is a consequence of more general
principles, that of energy and that of least action; this is what
we can affirm; this is the truth which will always remain the
same under all the costumes in which we may deem it useful to
deck it out.
Numerous theories of dispersion have been proposed; the
first was imperfect and contained only a small part of truth.
Afterwards came that of Helmholtz ; then it was modified in vari-
ous ways, and its author himself imagined another founded on
the principles of Maxwell. But, what is remarkable, all the sci-
entists who came after Helmholtz reached the same equations,
starting from points of departure in appearance very widely
separated. I will venture to say these theories are all true at
the same time, not only because they make us foresee the same
phenomena, but because they put in evidence a true relation, that
of absorption and anomalous dispersion. What is true in the
premises of these theories is what is common to all the authors;
this is the affirmation of this or that relation between certain
things which some call by one name, others by another.
The kinetic theory of gases has given rise to many objections,
which we could hardly answer if we pretended to see in it the
absolute truth. But all these objections will not preclude its
having been useful, and particularly so in revealing to us a
relation true and but for it profoundly hidden, that of the
gaseous pressure and the osmotic pressure. In this sense, then,
it may be said to be true.
When a physicist finds a contradiction between two theories
142 SCIENCE AND HYPOTHESIS
equally dear to him, he sometimes says: "We will not bother
about that, but hold firmly the two ends of the chain, though the
intermediate links are hidden from us." This argument of an
embarrassed theologian would be ridiculous if it were necessary
to attribute to physical theories the sense the laity give them.
In case of contradiction, one of them at least must then be re-
garded as false. It is no longer the same if in them be sought
only what should be sought. May be they both express true
relations and the contradiction is only in the images wherewith
we have clothed the reality.
To those who find we restrict too much the domain accessible
to the scientist, I answer: These questions which we interdict
to you and which you regret, are not only insoluble, they are
illusory and devoid of meaning.
Some philosopher pretends that all physics may be explained
by the mutual impacts of atoms. If he merely means there are
between physical phenomena the same relations as between the
mutual impacts of a great number of balls, well and good, that
is verifiable, that is perhaps true. But he means something
more ; and we think we understand it because we think we know
what impact is in itself; whyt Simply because we have often
seen games of billiards. Shall we think Qod, contemplating his
work, feels the same sensations as we in watching a billiard
match? If we do not wish to give this bizarre sense to his asser-
tion, if neither do we wish the restricted sense I have just ex-
plained, which is good sense, then it has none.
Hypotheses of this sort have therefore only a metaphorical
sense. The scientist should no more interdict them than the poet
does metaphors; but he ought to know what they are worth.
They may be useful to give a certain satisfaction to the mind,
and they will not be injurious provided they are only indifferent
hypotheses.
These considerations explain to us why certain theories, sup-
posed to be abandoned and finally condemned by experiment,
suddenly arise from their ashes and recommence a new life.
It is because they expressed true relations; and because they
had not ceased to do so when, for one reason or another, we
felt it necessary to enunciate the same relations in another
language. So they retained a sort of latent life.
THE THEORIES OF MODERN PHYSICS 143
Scarcely fifteen years ago was there anything more ridicnlons,
more naively antiquated, than Coulomb 's fluids t And yet here
they are reappearing under the name of electrons. Wherein do
these permanently electrified molecules differ from Coulomb's
electric molecules? It is true that in the electrons the electricity
is supported by a little, a very little matter ; in other words, they
have a mass (and yet this is now contested) ; but Coulomb did
not deny mass to his fluids, or, if he did, it was only with reluc-
tance. It would be rash to affirm that the belief in electrons
will not again suffer eclipse ; it was none the less curious to note
this unexpected resurrection.
But the most striking example is Camot's principle. Camot
set it up starting from false hypotheses ; when it was seen that
heat is not indestructible, but may be transformed into work, his
ideas were completely abandoned ; afterwards Clausius returned
to them and made them finally triumph. Camot's theory, under
its primitive form, expressed, aside from true relations, other
inexact relations, dihris of antiquated ideas ; but the presence of
these latter did not change the reality of the others. Clausius
had only to discard them as one lops off dead branches.
The result was the second fundamental law of thermodynamics.
There were always the same relations ; though these relations no
longer subsisted, at least in appearance, between the same ob-
jects. This was enough for the principle to retain its value.
And even the reasonings of Camot have not perished because
of that ; they were applied to a material tainted with error ; but
their form (that is to say, the essential) remained correct.
What I have just said illuminates at the same time the role
of general principles such as the principle of least action, or that
of the conservation of energy.
These principles have a very high value; they were obtained
in seeking what there was in common in the enunciation of nu-
merous physical laws; they represent therefore, as it were, the
quintessence of innumerable observations.
However, from their very generality a consequence results to
which I have called attention in Chapter VIII., namely, that
they can no longer be verified. As we can not give a general
definition of energy, the principle of the conservation of energy
144 SCIENCE AND HYPOTHESIS
signifies simply that there is something which remains constant
Well, whatever be the new notions that future experiments shall
give us about the world, we are sure in advance that there will
be something there which will remain constant and which may
be called energy.
Is this to say that the principle has no meaning and vanishes
in a tautology t Not at all ; it signifies that the different things
to which we give the name of energy are connected by a true kin-
ship ; it aflBrms a real relation between them. But then if this
principle has a meaning, it may be false ; it may be that we have
not the right to extend indefinitely its applications, and yet it is
certain beforehand to be verified in the strict acceptation of the
term ; how then shall we know when it shall have attained all the
extension which can legitimately be given it 1 Just simply when
it shall cease to be useful to us, that is, to make us correctly fore-
see new phenomena. We shall be sure in such a case that the
relation aflBrmed is no longer real; for otherwise it would be
fruitful; experiment, without directly contradicting a new ex-
tension of the principle, will yet have condemned it.
Physics and JVIechanism. — Most theorists have a constant
predilection for explanations borrowed from mechanics or dy-
namics. Some would be satisfied if they could explain all phe-
nomena by motions of molecules attracting each other according
to certain laws. Others are more exacting ; they would suppress
attractions at a distance ; their molecules should follow rectilinear
paths from which they could be made to deviate only by impacts.
Others again, like Hertz, suppress forces also, but suppose their
molecules subjected to geometric attachments analogous, for in-
stance, to those of our linkages ; they try thus to reduce dynamics
to a sort of kinematics.
In a word, all would bend nature into a certain form outside
of which their mind could not feel satisfied. Will nature be
sufficiently flexible for that?
We shall examine this question in Chapter XII., d propos of
Maxwell's theory. Whenever the principles of energy and of
least action are satisfied, we shall see not only that there is always
one possible mechanical explanation, but that there is always an
infinity of them. Thanks to a well-known theorem of Eonig's on
THE THEORIES OF MODERN PHYSICS 146
linkages, it could be shown that we can, in an infinity of ways,
explain everything by attachments after the manner of Hertz, or
also by central forces. Without doubt it could be demonstrated
just as easily that everything can always be explained by simple
impacts.
For that, of course, we need not be content with ordinary
matter, with that which falls under our senses and whose motions
we observe directly. Either we shall suppose that this common
matter is formed of atoms whose internal motions elude us, the
displacement of the totality alone remaining accessible to our
senses. Or else we shall imagine some one of those subtile fluids
which under the name of ether or under other names, have at all
times played so great a role in physical theories.
Often one goes further and regards the ether as the sole
primitive matter or even as the only true matter. The more
moderate consider common matter as condensed ether, which is
nothing startling; but others reduce still further its importance
and see in it nothing more than the geometric locus of the ether's
singularities. For instance, what we call matter is for Lord
Kelvin only the locus of points where the ether is animated by
vortex motions; for Biemann, it was the locus of points where
ether is constantly destroyed; for other more recent authors,
Wiechert or Larmor, it is the locus of points where the ether
undergoes a sort of torsion of a very particular nature. If the
attempt is made to occupy one of these points of view, I ask
myself by what right shall we extend to the ether, under pretext
that this is the true matter, mechanical properties observed in
ordinary matter, which is only false matter.
The ancient fluids, caloric, electricity, etc., were abandoned
when it was perceived that heat is not indestructible. But they
were abandoned for another reason also. In materializing them,
their individuality was, so to speak, emphasized, a sort of abyss
was opened between them. This had to be filled up on the coming
of a more vivid feeling of the unity of nature, and the perception
of the intimate relations which bind together all its parts. Not
only did the old physicists, in multiplying fluids, create entities
unnecessarily, but they broke real ties.
It is not su£Bcient for a theory to affirm no false relations, it
must not hide true relations.
11
146 SCIENCE AND HYPOTHESIS
And does our ether really exist f We know the origin of our
belief in the ether. If light reaches us from a distant star, dur-
ing several years it was no longer on the star and not yet on the
earth ; it must then be somewhere and sustained, so to speak, by
some material support.
The same idea may be expressed under a more mathematical
and more abstract form. What we ascertain are the changes un-
dergone by material molecules; we see, for instance, that our
photographic plate feels the consequences of phenomena of which
the incandescent mass of the star was the theater several years
before. Now, in ordinary mechanics the state of the system
studied depends only on its state at an instant immediately an-
terior; therefore the system satisfies differential equations. On
the contrary, if we should not believe in the ether, the state of the
material universe would depend not only on the state immedi-
ately preceding, but on states much older; the system would
satisfy equations of finite differences. It is to escape this deroga-
tion of the general laws of mechanics that we have invented the
ether.
That would still only oblige us to fill up, with the ether, the
interplanetary void, but not to make it penetrate the bosom of
the material media themselves. Fizeau's experiment goes fur-
ther. By the interference of rays which have traversed air or
water in motion, it seems to show us two different media inter-
penetrating and yet changing place one with regard to the other.
We seem to touch the ether with the finger.
Yet experiments may be conceived which would make us touch
it still more nearly. Suppose Newton's principle, of the equality
of action and reaction, no longer true if applied to matter alone,
and that we have established it. The geometric sum of all the
forces applied to all the material molecules would no longer be
null. It would be necessary then, if we did not wish to change
all mechanics, to introduce the ether, in order that this action
which matter appeared to experience should be counterbalanced
by the reaction of matter on something.
Or again, suppose we discover that optical and electrical
phenomena are influenced by the motion of the earth. We should
be led to conclude that these phenomena might reveal to us not
THE THEORIES OF MODERN PHT8IC8 147
only the relative motions of material bodies, but what would
seem to be their absolute motions. Again, an ether would be
necessary, that these so-called absolute motions should not be
their displacements with regard to a void space, but their dis-
placements with regard to something concrete.
Shall we ever arrive at that? I have not this hope, I shall
soon say why, and yet it is not so absurd, since others have
had it.
For instance, if the theory of Lorentz, of which I shall speak
in detail further on in Chapter XIII., were true, Newton's prin-
ciple would not apply to matter alone, and the difference would
not be very far from being accessible to experiment.
On the other hand, many researches have been made on the
influence of the earth's motion. The results have always been
negative. But these experiments were undertaken because the
outcome was not sure in advance, and, indeed, according to the
ruling theories, the compensation would be only approximate,
and one might expect to see precise methods give positive results.
I believe that such a hope is illusory; it was none the less
interesting to show that a success of this sort would open to us,
in some sort, a new world.
And now I must be permitted a digression ; I must explain, in
fact, why I do not believe, despite Lorentz, that more precise
observations can ever put in evidence anything else than the rela-
tive displacements of material bodies. Experiments have been
made which should have disclosed the terms of the first order;
the results have been negative; could that be by chance t No
one has assumed that ; a general explanation has been sought, and
Lorentz has found it; he has shown that the terms of the first
order must destroy each other, but not those of the second. Then
more precise experiments were made; they also were negative;
neither could this be the effect of chance; an explanation was
necessary; it was found; they always are found; of hypotheses
there is never lack.
But this is not enough ; who does not feel that this is still to
leave to chance too great a role? Would not that also be a
chance, this singular coincidence which brought it about that a
certain circumstance should come just in the nick of time to
148 SCIENCE AND HYPOTHESIS
destroy the terms of the first order, and that another circTun-
stance, wholly different, but just as opportnne, should take upon
itself to destroy those of the second order ? No, it is necessary to
find an explanation the same for the one as for the other, and
then everything leads us to think that this explanation vdll
hold good equally well for the terms of higher order, and that the
mutual destruction of these terms will be rigorous and absolute.
Present State op the Science. — ^In the history of the de-
velopment of physics we distinguish two inverse tendencies.
On the one hand, new bonds are continually being discovered
between objects which had seemed destined to remain forever
unconnected; scattered facts cease to be strangers to one another;
they tend to arrange themselves in an imposing i^mthesis.
Science advances toward unity and simplicity.
On the other hand, observation reveals to us every day new
phenomena ; they must long await their place and sometimes, to
make one for them, a comer of the edifice must be demolished.
In the known phenomena themselves, where our crude senses
showed us uniformity, we perceive details from day to day more
varied; what we believed simple becomes complex, and science
appears to advance toward variety and complexity.
Of these two inverse tendencies, which seem to triumph turn
about, which will wint If it be the first, science is possible;
but nothing proves this a priori, and it may well be feared that
after having made vain efforts to bend nature in spite of herself
to our ideal of unity, submerged by the ever-rising flood of our
new riches, we must renounce classifying them, abandon our
ideal, and reduce science to the registration of innumerable
recipes.
To this question we can not reply. All we can do is to ob-
serve the science of to-day and compare it with that of yesterday.
Prom this examination we may doubtless draw some encourage-
ment.
Half a century ago, hope ran high. The discovery of the
conservation of energy and of its transformations had revealed to
us the unity of force. Thus it showed that the phenomena of
heat could be explained by molecular motions. What was the
nature of these motions was not exactly known, but no one
THE THEORIES OF MODERN PHYSICS 149
doubted that it soon would be. For light, the task seemed com-
pletely accomplished. In what concerns electricity, things were
less advanced. Electricity had just annexed magnetism. This
was a considerable step toward unity, and a decisive step.
But how should electricity in its turn enter into the general
unity, how should it be reduced to the universal mechanism t
Of that no one had any idea. Yet the possibility o£ this reduc-
tion was doubted by none, there was faith. Finally, in what
concerns the molecular properties of material bodies, the reduc-
tion seemed still easier, but all the detail remained hazy. In
a word, the hopes were vast and animated, but vague. To-day,
what do we see t First of all, a prime progress, immense prog-
ress. The relations of electricity and light are now known ; the
three realms, of light, of electricity and of magnetism, previously
separated, form now but one ; and this annexation seems final.
This conquest, however, has cost us some sacrifices. The optical
phenomena subordinate themselves as particular cases under the
electrical phenomena ; so long as they remained isolated, it was
easy to explain them by motions that were supposed to be known
in all their details, that was a matter of course; but now an
explanation, to be acceptable, must be easily capable of extension
to the entire electric domain. Now that is a matter not without
difficulties.
The moat satisfactory theory we have is that of Lorentz, which,
as we shall see in the last chapter, explains electric currents by
the motions of little electrified particles ; it is unquestionbly the
one which best esplains the known facts, the one which illumi-
nates the greatest number of true relations, tlie one of which most
traces will be found in the final construction. Nevertheless, it
still has a serious defect, which I have indicated above; it is
contrary to Newton's law of the equality of action and reaction;
or rather, this principle, in the eyes of Lorentz, would not be
applicable to matter alone ; for it to be true, it would be necessary
to take account of the action of the ether on matter and of tb«
reaction of matter on the ether.
Now, from what we know at present, it seems probable that
things do not happen in this way.
However that may be, thanks to horentz, Fizeau's results on
150 SCIENCE AND HYPOTHESIS
the optics of moving bodies, the laws of normal and anomalous dis-
persion and of absorption find themselves linked to one another
and to the other properties of the ether by bonds which beyond
any doubt will never more be broken. See the facility with which
the new Zeeman effect has found its place already and has even
aided in classifying Faraday's magnetic rotation which had de-
fied Maxwell's efforts; this facility abundantly proves that the
theory of Lorentz is not an artificial assemblage destined to fall
asunder. It will probably have to be modified, but not destroyed.
But Lorentz had no aim beyond that of embracing in one
totality all the optics and electrodynamics of moving bodies; he
never pretended to give a mechanical explanation of them. Lar-
mor goes further; retaining the theory of Lorentz in essentials,
he grafts upon it, so to speak, MacGullagh's ideas on the direction
of the motions of the ether.
According to him, the velocity of the ether would have the
same direction and the same magnitude as the magnetic force.
However ingenious this attempt may be, the defect of the theory
of Lorentz remains and is even aggravated. With Lorentz, we do
not know what are the motions of the ether ; thanks to this igno-
rance, we may suppose them such that, compensating those of
matter, they reestablish the equality of action and reaction.
With Larmor, we know the motions of the ether, and we can
ascertain that the compensation does not take place.
If Larmor has failed, as it seems to me he has, does tiiat mean
that a mechanical explanation is impossible? Far from it: I
have said above that when a phenomenon obeys the two principles
of energy and of least action, it admits of an infinity of mechan-
ical explanations ; so it is, therefore, with the optical and electrical
phenomena.
But this is not enough: for a mechanical explanation to be
good, it must be simple ; for choosing it among all which are pos-
sible, there should be other reasons besides the necessity of mak-
ing a choice. Well, we have not as yet a theory satisfying this
condition and consequently good for something. Must we lament
this? That would be to forget what is the goal sought; this is
not mechanism ; the true, the sole aim is unity.
We must therefore set bounds to our ambition ; let us not try
THE THEORIES OF MODERN PET8ICB 151
to formulate a mechanical explanation; let us be content with
showing that we could always find one if we wished to. In this
regard we have been successful ; the principle of the conservation
of energy has received only confirmations ; a second principle has
come to join it, that of least action, put under the form which is
suitable for physics. It also has always been verified, at least
in so far as concerns reversible phenomena which thus obey the
equations of Lagrange, that is to say, the most general laws of
mechanics.
Irreversible phenomena are much more rebellious. Yet these
also are being coordinated, and tend to come into unity ; the light
which has illuminated them has come to us from Gamot's prin-
ciple. Long did thermodynamics confine itself to the study of
the dilatation of bodies and their changes of state. For some time
past it has been growing bolder and has considerably extended
its domain. We owe to it the theory of the galvanic battery, and
that of the thermoelectric phenomena ; there is not in all physics
a comer that it has not explored, and it has attacked chemistry
itself.
Everywhere the same laws reign ; everywhere, under the diver-
sity of appearances, is found again Gamot's principle; every-
where also is found that concept so prodigiously abstract of
entropy, which is as universal as that of energy and seems like it
to cover a reality. Radiant heat seemed destined to escape it; but
recently we have seen that submit to the same laws.
In this way fresh analogies are revealed to us, which may
often be followed into detail; ohmic resistance resembles the
viscosity of liquids ; hysteresis would resemble rather the friction
of solids. In all cases, friction would appear to be the type which
the most various irreversible phenomena copy, and this kinship
is real and profound.
Of these phenomena a mechanical explanation, properly so
called, has also been sought. They hardly lent themselves to it.
To find it, it was necessary to suppose that the irreversibility is
only apparent, that the elementary phenomena are reversible and
obey the known laws of dynamics. But the elements are extremely
numerous and blend more and more, so that to our crude sight all
appears to tend toward uniformity, that is, everything seems to
152 SCIENCE AND HYPOTHESIS
go forward in the same sense without hox>e of return. The ap-
parent irreversibility is thns only an effect of the law of greaX
numbers. But, only a being with infinitely subtile senses, like
Maxwell's imaginary demon, could disentangle this inextricable
skein and turn back the course of the universe.
This conception, which attaches itself to the kinetic theory
of gases, has cost great efforts and has not, on the whole, been
fruitful ; but it may become so. This is not the place to examine
whether it does not lead to contradictions and whether it is in
conformity with the true nature of things.
We signalize, however, M. Grouy 's original ideas on the Brownian
movement. According to this scientist, this singular motion
should escape Camot's principle. The particles which it puts in
swing would be smaller than the links of that so compacted skein;
they would therefore be fitted to disentangle them and hence to
make the world go backward. We should almost see Maxwell's
demon at work.
To summarize, the previously known phenomena are better and
better classified, but new phenomena come to claim their place;
most of these, like the Zeeman effect, have at once found it.
But we have the cathode rays, the X-rays, those of uranium
and of radium. Herein is a whole world which no one suspected.
How many unexpected guests must be stowed away 1
No one can yet foresee the place they will occupy. But I do
not believe they will destroy the general unity ; I think they will
rather complete it. On the one hand, in fact, the new radiations
seem connected with the phenomena of luminescence; not only
do they excite fluorescence, but they sometimes take birth in the
same conditions as it.
Nor are they without kinship with the causes which produce
the electric spark under the action of the ultra-violet light.
Finally, and above all, it is believed that in all these phenomena
are found true ions, animated, it is true, by velocities incom-
parably greater than in the electrolytes.
That is all very vague, but it will all become more precise.
Phosphorescence, the action of light on the spark, these were
regions rather isolated, and consequently somewhat neglected by
investigators. One may now hope that a new path wiU be con-
TBE THEORIES OF MODERN PHYSICS 153
stracted which will facilitate their communications with the rest
of science.
Not only do we discover new phenomena, but in those we
thought we knew, unforeseen aspects reveal themselves. In the
free ether, the laws retain their majestic simplicity ; but matter,
properly so called, seems more and more complex; all that is
said of it is never more than approximate, and at each instant
our formulas require new terms.
Nevertheless the frames are not broken ; the relations that we
have recognized between objects we thought simple still subsist
between these same objects when we know their complexity, and
it is that alone which is of importance. Our equations become, it
is true, more and more complicated, in order to embrace more
closely the complexity of nature ; but nothing is changed in the
relations which permit the deducing of these equations one from
another. In a word, the form of these equations has persisted.
Take, for example, the laws of reflection : Fresnel had estab-
lished them by a simple and seductive theory which experiment
seemed to confirm. Since then more precise researches have
proved that this verification was only approximate; they have
shown everywhere traces of elliptic polarization. But, thanks to
the help that the first approximation gave us, we found forthwith
the cause of these anomalies, which is the presence of a transition
layer; and Fresnel 's theory has subsisted in its essentials.
But there is a reflection we can not help making: All these
relations would have remained unperceived if one had at first
suspected the complexity of the objects they connect. It has long
been said: If Tycho had had instruments ten times more pre-
cise neither Kepler, nor Newton, nor astronomy would ever have
been. It is a misfortune for a science to be bom too late, when
the means of observation have become too perfect. This is to-day
the case with physical chemistry; its founders are embarrassed
in their general grasp by third and fourth decimals ; happily they
are men of a robust faith.
The better one knows the properties of matter the more one
sees continuity reign. Since the labors of Andrews and of van der
Wals, we get an idea of how the passage is made from the liquid
to the gaseous state and that this passage is not abrupt. Similarly,
154 SCIENCE AND HTPOTHESIS
there is no gap between the liquid and solid states^ and in tiie
proceedings of a recent congress is to be seen, alongside of a work
on the rigidity of liquids, a memoir on the flow of solids.
By this tendency no doubt simplicity loses ; some phenomenon
was formerly represented by several straight lines, now these
straights must be joined by curves more or less complicated. In
compensation unity gains notably. Those cut-off categories quieted
the mind, but they did not satisfy it.
Finally the methods of physics have invaded a new domain,
that of chemistry; physical chemistry is bom. It is still very
young, but we already see that it will enable us to connect such
phenomena as electrolysis, osmosis and the motions of ions.
From this rapid exposition, what shall we conclude ?
Everything considered, we have approached unity; we have
not been as quick as was hoped fifty years ago, we have not always
taken the predicted way; but, finally, we have gained ever so
much ground.
CHAPTER XI
The Calculus op Probabilities
Doubtless it will be astonishing to find here thoughts about
the calculus of probabilities. What has it to do with the method
of the physical sciences t And yet the questions I shall raise with-
out solving present themselves naturally to the philosopher who
is thinking about physics. So far is this the case that in the
two preceding chapters I have often been led to use the words
' probability ' and ' chance. '
'Predicted facts,' as I have said above, 'can only be probable.'
''However solidly founded a prediction may seem to us to be,
we are never absolutely sure that experiment will not prove it
false. But the probability is often so great that practically we
may be satisfied with it." And a little further on I have added :
"See what a role the belief in simplicity plays in our generaliza-
tions. We have verified a simple law in a great number of par-
ticular cases; we refuse to admit that this coincidence, so often
repeated, can be a mere effect of chance. ..."
Thus in a multitude of circumstances the physicist is in the
same position as the gambler who reckons up his chances. As
often as he reasons by induction, he requires more or less con-
sciously the calculus of probabilities, and this is why I am obliged
to introduce a parenthesis, and interrupt our study of method in
the physical sciences in order to examine a little more closely the
value of this calculus, and what confidence it merits.
The very name calculus of probabilities is a paradox. Prob-
ability opposed to certainty is what we do not know, and how can
we calculate what we do not know? Yet many eminent savants
have occupied themselves with this calculus, and it can not be
denied that science has drawn therefrom no small advantage.
How can we explain this apparent contradiction ?
Has probability been defined? Can it even be defined? And
if it can not, how dare we reason about it t The definition, it will
166
166 SCIENCE AND HYPOTHESIS
be said, is very simple : the probability of an event is the ratio of
the number of eases favorable to this event to the total number of
possible cases.
A simple example will show how incomplete this definition is.
I throw two dice. What is the probability that one of the two
at least turns up a sixf Each die can turn up in six different
ways; the number of possible cases is 6 X 6 = 36; the number
of favorable cases is 11 ; the probability is 11/36.
That is the correct solution. But could I not just as well say:
The points which turn up on the two dice can form 6 X 7/2=21
different combinations f Among these combioationa 6 are favor-
able ; the probability is 6/21.
Now why is the first method of enumerating the i)0ssible cases
more legitimate than the second f In any case it is not our
definition that tells us.
We are therefore obliged to complete this definition by saying :
' ... to the total number of possible cases provided these cases
are equally probable.' So, therefore, we are reduced to defining
the probable by the probable.
How can we know that two possible cases are eqvally probable f
Will it be by a convention 1 If we place at the beginning of each
problem an explicit convention, well and good. We shall then
have nothing to do but apply the rules of arithmetic and of
algebra, and we shall complete our calculation without our result
leaving room for doubt. But if we wish to make the slightest
application of this result, we must prove our convention was
legitimate, and we shall find ourselves in the presence of the very
difficulty we thought to escape.
Will it be said that good sense suffices to show us what con-
vention should be adopted 1 Alas ! M. Bertrand has amused him-
self by discussing the following simple problem: "What is the
probability that a chord of a circle may be greater than the side
of the inscribed equilateral triangle?" The illustrious geometer
successively adopted two conventions which good sense seemed
equally to dictate and with one he found 1/2, with the other 1/3.
The conclusion which seems to follow from all this is that the
calculus of probabilities is a useless science, and that the obscure
THE CALCULUS OF PROBABILITIES 167
instinct which we may call good sense, and to which we are wont
to appeal to legitimatize our conventions, must be distrusted.
But neither can we subscribe to this conclusion; we can not
do without this obscure instinct. Without it science would be
impossible, without it we could neither discover a law nor apply
it. Have we the right, for instance, to enunciate Newton's lawt
Without doubt, numerous observations are in accord with it ; but
is not this a simple effect of chance t Besides how do we know
whether this law, true for so many centuries, will still be true
next year! To this objection, you will find nothing to reply,
except: 'That is very improbable.'
But grant the law. Thanks to it, I believe myself able to
calculate the position of Jupiter a year from now. Have I the
right to believe this! Who can tell if a gigantic mass of enor-
mous velocity will not between now and that time pass near the
aolar system, and produce unforeseen perturbations! Here again
the only answer is: 'It is very improbable.'
Prom this point of view, all the sciences would be only uncon-
scious applications of the calculus of probabilities. To condemn
this calculus would be to condemn the whole of science.
I shall dwell lightly on the scientific problems in which the
intervention of the calculus of probabilities is more evident. In
the forefront of these is the problem of interpolation, in which,
knowing a certain number of values of a function, we seek to
divine the intermediate values.
I shall likewise mention: the celebrated theory of errors of
observation, to which I shall return later; the kinetic theory of
g^es, a well-known hypothesis, wherein each gaseous molecule is
supposed to describe an extremely complicated trajectory; but in
which, through the effect of great numbers, the mean phenomena,
alone observable, obey the simple laws of Mariotte and Gay-
Lussac.
All these theories are based on the laws of great numbers, and
the calculus of probabilities would evidently involve them in its
ruin. It is true that they have only a particular interest, and
that, save as far as interpolation is concerned, these are sacrifices
to which we might readily be resigned.
But, as I have said above, it would not be only these partial
158 SCIENCE AND HYPOTHESIS
sacrifices that would be in question ; it would be the legitimacy of
the whole of science that would be challenged.
I quite see that it might be said: ''We are ignorant, and yet
we must act. For action, we have not time to devote ourselves
to an inquiry sufficient to dispel our ignorance. Besides, such an
inquiry would demand an infinite time. We must therefore decide
without knowing ; we are obliged to do so, hit or miss, and we must
follow rules without quite believing them. What I know is not
that such and such a thing is true, but that the best course for me
is to act as if it were true." The calculus of probabilities, and
consequently science itself, would thenceforth have merely a prac-
tical value.
Unfortunately the difficulty does not thus disappear. A gam-
bler wants to try a coup; he asks my advice. If I give it to him,
I shall use the calculus of probabilities, but I shall not guarantee
success. This is what I shall call subjective probability. In this
case, we might be content with the explanation of which I have
just given a sketch. But suppose that an observer is present at
the game, that he notes all its coupsy and that the game goes on a
long time. When he makes a summary of his book, he will find
that events have taken place in conformity with the laws of the
calculus of probabilities. This is what I shall call objective
probability, and it is this phenomenon which has to be explained.
There are numerous insurance companies which apply the rules
of the calculus of probabilities, and they distribute to their share-
holders dividends whose objective reality can not be contested.
To invoke our ignorance and the necessity to act does not suffice
to explain them.
Thus absolute skepticism is not admissible. We may distrust,
but we can not condemn en bloc. Discussion is necessary.
I. Classification op the Problems op Probability. — In
order to classify the problems which present themselves d propos
of probabilities, we may look at them from many different points
of view, and, first, from the point of vieiv of generality. I have
said above that probability is the ratio of the number of favorable
cases to the number of possible cases. What for want of a better
term I call the generality will increase with the number of pos-
THE CALCULUS OF PROBABILITIES 159
sible cases. This number may be finite, as, for instance, if we
take a throw of the dice in which the number of possible cases is
36. That is the first degree of generality.
But if we ask, for example, what is the probability that a
point within a circle is within the inscribed square, there are as
many possible cases as there are points in the circle, that is to
say, an infinity: This is the second degree of generality. Gener-
ality can be pushed further still. We may ask the probability that
a function will satisfy a given condition. There are then as many
possible cases as one can imagine different functions. This is the
third degree of generality, to which we rise, for instance, when
we seek to find the most probable law in conformity with a finite
number of observations.
We may place ourselves at a point of view wholly different.
H we were not ignorant, there would be no probability, there
would be room for nothing but certainty. But our ignorance can
not be absolute, for then there would no longer be any probability
at all, since a little light is necessary to attain even this uncertain
science. Thus the problems of probability may be classed accord-
ing to the greater or less depth of this ignorance.
In mathematics even we may set ourselves problems of prob-
ability. What is the probability that the fifth decimal of a log-
arithm taken at random from a table is a *9't There is no
hesitation in answering that this probability is 1/10; here we
possess all the data of the problem. We can calculate our loga-
rithm without recourse to the table, but we do not wish to give
ourselves the trouble. This is the first degree of ignorance.
In the physical sciences our ignorance becomes greater. The
state of a system at a given instant depends on two things : Its
initial state, and the law according to which that state varies. If
we know both this law and this initial state, we shall have then
only a mathematical problem to solve, and we fall back upon the
first degree of ignorance.
But it often happens that we know the law, and do not know
the initial state. It may be asked, for instance, what is the
present distribution of the minor planets t We know that from
all time they have obeyed the laws of Kepler, but we do not know
what was their initial distribution.
160 SCIENCE AND HYPOTHESIS
In the kmetic theory of gases, we assume that the gaseous
molecules follow rectilinear trajectories, and obey the laws of
impact of elastic bodies. But, as we know nothing of their initial
velocities, we know nothing of their present velocities.
The calculus of probabilities only enables us to predict the
mean phenomena which will result from the combination of these
velocities. This is the second degree of ignorance.
Finally it is possible that not only the initial conditions but
the laws themselves are unknown. We then reach the third degree
of ignorance and in general we can no longer affirm anything at
all as to the probability of a phenomenon.
It often happens that instead of trying to guess an event, by
means of a more or less imperfect knowledge of the law, the
events may be known and we want to find the law ; or that instead
of deducing effects from causes, we wish to deduce the causes
from the effects. These are the problems called probability of
causes, the most interesting from the point of view of their sci-
entific applications.
I play 6carte with a gentleman I know to be perfectly honest
He is about to deal. What is the probability of his turning up
the kingt It is 1/8. This is a problem of the probability of
effects.
I play with a gentleman whom I do not know. He has dealt
ten times, and he has turned up the king six times. What is
the probability that he is a sharper! This is a problem in the
probability of causes.
It may be said that this is the essential problem of the experi-
mental method. I have observed n values of x and the corres-
ponding values of y. I have found that the ratio of the latter to
the former is practically constant. There is the event, what is
the cause?
Is it probable that there is a general law according to which y
would be proportional to re, and that the smaU divergencies are
due to errors of observation ? This is a type of question that one
is ever asking, and which we unconsciously solve whenever we are
engaged in scientific work.
I am now going to pass in review these different categories of
problems, discussing in snccession what I have called above sub-
jective and objective probability.
II, Probability in Mathematics. — The impossibility of squar-
ing the circle has been proved since 1882; but even before
that date all geometers considered that impossibility as so
'probable,' that the Academy of Sciences rejected without exami-
nation the alas! too numerous memoirs on this subject, that some
unhappy madmen sent in every year.
Was the Academy wrong I Evidently not, and it knew well
that in acting thus it did sot run the least risk of stilling a dis-
covery of moment. The Academy could not have proved that it
was right; but it knew quite well that its instinct was not mis-
taken. If you had asked the Academicians, they would have
answered: "We have compared the probability that an unknown
savant should have found out what has been vainly sought for so
long, with the probability that there is one madman the more
on the earth; the second appears to us the greater." These are
very good reasons, but there is nothing mathematical about them ;
Ibey are purely psychological.
And if you had pressed them further they would have added:
"Why do you suppose a particular value of a transcendental
function to be an algebraic number i and if jt were a root of aa
algebraic equation, why do yon suppose this root to be a period of
the function sin 2x, and not the same about the other roots of this
same equation?" To sum up, they would have invoked the prin-
ciple of sufficient reason in its vaguest form.
But what could they deduce from itT At most a rule of con-
duct for tile employment of their time, more usefully spent at
their ordinary work than in reading a lucubration that inspired
in them a legitimate distrust. But what I call above objective
probabili^ has nothing in common with this first problem.
It ia otherwise with the second problem.
Consider the first 10,000 logarithms that we find in a table.
Among these 10,000 logarithms I take one at random. What is
the probabQity that its third decimal is an even number? Ton
will not hesitate to answer 1/2; and in fact if you pick out in a
table the third decimals of these 10,000 numbers, you will find
nearly as many even digits aa odd.
162 SCIENCE AND HTPOTHESIS
Or if you prefer, let us write 10,000 numbers corresponding
to our 10,000 logarithms, each of these numbers being +1 ^
the third decimal of the corresponding logarithm is even, and
— 1 if odd. Then take the mean of these 10,000 numbers.
I do not hesitate to say that the mean of these 10,000 numbers
is probably 0, and if I were actually to calculate it I should
verify that it is extremely small.
But even this verification is needless. I might have rigorously
proved that this mean is less than 0.003. To prove this result, I
should have had to make a rather long calculation for which there
is no room here, and for which I confine myself to citing an article
I published in the Revue gSnerale des Sciences, April 15, 1899.
The only point to which I wish to call attention is the following:
in this calcula^tion, I should have needed only to rest my case on
two facts, to wit, that the first and second derivatives of the log-
arithm remain, in the interval considered, between certain limits.
Hence this important consequence that the property is true not
only of the logarithm, but of any continuous function whatever,
since the derivatives of every continuous function are limited.
If I was certain beforehand of the result, it is first, because I
had often observed analogous facts for other continuous func-
tions; and next, because I made in my mind, in a more or less
unconscious and imperfect manner, the reasoning which led me to
the preceding inequalities, just as a skilled calculator before
finishing his multiplication takes into account what it should
eome to approximately.
And besides, since what I call my intuition was only an in-
complete summary of a piece of true reasoning, it is clear why
observation has confirmed my predictions, and why the objective
probability has been in agreement with the subjective probability.
As a third example I shall choose the following problem: A
number u is taken at random, and n is a given very large integer.
What is the probable value of sin nu t This problem has no mean-
ing by itself. To give it one a convention is needed. We shall
agree that the probability for the number u to lie between a and
o + da is equal to <^(a) da ; that it is therefore proportional to the
infinitely small interval da, and equal to this multiplied by a
function <^(a) depending only on a. As for this function, I
THE CALCULUS OF PROBABILITIES 163
choose it arbitrarily^ but I must assume it to be continuous. The
value of sin nu remaining the same when u increases by 2^, I may
without loss of generality assume that u lies between and 2ir,
and I shall thus be led to suppose that <^(a) is a periodic function
whose period is 2ir.
The probable value sought is readily expressed by a simple
integral, and it is easy to show that this integral is less than
2TMfc/ii*,
Mjfc being the maximum value of the k^ derivative of if>{u). We
see then that if the k^ derivative is finite, our probable value will
tend toward when n increases indefinitely, and that more rapidly
than 1/n^K
The probable value of sin nu when n is very large is therefore
naught. To define this value I required a convention; but the
result remains the same whatever thai convention may be. I
have imposed upon myself only slight restrictions in assuming
that the function <^(a) is continuous and periodic, and these hy-
potheses are so natural that we may ask ourselves how they can
be escaped.
Examination of the three preceding examples, so different in
all respects, has already given us a glimpse, on the one hand,
of the role of what philosophers call the principle of sufficient
reason, and, on the other hand, of the importance of the fact that
certain properties are common to all continuous functions. The
study of probability in the physical sciences will lead us to the
same result.
III. Probability in the Physical Sciences. — ^We come now
to the problems connected with what I have called the second
degree of ignorance, those, namely, in which we know the law,
but do not know the initial state of the system. I could multiply
examples, but will take only one. What is the probable present
distribution of the minor planets on the zodiac t
We know they obey the laws of Kepler. We may even, with-
out at all changing the nature of the problem, suppose that their
orbits are all circular, and situated in the same plane, and that we
know this plane. On the other hand, we are in absolute ignorance
as to what was their initial distribution. However, we do not
164 SCIENCE AND HTPOTHESIS
hesitate to affirm that their distribution is now nearly nnifomu
Whyt
Let b be the longitade of a minor planet in the initial epoch,
that is to say y the epoch zero. Let a be its mean motion. Its longi-
tude at the present epoch, that is to say, at the epoch ty will be
at + b. To say that the present distribution is uniform is to say
that the mean value of the sines and cosines of multiples otat-\'b
is zero. Why do we assert this ?
Let us represent each minor planet by a point in a plane, to
wit, by a point whose coordinates are precisely a and b. All
these representative points will be contained in a certain region
of the plane, but as they are very numerous, this region will
appear dotted with points. We know nothing else about the dis-
tribution of these points.
What do we do when we wish to apply the calculus of proba-
bilities to such a question! What is the probability that one or
more representative points may be found in a certain portion of
the plane t In our ignorance, we are reduced to making an arbi-
trary hypothesis. To explain the nature of this hypothesis, allow
me to use, in lieu of a mathematical formula, a crude but con-
crete image. Let us suppose that over the surface of our plane
has been spread an imaginary substance, whose density is vari-
able, but varies continuously. We shall then agree to say that the
probable number of representative points to be found on a portion
of the plane is proportional to the quantity of fictitious matter
found there. If we have then two regions of the plane of the
same extent, the probabilities that a representative point of one
of our minor planets is found in one or the other of these regions
will be to one another as the mean densities of the fictitious matter
in the one and the other region.
Here then are two distributions, one real, in which the repre-
sentative points are very numerous, very close together, but dis-
crete like the molecules of matter in the atomic hypothesis; the
other remote from reality, in which our representative points are
replaced by continuous fictitious matter. We know that the latter
can not be real, but our ignorance forces us to adopt it.
If again we had some idea of the real distribution of the
representative points, we could arrange it so that in a region
THE CALCULUS OF PROBABILITIES
166
of some extent the density of thia imaginary continuous matter
would be nearly proportional to tlie number of the representative
points, or, if you wish, to the number of atoms which are con-
tained in that region. Even that is impossible, and our ignorance
is BO great that we are forced to choose arbitrarily the function
which de6nes the density of our imaginary matter. Only we shall
be forced to a hypothesis from which we can hardly get away,
we shall suppose that this function is continuous. That is suf-
ficient, as we shall see, to enable us to reach a conclusion.
What is at the instant t the probable distribution of the minor
planetst Or rather what is the probable value of the sine of the _
longitude at the instant (, that is to say of sin (ot + 6) I Wtf]
made at the outset an arbitrary convention, but if we adopt itM
this probable value is entirely defined. Divide the plane into ele-
ments of surface. Consider the value of sin (o( -\-h) at the cen-
ter of each of these elements; multiply this value by the surface
of the element, and by the corresponding density of the imaginary
matter. Take then the sum for alt the elements of the plane.
This 8um, by definition, will be the probable mean value we seek,
which will thus be expressed by a double integral. It may be
thought at first that this mean value depends on the choice of the
function which defines the density of the imaginary matter, and
that, as this function ^ is arbitrary, we can, according to the
arbitrary choice which we make, obtain any mean value. This
is not so,
A simple calculation shows that our double integral decreases
veiy rapidly when ( increases. Thus I could not quite tell what
liypothesis to make as to the probability of this or that initial
distribution 1 but whatever the hypothesis made, the result will
be the same, and this gets me out of my difficulty.
Whatever be the function <^, the mean value tends toward zero
as ( increases, and as the minor planets have certainly accom-
plished a very great number of revolutions, I may assert that this
mean value is very small.
I may clioose ^ as I wish, save always on« restriction: thia
function must be continuous ; and, in fact, from the poiat of v
of subjective probability, the choice of a discontinuous functiol
would have been unreasonable. For instance, what reason coal4
166 SCIENCE AND HYPOTHESIS
I have for supposing that the initial longitude might be exactly
0®, but that it could not lie between 0® and 1® t
But the diflBeulty reappears if we take the point of view of
objective probability, if we pass from our imaginary distribution
in which the fictitious matter was supposed continuous, to the real
distribution in which our representative points form, as it were,
discrete atoms.
The mean value of sin (a^ + &) will be represented quite
simply by
Izsin (a« + d),
n
n being the number of minor planets. In lieu of a double integral
referring to a continuous function, we shall have a sum of dis-
crete terms. And yet no one will seriously doubt that this mean
value is practically very small.
Our representative points being very close together, our dis-
crete sum will in general differ very little from an integral.
An integral is the limit toward which a sum of terms tends
when the number of these terms is indefinitely increased. If the
terms are very numerous, the sum will differ very little from
its limit, that is to say from the integral, and what I said of this
latter will still be true of the sum itself.
Nevertheless, there are exceptions. If, for instance, for all
the minor planets,
6 = ^— ot,
the longitude for all the planets at the time t would be 7r/2, and
the mean value would evidently be equal to unity. For this to
be the ease, it would be necessary that at the epoch 0, the minor
planets must have all been lying on a spiral of peculiar form, with
its spires very close together. Every one will admit that such an
initial distribution is extremely improbable (and, even supposing
it realized, the distribution would not be uniform at the present
time, for example, on January 1, 1913, but it would become so
a few years later) .
Why then do we think this initial distribution improbable t
This must be explained, because if we had no reason for rejecting
TRE CALCULUS OF PROBABILITIES 167
as improbable this absurd hypothesis everything would break
down, and we could no longer make any affirmation about the
probability of this or that present distribution.
Once more we shall invoke the principle of sufficient reason to
which we must always recur. We might admit that at the begin-
ning the planets were distributed almost in a straight line. We
might admit that they were irregularly distributed. But it seems
to us that there is no sufficient reason for the unknown cause that
gave them birth to have acted along a curve so regular and yet so
complicated, which would appear to have been expressely chosen
so that the present distribution would not be uniform.
rV. RouoE ET Nom. — The questions raised by games of chance,
such as roulette, are, fundamentally, entirely analogous to those
we have just treated. For example, a wheel is partitioned into
a great number of equal subdivisions, alternately red and black.
A needle is whirled with force, and after having made a great
number of revolutions, it stops before one of these subdivisions.
The probability that this division is red is evidently 1/2. The
needle describes an angle 0, including several complete revolu-
tions. I do not know what is the probability that the needle may
be whirled with a force such that this angle should lie between d
and 6 -{-dO ; but I can make a convention. I can suppose that this
probability is 4>{6)dB. As for the function <^(^), I can choose it
in an entirely arbitrary manner. There is nothing that can guide
me in my choice, but I am naturally led to suppose this function
continuous.
Let € be the length (measured on the circumference of radius
1) of each red and black subdivision. We have to calculate the
integral of <t>{0)d$, extending it, on the one hand, to all the red
divisions, and, on the other hand, to all the black divisions, and to
compare the results.
Consider an interval 2c, comprising a red division and a black
division which follows it. Let M and m be the greatest and least
values of the function <f>{0) in this interval. The integral extended
to the red divisions will be smaller than 2Mc ; the integral extended
to the black divisions will be greater than ^m€\ the diflPerence
will therefore be less than 2(M — m)€. But, if the function $
is supposed continuous; if, besides, the interval c is very
168 SCIENCE AND HYPOTHESIS
small with respect to the total angle described by the needle,
the difference M — m will be very small. The difference of the
two integrals will therefore be very small, and the probability
will be very nearly 1/2.
We see that without knowing anything of the function 0, I
must act as if the probability were 1/2. We understand, on the
other hand, why, if, placing myself at the objective point of
view, I observe a certain number of coups, observation will give
me about as many black coups as red.
All players know this objective law; but it leads them into a
remarkable error, which has been often exposed, but into which
they always fall again. When the red has won, for instance, six
times running, they bet on the black, thinking they are playing a
safe game ; because, say they, it is very rare that red wins seven
times running.
In reality their probability of winning remains 1/2. Observa-
tion shows, it is true, that series of seven consecutive reds are very
rare, but series of six reds followed by a black are just as rare.
They have noticed the rarity of the series of seven reds; if
they have not remarked the rarity of six reds and a black, it is
only because such series strike the attention less.
V. The ProbabiltITy op Causes. — ^We now come to the prob-
lems of the probability of causes, the most important from the
point of view of scientific applications. Two stars, for instance,
are very close together on the celestial sphere. Is this apparent
contiguity a mere effect of chance? Are these stars, although on
almost the same visual ray, situated at very different distances
from the earth, and consequently very far from one another!
Or, perhaps, does the apparent correspond to a real contiguity!
This is a problem on the probability of causes.
I recall first that at the outset of all problems of the proba-
bility of effects that have hitherto occupied us, we have always
had to make a convention, more or less justified. And if in most
cases the result was, in a certain measure, independent of this
convention, this was only because of certain hypotheses which
permitted us to reject a priori discontinuous functions, for ex-
ample, or certain absurd conventions.
We shall find something analogous when we deal with the
' |»*obBbiUty of causes. An effect may be produced by the cause
A or by the cause B, The effect has just been observed. We
ask the probability that it is due to the cause A. This is an a
posteriori probability of cause. But I could not calculate it, if
a convention more or less justified did not tell me in advance
irbat is the a priori probability for the cause A to come into
play; I mean the probability of this event for some one who had
not observed the effect.
The better to explain myself I go back to the example of the
game of ^carte mentioned above. My adversary deals for the
first time and he turns up a king. What is the probability that he
is a sharper? The formulas ordinarily taught give 8/9, a result
evidently rather surprising. If we look at it closer, we see that
the calculation is made as if, before sittirtg down at the tahle, I
had considered that there was one chance in two that my adver-
sary was not honest. An absurd hypothesis, because in that case
I should have certainly not played with him, and this explains
the absurdity of the conclusion.
The convention about the a priori probability was unjustitied,
and that is why the calculation of the a posteriori probnbility led
me to an inadmissible result. We see the importance of this pre-
liminary convention. I shall even add that if none were made,
the problem of the a posteriori probability would have no mean-
ii^. It must always be made either explicitly or tacitly.
Pass to an example of a more scientific character. I wish to
determine an experimental law. This law, when 1 know it, can
be represented by a curve. I make a certain number of isolated
observations; each of these will be represented by a point. When
I have obtained these different points, I draw a curve between
them, striving to pass as near to them as possible and yet preserve
for my curve a regular form, without angular points, or inflec-
tions too accentuated, or brusque variation of the radius of curva-
ture. This curve will represent for me the probable law, and I
assnme not only that it will tell me the values of the function
intermediate between those which have been observed, but also
that it will give me the observed values themselves more exactly J
than direct observation. This is why I make it pass near t
I, and not through the points themselves.
170 SCIENCE AND HYPOTHESIS
Here is a problem in the probability of causes. The effects
are the measurements I have recorded ; they depend on a combina-
tion of two causes: the true law of the phenomenon and the
errors of observation. Knowing the effects, we have to seek the
probability that the phenomenon obeys this law or that, and that
the observations have been affected by this or that error. The
most probable law then corresponds to the curve traced, and the
most probable error of an observation is represented by the dis-
tance of the corresponding point from this curve.
But the problem would have no meaning if, before any obser-
vation, I had not fashioned an a priori idea of the probability of
this or that law, and of the chances of error to which I am exposed.
If my instruments are good (and that I knew before making
the observations), I shall not permit my curve to depart much
from the points which represent the rough measurements. If
they are bad, I may go a little further away from them in order
to obtain a less sinuous curve ; I shall sacrifice more to regularity.
Why then is it that I seek to trace a curve without sinuosities!
It is because I consider a priori a law represented by a continu-
ous function (or by a function whose derivatives of high order
are small), as more probable than a law not satisfying these con-
ditions. Without this belief, the problem of which we speak
would have no meaning; interpolation would be impossible; no
law could be deduced from a finite number of observations;
science would not exist.
Fifty years ago physicists considered, other things being equal,
a simple law as more probable than a complicated law. They
even invoked this principle in favor of Mariotte's law as against
the experiments of Eegnault. To-day they have repudiated this
belief; and yet, how many times are they compelled to act as
though they still held it ! However that may be, what remains of
this tendency is the belief in continuity, and we have just seen
that if this belief were to disappear in its turn, experimental
science would become impossible.
VI. The Theory op Errors. — ^We are thus led to speak of
the theory of errors, which is directly connected with the problem
of the probability of causes. Here again we find effects, to wit,
a certain number of discordant observations, and we seek to
THE CALCULUS OF PROBABILITIES 171
divine the causes, which are, on the one hand, the real value of the
quantity to be measured; on the other hand, the error made in
each isolated observation. It is necessary to calculate what is
a posteriori the probable magnitude of each error, and conse-
quently the probable value of the quantity to be measured.
But as I have just explained, we should not know how to un-
dertake this calculation if we did not admit a priori, that is to
say, before all observation, a law of probability of errors. Is
there a law of errors t
The law of errors admitted by all calculators is Gauss's law,
which is represented by a certain transcendental curve known
under the name of 'the bell.'
But first it is proper to recall the classic distinction between
systematic and accidental errors. If we measure a length with
too long a meter, we shall always find too small a number, and
it will be of no use to measure several times; this is a systematic
error. If we measure with an accurate meter, we may, however,
make a mistake ; but we go wrong, now too much, now too little,
and when we take the mean of a great number of measurements,
the error will tend to grow small. These are accidental errors.
It is evident from the first that systematic errors can not
satisfy Gauss's law; but do the accidental errors satisfy itt A
great number of demonstrations have been attempted ; almost all
are crude paralogisms. Nevertheless, we may demonstrate
Gauss's law by starting from the following hypotheses: the error
committed is the result of a great number of partial and inde-
pendent errors; each of the partial errors is very little and
besides, obeys any law of probability, provided that the prob-
ability of a positive error is the same as that of an equal negative
error. It is evident that these conditions will be often but not
always fulfilled, and we may reserve the name of accidental for
errors which satisfy them.
We see that the method of least squares is not legitimate in
every case; in general the physicists are more distrustful of it
than the astronomers. This is, no doubt, because the latter, be-
sides the systematic errors to which they and the physicists are
subject alike, have to contend with an extremely important source
of error which is wholly accidental ; I mean atmospheric undula-
172 SCIENCE AND HTP0THE8I8
lions. So it is very curious to hear a physicist discuss with an
astronomer about a method of observation. The physicist, per-
suaded that one good measurement is worth more than many
bad ones, is before all concerned with eliminating by dint of
precautions the least rystematic errors, and the astronomer says
to him: 'But thus you can observe only a small number of stars;
the accidental errors will not disappear. '
What should we conclude f Must we continue to use the
method of least squares f We must distinguish. We have elimi-
nated all the systematic errors we could suspect ; we know well
there are still others, but we can not detect them; yet it is
necessary to make up our mind and adopt a definitive value
which will be regarded as the probable value ; and for that it is
evident the best thing to do is to apply Gauss's method. We
have only applied a practical rule referring to subjective prob-
ability. There is nothing more to be said.
But we wish to go farther and affirm that not only is the
probable value so much, but that the probable error in the re-
sult is so much. This is absolutely illegitimate; it would be true
only if we were sure that all the systematic errors were elimi-
nated, and of that we know absolutely nothing. We have two
series of observations ; by applying the rule of least squares, we
find that the probable error in the first series is twice as small
as in the second. The second series may, however, be better than
the first, because the first perhaps is affected by a large system-
atic error. All we can say is that the first series is probably
better than the second, since its accidental error is smaller, and
we have no reason to affirm that the systematic error is greater
for one of the series ^han for the other, our ignorance on this
point being absolute.
VII. Conclusions. — In the lines which precede, I have set
many problems without solving any of them. Yet I do not regret
having written them, because they will perhaps invite the reader
to reflect on these delicate questions.
However that may be, there are certain points which seem
well established. To undertake any calculation of probability,
and even for that calculation to have any meaning, it is neces-
THE CALCULUS OF PROBABILITIES 173
sary to admit, as point of departure, a hypothesis or convention
which has always something arbitrary about it. In the choice
of this convention, we can be guided only by the principle of
sufBcient reason. Unfortunately this principle is very vague
and very elastic, and in the cursory examination we have just
made, we have seen it take many different forms. The form un-
der which we have met it most often is the belief in continuity, a
belief which it would be difficult to justify by apodeictic reason-
ing, but without which all science would be impossible. Finally
the problems to which the calculus of probabilities may be applied
with profit are those in which the result is independent of the
hypothesis made at the outset, provided only that this hypothesis
satisfies the condition of continuity.
CHAPTER XII
Optics and Electricity
Fresnel's Theory. — The best example^ that can be chosen
of physics in the making is the theory of light and its relations to
the theory of electricity. Thanks to Fresnel, optics is the best
developed part of physics; the so-called wave-theory forms a
whole truly satisfying to the mind. We must not, however, ask
of it what it can not give us.
The object of mathematical theories is not to reveal to us the
true nature of things ; this would be an unreasonable pretension.
Their sole aim is to coordinate the physical laws which experi-
ment reveals to us, but which, without the help of mathematics,
we should not be able even to state.
It matters little whether the ether really exists; that is the
affair of metaphysicians. The essential thing for us is that
everything happens as if it existed, and that this hypothesis is
convenient for the explanation of phenomena. After all, have
we any other reason to believe in the existence of material
objects? That, too, is only a convenient hypothesis; only this
will never cease to be so, whereas, no doubt, some day the ether
will be thrown aside as useless. But even at that day, the laws
of optics and the equations which translate them analytically
will remain true, at least as a first approximation. It will always
be useful, then, to study a doctrine that unites all these equations.
The undulatory theory rests on a molecular hypothesis. For
those who think they have thus discovered the cause under the
law, this is an advantage. For the others it is a reason for dis-
trust. But this distrust seems to me as little justified as the
illusion of the former.
These hypotheses play only a secondary part. They might be
sacrificed. They usually are not, because then the explanation
would lose in clearness ; but that is the only reason.
1 This chapter is a partial reproduction of the prefaces of two of my
works: TMorie math&matique de la lumi^e (Paris, Naud, 1889), and Eleo-
iriciU et optigue (Paris, Naud, 1901).
174
OPTICS AND ELECTRICITY 175
In fact, if we looked closer we should see that only two things
are borrowed from the molecular hypotheses : the principle of the
consecration of energy, and the linear form of the equations,
which is the general law of small movements, as of all small
variations.
This explains why most of Fresnel's conclusions remain un-
changed when we adopt the electromagnetic theory of light.
Maxwell's Theory. — ^Maxwell, we know, connected by a
close bond two parts of physics until then entirely foreign to one
another, optics and electricity. By blending thus in a vaster
whole, in a higher harmony, the optics of Fresnel has not ceased
to be alive. Its various parts subsist, and their mutual relations
are stiU the same. Only the language we used to express them
has changed ; and, on the other hand, MaxweU has revealed to us
other relations, before unsuspected, between the different parts
of optics and the domain of electricity.
When a French reader first opens Maxwell's book, a feeling
of uneasiness and often even of mistrust mingles at first with his
admiration. Only after a prolonged acquaintance and at the
cost of many efforts does this feeling disappear. There are even
some eminent minds that never lose it.
Why are the English scientist's ideas with such difficulty
acclimatized among usT It is, no doubt, because the education
received by the majority of enlightened Frenchmen predisposes
them to appreciate precision and logic above every other quality.
The old theories of mathematical physics gave us in this re-
spect complete satisfaction. All our masters, from Laplace to
Cauchy, have proceeded in the same way. Starting from clearly
stated hypotheses, they deduced all their consequences with
mathematical rigor, and then compared them with experiment.
It seemed their aim to give every branch of physics the same pre-
cision as celestial mechanics.
A mind accustomed to admire such models is hard to suit with
a theory. Not only will it not tolerate the least appearance of
contradiction, but it will demand that the various parts be
logically connected with one another, and that the number of
distinct hypotheses be reduced to minimum.
This is not all ; it will have still other demands, which seem to
176 SCIENCE AND HYPOTHESIS
me less reasonable. Behind the matter which our senses can
reach, and which experiment tells us of, it will desire to see
another, and in its eyes the only real, matter, which wiU have
only purely geometric properties, and whose atoms wiU be noth-
ing but mathematical points, subject to the laws of dynamics
alone. And yet these atoms, invisible and without color, it will
seek by an unconscious contradiction to represent to itself and
consequently to identify as closely as possible with common
matter.
Then only will it be fully satisfied and imagine that it has
penetrated the secret of the universe. If this satisfaction is de-
ceitful, it is none the less difficult to renounce.
Thus, on opening Maxwell, a Frenchman expects to find a
theoretical whole as logical and precise as the physical optics
based on the hypothesis of the ether; he thus prepares for him-
self a disappointment which I should Uke to spare the reader by
informing him immediately of what he must look for in Maxwell,
and what he can not find there.
Maxwell does not give a mechanical explanation of electricity
and magnetism; he confines himself to demonstrating that such
an explanation is possible.
He shows also that optical phenomena are only a special case
of electromagnetic phenomena. From every theory of electri-
city, one can therefore deduce immediately a theory of light.
The converse unfortunately is not true; from a complete ex-
planation of light, it is not always easy to derive a complete ex-
planation of electric phenomena. This is not easy, in particular,
if we wish to start from Fresnel's theory. Doubtless it would
not be impossible ; but nevertheless we must ask whether we are
not going to be forced to renounce admirable results that we
thought definitely acquired. That seems a step backward; and
many good minds are not willing to submit to it.
When the reader shall have consented to limit his hopes, he
will still encounter other difficulties. The English scientist does
not try to construct a single edifice, final and well ordered; he
seems rather to erect a great number of provisional and inde-
pendent constructions, between which communication is difficult
and sometimes impossible.
OPTICS AND ELECTRICITY
177
Take as example the chapter in which he explains electrostatie
attractionB by pressures and tensions in the dielectric medium.
Thia chapter might be omitted without making thereby the rest
of the book less clear or complete ; and, on the other hand, it con-
tains a theory complete in itself which one could understand with-
out having read a single line that precedes or follows. But it
is not only independent of the rest of the work; it is difGcult to
reconcile with the fundamental ideas of the book. Maxwell does
not even attempt thia reconciliation; he merely says: "I have
not been able to make the nest step, namely, to account by
mechanical considerations for these stresses in the dielectric."
This example will suEBce to make my thought understood ; I
could cite many others. Thua who would suspect, in reading
the pages devoted to magnetic rotary polarization, that there is
an identity between optical and magnetic phenomena!
One must not then datter himself that he can avoid all con-
tradiction; to that it is necessary to be resigned. In fact, two
contradictory theories, provided one does not mingle them, and
if one does not seek in them the basis of things, may both be
useful instruments of research; and perhaps the reading of
Maxwell would be less suggestive if he had not opened up to us
so many new and divergent paths.
The fundamental idea, however, is thus a little obscured. So
far ie this the case that in the majority of popularized versions
it is the only point completely left aside.
I feel, then, that the better to make its importance stand out.
I ought to explain in what this fundamental idea consists. Bnt
for that a short digression is necessary.
The Mechanical Explanation op Physical Phenouena. —
Th«re is in every physical phenomenon a certain number of
parameters which experiment reaches directly and allows us to
measure. I shall call these the parameters q.
Observation then teaches us the laws of the variations of these
parameters; and these laws can generally be put in the form
of differential equations, which connect the parameters q with the
time.
What is it necessary to do to give a mechanical interpretation
of Baeb a phenomenon!
178 SCIENCE AND HTP0THE8I8
One will try to explain it either by the motions of ordinary
matter, or by those of one or more hypothetical fluids.
These fluids will be considered as formed of a very great num-
ber of isolated molecules m.
When shall we say, then, that we have a complete mechanical
explanation of the phenomenon f It will be, on the one hand,
when we know the differential equations satisfied by the coordi-
nates of these hypothetical molecules m, equations which, more-
over, must conform to the principles of dynamics; and, on the
other hand, when we know the relations that define the coordi-
nates of the molecules m as functions of the parameters q acces-
sible to experiment.
These equations, as I have said, must conform to the prin-
ciples of dynamics, and, in particular, to the principle of the
conservation of energy and the principle of least action.
The first of these two principles teaches us that the total energy
is constant and that this energy is divided into two parts :
1® The kinetic energy, or vis viva, which depends on the
masses of the hypothetical molecules m, and their velocities, and
which I shall call T.
2° The potential energy, which depends only on the coordi-
nates of these molecules and which I shall call U. It is the sum
of the two energies T and U which is constant.
"What now does the principle of least action tell us? It tells
us that to pass from the initial position occupied at the instant ^o
to the final position occupied at the instant ^i, the system must
take such a path that, in the interval of time that elapses be-
tween the two instants t^ and ^i, the average value of *the
action' (that is to say, of the differ erice between the two energies
T and U) shall be as small as possible.
If the two functions T and U are known, this principle suffices
to determine the equations of motion.
Among all the possible ways of passing from one position to
another, there is evidently one for which the average value of
the action is less than for any other. There is, moreover, only
one; and it results from this that the principle of least action
suffices to determine the path followed and consequently the
equations of motion.
OPTICS AND ELECTBICITY 179
Thus we obtain what are called the equations of Lagrange.
In these equations, the independent variables are the coordi-
nates of the hypothetical molecules m; but I now suppose that
one takes as variables the parameters q directly accessible to ex-
periment.
The two parts of the energy must then be expressed as func-
tions of the parameters q and of their derivatives. They will
evidently appear under this form to the experimenter. The
latter will naturally try to define the potential and the kinetic
energy by the aid of quantities that he can directly observe.^
That granted, the system will always go from one position to
another by a path such that the average action shall be a mini-
mum*
It matters little that T and U are now expressed by the aid
of the parameters q and their derivatives ; it matters little that it
is also by means of these parameters that we define the initial and
final positions ; the principle of least action remains always true.
Now here again, of all the paths that lead from one position
to another, there is one for which the average action is a mini-
mum, and there is only one. The principle of least action
suffices, then, to determine the differential equations which de-
fine the variations of the parameters q.
The equations thus obtained are another form of the equa-
tions of Lagrange.
To form these equations we need to know neither the relations
that connect the parameters q with the coordinates of the
hypothetical molecules, nor the masses of these molecules, nor
the expression of 17 as a function of the coordinates of these
molecules.
All we need to know is the expression of Z7 as a function of
the parameters, and that of T as a function of the parameters q
and their derivatives, that is, the expressions of the kinetic and
of the potential energy as functions of the experimental data.
Then we shall have one of two things: either for a suitable
s We add that U wiU depend only on the parameters q, that T will depend
on the parameters q and their derivatives with respect to the time and wiU
be a homogeneous polynomial of the second degree with respect to these
derivatiyes.
180 SCIENCE AND HYPOTHESIS
choice of the functions T and U, the equations of Lagrange, con-
structed as we have just said, will be identical with the differ-
ential equations deduced from experiments; or else there will
exist no functions T and U, for which this agreement takes place.
In the latter case it is clear that no mechanical explanation i^
possible.
The necessary condition for a mechanical explanation to be
possible is therefore that we can choose the functions T and U
in such a way as to satisfy the principle of least action, which in-
volves that of the conservation of energy.
This condition, moreover, is sufficient. Suppose, in fact, that
we have found a function Z7 of the parameters q, which repre-
sents one of the parts of the energy; that another part of the
energy, which we shall represent by T, is a function of the
parameters q and their derivatives, and that it is a homogeneous
polynomial of the second degree with respect to these derivatives;
and finally that the equations of Lagrange, formed by means of
these two functions, T and U, conform to the data of the
experiment.
What is necessary in order to deduce from this a mechanical
explanation? It is necessary that U can be regarded as the po-
tential energy of a system and T as the vis viva of the same
system.
There is no diflSculty as to U, but can T be regarded as the
vis viva of a material system ?
It is easy to show that this is always possible, and even in
an infinity of ways. I will confine myself to referring for more
details to the preface of my work, 'Electricity et optique.'
Thus if the principle of least action can not be satisfied, no
mechanical explanation is possible ; if it can be satisfied, there is
not only one, but an infinity, whence it follows that as soon as
there is one there is an infinity of others.
One more observation.
Among the quantities that experiment gives us directly, we
shall regard some as functions of the coordinates of our hypo-
thetical molecules; these are our parameters g. We shall look
upon the others as dependent not only on the coordinates, but on
the velocities, or, what comes to the same thing, on the derivatives
L '4iCtiie parameters g, or as combinations of these parameters and
their derivatives.
And then a question presents itself: among all these quantities
measured experimentally, which shall we choose to represent the
parameters gT Which sliall we prefer to regard aa the deriva-
tives of these parameters'? This choice remains arbitrary to a
very large extent ; but, for a mechanical explanation to be possi-
ble, it suffices if we can make tlie choice in such a way as to
accord with the prinei]jle of least action.
And then Maxwell asked himself whether he could make this
choice and that of the two energies T and U, in such a way
that the electrical phenomena would satisfy this principle. Ex-
periment shows us that the energy of an electromagnetic field is
decomposed into two parts, the electrostatic energy and the elee-
trodynamic energy. Maxwell observed that if we regard the
first as representing the potential energy U, the second as repre-
senting the kinetic energy T; if, moreover, the electrostatic
charges of the conductors are considered as parameters q and
the intensities of the currents aa the derivatives of other para-
meters q ; under these conditions, I say. Maxwell observed that the
electric phenomena satisfy the principle of least action. Thence-
forth he was certain of the possibility of a mechanical ex-
planation.
If he had explained this idea at the beginning of his book
instead of relegating it to an obscure part of the second volume,
it would not have escaped thp majority of readers.
If, then, a phenomenon admits of a complete mechanical ex-
planation, it will admit of an infinity of others, that will render
an acoount equally well of all the particulars revealed by ex-
periment.
And this is confirmed by the history of every branch of
physics ; in optics, for instance, Presnel believed vibration to be
perpendicular to the plane of polarization; Neumann regarded
it as parallel to this plane. An ' experimentnm erucis' has long
been sought which would enable us to decide between these two
theories, but it has not been found.
In the same way, without leaving the domain of electricity,
we may ascertain that the theory of two fluids and that of the
182 SCIENCE AND HYPOTHESIS
single fluid both acconnt in a fashion equally satisfactory for all
the observed laws of electrostatics.
All these facts are easily explicable, thanks to the properties
of the equations of Lagrange which I have just recalled.
It is easy now to comprehend what is Maxwell's fundamental
idea.
To demonstrate the possibility of a mechanical explanation of
electricity, we need not preoccupy ourselves with finding this
explanation itself; it suffices us to know the expression of the
two functions T and U, which are the two parts of energy, to
form with these two functions the equations of Lagrange and
then to compare these equations with the experimental laws.
Among all these possible explanations, how make a choice for
which the aid of experiment fails usf A day will come perhaps
when physicists will not interest themselves in these questions,
inaccessible to x>ositive methods, and will abandon them to the
metaphysicians. This day has not yet arrived; man does not
resign himself so easily to be forever ignorant of the foundation
of things.
Our choice can therefore be further guided only by considera-
tions where the part of personal appreciation is very great ; there
are, however, solutions that all the world will reject because of
their whimsicaUty, and others that all the world wiU prefer be-
cause of their simplicity.
In what concerns electricity and magnetism, Maxwell abstains
from making any choice. It is not that he systematically dis-
dains all that is unattainable by positive methods; the time he
has devoted to the kinetic theory of gases sufficiently proves that.
I will add that if, in his great work, he develops no complete
explanation, he had previously attempted to give one in an article
in the Philosophical Magazine. The strangeness and the com-
plexity of the hypotheses he had been obliged to make had led
him afterwards to give this up.
The same spirit is found throughout the whole work. What
is essential, that is to say what must remain common to all
theories, is made prominent; all that would only be suitable to
a particular theory is nearly always passed over in silence. Thus
the reader finds himself in the presence of a form almost devoid
OPTICS AND ELECTBICITT 183
of matter, which he is at first tempted to take for a fugitive
shadow not to be grasped. But the efforts to which he is thus
condemned force him to think and he ends by comprehending
what was often rather artificial in the theoretic constructs he
had previously only wondered at.
CHAPTER XIII
Electrodynamics
The history of electrodynamics is particularly instructive from
our point of view.
Ampere entitled his immortal work, 'Thdorie des ph6nom^nes
^lectrodynamiques, uniquement fond^ sur Inexperience/ He
therefore imagined that he had made no hypothesis, but he had
made them, as we shall soon see; only he made them without
being conscious of it.
His successors, on the other hand, perceived them, since their
attention was attracted by the weak points in Ampere's solution.
They made new hypotheses, of which this time they were fully
conscious ; but how many times it was necessary to change them
before arriving at the classic system of to-day which is perhaps
not yet final ; this we shall see.
I. Amp&ke's Theory. — ^When Ampere studied experimentally
the mutual actions of currents, he operated and he only could
operate with closed currents.
It was not that he denied the possibility of open currents.
If two conductors are charged with positive and negative elec-
tricity and brought into communication by a wire, a current is
established going from one to the other, which continues until the
two potentials are equal. According to the ideas of Ampere's
time this was an open current; the current was known to go
from the first conductor to the second, it was not seen to return
from the second to the first.
So Ampere considered as open currents of this nature, for ex-
ample, the currents of discharge of condensers; but he could
not make them the objects of his experiments because their
duration is too short.
Another sort of open current may also be imagined. I sup-
pose two conductors, A and B, connected by a wire AMB. Small
conducting masses in motion first come in contact with the
184
ELECTRODYNAMICS 185
conductor B, take from it an electric charge, leave contact with
B and move along the path BNA, and, transporting with them
their charge, come into contact with A and give to it their charge,
which returns then to B along the wire AMB,
Now there we have in a sense a closed circuit, since the elec-
tricity describes the closed circuit BNAMB; but the two parts
of this current are very different. In the wire AMB, the elec-
tricity is displaced through a fixed conductor, like a voltaic cur-
rent, overcoming an ohmic resistance and developing heat; we
say that it is displaced by conduction. In the part BNA, the
electricity is carried by a moving conductor ; it is said to be dis-
placed by convection.
If then the current of convection is considered as altogether
analogous to the current of conduction, the circuit BNAMB is
closed; if, on the contrary, the convection current is not 'a true
current,' and, for example, does not act on the magnet, there
remains only the conduction current AMB, which is open.
For example, if we connect by a wire the two poles of a Holtz
machine, the charged rotating disc transfers the electricity by
convection from one pole to the other, and it returns to the first
pole by conduction through the wire.
But currents of this sort are very difficult to produce with ap-
preciable intensity. With the means at Ampere's disposal, we
may say that this was impossible.
To sum up, Ampere could conceive of the existence of two
kinds of open currents, but he could operate on neither because
they were not strong enough or because their duration was too
short
Experiment therefore could only show him the action of a
closed current on a closed current, or, more accurately, the action
of a closed current on a portion of a current, because a current
can be made to describe a closed circuit composed of a moving
part and a fixed part. It is possible then to study the displace-
ments of the moving part under the action of another closed
current.
On the other hand, Ampfere had no means of studying the
action of an open current, either on a closed current or another
open current.
186 SCIENCE AND HYPOTHESIS
1. The Case of Closed Currents. — ^In the case of the mntaal
action of two closed currents, experiment revealed to Ampere re-
markably simple laws.
I recall rapidly here those which will be useful to us in the
sequel :
V If the intensity of the currents is kept constant, and if
the two circuits, after having undergone any deformations and
displacements whatsoever, return finally to their initial positions,
the total work of the electrodynamic actions will be nulL
In other words, there is an electrodynamic potential of the
two circuits, proportional to the product of the intensities, and
depending on. the form and relative position of the circuits; the
work of the electrodynamic actions is equal to the variation of
this potential:
2^ The action of a closed solenoid is nulL
3° The action of a circuit C on another voltaic circuit C de-
pends only on the 'magnetic field' developed by this circuit. At
each point in space we can in fact define in magnitude and direc-
tion a certain force called magnetic force, which enjoys the fol-
lowing properties :
(a) The force exercised by C on a magnetic pole is applied to
that pole and is equal to the magnetic force multiplied by the
magnetic mass of that pole ;
(b) A very short magnetic needle tends to take the direction
of the magnetic force, and the couple to which it tends to reduce
is proportional to the magnetic force, the magnetic moment of
the needle and the sine of the dip of the needle ;
(c) If the circuit C is displaced, the work of the electrody-
namic action exercised by C on C will be equal to the increment
of the 'flow of magnetic force' which passes through the circuit.
2. Action of a Closed Current on a Portion of Current. —
Ampere not having been able to produce an open current, prop-
erly so called, had only one way of studying the action of a
closed current on a portion of current.
This was by operating on a circuit C composed of two parts,
the one fixed, the other movable. The movable part was, for
instance, a movable wire ap whose extremities a and p could
ELECTRODYNAMICS 187
dide along a fixed wire. In one of the positions of the movable
wire, the end a rested on the A of the fixed wire and the extrem-
ity p on the point B of the fixed wire. The current circulated
from a to p^ that is to say, from Aio B along the movable wire,
and then it returned from B io A along the fixed wire. This
current was therefore closed.
In a second position, the movable wire having slipped, the ex-
tremity a rested on another point A' of the fixed wire, and the
extremity p on another point B' of the fixed wire. The current
circulated then from a to Pj that is to say from A' to B' along the
movable wire, and it afterwards returned from B' to B, then from
BXo Aj then finally from A to A'^ always following the fixed wire.
The current was therefore also closed.
If a like current is subjected to the action of a closed current
C, the movable part will be displaced just as if it were acted
upon by a force. Ampere assumes that the apparent force to
which this movable part AB seems thus subjected, representing
the action of the C on the portion ap of the current, is the same
Bsitap were traversed by an open current, stopping at a and p,
in place of being traversed by a closed current which after arriv-
ing at p returns to a through the fixed part of the circuit.
This hypothesis seems natural enough, and Ampere made it
unconsciously ; nevertheless it is not necessary, since we shall see
further on that Helmholtz rejected it. However that may be, it
permitted Ampere, though he had never been able to produce an
open current, to enunciate the laws of the action of a closed cur-
rent on an open current, or even on an element of current.
The laws are simple:
1^ The force which acts on an element of current is applied
to this element; it is normal to the element and to the magnetic
force, and proportional to the component of this magnetic force
which is normal to the element.
2° The action of a closed solenoid on an element of current is
null.
But the electrodynamic potential has disappeared, that is to
say that, when a closed current and an open current, whose in-
tensities have been maintained constant, return to their initial
positions, the total work is not null.
188 SCIENCE AND HYPOTHESIS
3. Continuous Rotations. — ^Among electrodynamic experiments,
the most remarkable are those in which continuous rotations are
produced and which are sometimes called unipolar induction ex-
periments. A magnet may turn about its axis; a current passes
j5rst through a fixed wire, enters the magnet by the pole N, for
example, passes through half the magnet, emerges by a sliding
contact and reenters the fixed wire.
The magnet then begins to rotate continuously without being
able ever to attain equilibrium; this is Faraday's experiment
How is it possible? If it were a question of two circuits of
invariable form, the one C fixed, the other C movable about an
axis, this latter could never take on continuous rotation ; in fact
there is an electrodynamic potential; there must therefore be
necessarily a position of equilibrium when this potential is a
maximum.
Continuous rotations are therefore possible only when the cir-
cuit C is composed of two parts: one fixed, the other movable
about an axis, as is the case in Faraday's experiment. Here
again it is convenient to draw a distinction. The passage from
the fixed to the movable part, or inversely, may take place either
by simple contact (the same point of the movable part remaining
constantly in contact with the same point of the fixed part) , or by
a sliding contact (the same point of the movable part coming
successively in contact with diverse points of the fixed part).
It is only in the second case that there can be continuous rota-
tion. This is what then happens: The system tends to take a
position of equilibrium; but, when at the point of reaching that
position, the sliding contact puts the movable part in communi-
cation with a new point of the fixed part; it changes the con-
nections, it changes therefore the conditions of equilibrium, so
that the position of equilibrium fleeing, so to say, before the
system which seeks to attain it, rotation may take place indefi-
nitely.
Ampere assumes that the action of the circuit on the movable
part of C is the same as if the fixed part of C did not exist, and
therefore as if the current passing through the movable part were
open.
ELECTRODTNAMJCS
18d
He concludes therefore that the action of a closed on an open
cnrreot, or inversely that of an open current on a closed current,
may g^ve rbe to a continuoas rotation.
But this conclusion depends on the hypotJiesis I have enun-
ciated and which, as I said above, is not admitted by Helmholtz.
4. Mutual Action of Two Open Currents, — In what concerns
the mutual actions of two open currents, and in particular that
of two elements of current, all experiment breaks down. Am-
pere has recourse to hypothesis. He supposes :
1" That the mutual action of two elements reduces to a force
acting along their join;
2" That the action of two closed currents is the resultant of
the mutual actions of their diverse elements, which are besides
the same as if these elements were isolated.
What is remarkable is that here again Ampere makes these
hypotheses unconsciously.
However that may be, these two hypotheses, together with the
experiments on closed currents, suffice to determine completely
the law of the mutual action of two elements. But then most
of the simple laws we have met in the case of closed currents are
no longer true.
In the first place, there is no electrodynamic potential ; nor wag
there any, as we have seen, in tlie case of a closed current acting
on an open current.
Next there is, properly speaking, no magnetic force.
And, in fact, we have given above three different definitions
of this force:
1° By the action on a magnetic pole;
2° By the director couple which orientates the magnetic
needle;
3° By the action on an element of current.
But in the case which now occupies us, not only these three
de&utions are no longer in harmony, but each has lost its mean-
ing, and in fact:
1° A magnetic pole is no longer acted upon simply by a ungle
force applied to this pole. We have seen in fact that the force
due to the action of an element of current on a pole is not applied
to the pole, but to the element ; it may moreover be replaced by
a force applied to the pole and by a couple ;
190 SCIENCE AND HYPOTHESIS
2^ The couple which acts on the magnetic needle is no longer
a simple director couple, for its moment with respect to the axis
of the needle is not null. It breaks up into a director couple,
properly so called, and a supplementary couple which tends to
produce the continuous rotation of which we have above spoken;
3"* Finally the force acting on an element of current is not
normal to this element.
In other words, the unity of th6 magnetic force has disap
peared.
Let us see in what this unity consists. Two systems which
exercise the same action on a magnetic pole will exert also the
same action on an indefinitely small magnetic needle, or on an
element of current placed at the same point of space as this pole.
Well, this is true if these two systems contain only closed
currents ; this would no longer be true if these two systems con-
tained open currents.
It su£5ces to remark, for instance, that, if a magnetic pole is
placed at A and an element at B, the direction of the element
being along the prolongation of the sect AB^ this element which
will exercise no action on this pole will, on the other hand, exer-
cise an action either on a magnetic needle placed at the point A,
or on an element of current placed at the point A.
5. Induction. — ^We know that the discovery of electrodynamic
induction soon followed the immortal work of AmpSre.
As long as it is only a question of closed currents there is no
difficulty, and Helmholtz has even remarked that the principle of
the conservation of energy is sufficient for deducing the laws
of induction from the electrodynamic laws of Ampfere. But
always on one condition, as Bertrand has well shown; that we
make besides a certain number of hypotheses.
The same principle again permits this deduction in the case of
open currents, although of course we can not submit the result
to the test of experiment, since we can not produce such currents.
If we try to apply this mode of analysis to Ampere's theory
of open currents, we reach results calculated to surprise us.
In the first place, induction can not be deduced from the
variation of the magnetic field by the formula well known to
savants and practicians, and, in fact, as we have said, properly
speaking there is no longer a magnetic field.
ELECTRODYNAMICS 191
Bat, further; if a circuit C is subjected to the induction of a
variable voltaic system 8, if this system 8 be displaced and de-
formed in any way whatever, so that the intensity of the currents
of this system varies according to any law whatever, but that
after these variations the system finally returns to its initial sit-
uation, it seems natural to suppose that the mean electromotive
force induced in the circuit C is null.
This is true if the circuit C is closed and if the system 8 con-
tains only closed currents. This would no longer be true, if one
accepts the theory of AmpSre, if there were open currents. So
that not only induction will no longer be the variation of the
flow of magnetic force, in any of the usual senses of the word, but
it can not be represented by the variation of anything whatever.
11. Theory op Helmholtz. — I have dwelt upon the conse-
quences of Ampere's theory, and of his method of explaining
open currents.
It is difficult to overlook the paradoxical and artificial char-
acter of the propositions to which we are thus led. One can not
help thinking 'that can not be so.'
We understand therefore why Helmholtz was led to seek some-
thing else.
Helmholtz rejects Ampere's fundamental hypothesis, to wit,
that the mutual action of two elements of current reduces to a
force along their join. He assumes that an element of current is
not subjected to a single force, but to a force and a couple. It is
just this which gave rise to the celebrated polemic between Ber-
trand and Helmholtz.
Helmholtz replaces Ampere's hypothesis by the following: two
elements always admit of an electrodynamic potential depend-
ing solely on their position and orientation ; and the work of the
forces that they exercise, one on the other, is equal to the varia-
tion of this potential. Thus Helmholtz can no more do without
hypothesis than Ampere ; but at least he does not make one with-
out explicitly announcing it.
In the case of closed currents, which are alone accessible to
experiment, the two theories agree.
In all other cases they differ.
In the first place, contrary to what Ampere supposed, the force
192 SCIENCE AND HYPOTHESIS
which seems to act on the movable portion of a closed current
is not the same as would act upon this movable portion if it
were isolated and constituted an open current.
Let us return to the circuit C, of which we spoke above, and
which was formed of a movable wire ap sliding on a fixed wire.
In the only experiment that can be made, the movable portion afi
is not isolated, but is part of a closed circuit. When it passes
from AB to A'B', the total electrodynamic potential varies for
two reasons:
l"* It undergoes a first increase because the potential of A'V
with respect to the circuit C is not the same as that of AB\
2^ It takes a second increment because it must be increased
by the potentials of the elements AA', BB' with respect to C.
It is this double increment which represents the work of the
force to which the portion AB seems subjected.
If, on the contrary, ap were isolated, the potential would
undergo only the first increase, and this first increment alone
would measure the work of the force which acts on AB.
In the second place, there could be no continuous rotation
without sliding contact, and, in fact, that, as we have seen i
propos of closed currents, is an immediate consequence of the
existence of an electrodynamic potential.
In Faraday's experiment, if the magnet is fixed and if the
part of the current exterior to the magnet runs along a movable
wire, that movable part may undergo a continuous rotation.
But this does not mean to say that if the contacts of the wire
with the magnet were suppressed, and an open current were to
run along the Avire, the wire would still take a movement of con-
tinuous rotation.
I have just said in fact that an isolated element is not acted
upon in the same way as a movable element making part of a
closed circuit.
Another diflference: The action of a closed solenoid on a
closed current is null according to experiment and according to
the two theories. Its action on an open current would be null
according to AmpSre; it would not be null according to Helm-
holtz. From this follows an important consequence. We have
given above three definitions of magnetic force. The third has
ELECTROD 7NAMICS
no meaning here aince an element of current is no longer acted
upon by a single force. No more has the first any meaning.
What, in fact, is a magnetic pole? It is the extremity of an
indefinite linear magnet. This magnet may be replaced by an
indefinite solenoid. For the definition of magnetic force to have
any meaning, it would be necessary that the action exercised by
an open current on an indefinite solenoid should depend only on
the position of the extremity of this solenoid, that is to say, that
the action on a closed solenoid should be null. Now we have
just seen that such is not tlie case.
On the other hand, nothing prevents our adopting the second
definition, which is founded on the measurement of the director
couple which tends to orientate the magnetic needle.
But if it is adopted, neither the effects of induction nor the
electrodynamic effects will depend solely on the distribution of
the lines of force in this magnetic field.
1X1. Difficulties Raised by These Theories. — The theory
of Helmholtz is in advance of that of Ampere ; it Js necessary,
however, that all the difficulties should be smoothed away. In
the one as in the other, the phrase 'magnetic field' has no mean-
ing, or, if we give it one, by a more or less artificial convention,
the ordinary laws so familiar to all electricians no longer apply i
thus the electromotive force induced in a wire is no longer
measured by the number of lines of force met by this wire.
And our repugnance does not come alone from the difficulty
of renouncing inveterate habits of language iind of thought.
There is something more. If we do not believe in action at a di»-
tanee, electrodynamic phenomena must be explained by a modi-
fication of the medium. It is precisely this modification that we
call 'magnetic field.' And then the electrodynamic effects must
depend only on this field.
All these difficulties arise from the hypothesis of open currents.
rV. Maxwell's Theoby. — Such were the difficulties raised
by the dominant theories when Maxwell appeared, who with a
ftroke of the pen made them all vanish. To his mind, in fact,
all currents are closed currents. Maxwell assumes that if in
8 dielectric the electric field happens to vary, this dielectric
becomes the seat of a particular phenomenon, acting on the gal-
194 SCIENCE AND HYPOTHESIS
vanometer like a current, and which he calls current of dis-
placement.
If then two conductors bearing contrary charges are put in
communication by a wire, in this wire during the discharge there
is an open current of conduction; but there are produced at the
same time in the surrounding dielectric, currents of displacement
which close this current of conduction.
We know that Maxwell's theory leads to the explanation of
optical phenomena, which would be due to extremely rapid elec-
trical oscillations.
At that epoch such a conception was only a bold hypothesis,
which could be supported by no experiment.
At the end of twenty years, Maxwell's ideas received the con-
firmation of experiment. Hertz succeeded in producing sys-
tems of electric oscillations which reproduce all the properties
of light, and only differ from it by the length of their wave ; that
is to say as violet differs from red. In some measure he made
the synthesis of light.
It might be said that Hertz has not demonstrated directly
Maxwell's fundamental idea, the action of the current of dis-
placement on the galvanometer. This is true in a sense. What
he has shown in sum is that electromagnetic induction is not
propagated instantaneously as was supposed; but with the speed
of light.
But to suppose there is no current of displacement, and induc-
tion is propagated with the speed of light ; or to suppose that the
currents of displacement produce effects of induction, and that
the induction is propagated instantaneously, comes to the samA
thing.
This can not be seen at the first glance, but it is proved by an
analysis of which I must not think of giving even a summary
here.
V. Rowland's Experiment. — But as I have said above, there
are two kinds of open conduction currents. There are first the
currents of discharge of a condenser or of any conductor what-
ever.
There are also the cases in which electric discharges describe
a closed contour, being displaced by conduction in one part of
the circuit and by convection in the other part.
For open currents of the first sort, the question might be con-
sidered as solved; they were closed by the currents of displace-
ment.
For open currents of the second sort, the solution appeared
still more simple. It seemed that if the current were closed, it
could only be by the current of convection itself. For that it
snfficed to assume that a 'convection current,' that is to say a
charged conductor in motion, could act on the galvanometer.
But experimental confirmation was lacking. It appeared diffi-
cult in fact to obtain a sufReient intensity even by augmenting as
much as possible the charge and the velocity of the conductors. It
was Rowland, an extremely skillful experimenter, who first tri-
umphed over these difficulties. A disc received a strong electro-
static charge and a very great speed of rotation. An astatic mag-
netic system placed beside the disc underwent deviations.
The experiment was made twice by Rowland, once in Berlin,
once in Baltimore. It was afterwards repeated by Himetedt.
These physicists even announced that they had succeeded in mak-
ing quantitative measurements.
In fact, for twenty years Rowland's law was admitted without
objection by all physicists. Besides everything seemed to confirm
it. The spark certainly does produce a magnetic effect. Now does
it not aeem probable that the discharge by spark is due to particles
taken from one of the electrodes and transferred to the other elec-
trode with their charge t Is not the very spectrum of the spark,
in which we recognize the lines of the raetal of the electrode, a
proof of itt The spark would then be a veritable current of
etmreotion.
On the other hand, it is also admitted that in an electrolyte
tfas electricity is carried by the iona in motion. The current in
•n electrolj^e would therefore be also a current of convection;
BOW, it acts on the magnetic needle.
^nie Bame for cathode rays. Crookes attributed these raya
to a very subtile matter charged with electricity and movi
irith a very great velocity. He regarded them, in other,
words, as currents of convection. Now these cathode rays
■ayi ^^^H
ther^^^^H
ate ^^^H
196 SCIENCE AND HYPOTHESIS
deviated by the magnet. In virtue of the principle of action and
reaction, they should in turn deviate the magnetic needle. It is
true that Hertz believed he had demonstrated that the cathode
rays do not carry electricity, and that they do not act on the
magnetic needle. But Hertz was mistaken. First of all, Perrin
succeeded in collecting the electricity carried by these rays, elec-
tricity of which Hertz denied the existence ; the Oerman scientist
appears to have been deceived by effects due to the action of
X-rays, which were not yet discovered. Afterwards, and quite
recently, the action of the cathode rays on the magnetic needle
has been put in evidence.
Thus all these phenomena regarded as currents of convection,
sparks, electrolytic currents, cathode rays, act in the same manner
on the galvanometer and in conformity with Bowland's law.
VI. Theory op Lokentz. — ^We soon went further. Accord-
ing to the theory of Lorentz, currents of conduction themselves
would be true currents of convection. Electricity would remain
inseparably connected with certain material particles called elec-
trons. The circulation of these electrons through bodies would
produce voltaic currents. And what would distinguish con-
ductors from insulators would be that the one could be traversed
by these electrons while the others would arrest their movements.
The theory of Lorentz is very attractive. It gives a very
simple explanation of certain phenomena which the earlier the-
ories, even Maxwell's in its primitive form, could not explain in a
satisfactory way; for example, the aberration of light, the par-
tial carrying away of luminous waves, magnetic polarization and
the Zeeman effect.
Some objections still remained. The phenomena of an elec-
tric system seemed to depend on the absolute velocity of transla-
tion of the center of gravity of this system, which is contrary to
the idea we have of the relativity of space. Supported by M.
Cremieu, M. Lippmann has presented this objection in a striking
form. Imagine two charged conductors with the same velocity
of translation; they are relatively at rest. However, each of
them being equivalent to a current of convection, they ought to
attract one another, and by measuring this attraction we could
measure their absolute velocity.
ELECTRODYNAMICS 197
"No!" replied the partisans of Lorentz. ''What we could
measure in tbat way is not their absolute velocity, but their rela-
tive velocity wiih respect to the ether, so that the principle of
relativity is safe."
Whatever there may be in these latter objections, the edifice of
electrodynamics, at least in its broad lines, seemed definitively
constructed. Everything was presented under the most satis-
factory aspect. The theories of Ampere and of Helmholtz, made
for open currents which no longer existed, seemed to have no
longer anything but a purely historic interest, and the inextricable
complications to which these theories led were almost forgotten.
This quiescence has been recently disturbed by the experi-
ments of M. Cr^mieu, which for a moment seemed to contradict
the result previously obtained by Eowland.
But fresh researches have not confirmed them, and the theory
of Lorentz has victoriously stood the test.
The history of these variations will be none the less instruct-
ive; it will teach us to what pitfalls the scientist is exposed, and
how he may hope to escape them.
THE VAX-TJE OF SCIENCE.
TRANSLATOR'S INTRODUCTION
1. Does the Scientist create Science f — ^Professor Rados of Buda-
pest in his report to the Hungarian Academy of Science on the
award to Poincar^ of the Bolyai prize of ten thousand crowns,
speaking of him as unquestionably the most powerful investiga-
tor in the domain of mathematics and mathematical physics,
characterized him as the intuitive genius drawing the inspiration
for his wide-reaching researches from the exhaustless fountain
of geometric and physical intuition, yet working this inspira*
tion out in detail with marvelous logical keenness. With his
brilliant creative genius was combined the capacity for sharp
and successful generalization, pushing far out the boundaries of
thought in the most widely different domains, so that his works
must be ranked with the greatest mathematical achievements of
all time. *' Finally, '* says Bados, *' permit me to make especial
mention of his intensely interesting book, 'The Value of Science,'
in which he in a way has laid down the scientist's creed." Now
what is this creed f
Sense may act as stimulus, as suggestive, yet not to awaken a
dormant depiction, or to educe the conception of an archetypal
form, but rather to strike the hour for creation, to sununon to
work a sculptor capable of smoothing a Venus of Milo out of the
formless clay. Knowledge is not a gift of bare experience, nor
even made solely out of experience. The creative activity of
mind is in mathematics particularly clear. The axioms of geom-
etry are conventions, disguised definitions or unprovable hy-
potheses precreated by auto-active animal and human minds.
Bertrand Russell says of projective geometry: **It takes nothing
from experience, and has, like arithmetic, a creature of the pure
intellect for its object. It deals with an object whose properties
are logically deduced from its definition, not empirically dis-
covered from data." Then does the scientist create science?
This is a question Poincare here dissects with a master hand.
The physiologic-psychologic investigation of the space problem
201
202 THE VALUE OF SCIENCE
must give the meaning of the words geometric fact, geometric
reality. Poincarg here subjects to the most successful analysis
ever made the tridimensionality of our space.
2. The Mind Dispelling Optical Illusions. — ^Actual perception
of spatial properties is accompanied by movements correspond-
ing to its character. In the case of optical illusions, with the so-
called false perceptions eye-movements are closely related. But
though the perceived object and its environment remain constant,
the sufficiently powerful mind can, as we say, dispel these illu-
sions, the perception itself being creatively changed. Photo-
graphs taken at intervals during the presence of these optical
illusions, during the change, perhaps gradual and unconscious,
in the perception, and after these illusions have, as the phrase is,
finally disappeared, show quite clearly that changes in eye-
movements corresponding to those internally created in percep-
tion itself successively occur. What is called accuracy of move-
ment is created by what is called correctness of perception. The
higher creation in the perception is the determining cause of an
improvement, a precision in the motion. Thus we see correct per-
ception in the individual helping to make that cerebral organiza-
tion and accurate motor adjustment on which its possibility and
permanence seem in so far to depend. So-called correct percep-
tion is connected with a long-continued process of perceptual
education motived and initiated from within. How this may
take place is here illustrated at length by our author.
3. Euclid not Necessary. — Geometry is a construction of the
intellect, in application not certain but convenient. As Schiller
says, when we see these facts as clearly as the development of
metageometry has compelled us to see them, we must surely con-
fess that the Kantian account of space is hopelessly and demon-
strably antiquated. As Royce says in * Kant's Doctrine of the
Basis of Mathematics, ' * * That very use of intuition which Kant
regarded as geometrically ideal, the modem geometer regards
as scientifically defective, because surreptitious. No mathemat-
ical exactness without explicit proof from assumed principles —
such is the motto of the modem geometer. But suppose the
reasoning of Euclid purified of this comparatively surreptitious
TBANSLATOB'S INTRODUCTION 208
appeal to intuition. Suppose that the principles of geometry are
made quite explicit at the outset of the treatise, as Fieri and
Hilbert or Professor Halsted or Dr. Yeblen makes his principles
explicit in his recent treatment of geometry. Then, indeed, geom-
etry becomes for the modem mathematician a purely rational
scienee. But very few students of the logic of mathematics at the
present time can see any warrant in the analysis of geometrical
truth for regarding just the Euclidean system of principles as
possessing any discoverable necessity." Yet the environmental
and perhaps hereditary premiums on Euclid still make even the
scientist think Euclid most convenient.
4. Without Hypotheses, no Science. — ^Nobody ever observed an
equidistantial, but also nobody ever observed a straight line.
Emerson's Uriel
''Gave hlB sentiment divine
Against the being of a line.
Line in Nature is not found. ' '
Clearly not, being an eject from man's mind. What is called 'a
knowledge of facts' is usually merely a subjective realization that
the old hypotheses are still sufficiently elastic to serve in some
domain; that is, with a sufficiency of conscious or unconscious
omissions and doctorings and fudgings more or less wilful. In
the present book we see the very foundation rocks of science, the
conservation of energy and the indestructibility of matter, beat-
ing against the bars of their cages, seemingly anxious to take
wing away into the empyrean, to chase the once divine parallel
postulate broken loose from Euclid and Kant.
5. What Outcome? — ^What now is the definite, the permanent
outcome ? What new islets raise their f ronded palms in air within
thought's musical domain? Over what age-gray barriers rise the
fragrant floods of this new spring-tide, redolent of the wolf-
haunted forest of Transylvania, of far Erd^ly's plunging river,
llaros the bitter, or broad mother Volga at Kazan ? What victory
heralded the great rocket for which young Lobachevski, the
widow's son, was cast into prison? What severing of age-old
mental fetters symbolized young Bolyai's cutting-oflf with his
204
THE VALUE OF SCIENCE
Dam&scos blade the apikes driven into his door-post, and atrew-
ing over the aod the tliirteen Anatrian cavalry officerat This
boob b7 the greatest mathematician of our time gives weightiest
and most charming answer.
QEOBas Bruce Hai£ted.
INTEODUCTION
The search for truth should be the goal of our activities ; it is
the sole end worthy of them. Doubtless we should first bend our
efforts to assuage human suffering, but whyf Not to suffer is a
negative ideal more surely attained by the annihilation of the
world. If we wish more and more to free man from material
cares, it is that he may be able to employ the liberty obtained in
the study and contemplation of truth.
But sometimes truth frightens ns. And in fact we know that it
is sometimes deceptive, that it is a phantom never showing itself
for a moment except to ceaselessly fiee, that it must be pursued
further and ever further without ever being attained. Yet to
work one must stop, as some Qreek, Aristotle or another, has said.
We also know how cruel the truth often is, and we wonder
whether illusion is not more consoling, yea, even more bracing,
for illusion it is which gives confidence. When it shall have
vanished, will hope remain and shall we have the courage to
achieve? Thus would not the horse harnessed to his treadmill
refuse to go, were his eyes not bandaged? And then to seek
truth it is necessary to be independent, wholly independent. If,
on the contrary, we wish to act, to be strong, we should be united.
This is why many of us fear truth; we consider it a cause of
weakness. Yet truth should not be feared, for it alone is beautiful.
When I speak here of truth, assuredly I refer first to scientific
truth ; but I also mean moral truth, of which what we call justice
is only one aspect. It may seem that I am misusing words, that
I combine thus under the same name two things having nothing
in common ; that scientific truth, which is demonstrated, can in no
way be likened to moral truth, which is felt. And yet I can not
separate them, and whosoever loves the one can not help loving
the other. To find the one, as well as to find the other, it is neces-
sary to free the soul completely from prejudice and from passion ;
it is necessary to attain absolute sincerity. These two sorts of
205
206 TRE VALUE OF SCIENCE
truth when discovered give the same joy; each when perceived
beams with the same splendor, so that we must see it or close our
eyes. Lastly, both attract us and flee from us; they are never
fixed : when we think to have reached them, we find that we have
still to advance, and he who pursues them is condemned never to
know repose. It must be added that those who fear the one will
also fear the other; for they are the ones who in everything are
concerned above all with consequences. In a word, I liken the
two truths, because the same reasons make ns love them and
because the same reasons make ns fear them.
If we ought not to fear moral truth, still less should we dread
scientific truth. In the first place it can not confiict with ethics.
Ethics and science have their own domains, which touch but do
not interpenetrate. The one shows ns to what goal we should
aspire, the other, given the goal, teaches us how to attain it. So
they can never conflict since they can never meet. There can no
more be immoral science than there can be scientific morals.
But if science is feared, it is above all because it can not give us
happiness. Of course it can not. We may even ask whether the
beast does not suffer less than man. But can we regret that
earthly paradise where man brute-like was really immortal in
knowing not that he must die ? When we have tasted the apple,
no suffering can make us forget its savor. We always come back
to it. Could it be otherwise? As well ask if one who has seen
and is blind will not long for the light. Man, then, can not be
happy through science, but to-day he can much less be happy
without it.
But if truth be the sole aim worth pursuing, may we hope to
attain it? It may well be doubted. Readers of my little book
* Science and Hypothesis' already know what I think about the
question. The truth we are permitted to glimpse is not alto-
gether what most men call by that name. Does this mean that
our most legitimate, most imperative aspiration is at the same
time the most vain? Or can we, despite all, approach truth on
some side ? This it is which must be investigated.
In the first place, what instrument have we at our disposal for
this conquest? Is not human intelligence, more specificaUy the
INTRODUCTION 207
intelligence of the scientist, susceptible of infinite variation f
Volumes could be written without exhausting this subject ; I, in
a few brief pages, have only touched it lightly. That the geom-
eter's mind is not like the physicist's or the naturalist's, aU the
world would agree; but mathematicians themselves do not re-
semble each other; some recognize only implacable logic, others
appeal to intuition and see in it the only source of discovery.
And this would be a reason for distrust. To minds so unlike can
the mathematical theorems themselves appear in the same light t
Truth which is not the same for all, is it truth f But looking
at things more closely, we see how these very different workers
collaborate in a common task which could not be achieved without
their cooperation. And that already reassures us.
Next must be examined the frames in which nature seems en-
closed and which are called time and space. In 'Science and
Hypothesis' I have already shown how relative their value is;
it is not nature which imposes them upon us, it is we who impose
them upon nature because we find them convenient. But I have
spoken of scarcely more than space, and particularly quanti-
tative space, so to say, that is of the mathematical relations whose
aggregate constitutes geometry. I should have shown that it is
the same with time as with space and still the same with 'qualita-
tive space'; in particular, I should have investigated why we
attribute three dimensions to space. I may be pardoned then for
taking up again these important questions.
Is mathematical analysis, then, whose principal object is the
study of these empty frames, only a vain play of the mind ? It
can give to the physicist only a convenient language ; is this not
a mediocre service, which, strictly speaking, could be done with-
out ; and even is it not to be feared that this artificial language
may be a veil interposed between reality and the eye of the
physicist ? Far from it ; without this language most of the inti-
mate analogies of things would have remained forever unknown
to us ; and we should forever have been ignorant of the internal
harmony of the world, which is, we shall see, the only true
objective reality.
The best expression of this harmony is law. Law is one of the
208 TRE VALUE OF SCIENCE
most recent conquests of the human mind; there still are people
who live in the presence of a perpetual miracle and are not
astonished at it. On the contrary, we it is who should be aston-
ished at nature's regularity. Men demand of their gods to prove
their existence by miracles ; but the eternal marvel is that there
are not miracles without cease. The world is divine because it is
a harmony. If it were ruled by caprice, what could prove to us
it was not ruled by chance ?
This conquest of law we owe to astronomy, and just this makes
the grandeur of the science rather than the material grandeur of
the objects it considers. It was altogether natural, then, that
celestial mechanics should be the first model of mathematical
physics; but since then this science has developed; it is still
developing, even rapidly developing. And it is already neces-
sary to modify in certain points the scheme from which I drew
two chapters of ' Science and Hypothesis. ' In an address at the
St. Louis exposition, I sought to survey the road traveled; the
result of this investigation the reader shall see farther on.
The progress of science has seemed to imperil the best estab-
lished principles, those even which were regarded as fundamental.
Yet nothing shows they will not be saved ; and if this comes about
only imperfectly, they will still subsist even though they are
modified. The advance of science is not comparable to the changes
of a city, where old edifices are pitilessly torn down to give place
to new, but to the continuous evolution of zoologic types which
develop ceaselessly and end by becoming unrecognizable to the
common sight, but where an expert eye finds always traces of the
prior work of the centuries past. One must not think then that
the old-fashioned theories have been sterile and vain.
Were we to stop there, we should find in these pages some
reasons for confidence in the value of science, but many more for
distrusting it ; an impression of doubt would remain ; it is need-
ful now to set things to rights.
Some people have exaggerated the role of convention in science ;
they have even gone so far as to say that law, that scientific fact
itself, was created by the scientist. This is going much too far
in the direction of nominalism. No, scientific laws are not arti-
INTRODUCTION 209
ficial creations; we have no reason to regard them as accidental,
though it be impossible to prove they are not.
Does the harmony the human intelligence thinks it discovers
in nature eidst outside of this intelligence f No, beyond doubt
a reality completely independent of the mind which conceives it,
sees or feels it, is an impossibility. A world as exterior as that,
even if it existed, would for us be forever inaccessible. But what
we call objective reality is, in the last analysis, what is common
to many thinking beings, and could be common to all ; this com-
mon part, we shall see, can only be the harmony expressed by
mathematical laws. It is this harmony then which is the sole
objective reality, the only truth we can attain ; and when I add
that the universal harmony of the world is the source of all
beauty, it will be understood what price we should attach to the
slow and difficult progress whieh little by little enables us to know
it better.
15
PART I
THE MATHEMATICAL SCIENCES
CHAPTER I
Intuition and Logic in Mathematics
I
It is impossible to study the works of the great matheniaticiaiis»
or even those of the lesser, without noticing and distinguishing
two opposite tendencies, or rather two entirely different kinds of
minds. The one sort are above all preoccupied with logic; to
read their works, one is tempted to believe they have advanced
only step by step, after the manner of a Vauban who pushes
on his trenches against the place besieged, leaving nothing to
chance. The other sort are guided by intuition and at the first
stroke make quick but sometimes precarious conquests, like bold
cavalrymen of the advance guard.
The method is not imposed by the matter treated. Though one
often says of the first that they are analysts and calls the others
geometers, that does not prevent the one sort from remaining
analysts even when they work at geometry, while the others are
still geometers even when they occupy themselves with pure
analysis. It is the very nature of their mind which makes them
logicians or intuitionalists, and they can not lay it aside when
they approach a new subject.
Nor is it education which has developed in them one of the two
tendencies and stifled the other. The mathematician is bom, not
made, and it seems he is bom a geometer or an analyst. I should
like to cite examples and there are surely plenty; but to accentu-
ate the contrast I shall begin with an extreme example, taking the
liberty of seeking it in two living mathematicians.
210
mWlTION AND LOGIC IN MATUEUATICS
211
M. M^ray wants to prove that a binomial equation alwaye haa
a root, or, in ordinary words, that an angle may always be sub-
divided. If there is any truth that we think we know by direct
intuition, it is this. Wlio eould doubt that an angle may always
be divided into any number of equal parts T M, Meray does not
look at it that way; in his eyes tJiis proposition is not at all
evident and to prove it he needs several pages.
On the other hand, look at Professor Klein : he is studying one
of the most abstract questions of the theory of functions : to deter-
mine whether on a given Kiemann surface there always exists a
function admitting of given singularities. What does the cele-
brated Qerman geometer doT He replaces his Riemann surface
by a metallic surface whose electric conductivity varies according
to certain laws. He connects two of its points with the two pole
of a battery. The current, says he, must pass, and the distribn*!
lion of this current on the surface will define a function whose '
singularities will be precisely those called for by the enunciation.
Doubtless Professor Klein wcU knows he has given here only
a sketch; nevertheless he has not hesitated to publish it; and he
would probably believe he finds in it, if not a rigorous demon-
stration, at least a kind of moral certainty. A logician would
have rejected with horror such a conception, or rather he would
not have had to reject it, because in his mind it would never have
originated.
Again, permit me to compare two men, the honor of French
science, who have recently been taken from us, but who both
entered long ago into immortality. I speak of M, Bertrand and J
il. Hermite. They were scholars of the same school at the s
time; they bad the same education, were under the same inflo- "
ences; and yet what a difference 1 Not only does it blaze forth
in their writings ; it is in their teaching, in their way of speaking,
in their very look. In the memory of all their pupils these two
faees are stamped in deathless lines; for all who have had the
pleasure of following their teaching, this remembrance is still
fresh 1 it is easy for us to evoke it.
While speaking, ^I. Bertrand is always in motion ; now he seems
in combat with some outside enemy, now he outlines with a gesture
of the hand the figures he studies. Plainly he sees and he is
212 TRE VALVE OF SCIENCE
eager to paint, this is why he calls gesture to his aid. With M.
Hermite, it is just the opposite; his eyes seem to shun contact
with the world ; it is not without, it is within he seeks the vision
of truth.
Among the Qerman geometers of this century, two names above
all are illustrious, those of the two scientists who founded the
general theory of functions, Weierstrass and Biemann. Weier-
strass leads everything back to the consideration of series and
their analytic transformations; to express it better, he reduces
analysis to a sort of prolongation of arithmetic ; you may turn
through all his books without finding a figure. Biemann, on the
contrary, at once calls geometry to his aid; each of his concei>-
tions is an image that no one can forget, once he has caught its
meaning.
More recently. Lie was an intuitionalist; this might have been
doubted in reading his books, no one could doubt it after talking
¥dth him ; you saw at once that he thought in pictures. Madame
Eovalevski was a logician.
Among our students we notice the same differences; some prefer
to treat their problems 'by analysis,' others 'by geometry.* The
first are incapable of 'seeing in space,' the others are quickly
tired of long calculations and become perplexed.
The two sorts of minds are equally necessary for the progress
of science ; both the logicians and the intuitionalists have achieved
great things that others could not have done. Who would ven-
ture to say whether he preferred that Weierstrass had never
written or that there had never been a Biemann t Analysis and
synthesis have then both their legitimate roles. But it is inter-
esting to study more closely in the history of science the part
which belongs to each.
II
Strange! If we read over the works of the ancients we are
tempted to class them all among the intuitionalists. And yet
nature is always the same ; it is hardly probable that it has begun
in this century to create minds devoted to logic. If we could put
ourselves into the flow of ideas which reigned in their time, we
should recognize that many of the old geometers were in tendency
f
INTUITION AND LOGIC IN MATHEMATICS 218
analysts. Euclid, for example, erected a scientific stractore
wherein his contemporaries could find no fault. In this vast
oonstructiony of which each piece however is due to intuition, we
may still to-day, without much effort, recognize the work of a
logician.
It is not minds that have changed, it is ideas; the intuitional
minds have remained the same; but their readers have required
of them greater concessions.
What is the -cause of this evolution t It is not hard to find.
Intuition can not give us rigor, nor even certainty; this has been
recognized more and more. Let us cite some examples. We know
there exist continuous functions lacking derivatives. Nothing is
more shocking to intuition than this proposition which is imposed
upon us by logic. Our fathers would not have failed to say : ''It
is evident that every continuous function has a derivative, since
every curve has a tangent.'*
How can intuition deceive us on this point f It is because when
we seek to imagine a curve we can not represent it to ourselves
without width ; just so, when we represent to ourselves a straight
line, we see it under the form of a rectilinear band of a certain
breadth. We well know these lines have no width; we try to
imagine them narrower and narrower and thus to approach the
limit; so we do in a certain measure, but we shall never attain
this limit. And then it is clear we can always picture these two
narrow bands, one straight, one curved, in a position such that
they encroach slightly one upon the other without crossing. We
shall thus be led, unless warned by a rigorous analysis, to con-
clude that a curve always has a tangent.
I shall take as second example Dirichlet's principle on which
rest so many theorems of mathematical physics ; to-day we estab-
lish it by reasoning very rigorous but very long; heretofore, on
the contrary, we were content with a very summary proof. A
certain integral depending on an arbitrary function can never
vanish. Hence it is concluded that it must have a minimum. The
flaw in this reasoning strikes us immediately, since we use the
abstract term function and are familiar with all the singularities
functions can present when the word is understood in the most
general sense.
214 THE VALUE OF SCIENCE
But it would not be the same had we used concrete images,
had we, for example, considered this function as an electric poten-
tial ; it would have been thought legitimate to affirm that electro-
static equilibrium can be attained. Yet perhaps a physical com-
parison would have awakened some vague distrust. But if care
had been taken to translate the reasoning into the language of
geometry, intermediate between that of analysis and that of
physics, doubtless this distrust would not have been produced,
and perhaps one might thus, even to-day, still deceive many
readers not forewarned.
Intuition, therefore, does not give us certainty. This is why
the evolution had to happen ; let us now see how it happened.
It was not slow in being noticed that rigor could not be intro-
duced in the reasoning unless first made to enter into the defini-
tions. For the most part the objects treated of by mathemati-
cians were long ill defined; they were supposed to be known
because represented by means of the senses or the imagination;
but one had only a crude image of them and not a precise idea
on which reasoning could take hold. It wists there first that the
logicians had to direct their efforts.
So, in the case of incommensurable numbers. The vague idea
of continuity, which we owe to intuition, resolved itself into a
complicated system of inequalities referring to whole numbers.
By that means the difficulties arising from passing to the limit,
or from the consideration of infinitesimals, are finally removed.
To-day in analysis only whole numbers are left or systems, finite
or infinite, of whole numbers bound together by a net of equality
or inequality relations. Mathematics, as they say, is arithmetized.
Ill
A first question presents itself. Is this evolution ended ? Have
we finally attained absolute rigor? At each stage of the evolu-
tion our fathers also thought they had reached it. If they deceived
themselves, do we not likewise cheat ourselves?
We believe that in our reasonings we no longer appeal to
intuition ; the philosophers will tell us this is an illusion. Pure
logic could never lead us to anything but tautologies; it could
INTUITION AND LOGIC IN MATHEMATICS 215
•
create nothing new; not from it alone can any science issue. In
one sense these philosopers are right; to make arithmetic, as to
make geometry, or to make any science, something else than pure
logic is necessary. To designate this something else we have no
word other than intuition. But how many different ideas are
hidden under this same wordf
Compare these four axioms: (1) Two quantities equal to a
third are equal to one another; (2) if a theorem is true of the
number 1 and if we prove that it is true of n + 1 if true for n,
then will it be true of all whole numbers; (3) if on a straight
the point C is between A and B and the point D between A and
C, then the point D will be between A and B ; (4) through a given
point there is not more than one parallel to a given straight.
All four are attributed to intuition, and yet the first is the
enunciation of one of the rules of formal logic ; the second is a
real synthetic a priori judgment, it is the foundation of rigorous
mathematical induction ; the third is an appeal to the imagina-
tion; the fourth is a disguised definition.
Intuition is not necessarily founded on the evidence of the
senses ; the senses would soon become powerless ; for example, we
can not represent to ourselves a chiliagon, and yet we reason by
intuition on polygons in general, which include the chiliagon as
a particular case.
You know what Poncelet understood by the principle of con-
tinuity. What is true of a real quantity, said Poncelet, should
be true of an imaginary quantity ; what is true of the hyperbola
whose asymptotes are real, should then be true of the ellipse
whose asymptotes are imaginary. Poncelet was one of the most
intuitive minds of this century; he was passionately, almost
ostentatiously, so ; he regarded the principle of continuity as one
of his boldest conceptions, and yet this principle did not rest on
the evidence of the senses. To assimilate the hyperbola to the
ellipse was rather to contradict this evidence. It was only a sort
of precocious and instinctive generalization which, moreover, I
have no desire to defend.
We have then many kinds of intuition ; first, the appeal to the
senses and the imagination; next generalization by induction,
copied, so to speak, from the procedures of the experimental sci-
216 THE VALUE OF SCIENCE
ences; finally, we have the intuition of pnre number, whence
arose the second of the axioms just enunciated, which is able to
create the real mathematical reasoning. I have shown above by
examples that the first two can not give us certainty ; but who will
seriously doubt the third, who will doubt arithmetic t
Now in the anal3rsis of to-day, when one cares to take the
trouble to be rigorous, there can be nothing but EQrllogisms or
appeals to this intuition of pure number, the only intuition which
can not deceive us. It may be said that to^ay absolute rigor is
attained.
IV
The philosophers make still another objection : ''What you gain
in rigor, ' ' they say, ' ' you lose in objectivity. You can risetoward
your logical ideal only by cutting the bonds which attach you to
reality- Your science is infallible, but it can only remain so by
imprisoning itself in an ivory tower and renouncing all relation
with the external world. From this seclusion it must go out when
it would attempt the slightest application."
For example, I seek to show that some property pertains to
some object whose concept seems to me at first indefinable, be-
cause it is intuitive. At first I fail or must content myself with
approximate proofs; finally I decide to give to my object a pre-
cise definition, and this enables me to establish this property in an
irreproachable manner.
'*And then," say the philosophers, **it still remains to show
that the object which corresponds to this definition is indeed the
same made known to you by intuition ; or else that some real and
concrete object whose conformity with your intuitive idea you
believe you immediately recognize corresponds to your new defi-
nition. Only then could you affirm that it has the property in
question. You have only displaced the difficulty."
That is not exactly so ; the difficulty has not been displaced, it
has been divided. The proposition to be established was in reality
composed of two different truths, at first not distinguished. The
first was a mathematical truth, and it is now rigorously estab-
lished. The second was an experimental verity. Experience alone
can teach us that some real and concrete object corresponds or
INTUITION AND LOGIC IN MATBEMATICS 217
does not correspond to some abstract definition. This second
verity is not mathematically demonstrated, but neither can it be,
no more than can the empirical laws of the physical and natural
sciences. It would be unreasonable to ask more.
Well, is it not a great advance to have distinguished what long
was wrongly confused f Does this mean that nothing is left of
this objection of the philosophers? That I do not intend to say;
in becoming rigorous, mathematical science takes a character so
artificial as to strike every one ; it forgets its historical origins ;
we see how the questions can be answered, we no longer see how
and why they are put.
This shows us that logic is not enough; that the science of
demonstration is not all science and that intuition must retain its
role as complement, I was about to say as counterpoise or as
antidote of logic.
I have already had occasion to insist on the place intuition
should hold in the teaching of the mathematical sciences. With-
out it young minds could not make a beginning in the under-
standing of mathematics; they could not learn to love it and
would see in it only a vain logomachy ; above all, without intui-
tion they would never become capable of applying mathematics.
But now I wish before all to speak of the role of intuition in
science itself. If it is useful to the student it is still more so to
the creative scientist.
We seek reality, but what is reality! The physiologists tell us
that organisms are formed of cells; the chemists add that cells
themselves are formed of atoms. Does this mean that these atoms
or these cells constitute reality, or rather the sole reality! The
way in which these cells are arranged and from which results the
unity of the individual, is not it also a reality much more inter-
esting than that of the isolated elements, and should a naturalist
who had never studied the elephant except by means of the micro-
scope think himself sufficiently acquainted with that animal f
Well, there is something analogous to this in mathematics. The
logician cuts up, so to speak, each demonstration into a very great
number of elementary operations ; when we have examined these
\
218 THE VALUE OF SCIENCE
m
operations one after the other and ascertained that each is correct,
are we to think we have grasped the real meaning of the demon-
stration f Shall we have understood it even when, by an effort of
memory, we have become able to repeat this proof by reproducing
all these elementary operations in just the order in which the
inventor had arranged them Y Evidently not ; we shall not yet
possess the entire reality ; that I know not what, which makes the
unity of the demonstration, will completely elude us.
Pure analysis puts at our disposal a multitude of procedures
whose infallibility it guarantees; it opens to us a thousand dif-
ferent ways on which we can embark in all confidence; we are
assured of meeting there no obstacles; but of all these ways,
which will lead us most promptly to our goalf Who shall tell
us which to choose f We need a faculty which makes us see the
the end from afar, and intuition is this faculty. It is necessary
to the explorer for choosing his route ; it is not less so to the one
following his trail who wants to know why he chose it*
If you are present at a game of chess, it will not suffice, for the
understanding of the game, to know the rules for moving the
pieces. That will only enable you to recognize that each move has
been made conformably to these rules, and this knowledge will
truly have very little value. Yet this is what the reader of a
book on mathematics would do if he were a logician only. To
understand the game is wholly another matter; it is to know why
the player moves this piece rather than that other which he could
have moved without breaking the rules of the game. It is to
perceive the inward reason which makes of this series of succes-
sive moves a sort of organized whole. This faculty is still more
necessary for the player himself, that is, for the inventor.
Let us drop this comparison and return to mathematics. For
example, see what has happened to the idea of continuous func-
tion. At the outset this was only a sensible image, for example,
that of a continuous mark traced by the chalk on a blackboard.
Then it became little by little more refined ; ere long it was used
to construct a complicated system of inequalities, which repro-
duced, so to speak, all the lines of the original image ; this con-
struction finished, the centering of the arch, so to say, was
removed, that crude representation which had temporarily served
INTUITION AND LOGIC IN MATHEMATICS 219
as gapport and which was afterward useless was rejected; there
remained only the construction itself, irreproachable in the eyes
of the logician. And yet if the primitive image had totally dis-
appeared from our recollection, how could we divine by what
caprice all these inequalities were erected in this fashion one
upon another?
Perhaps you think I use too many comparisons ; yet pardon still
another. You have doubtless seen those delicate assemblages of
silicious needles which form the skeleton of certain sponges.
When the organic matter has disappeared, there remains only a
frail and elegant lace-work. True, nothing is there except silica,
but what is interesting is the form this silica has taken, and we
could not understand it if we did not know the living sponge
which has given it precisely this form. Thus it is that the old
intuitive notions of our fathers, even when we have abandoned
them, still imprint their form upon the logical constructions we
have put in their place.
This view of the aggregate is necessary for the inventor ; it is
equally necessary for whoever wishes really to comprehend the
inventor. Can logic give it to us Y No ; the name mathematicians
give it would suffice to prove this. In mathematics logic is called
analysis and analysis means division, dissection. It can have,
therefore, no tool other than the scalpel and the microscope.
Thus logic and intuition have each their necessary role. Each
is indispensable. Logic, which alone can give certainty, is the
instrument of demonstration; intuition is the instrument of
invention.
VI
But at the moment of formulating this conclusion I am seized
with scruples. At the outset I distinguished two kinds of mathe-
matical minds, the one sort logicians and analysts, the others
intuitionalists and geometers. Well, the analysts also have been
inventors. The names I have just cited make my insistence on
this unnecessary.
Here is a contradiction, at least apparently, which needs expla-
nation. And first, do you think these logicians have always pro-
ceeded from the general to the particular, as the rules of formal
220 THE VALUE OF SCIENCE
logic would seem to require of themf Not thus could they have
extended the boundaries of science; scientific conquest is to be
made only by generalization.
In one of the chapters of ' Science and Hypothesis, * I have had
occasion to study the nature of mathematical reasoning, and I
have shown how this reasoning, without ceasing to be absolutely
rigorous, could lift us from the particular to the general by a
procedure I have called mathematical induction. It is by this
procedure that the analysts have made science progress, and if we
examine the detail itself of their demonstrations, we shall find it
there at each instant beside the classic syllogism of Aristotle.
We, therefore, see already that the analysts are not simply
makers of syllogisms after the fashion of the scholastics.
Besides, do you think they have always marched step by step
with no vision of the goal they wished to attain f They must have
divined the way leading thither, and for that they needed a guide.
This guide is, first, analogy. For example, one of the methods of
demonstration dear to analysts is that founded on the employ-
ment of dominant functions. We know it has already served to
solve a multitude of problems; in what consists then the role of
the inventor who wishes to apply it to a new problem f At the
outset he must recognize the analogy of this question with those
which have already been solved by this method; then he must
perceive in what way this new question differs from the others,
and thence deduce the modifications necessary to apply to the
method.
But how does one perceive these analogies and these differences Y
In the example just cited they are almost always evident, but I
could have found others where they would have been much more
deeply hidden ; often a very uncommon penetration is necessary
for their discovery. The analysts, not to let these hidden analo-
gies escape them, that is, in order to be inventors, must, without
the aid of the senses and imagination, have a direct sense of what
constitutes the unity of a piece of reasoning, of what makes, so
to speak, its soul and inmost life.
When one talked with M. Hermite, he never evoked a sensuous
image, and yet you soon perceived that the most abstract entities
were for him like living beings. He did not see them, but he per-
INTUITION AND LOGIC IN MATHEMATICS 221
ceived that they are not an artificial assemblage, and that they
have some principle of internal unity.
But, one will say, that still is intuition. Shall we conclude that
the distinction made at the outset was only apparent, that there is
only one sort of mind and that all the mathematicians are intui-
tionalists, at least those who are capable of inventing f
No, our distinction corresponds to something real. I have said
above that there are many kinds of intuition. I have said how
much the intuition of pure number, whence comes rigorous mathe-
matical induction, differs from sensible intuition to which the
imagination, properly so called, is the principal contributor.
Is the abyss which separates them less profound than it at first
appeared? Could we recognize with a little attention that this
pure intuition itself could not do without the aid of the senses f
This is the affair of the psychologist and the metaphysician and
I shall not discuss the question. But the thing's being doubtful
is enough to justify me in recognizing and affirming an essen-
tial difference between the two kinds of intuition ; they have not
the same object and seem to call into play two different faculties
of our soul ; one would think of two search-lights directed upon
two worlds strangers to one another.
It is the intuition of pure number, that of pure logical forms,
which illumines and directs those we have called analysts. This
it is which enables them not alone to demonstrate, but also to
invent. By it they perceive at a glance the general plan of a
logical edifice, and that too without the senses appearing to inter-
vene. In rejecting the aid of the imagination, which, as we have
seen, is not always infallible, they can advance without fear of
deceiving themselves. Happy, therefore, are those who can do
without this aid ! We must admire them ; but how rare they are !
Among the analysts there will then be inventors, but they will
be few. The majority of us, if we wished to see afar by pure intu-
ition alone, would soon feel ourselves seized with vertigo. Our
weakness has need of a staff more solid, and, despite the excep-
tions of which we have just spoken, it is none the less true that
sensible intuition is in mathematics the most usual instrument of
invention.
Apropos of these reflections, a question comes up that I have
222 THE VALUE OF SCIESCE
not the time either to solve or even to enuxieiate with the derelop-
mentA it would admit of. Is there room for a new disdnctioii, for
distingaishing among the analysts those who above all nse poie
intuition and those who are first of all preoccupied with formal
logic f
M. Ilf-rEiite, for example, whcHn I have just cited, can not be
classed among the geometers who make use of the sensible intui-
tion ; but neither is he a logician, properly so called. He does not
conceal his aversion to purely deductive procedures which start
from the general and end in the particular.
CHAPTER II
The Measure of Time
I
So long as we do not go outside the domain of consciousness,
the notion of time is relatively dear. Not only do we distinguish
without difficulty present sensation from the remembrance of past
sensations or the anticipation of future sensations, but we know
perfectly well what we mean when we say that of two conscious
phenomena which we remember, one was anterior to the other;
or that, of two foreseen conscious phenomena, one will be ante-
rior to the other.
When we say that two conscious facts are simultaneous, we
mean that they profoundly interpenetrate, so that analysis can
not separate them without mutilating them.
The order in which we arrange conscious phenomena does not
admit of any arbitrariness. It is imposed upon us and of it
we can change nothing.
I have only a single observation to add. For an aggregate of
sensations to have become a remembrance capable of classifica-
tion in time, it must have ceased to be actual, we must have
lost the sense of its infinite complexity, otherwise it would have
remained present. It must, so to speak, have crystallized around
a center of associations of ideas which will be a sort of label. It
is only when they thus have lost all life that we can classify our
memories in time as a botanist arranges dried flowers in his
herbarium.
But these labels can only be finite in number. On that score,
psychologic time should be discontinuous. Whence comes the
feeling that between any two instants there are others! We
arrange our recollections in time, but we know that there remain
empty compartments. How could that be, if time were not a
form pre-existent in our minds f How could we know there were
empty compartments, if these compartments were revealed to us
only by their content!
223
224 THE VALUE OF SCIENCE
II
But that is not all ; into this form we wish to put not only the
phenomena of our own consciousness, but those of which other
consciousnesses are the theater. But more, we wish to put there
physical facts, these I know not what with which we people space
and which no consciousness sees directly. This is necessary be-
cause without it science could not exist. In a word, psychologic
time is given to us and must needs create scientific and physical
time. There the difficulty begins, or rather the difficulties, for
there are two.
Think of two consciousnesses, which are like two worlds im-
penetrable one to the other. By what right do we strive to put
them into the same mold, to measure them by the same standard!
Is it not as if one strove to measure length with a gram or
weight with a meter f And besides, why do we speak of measur-
ing f We know perhaps that some fact is anterior to some other,
but not hy how much it is anterior.
Therefore two difficulties: (1) Can we transform psychologic
time, which is qualitative, into a quantitative timef (2) Can
we reduce to one and the same measure facts which transpire in
different worlds!
Ill
The first difficulty has long been noticed ; it has been the sub-
ject of long discussions and one may say the question is settled.
We have not a direct intuition of the eqvMity of two intervals
of time. The persons who believe they possess this intuition are
dupes of an illusion. When I say, from noon to one the same
time passes as from two to three, what meaning has this affir-
mation!
The least reflection shows that by itself it has none at all. It
will only have that which I choose to give it, by a definition which
will certainly possess a certain degree of arbitrariness. Psy-
chologists could have done without this definition ; physicists and
astronomers could not ; let us see how they have managed.
To measure time they use the pendulum and they suppose by
definition that all the beats of this pendulum are of equal dura-
tion. But this is only a first approximation; the temperature,
the resistance of the air, the barometric pressure, make the pace
TEE MEASURE OF TIME 225
of the pendulum vary. If we could escape these sources of error,
we should obtain a much closer approximation, but it would still
be only an approximation. New causes, hitherto neglected, elec-
tric, magnetic or others, would introduce minute perturbations.
In fact, the best chronometers must be corrected from time to
time, and the corrections are made by the aid of astronomic
observations; arrangements are made so that the sidereal clock
marks the same hour when the same star passes the meridian*
In other words, it is the sidereal day, that is, the duration of the
rotation of the earth, which is the constant unit of time. It is
supposed, by a new definition substituted for that based on the
beats of the pendulum, that two complete rotations of the earth
about its axis have the same duration.
However, the astronomers are still not content with this defi-
nition. Many of them think that the tides act as a check on our
globe, and that the rotation of the earth is becoming slower and
slower. Thus would be explained the apparent acceleration of
the motion of the moon, which would seem to be going more
rapidly than theory permits because our watch, which is the
earth, is going slow.
IV
All this is unimportant, one will say ; doubtless our instruments
of measurement are imperfect, but it suflSces that we can conceive
a perfect instrument. This ideal can not be reached, but it is
enough to have conceived it and so to have put rigor into the
definition of the unit of time.
The trouble is that there is no rigor in the definition. When
we use the pendulum to measure time, what postulate do we
implicitly admit? It is that the duration of two identical phe-
nomena is the same; or, if you prefer, that the same causes take
the same time to produce the same effects.
And at first blush, this is a good definition of the equality of
two durations. But take care. Is it impossible that experiment
may some day contradict our postulate?
Let me explain myself. I suppose that at a certain place in the
world the phenomenon a happens, causing as consequence at the
end of a certain time the effect a\ At another place in the world
16
226 TEE VALUE OF SCIENCE
very far away from the first, happens the phenomenon )8, which
causes as consequence the effect p'. The phenomena a and p are
simultaneous, as are also the effects a' and p'.
Later, the phenomenon a is reproduced under approximately
the same conditions as before, and simultaneously the phenom-
enon p is also reproduced at a very distant place in the world
and almost under the same circumstances. The effects cf and fi^
also take place. Let us suppose that the effect a' happens per-
ceptibly before the effect p'.
If experience made us witness such a sight, our postulate
would be contradicted. For experience would tell us that the
first duration aa' is equal to the first duration pp' and that the
second duration aa' is less than the second duration pp'. On the
other hand, our postulate would require that the two durations
aa' should be equal to each other, as likewise the two durations
pp'. The equality and the inequality deduced from experience
would be incompatible with the two equalities deduced from the
postulate.
Now can we affirm that the hypotheses I have just made are
absurd? They are in no wise contrary to the principle of con-
tradiction. Doubtless they could not happen without the prin-
ciple of sufficient reason seeming violated. But to justify a
definition so fundamental I should prefer some other guarantee.
But that is not all. In physical reality one cause does not pro-
duce a given effect, but a multitude of distinct causes contribute
to produce it, without our having any means of discriminating
the part of each of them.
Physicists seek to make this distinction ; but they make it only
approximately, and, however they progress, they never will
make it except approximately. It is approximately true that the
motion of the pendulum is due solely to the earth's attraction;
but in all rigor every attraction, even of Sirius, acts on the pen-
dulum.
Under these conditions, it is clear that the causes which have
produced a certain effect will never be reproduced except ap-
proximately. Then we should modify our postulate and our
TEE MEASURE OF TIME 227
definition. Instead of saying: 'The same causes take the same
time to produce the same effects, * we should say : * Causes almost
identical take almost the same time to produce almost the same
effects.'
Our definition therefore is no longer anything but approxi-
mate. Besides, as M. Calinon very justly remarks in a recent
memoir :^
One of the circunuitancee of any phenomenon is the velocity of the earth's
rotation; if this velocity of rotation varies, it constitutes in the reproduction
of this phenomenon a circumstance which no longer remains the same. But
to suppose this velocity of rotation constant is to suppose that we know how
to measure time.
Our definition is therefore not yet satisfactory; it is certainly
not that which the astronomers of whom I spoke above implicitly
adopt, when they afiBlrm that the terrestrial rotation is slowing
down.
What meaning according to them has this affirmation f We
can only understand it by analyzing the proofs they give of their
proposition. They say first that the friction of the tides pro-
ducing heat must destroy vis viva. They invoke therefore the
principle of vis viva, or of the conservation of energy.
They say next that the secular acceleration of the moon, cal-
culated according to Newton's law, would be less than that de-
duced from observations unless the correction relative to the
slowing down of the terrestrial rotation were made. They invoke
therefore Newton's law. In other words, they define duration
in the following way: time should be so defined that Newton's
law and that of vis viva may be verified. Newton's law is an
experimental truth ; as such it is only approximate, which shows
that we still have only a definition by approximation.
If now it be supposed that another way of measuring time is
adopted, the experiments on which Newton's law is founded
would none the less have the same meaning. Only the enun-
ciation of the law would be different, because it would be trans-
lated into another language; it would evidently be much less
simple. So that the definition implicitly adopted by the astron-
omers may be summed up thus : Time should be so defined that
1 Etude sur les diverses grandeurs, Paris, Gauthier-Yillars, 1897.
228 THE VALVE OF SCIENCE
the equations of mechanics may be as simple as possible. In
other words, there is not one way of measuring time more true
than another; that which is generally adopted is only more
convenient. Of two watches, we have no right to say that the
one goes true, the other wrong; we can only say that it is ad-
vantageous to conform to the indications of the first.
The diflSculty which has just occupied us has been, as I have
said, often pointed out; among the most recent works in which
it is considered, I may mention, besides M. Calinon's little book,
the treatise on mechanics of Andrade.
VI
The second diflSculty has up to the present attracted much
less attention; yet it is altogether analogous to the preceding;
and even, logically, I should have spoken of it first.
Two psychological phenomena happen in two different con-
sciousnesses; when I say they are simultaneous, what do I meanf
When I say that a physical phenomenon, which happens outside
of every consciousness, is before or after a psychological phenom-
enon, what do I mean 1
In 1572, Tycho Brahe noticed in the heavens a new star. An
immense conflagration had happened in some far distant heavenly
body; but it had happened long before; at least two hundred
years were necessary for the light from that star to reach our
earth. This conflagration therefore happened before the discov-
ery of America. Well, when I say that ; when, considering this
gigantic phenomenon, which perhaps had no witness, since the
satellites of that star were perhaps uninhabited, I say this phe-
nomenon is anterior to the formation of the visual image of the
isle of Espanola in the consciousness of Christopher Columbus,
what do I mean?
A little reflection is suflScient to understand that all these
aflBrmations have by themselves no meaning. They can have one
only as the outcome of a convention.
VII
We should first ask ourselves how one could have had the idea
of putting into the same frame so many worlds impenetrable to
THE MEASURE OF TIME 229
one another. We should like to represent to ourselves the ex-
ternal universe, and only by so doing could we feel that we un-
derstood it. We know we never can attain this representation :
our weakness is too great. But at least we desire the ability to
conceive an infinite intelligence for which this representation
could be possible, a sort of great consciousness which should see
all, and which should classify all in its time, as we classify, in
our time, the little we see.
This hypothesis is indeed crude and incomplete, because this
supreme intelligence would be only a demigod; infinite in one
sense, it would be limited in another, since it would have only an
imperfect recollection of the past; and it could have no other,
since otherwise all recollections would be equally present to it
and for it there would be no time. And yet when we speak of
time, for all which happens outside of us, do we not uncon-
sciously adopt this hypothesis; do we not put ourselves in the
place of this imperfect god; and do not even the atheists put
themselves in the place where god would be if he existed f
What I have just said shows us, perhaps, why we have tried
to put all physical phenomena into the same frame. But that
can not pass for a definition of simultaneity, since this hypo-
thetical intelligence, even if it existed, would be for us impene-
trable. It is therefore necessary to seek something else.
VIII
The ordinary definitions which are proper for psychologic time
would suflBce us no more. Two simultaneous psychologic facts
are so closely bound together that analysis can not separate with-
out mutilating them. Is it the same with two physical facts 1 Is
not my present nearer my past of yesterday than the present of
Sirius 1
It has also been said that two facts should be regarded as
simultaneous when the order of their succession may be inverted
at will. It is evident that this definition would not suit two
physical facts which happen far from one another, and that, in
what concerns them, we no longer even understand what this
reversibility would be; besides, succession itself must first be
defined.
230 THE VALUE OF SCIENCE
IX
Let us then seek to give an account of what is understood by
simultaneity or antecedence, and for this let us analyze some
examples.
I write a letter; it is afterward read by the friend to whom I
have addressed it. There are two facts which have had for their
theater two different consciousnesses. In writing this letter I
have had the visual image of it, and my friend has had in his turn
this same visual image in reading the letter. Though these two
facts happen in impenetrable worlds, I do not hesitate to regard
the first as anterior to the second, because I believe it is its cause.
I hear thunder, and I conclude there has been an electric dis-
charge; I do not hesitate to consider the physical phenomenon
as anterior to the auditory image perceived in my consciousness,
because I believe it is its cause.
Behold then the rule we follow, and the only one we can follow :
when a phenomenon appears to us as the cause of another, we
regard it as anterior. It is therefore by cause that we define
time; but most often, when two facts appear to us bound by a
constant relation, how do we recognize which is the cause and
which the effect ? We assume that the anterior fact, the antece-
dent, is the cause of the other, of the consequent. It is then by
time that we define cause. How save ourselves from this petitio
principiit
We say now post hoc, ergo propter hoc; now propter hoc, ergo
post hoc; shall we escape from this vicious circlet
Let us see, not how we succeed in escaping, for we do not
completely succeed, but how we try to escape.
I execute a voluntary act A and I feel afterward a sensation D,
which I regard as a consequence of the act A ; on the other hand,
for whatever reason, I infer that this consequence is not imme-
diate, but that outside my consciousness two facts B and C, which
I have not witnessed, have happened, and in such a way that
B is the effect of A, that C is the effect of 5, and D of C.
But why? If I think I have reason to regard the four facts
A, B, C, D, as bound to one another by a causal connection, why
TBE "51EASUBE OF TIME 231
rejige them in the causal order A B C D, and at the same time
in the chronologic order A B C D, rather than in any other
order!
I clearly see that in the act A I have the feeling of having
been active, while in undergoing the sensation D I have that of
having been passive. This is why I regard A as the initial cause
and D as the ultimate effect; this is why I put A at the beginning
of the chain and D at the end; but why put B before C rather
than C before B1
If this question is put, the reply ordinarily is : we know that it
is B which is the cause of C because we always see B happen
before C. These two phenomena, when witnessed, happen in a
certain order; when analogous phenomena happen without wit-
ness, there is no reason to invert this order.
Doubtless, but take care ; we never know directly the physical
phenomena B and C. What we know are sensations B' and C
produced respectively by B and C. Our consciousness tells us
immediately that B' precedes C and we suppose that B and C
succeed one another in the same order.
This rule appears in fact very natural, and yet we are often
led to depart from it. We hear the sound of the thunder only
some seconds after the electric discharge of the cloud. Of two
flashes of lightning, the one distant, the other near, can not the
first be anterior to the second, even though the sound of the
second comes to us before that of the first!
Another difBculty; have we really the right to speak of the
cause of a phenomenon ! If all the parts of the universe are inter-
chained in a certain measure, any one phenomenon will not be
the effect of a single cause, but the resultant of causes infinitely
numerous; it is, one often says, the consequence of the state of
the universe a moment before. How enunciate rules applicable
to circumstances so complex f And yet it is only thus that these
rules can be general and rigorous.
Not to lose ourselves in this infinite complexity, let us make a
simpler hypothesis. Consider three stars, for example, the sun,
Jupiter and Saturn ; but, for greater simplicity, regard them as
232 THE VALVE OF SCIENCE
reduced to material points and isolated from the rest of the
world. The positions and the velocities of three bodies at a
given instant sufSce to determine their positions and velocities at
the following instant, and consequently at any instant. Their
positions at the instant t determine their positions at the instant
/ + fc as weU as their positions at the instant t — h.
Even more; the position of Jupiter at the instant t, together
with that of Saturn at the instant t + a, determines the position
of Jupiter at any instant and that of Saturn at any instant
The aggregate of positions occupied by Jupiter at the instant
t + e and Saturn at the instant ^ + a + e is bound to the aggre-
gate of positions occupied by Jupiter at the instant t and Saturn
at the instant / -f a, by laws as precise as that of Newton, though
more complicated. Then why not regard one of these aggre-
gates as the cause of the other, which would lead to considering
as simultaneous the instant t of Jupiter and the instant ^ -j- a of
Saturn t
In answer there can only be reasons, very strong, it is true, of
convenience and simplicity.
XII
But let us pass to examples less artificial; to understand the
definition implicitly supposed by the savants, let us watch them at
work and look for the rules by which they investigate simul-
taneity.
I will take two simple examples, the measurement of the
velocity of light and the determination of longitude.
When an astronomer tells me that some stellar phenomenon,
which his telescope reveals to him at this moment, happened,
nevertheless, fifty years ago, I seek his meaning, and to that
end I shall ask him first how he knows it, that is, how he has
measured the velocity of light.
He has begun by supposing that light has a constant velocity,
and in particular that its velocity is the same in all directions.
That is a postulate without which no measurement of this veloc-
ity could be attempted. This postulate could never be verified
directly by experiment; it might be contradicted by it if the
results of different measurements were not concordant. We
THE MEASURE OF TIME 233
should think ourselves fortunate that this contradiction has
not happened and that the slight discordances which may happen
can be readily explained.
The postulate, at all events, resembling the principle of suffi-
cient reason, has been accepted by everybody ; what I wish to em-
phasize is that it furnishes us with a new rule for the investi-
gation of simultaneity, entirely different from that which we
have enunciated above.
This postulate assumed, let us see how the velocity of light has
been measured. You know that Boemer used eclipses of the
satellites of Jupiter, and sought how much the event fell behind
its prediction. But how is this prediction madef It is by the
aid of astronomic laws; for instance Newton's law.
Could not the observed facts be just as weU explained if we at-
tributed to the velocity of light a little different value from that
adopted, and supposed Newton's law only approximate? Only
this would lead to replacing Newton's law by another more com-
plicated. So for the velocity of light a value is adopted, such
that the astronomic laws compatible with this value may be as
simple as possible. When navigators or geographers determine
a longitude, they have to solve just the problem we are discuss-
ing; they must, without being at Paris, calculate Paris time.
How do they accomplish itt They carry a chronometer set for
Paris. The qualitative problem of simultaneity is made to de-
pend upon the quantitative problem of the measurement of
time. I need not take up the difficulties relative to this latter
problem, since above I have emphasized them at length.
Or else they observe an astronomic phenomenon, such as an
eclipse of the moon, and they suppose that this phenomenon is
perceived simultaneously from aU points of the earth. That is
not altogether true, since the propagation of light is not instan-
taneous; if absolute exactitude were desired, there would be a
correction to make according to a complicated rule.
Or else finally they use the telegraph. It is clear first that the
reception of the signal at Berlin, for instance, is after the send-
ing of this same signal from Paris. This is the rule of cause and
effect analyzed above. But how much after t In general, the
duration of the transmission is neglected and the two events are
234 THE VALUE OF SCIENCE
regarded as simultaneous. But, to be rigorous, a little correc-
tion would still have to be made by a complicated calculation;
in practise it is not made, because it would be well within the
errors of observation; its theoretic necessity is none the less
from our point of view, which is that of a rigorous definition.
From this discussion, I wish to emphasize two things: (1) The
rules applied are exceedingly various. (2) It is difficult to sep-
arate the qualitative problem of simultaneity from the quanti-
tative problem of the measurement of time ; no matter whether a
chronometer is used, or whether account must be taken of a
velocity of transmission, as that of light, because such a velocity
could not be measured without measuring a time.
XIII
To conclude : We have not a direct intuition of simultaneityi
nor of the equality of two durations. If we think we have this
intuition, this is an illusion. We replace it by the aid of certain
rules which we apply almost always without taking count of
them.
But what is the nature of these rules t No general rule, no
rigorous rule ,* a multitude of little rules applicable to each par-
ticular case.
These rules are not imposed upon us and we might amuse our-
selves in inventing others ; but they could not be cast aside with-
out greatly complicating the enunciation of the laws of physics,
mechanics and astronomy.
We therefore choose these rules, not because they are true,
but because they are the most convenient, and we may recapitu-
late them as foUows: " The simultaneity of two events, or the
order of their succession, the equality of two durations, are to be
so defined that the enunciation of the natural laws may be as
simple as possible. In other words, aU these rules, all these
definitions are only the fruit of an unconscious opportunism.*'
CHAPTER III
The Notion op Space
1. Introduction
In the articles I have heretofore devoted to space I have above
all emphasized the problems raised by non-Euclidean geometry,
while leaving almost completely aside other questions more diffi-
cult of approach, such as those which pertain to the number of
dimensions. All the geometries I considered had thus a common
basis, that tridimensional continuum which was the same for all
and which differentiated itself only by the figures one drew in
it or when one aspired to measure it.
In this continuum, primitively amorphous, we may imagine a
network of lines and surfaces, we may then convene to regard
the meshes of this net as equal to one another, and it is only
after this convention that this continuum, become measurable,
becomes Euclidean or non-Euclidean space. From this amor-
phous continuum can therefore arise indifferently one or the
other of the two spaces, just as on a blank sheet of paper may
be traced indifferently a straight or a circle.
In space we know rectilinear triangles the sum of whose angles
is equal to two right angles; but equally we know curvilinear
triangles the sum of whose angles is less than two right angles.
The existence of the one sort is not more doubtful than that of
the other. To give the name of straights to the sides of the first
is to adopt Euclidean geometry ; to give the name of straights to
the sides of the latter is to adopt the non-Euclidean geometry.
So that to ask what geometry it is proper to adopt is to ask, to
what line is it proper to give the name straight t
It is evident that experiment can not settle such a question;
one would not ask, for instance, experiment to decide whether I
should call AB or CD a straight. On the other hand, neither
can I say that I have not the right to give the name of straights
to the sides of non-Euclidean triangles because they are not in
235
236 THE VALUE OF. SCIENCE
conformity with the eternal idea of straight which I have by
intuition. I grant, indeed, that I have the intuitive idea of the
side of the Euclidean triangle, but I have equally the intuitive
idea of the side of the non-Euclidean triangle. Why should I
have the right to apply the name of straight to the first of these
ideas and not to the second f Wherein does this syllable form
an integrant part of this intuitive idea t Evidently when we say
that the Euclidean straight is a true straight and that the non-
Euclidean straight is not a true straight, we simply mean that
the first intuitive idea corresponds to a more noteworthy object
than the second. But how do we decide that this object is more
noteworthy? This question I have investigated in 'Science and
Hypothesis.'
It is here that we saw experience come in. If the Euclidean
straight is more noteworthy than the non-Euclidean straight, it
is so chiefly because it differs little from certain noteworthy
natural objects from which the non-Euclidean straight differs
greatly. But, it wiU be said, the definition of the non-Euclidean
straight is artificial; if we for a moment adopt it, we shall see
that two circles of different radius both receive the name of
non-Euclidean straights, while of two circles of the same radius
one can satisfy the definition without the other being able to sat-
isfy it, and then if we transport one of these so-called straights
without deforming it, it will cease to be a straight. But by
what right do we consider as equal these two figures which the
Euclidean geometers call two circles with the same radius t It is
because by transporting one of them without deforming it we
can make it coincide with the other. And why do we say this
transportation is effected without deformation t It is impossible
to give a good reason for it. Among all the motions conceiv-
able, there are some of which the Euclidean geometers say that
they are not accompanied by deformation ; but there are others of
which the non-Euclidean geometers would say that they are not
accompanied by deformation. In the first, called Euclidean mo-
tions, the Euclidean straights remain Euclidean straights and the
non-Euclidean straights do not remain non-Euclidean straights;
in the motions of the second sort, or non-Euclidean motions,
the non-Euclidean straights remain non-Euclidean straights
and the Euclidean straights do not remain Euclidean
straights. It has, therefore, not been demonstrated that it was
unreasonable to call straights the sides of non-Euclidean tri-
angles; it has only been shown that that would be unreasonable
if one continued to call the Euclidean motions motions witJiout
deformation; but it has at the same time been shown that it
would be just as unreasonable to call straights the sides of Eu-
clidean triangles if the non-Euclidean motions were called mo-
tions without deformation.
Now when we say that the Euclidean motions are the (rue
motions without deformation, what do we meanT We simply
mean that they are more noteworthy than the others. And why
are they more noteworthy! It is because certain noteworthy
natural bodies, the solid bodies, undergo motions almost similar.
And then when we ask: Can one imagme non-Euclidean space t
that means : Can we imagine a world where there would he note-
worthy natural objects affecting almost the form of non-Euclid-
ean straights, and noteworthy natural bodies frequently under-
going motions almost similar to the non-EucUdean motions T I
have shown in 'Science and Hypothesis' that to this question we
must answer yes.
It has often been observed that if all the bodies in the universe
were dilated simultaneously and in the same proportion, we
shoold have no means of perceiving it, since all our measuring
instruments would grow at the same time as the objects them-
selves which they sen'e to measure. The world, after this dila-
tation, would continue on its course without anything appris-
ing us of so considerable an event. In other words, two worlds
similar to one another (understanding the word similitude in
the sense of Euclid, Book VI,) would be absolutely indistin-
guishable. But more; worlds will be indistinguishable not only
if they are equal or similar, that is, if we can pass from one to
the other by changing the axes of coordinates, or by changing
the scale to which lengths are referred; but they will still b«
indistinguishable if we can pass from one to the other by any
'point-transformation' whatever. I will explain my meaning. I
suppose that to each point of one corresponds one point of the
other and only one, and inversely; and besides that the coordi-
288 THE VALUE OF SCIENCE
m
nates of a point are eontinuous fonctionsy otherwise altogetker
arbitrary, of the corresponding point I suppose besides that to
each object of the first world corresponds in the second an object
of the same nature placed precisely at the corresponding x>oint
I suppose finally that this correspondence fulfilled at the initial
instant is maintained indefinitely. We should have no means
of distinguishing these two worlds one from the other. The rela-
tivity of space is not ordinarily understood in so broad a sense;
it is thus, however, that it would be proper to understand it.
If one of these universes is our Euclidean world, what its in-
habitants will call straight will be our Euclidean straight; but
what the inhabitants of the second world will call straight will
be a curve which will have the same properties in relation to the
world they inhabit and in relation to the motions that they will
call motions without deformation. Their geometry wiD, there-
fore, be Euclidean geometry, but their straight will not be our
Euclidean straight. It will be its transform by the point-trans-
formation which carries over from our world to theirs. The
straights of these men will not be our straights, but they will
have among themselves the same relations as our straights to one
another. It is in this sense I say their geometry wiU be ours.
If then we wish after all to proclaim that they deceive them-
selves, that their straight is not the true straight, if we still are
unwilling to admit that such an afiSrmation has no meaning, at
least we must confess that these people have no means whatever
of recognizing their error.
2. Qualitative Oeometry
All that is relatively easy to understand, and I have already so
often repeated it that I think it needless to expatiate further on
the matter. Euclidean space is not a form imposed upon our
sensibility, since we can imagine non-Euclidean space; but the
two spaces, Euclidean and non-Euclidean, have a common basis,
that amorphous continuum of which I spoke in the beginning.
From this continuum we can get either Euclidean space or
Lobachevskian space, just as we can, by tracing upon it a proper
graduation, transform an ungraduated thermometer into a Fahr-
enheit or a Reaumur thermometer.
And then comes a question : Ib not this amorphous continuum,
that our analysis has allowed to survive, a form imposed upon
oar sensibility T If so, we should have enlarged the prison in
which this sensibility is confined, but it would always be a
prison.
This continuum has a certain number of properties, exempt
from all idea of measurement. The study of these properties is
the object of a science which has' been cultivated by many great
geometers and in particular by Riemann and Betti and which
has received the name of analysts situs. In this science abstrac-
tion is made of every quantitative idea and, for example, if we
ascertain that on a line the point B is between the points A and
C, we shall be content with this ascertainment and shall not
trouble to know whether the line ABC is straight or curved, nor
whether the length AB is equal to the length BC, or whether it
ia twice as great.
The theorems of analysis situs have, therefore, this peculiarity,
that they would remain true if the figures were copied by an
inexpert draftsman who should grossly change all the propor-
tions and replace the straights by lines more or less sinuous. In
mathematical terms, they are not altered by any 'poinMrans-
formation' whatsoever. It has often been said that metric geom-
etry was quantitative, while projective geometry was purely qual-
itative. That is not altogether true. The straight is still dis-
tinguished from other lines by properties which remain quanti-
tative in some respects. The real qualitative geometry is, there-
fore, analysis situs.
The same questions wtiich came up apropos of the truths of
Euclidean geometry, come up anew apropos of the theorems of
aualyds situs. Are they obtainable by deductive reasoning!
Are they disguised conventions! Are they experimental veri-
tieat Are they the characteristics of a form imposed either
upon our sensibility or upon our understanding t
I wish simply to observe that the last two solutions exclude
each other. We can not admit at the same time that it is impos-
sihle to imagine space of four dimensions and that experience
proves to us that space has three dimensions. The experimenter
pnta to nature a question : Is it this or that f and he can not put
240 THE VALVE OF SCIENCE
it without imagining the two terms of the alternative. If it were
impossible to imagine one of these terms, it would be futile and
besides impossible to consult experience. There is no need of ob-
servation to know that the hand of a watch is not marking the
hour 15 on the dial, because we know beforehand that there are
only 12, and we could not look at the mark 15 to see if the hand
is there, because this mark does not exist.
Note likewise that in analysis situs the empiricists are disem-
barrassed of one of the gravest objections that can be leveled
against them, of that which renders absolutely vain in advance
all their efforts to apply their thesis to the verities of Euclidean
geometry. These verities are rigorous and all experimentation
can only be approximate. In analysis situs approximate exper-
iments may suffice to give a rigorous theorem and, for instance,
if it is seen that space can not have either two or less than two
dimensions, nor four or more than four, we are certain that it has
exactly three, since it could not have two and a half or three
and a half.
Of all the theorems of analysis situs, the most important is
that which is expressed in sajdng that space has three dimen-
sions. This it is that we are about to consider, and we shall put
the question in these terms: "When we say that space has three
dimensions, what do we meant
3. The Physical Continuum of Several Dimensions
I have explained in 'Science and Hypothesis' whence we
derive the notion of physical continuity and how that of mathe-
matical continuity has arisen from it. It happens that we are
capable of distinguishing two impressions one from the other,
while each is indistinguishable from a third. Thus we can read-
ily distinguish a weight of 12 grams from a weight of 10 grams,
while a weight of 11 grams could be distinguished from neither
the one nor the other. Such a statement, translated into sym-
bols, may be written :
A=B, B = C, A<C,
This would be the formula of the physical continuum, as crude
experience gives it to us, whence arises an intolerable contradic-
THE NOTION OF SPACE 241
tion that has been obviated by the introduction of the mathe-
matical continuum. This is a scale of which the steps (com-
mensurable or incommensurable numbers) are infinite in number
but are exterior to one another, instead of encroaching on one
another as do the elements of the physical continuum, in con-
formity with the preceding formula.
The physical continuum is, so to speak, a nebula not resolved ;
the most perfect instruments could not attain to its resolution.
Doubtless if we measured the weights with a good balance instead
of judging them by the hand, we could distinguish the weight of
11 grams from those of 10 and 12 grams, and our formula would
become :
A<B, B<C, A<C.
But we should always find between A and B and between B
and C new elements D and £, such that
A=D, D = B, A<B; B = E, B = C, B<C,
and the difficulty would only have receded and the nebula would
always remain unresolved ; the mind alone can resolve it and the
mathematical continuum it is which is the nebula resolved into
stars.
Yet up to this point we have not introduced the notion of the
number of dimensions. What is meant when we say that a math-
ematical continuum or that a physical continuum has two or
three dimensions?
First we must introduce the notion of cut, studying first phys-
ical continua. We have seen what characterizes the physical con-
tinuum. Each of the elements of this continuum consists of a
manifold of impressions ; and it may happen either that an ele-
ment can not be discriminated from another element of the same
continuum, if this new element corresponds to a manifold of
impressions not sufficiently different, or, on the contrary, that
the discrimination is possible; finally it may happen that two
elements indistinguishable from a third may, nevertheless, be
distinguished one from the other.
That postulated, if A and B are two distinguishable elements of
a continuum C, a series of elements may be found, E^, E^, • • • , Eny
all belonging to this same continuum C and such that each of
17
242 THE VALUE OF SCIENCE
them is indistin^shable from the preceding, that E^ is indis-
tinguishable from A, and En indistinguishable from B. There-
fore we can go from A to B by a continuous route and without
quitting C. If this condition is fulfilled for any two elements
A and B of the continuum C, we may say that this continuum C
is all in one piece. Now let us distinguish certain of the elements
of C which may either be all distinguishable from one another,
or themselves form one or several continua. The assemblage of
the elements thus chosen arbitrarily among all those of C will
form what I shall call the cut or the cuts.
Take on C any two elements A and B, Either we can also find
a series of elements E^, E^, • • •, En, such : (1) that they all belong
to C; (2) that each of them is indistinguishable from the follow-
ing, E^ indistinguishable from A and En from B ; (3) and besides
that none of the elements E is indistinguishable from any element
of the cut. Or else, on the contrary, in each of the series E^^ E^,
"-y En satisfying the first two conditions, there will be an element
E indistinguishable from one of the elements of the cut. In the
first case we can go from A to 5 by a continuous route without
quitting C and without meeting the cuts; in the second case that
is impossible.
If then for any two elements A and B of the continuum C, it is
always the first case which presents itself, we shall say that C
remains all in one piece despite the cuts.
Thus, if we choose the cuts in a certain way, otherwise arbi-
trary, it may happen either that the continuum remains all in one
piece or that it does not remain all in one piece; in this latter
hypothesis we shall then say that it is divided by the cuts.
It will be noticed that all these definitions are constructed in
setting out solely from this very simple fact, that two manifolds
of impressions sometimes can be discriminated, sometimes can
not be. That postulated, if, to divide a continuum, it suffices to
consider as cuts a certain number of elements all distinguishable
from one another, we say that this continuum is of one dimen-
sion; if, on the contrary, to divide a continuum, it is necessary to
consider as cuts a system of elements themselves forming one or
several continua, we shall say that this continuum is of several
dimensions.
If to divide a continuum C, cute forming one or several con-
tinus of one dimension suffice, we shall say that C is a continuum
of two dimensions; if cuts sufBce wbicli form one or several con-
tinua of two dimensions at most, we shall say that (7 is a con-
tinuum of three dimensions; and so on.
To justify tliia definition it is proper to see whether it is in this
way that geometers introduce the notion of three dimensions at
the beginning of their works. Now, what do we see! Usually
they begin by defining surfaces as the boundaries of solids or
pieces of space, lines as the boundaries of surfaces, points as the
boundaries of lines, and they affirm that the same procedure can
not be pushed further.
This is just the idea given above : to divide space, cuts that are
called surfaces are necessary; to divide surfaces, cuts that are
called lines are necessary; to divide lines, cuts that are called
points are necessary; we can go no further, the point can not be
divided, so the point is not a continuum. Then lines which can be
divided by cuts which are not continua will be continua of one
dimension; surfaces which can be divided by continuous cuts of
one dimension will be continua of two dimensions ; finally, space
which can be divided by continuous cuts of two dimensions will
be a continuum of three dimensions.
Thus the definition I have just given does not differ essentially
from the usual definitions; I have only endeavored to give it a
form applicable not to the mathematical continuum, but to the
physical continuum, which alone is susceplible of representation,
and yet to retain all its precision. Moreover, we see that this
definition applies not alone to space ; that in all which falls under
our senses we find the characteristics of the physical continuum,
which would allow of the same classification; that it would be
easy to find there esamples of continua of four, of five, dimen-
sions, in the sense of the preceding definition; such examples
occur of themselves to the mind.
I should explain finally, if I had the time, that this science,
of which I spoke above and to which Riemann gave the name of
atulysis situs, teaches us to make distinctions among continua of
th« same number of dimensions and that the classification of these
continua rests also on the consideration of cuts.
244 THE VALUE OF SCIENCE
From this notion has arisen that of the mathematical con-
tinuum of several dimensions in the same way that the physical
continuum of one dimension engendered the mathematical con-
tinuum of one dimension. The formula
A>C, A = B, B = C,
which summed up the data of crude experience, implied an in-
tolerable contradiction. To get free from it, it was necessary to
introduce a new notion while still respecting the essential char-
acteristics of the physical continuum of several dimensions. The
mathematical continuum of one dimension admitted of a scale
whose divisions, infinite in number, corresponded to the different
values, commensurable or not, of one same magnitude. To have
the mathematical continuum of n dimensions, it will suffice to
take n like scales whose divisions correspond to different values
of n independent magnitudes called coordinates. We thus shall
have an image of the physical continuum of n dimensions, and
this image will be as faithful as it can be after the determina-
tion not to allow the contradiction of which I spoke above.
4. The Notion of Point
It seems now that the question we put to ourselves at the start
is answered. When we say that space has three dimensions, it
will be said, we mean that the manifold of points of space satis-
fies the definition we have just given of the physical continuum
of three dimensions. To be content with that would be to sup-
pose that we know what is the manifold of points of space, or even
one point of space.
Now that is not as simple as one might think. Every one
believes he knows what a point is, and it is just because we know
it too well that we think there is no need of defining it. Surely
we can not be required to know how to define it, because in going
back from definition to definition a time must come when we must
stop. But at what moment should we stop ?
We shall stop first when we reach an object which falls under
our senses or that we can represent to ourselves; definition then
will become useless; we do not define the sheep to a child; we
say to him : See the sheep.
So, then, we should ask ourselves if it ia possible to represent
to ourselves a point of space. Those who answer yes do not refleet
that they represent to themselves in reality a white spot made
with the clialk on a blackboan.1 or a black spot made with a pen
on white paper, and that they can represent to themselves only
an object or rather the impressions that this object made on their
When they try to represent to themselves a point, they repre-
sent the impressions that very little objects made them feeL It
is needless to add that two different objects, though both very
little, may produce extremely different impressions, but I
shall not dwell on this difficulty, which would still require some
discussion.
But it is not a question of that ; it does not suffice to represent
one point, it is necessary to represent a certain point and to have
the means of distingruishing it from an other point. And in fact,
that we may be able to apply to a continuum the rule 1 have above
expounded and by which one may recognize the niunber of its
diuiensions, we must rely upon the fact that two elements of this
continuum sometimes can and sometimes cannot be distinguished.
It is necessary therefore that we should in certain eases know how
to represent to ourselves a specific element and to cUstinguiah it
from an other element.
The question ia to know whether the point that I represented
to myself an hour ago is the same as this that I now represent
to myself, or whether it is a different point. In other words,
how do we know whether the point occupied by the object A at
the instant a is the same as the point occupied by the object B at
the instant jff, or still better, what this means I
I am seated in my room ; an object is placed on my table ; dur-
ing a second I do not move, do one touches the object. I am
tempted to say that the point A which this object occupied at the
beginnint; of this second is idcntienl with the poiot B which it
occupies at its end. Not at all ; from the point ,4 to the point B
is 30 kilometers, because the object has been carried along in the
motion of the earth. We can not know whether an object, be it
large or small, has not changed its absolute position in space,
and not only can we not affirm it. but this affirmation has no
h
246 THE VALUE OF SCIENCE
meaning and in any case can not correspond to any representation.
But then we may ask ourselves if the relative position of an
object with regard to other objects has changed or not^ and first
whether the relative position of this object with regard to our
body has changed. If the impressions this object makes upon us
have not changed, we shall be inclined to judge that neither has
this relative position changed; if they have changed, we shall
judge that this object has changed either in state or in relative
position. It remains to decide which of the two. I have explained
in 'Science and Hypothesis' how we have been led to distinguish
the changes of position. Moreover, I shall return to that further
on. We come to know, therefore, whether the relative position
of an object with regard to our body has or has not remained
the same.
If now we see that two objects have retained their relative posi-
tion with regard to our body, we conclude that the relative posi-
tion of these two objects with regard to one another has not
changed ; but we reach this conclusion only by indirect reasoning.
The only thing that we know directly is the relative position of
the objects with regard to our body. A fortiori it is only by
indirect reasoning that we think we know ("and, moreover, this
belief is delusive) whether the absolute position of the object has
changed.
In a word, the system of coordinate axes to which we naturally
refer all exterior objects is a system of axes invariably bound to
our body, and carried around with us.
It is impossible to represent to oneself absolute space ; when I
try to represent to myself simultaneously objects and myself in
motion in absolute space, in reality I represent to myself my own
self montionless and seeing move around me different objects and
a man that is exterior to me, but that I convene to call me.
Will the diflSculty be solved if we agree to refer everything to
these axes bound to our body? Shall we know then what is a
point thus defined by its relative position with regard to our-
selves? Many persons will answer yes and will say that they
^localize' exterior objects.
What does this mean ? To localize an object simply means to
represent to oneself the movements that would be necessary to
THE NOTION OF SPACE 247
reach it. I will explain myself. It is not a question of repre-
senting the movements themselves in space, but solely of repre-
senting to oneself the muscular sensations which accompany these
movements and which do not presuppose the preexistence of the
notion of space.
If we suppose two different objects which successively occupy
the same relative position with regard to ourselves, the impres-
sions that these two objects make upon us will be very different;
if we localize them at the same point, this is simply because it is
necessary to make the same movements to reach them ; apart from
that, one can not just see what they could have in common.
But, given an object, we can conceive many different series of
movements which equally enable us to reach it. If then we repre-
sent to ourselves a point by representing to ourselves the series
of muscular sensations which accompany the movements which
enable us to reach this point, there will be many ways entirely
different of representing to oneself the same point. If one is not
satisfied with this solution, but wishes, for instance, to bring in
the visual sensations along with the muscular sensations, there
will be one or two more ways of representing to oneself this same
point and the difficulty will only be increased. In any case the
following question comes up: Why do we think that all these
representations so different from one another still represent the
same point?
Another remark: I have just said that it is to our own body
that we naturally refer exterior objects ; that we carry about every-
where with us a system of axes to which we refer all the points
of space, and that this system of axes seems to be invariably
bound to our body. It should be noticed that rigorously we could
not speak of axes invariably bound to the body unless the dif-
ferent parts of this body were themselves invariably bound to
one another. As this is not the case, we ought, before referring
exterior objects to these fictitious axes, to suppose our body
brought back to the initial attitude.
5. The Notion of Displacement
I have shown in 'Science and Hypothesis' the preponderant
role played by the movements of our body in the genesis of the
248 THE VALUE OF SCIENCE
notion of space. For a being completely immovable there would
be neither space nor geometry ; in vain would exterior objects be
displaced about him, the variations which these displacements
would make in his impressions would not be attributed by this
being to changes of position, but to simple changes of state;
this being would have no means of distinguishing these two sorts
of changes, and this distinction, fundamental for us, would have
no meaning for him.
The movements that we impress upon our members have as
effect the varying of the impressions produced on our senses by
external objects ; other causes may likewise make them vary ; but
we are led to distinguish the changes produced by our own
motions and we easily discriminate them for two reasons: (1)
because they are voluntary; (2) because they are accompanied
by muscular sensations.
So we naturally divide the changes that our impressions may
undergo into two categories to which perhaps I have given an
inappropriate designation: (1) the internal changes, which are
voluntary and accompanied by muscular sensations; (2) the
external changes, having the opposite characteristics.
We then observe that among the external changes are some
which can be corrected, thanks to an internal change which brings
everything back to the primitive state ; others can not be corrected
in this way (it is thus that, when an exterior object is displaced,
we may then by changing our own position replace ourselves
as regards this object in the same relative position as before, so
as to reestablish the original aggregate of impressions; if this
object was not displaced, but changed its state, that is impos-
sible). Thence comes a new distinction among external changes:
those which may be so corrected we call changes of position;
and the others, changes of state.
Think, for example, of a sphere with one hemisphere blue and
the other red ; it first presents to us the blue hemisphere, then it
so revolves as to present the red hemisphere. Now think of a
spherical vase containing a blue liquid which becomes red in
consequence of a chemical reaction. In both cases the sensation
of red has replaced that of blue ; our senses have experienced the
same impressions which have succeeded each other in the same
THE NOTION OF SPACE 249
order, and yet these two changes are regarded by iib as very
different ; the first is a displacement, the second a change of state.
"Wbyl Because in the first case it is sufficient for me to go around
the sphere to place myself opposit« the blue hemisphere and
reestablish tlie original blue sensation.
Still more ; if the two hemispheres, in place of being red and
blue, had been yellow and green, how sliould I liave interpreted ]
the revolution of the splicre ? Before, the red succeeded the blue,
now the green succeeds the yellow; and yet I say that the two
^heres have undergone the same revolution, that each has turned
about its axis ; yet I can not say that the green is to yellow as
the red is to blue; how then am I led to decide that the two
spheres have undei^one the same displacement T Evidently be-
eause, in one case as in the other, I am able to reestablish the
original sensation by going around the sphere, by making the
same movements, and I know that 1 have made the same move-
ments because I have felt the same muscular sensations; to know
it, I do not need, therefore, to know geometry in advance and to
represent to myself the movements of my body in geometric space.
Another example: An object is displaced before my eye; its
image was first formed at the center of the retina; then it is
formed at the border; the old sensation was carried to me by a
nerve fiber ending at the center of the retina ; the new sensation
is carried to me by another nerve fiber starting from the border
■ of the retina; these two sensations are qualitatively different;
otherwise, how could I distinguish them I
Why then am I led to decide that these two sensations, quali-
tatively different, represent the same image, which has been dis-
plac«d t It is because I can follow the object vnth th-e eye and by
a displacement of the eye, voluntary and aocompanied by muscu-
lar sensations, bring back the image to the center of the retina
and reestablish the primitive sensation.
I suppose that the image of a red object has gone from the
center A to the border B of the retina, then that the image of a
bine object goes in its turn from the center A to the border B
of the retina ; I shall decide that these two objects have under-
gone the same displacement. Why! Because in botli cases I
■hall have been able to reestablish the primitive sensation, and
260 THE VALUE OF SCIENCE
that to do it I shall have had to execute the same movement of
the eye, and I shall know that my eye has executed the same
movement because I shall have felt the same muscular sensations.
If I could not move my eye, should I have any reason to sup-
pose that the sensation of red at the center of the retina is to the
sensation of red at the border of the retina as that of blue at the
center is to that of blue at the border? I should only have four
sensations qualitatively different, and if I were asked if they
are connected by the proportion I have just stated, the question
would seem to me ridiculous, just as if I were asked if there is an
analogous proportion between an auditory sensation, a tactile
sensation and an olfactory sensation.
Let us now consider the internal changes, that is, those which
are produced by the voluntary movements of our body and which
are accompanied by muscular changes. They give rise to the
two following observations, analogous to those we have just made
on the subject of external changes.
1. I may suppose that my body has moved from one point to
another, but that the same attitude is retained ; all the parts of
the body have therefore retained or resumed the same relative
situation, although their absolute situation in space may have
varied. I may suppose that not only has the position of my body
changed, but that its attitude is no longer the same, that, for
instance, my arms which before were folded are now stretched out.
I should therefore distinguish the simple changes of position
without change of attitude, and the changes of attitude. Both
would appear to me under form of muscular sensations. How
then am I led to distinguish them ? It is that the first may serve
to correct an external change, and that the others can not, or at
least can only give an imperfect correction.
This fact I proceed to explain as I would explain it to some one
who already knew geometry, but it need not thence be concluded
that it is necessary already to know geometry to make this dis-
tinction; before knowing geometry I ascertain the fact (experi-
mentally, so to speak), without being able to explain it. But
merely to make the distinction between the two kinds of change,
I do not need to explain the fact, it suflBces me to ascertain it.
However that may be, the explanation is easy. Suppose that
an exterior object is displaced ; if we wish the different parts of
our body to resume with regard to this object their initial relative
position, it is necessary that these different parts should have
resumed likewise their initial relative position with regard to
one another. Only the internal changes which satisfy this latter
condition will be capable of correcting the external change pro-
duced by the displacement of that object. If, therefore, the
relative position of my eye with regard to my finger has changed,
I shall still be able to replace the eye in its initial relative situa-
tion with regard to the object and reestablish thus the primitive
visual sensations, but then the relative position of the finger with
regard to the object will have changed and the tactile gensatianEi
will not be reestablished.
2. We ascertain likewise that the same external change raay be
corrected by two internal changes correHponding to different
inascular sensations. Here again I can ascertain this without
knowing geometry; and I have no need of anything else; but I
proceed to give the explanation of the fact, employing geometrical
language. To go from the position A to the position B I may
take several routes. To the first of these routes will correspond
a series S of muscular sensations; to a second route will corre-
spond another series S", of muscular sensations which generally
will be completely different, since other muscles will be used.
How am I led to regard these two series S and S" as corre-
sponding to the same displacement AB t It is because these two
series are capable of correcting the same external change. Apart
from that, they have nothing in common.
Let us now consider two external changes : a and ^, which shall
be, for instance, the rotation of a sphere half blue, half red, and
that of a sphere half yellow, half green ; these two changes have
nothing in common, since the one is for us the passing of blue
into red and the other the passing of yellow into green. Con-
sider, on the other hand, two series of internal changes S and S";
like the others, they will have nothing in common. And yet I say
that a and ^ correspond to the some displacement, and that >S^ and
8" correspond also to the same displacement. WhyT Simply
because S can correct a as well as /9 and because a con be cor-
i by S" as well as by S. And then a question su^^sts itself :
252 THE VALUE OF SCIENCE
If I have ascertained that 8 corrects a and p and that 8" corrects
a, am I certain that 8" likewise corrects pi Experiment alone
can teach us whether this law is verified. If it were not verified,
at least approximately, there would be no geometry, there wonld
be no space, because we should have no more interest in classi-
fying the internal and external changes as I have just done, and,
for instance, in distinguishing changes of state from changes of
position.
It is interesting to see what has been the role of experience in
all this. It has shown me that a certain law is approximately
verified. It has not told me how space is, and that it satis-
fies the condition in question. I knew, in fact, before all exi>eri-
ence, that space satisfied this condition or that it would not be;
nor have I any right to say that experience told me that geometry
is possible ; I very well see that geometry is possible, since it does
not imply contradiction ; experience only tells me that geometry
is useful.
6. Vistuil 8pace
Although motor impressions have had, as I have just explained,
an altogether preponderant influence in the genesis of the notion
of space, which never would have taken birth without them, it
will not be without interest to examine also the role of visual
impressions and to investigate how many dimensions 'visual
space' has, and for that purpose to apply to these impressions the
definition of § 3.
A first difficulty presents itself : consider a red color sensation
affecting a certain point of the retina ; and on the other hand a
blue color sensation affecting the same point of the retina. It is
necessary that we have some means of recognizing that these two
sensations, qualitatively different, have something in common.
Now, according to the considerations expounded in the preceding
paragraph, we have been able to recognize this only by the move-
ments of the eye and the observations to which they have given
rise. If the eye were immovable, or if we were unconscious of
its movements, we should not have been able to recognize that
these two sensations, of different quality, had something in com-
mon ; we should not have been able to disengage from them what
gives them a geometric character. The visual sensations, without
the muscular sensations, would have nothing geometric, so that
it may be said there is no pure visual space.
To do away with this difficulty, consider only sensations of the
same nature, red sensations, for instance, differing one from
another only as regards the point of the retina that they affect.
It is clear that I have no reason for making such an arbitrary
choice among all the possible visual sensations, for the purpose
of uniting in the same class all the sensations of the same color,
whatever may be the point of the retina affected, I should never
have dreamt of it, had I not before learned, by the means we
have just seen, to distinguish changes of state from changes of
position, that is, if my eye were immovable. Two sensations of
the same color affecting two different parts of the retina would
have appeared to me as qualitatively distinct, just as two sensa-
tions of different color.
la restricting myself to red sensations, I therefore impose upon
myself an artificial limitation and I neglect systematically one
whole side of the question ; but it is only by this artifice that I am
able to analyze visual space without mingling any motor sensation.
Imagine a line traced on the retina and dividing in two its
surface; and set apart the red sensations affecting a point of this
line, or those differing from them too little to be distinguished
from them. The aggregate of these sensations will form a sort of
cnt that I shall call C, and it is clear that this cut suffices to
divide the manifold of possible red sensations, and that if I take
two red sensations affecting two points situated on one side and
the other of the line, I can not pass from one of these sensations to
the other in a continuous way without passing at a certain
moment through a sensation belonging to the cut.
If, therefore, the cut has n dimensions, the total manifold of my
I red sensations, or if you wish, the whole visual space, will have
.+1.
Now, I distinguish the red sensations affecting a point of the
cut C. The assemblage of these sensations will form a new cut
C. It is clear that this will divide the cut C, always giving to the
divide the same meaning.
264 THE VALUE OF SCIENCE
If, therefore, the cut C has n dimensions, the cut C will have
n + 1 and the whole of visual space n + 2.
If all the red sensations affecting the same point of the retina
were regarded as identical, tiie cut C reducing to a single ele-
ment would have dimension^ and visual space would have 2.
And yet most often it is said that the eye gives us the sense of
a third dimension, and enables us in a certaiil measure to recog-
nize the distance of objects. When we seek to analyze this feel-
ing, we ascertain that it reduces either to the consciousness of the
convergence of the eyes, or to that of the effort of accommodation
which the ciliary muscle makes to focus the image.
Two red sensations affecting the same point of the retina will
therefore be regarded as identical only if they are accompanied
by the same sensation of convergence and also by the same sensa-
tion of effort of accommodation or at least by sensations of
convergence and accommodation so slightly different as to be
indistinguishable.
On this account the cut C is itself a continuum and the cut C
has more than one dimension.
But it happens precisely that experience teaches us that when
two visual sensations are accompanied by the same sensation of
convergence, they are likewise accompanied by the same sensa-
tion of accommodation. If then we form a new cut C" with all
those of the sensations of the cut C", which are accompanied by a
certain sensation of convergence, in accordance with the preced-
ing law they will all be indistinguishable and may be regarded
as identical. Therefore C" will not be a continuum and will
have dimension ; and as C" divides C" it will thence result that
C has one, C two and the whole visual space three dimeyisions.
But would it be the same if experience had taught us the con-
trary and if a certain sensation of convergence were not always
accompanied by the same sensation of accommodation? In this
case two sensations affecting the same point of the retina and
accompanied by the same sense of convergence, two sensations
which consequently would both appertain to the cut C", could
nevertheless be distinguished since they would be accompanied by
two different sensations of accommodation. Therefore C" would
be in its turn a continuum and would have one dimension (at
THE NOTION OF SPACE 255
least) ; then C would have two, C three and the whole vistial
space would have four dimensions.
Will it then be said that it is experience which teaches us that
space has three dimensions, since it is in setting out from an
experimental law that we have conie to attribute three to it ? But
we have therein performed, so to speak, only an experiment in
physiology ; and as also it would suffice to fit over the eyes glasses
of suitable construction to put an end to the accord between the
feelings of convergence and of accommodation, are we to say that
putting on spectacles is enough to make space have four dimen-
sions and that the optician who constructed them has given one
more dimension to space ? Evidently not ; all we can say is that
experience has taught us that it is convenient to attribute three
dimensions to space.
But visual space is only one part of space, and in even the
notion of this space there is something artificial, as I have ex-
plained at the beginning. The real space is motor space and this
it is that we shall examine in the following chapter.
CHAPTER IV
Space and its Three Dimensions
1. The Oroup of Displacements
Let us sum up briefly the results obtained. We proposed to
investigate what was meant in saying that space has three dimen-
sions and we have asked first what is a physical continuum and
when it may be said to have n dimensions. If we consider dif-
ferent systems of impressions and compare them with one another,
we often recognize that two of these systems of impressions are
indistinguishable (which is ordinarily expressed in saying that
they are too close to one another, and that our senses are too
crude, for us to distinguish them) and we ascertain besides that
two of these systems can sometimes be discriminated from one
another though indistinguishable from a third system. In that
case we say the manifold of these systems of impressions forms
a physical continuum C. And each of these systems is called an
element of the continuum C
How many dimensions has this continuum? Take first two
elements A and B of C, and suppose there exists a series S of
elements, all belonging to the continuum C, of such a sort that A
and B are the two extreme terms of this series and that each
term of the series is indistinguishable from the preceding. If
such a series 2 can be found, we say that A and B. are joined to
one another; and if any two elements of C are joined to one
another, we say that C is all of one piece.
Now take on the continuum C a certain number of elements in
a way altogether arbitrary. The aggregate of these elements will
be called a cut. Among the various series 5 which join A to B,
we shall distinguish those of which an element is indistinguish-
able from one of the elements of the cut (we shall say that these
are they which c\it the cut) and those of which all the elements
are distinguishable from all those of the cut. If all the series S
which join A to B cut the cut, we shall say that A and B are
256
SPACE AND ITS THREE DIMENSIONS
257
' separated by the cut, and that the cut divides G. If we can not
find on C two elements which are separated by the cut, we shall
say that the cut does not divide C.
These definitions laid down, if the continuum C can be divided
by cuts which do not themselves form a continuum, this con-
tinuum C has only one dimension; in the contraiy ease it has
several. If a cut forming a continuum of 1 dimension suffices
to divide C, C will have 2 dimensions; if a cut forming a con-
tinuum of 2 dimensions suffices, C will have 3 dimensions, et<:.
Thanks to these definitions, we can always recognize how many
dimensions any physical continuum has. It only remains to find
a physical continuum which is, so to speak, equivalent to space,
of such a sort that to every point of space corresponds an ele-
ment of this continuum, and that to points of space very near one
another correspond indistinguishable elements. Space will have
then as many dimensions as this continuum.
The intermediation of this physical continuum, capable of
representation, is indispensable; because we can not represent
space to ourselves, and that for a multitude of reasons. Space
ia a matliemattcal continuum, it is infinite, and we can represent
to ourselves only physical continua and finite objects. The dif-
ferent elements of space, which we call points, are all alike, and,
to apply our definition, it ia necessary that we know how to dis-
tinguish the elements from one another, at least if they are not
too close. Finally absolute space is nonsense, and it is necessary
for as to begin by referring space to a system of axes invariably
boond to our body (which we must always suppose put back in
tile initial attitude).
Then I have sought to form with our visual sensations a phys-
ical continuum equivalent to apace ; that certainly is easy and this
example ia particularly appropriate for the discussion of the
number of dimensions; this discussion has enabled us to see in
what measure it is allowable to say that 'visual space' has three
dimensions. Only this solution is incomplete and artificial. I
have explained why, and it is not on visual space, but on motor
space that it is necessary to bring our efforts to bear, I have then
recalled what is the origin of the distinction we make between
18
268 THE VALUE OF SCIENCE
changes of position and changes of state. Among the changes
which occur in our impressions, we distinguish, first the ifUermU
changes, voluntary and accompanied hy muscular sensations, and
the external changes, having opposite characteristics. We ascer-
tain that it may happen that an external change may be corrected
by an internal change which reestablishes the primitive sensa-
tions. The external changes, capable of being corrected by an
internal change are called changes of position, those not capable
of it are called changes of staie. The internal changes capable
of correcting an external change are called displacements of the
whole body; the others are called changes of attitude.
Now let a and p be two external changes, a' and fi' two internal
changes. Suppose that a may be corrected either by a' or by )9',
and that a' can correct either aor p; experience tells us then that
fi' can likewise correct p. In this case we say that a and p cor-
respond to the same displacement and also that of and p' cor-
respond to the same displacement. That postulated, we can
imagine a physical continuum which we shall call the continuum
or group of displacements and which we shall define in the fol-
lowing manner. The elements of this continuum shall be the in-
ternal changes capable of correcting an external change. Two of
these internal changes a' and p' shall be regarded as indis-
tinguishable : (1) if they are so naturally, that is, if they are
too close to one another; (2) if a' is capable of correcting
the same external change as a third internal change natu-
rally indistinguishable from p'. In this second case, they will
be, so to speak, indistinguishable by convention, I mean by agree-
ing to disregard circumstances which might distinguish them.
Our continuum is now entirely defined, since we know its ele-
ments and have fixed under what conditions they may be re-
garded as indistinguishable. We thus have all that is necessary
to apply our definition and determine how many dimensions this
continuum has. We shall recognize that it has six. The con-
tinuum of displacements is, therefore, not equivalent to space,
since the number of dimensions is not the same ; it is only related
to space. Now how do we know that this continuum of displace-
ments has six dimensions? We know it by experience.
It would be easy to describe the experiments by which we
could arrive at this reBult. It would be seeu that in this con-
tinuuni cuts can be made which divide it and which are con-
tinua; that these cuts themselves can be divided by other cuts
of the second order which yet are eontinua, and that this would
stop only after cuts of the sixth order which would no longer be
eontinua. From our definitions that would mean that the group
of displaeeiiients has six dimensions.
That would be easy, I have said, but that would be rather long;
and would it not be a little superficial J This group of displace-
ments, we have seen, is related to space, and space coiUd be de-
duced from it, but it is not equivalent to space, since it has not
the same number of dimensions; and when we shall have shown
how the notion of this continuum can be formed and how that of
space may be deduced from it, it might always be asked why
space of three dimensions is much more familiar to us than this
continuum of six dimensions, and consequently doubted whether
it was by this detour that the notion of space was formed in the
human mind.
2. Identity of Two Points
"What is a point! How do we know whether two points of
space are identical or different T Or, in other words, when I say :
The object A occupied at the instant a the point which the object
B occupies at the instant fi, what does that meant
Such is the problem we set ourselves in the preceding chapter,
§4 ^ I have explained it, it is not a question of comparing the
positions of the objects A and B in absolute space ; the question
then would manifestly have no meaning. It is a question of
comparing the positions of these two objects with regard to axes
invariably bound to my body, supposing always this body re-
placed in the same attitude.
I suppose that between the instants a and fi I have moved
neither my body nor my eye, as 1 know from my muscular sense.
Nop have I moved either my head, my arm or my hand. I ascer-
tain that at the instant a impressions that I attributed to the
object A were transmitted to me, some by one of the fibers of
my optic neri-e, the others by one of the sensitive tactile nerves
of my finger; I ascertain that at the instant P other impressions
I Wfaieb I attributf> to the object B are transmitted to me, some by
260 THE VALUE OF SCIENCE
this same fiber of the optic nerve, the others by this same tactile
nerve.
Here I must pause for an explanation ; how am I told that this
impression which I attribute to A, and that which I attribute to
B, impressions which are qualitatively different, are transmitted
to me by the same nerve ? Must we suppose, to take for example
the visual sensations, that A produces two simultaneous sensa-
tions, a sensation purely luminous a and a colored sensation a',
that B produces in the same way simultaneously a luminous sen-
sation b and a colored sensation V, that if these different sensa-
tions are transmitted to me by the same retinal fiber, a is iden-
tical with b, but that in general the colored sensations a' and V
produced by different bodies are different ? In that case it would
be the identity of the sensation a which accompanies a' with the
sensation b which accompanies &', which would teU that all these
sensations are transmitted to me by the same fiber.
However it may be with this hypothesis and although I am
led to prefer to it others considerably more complicated, it is
certain that we are told in some way that there is something in
common between these sensations a -{-a' and b + 5', without
which we should have no means of recognizing that the object B
has taken the place of the object A.
Therefore I do not further insist and I recall the hypothesis I
have just made: I suppose that I have ascertained that the im-
pressions which I attribute to B are transmitted to me at the
instant p by the same fibers, optic as well as tactile, which, at the
instant a, had transmitted to me the impressions that I attributed
to A. If it is so, we shall not hesitate to declare that the point
occupied by B at the instant p is identical with the point occu-
pied by A at the instant a.
I have just enunciated two conditions for these points being
identical ; one is relative to sight, the other to touch. Let us con-
sider them separately. The first is necessary, but is not suffi-
cient. The second is at once necessary and sufficient. A person
knowing geometry could easily explain this in the following
manner : Let be the point of the retina where is formed at the
instant a the image of the body A ; let ilf be the point of space
occupied at the instant a by this body A ; let M' be the point of
space occupied at the instant p by the body B. For this body B
to form its image in O, it is not necessary that tlie points U and
M' coincide; since vision acts at a distance, it suffices for the
three points M M' to be in a straight line. This condition that
the two objects form their image on is therefore necessary, but
not sufBcient for the points .¥ and M' to coincide. Let now P be
the point occupied by my finger and where it remains, sinee it
does not budge. As touch docs not act at a distance, if the
body A touches my finger at the instant a, it is because M and
P coincide; if B touches my finger at the instant p, it is because
Jtf' and P coincide. Therefore M and M' coincide. Thus this
condition that if A touches my finger at the instant a, B touches
it at the instant fi, is at once necessary and sufficient for M and
M' to coincide.
But we who, as yet, do not know geometry can not reason
thus; all that we can do is to ascertain experimentally that tlie
first condition relative to sight may be fulfilled without the
second, which is relative to touch, but that the second can not
be fulfilled without the first.
Suppose experience had taught us the contrary, as might well
be ; this hypothesis contains nothing absurd. Suppose, therefore,
that we had ascertained experimentally that the condition rela-
tive to touch may be fulfilled without that of sight being fulfilled
and that, on the contrary, that of sight can not be fulfilled i\-ith-
ont that of touch being also. It is clear that if this were so we
should conclude that it is touch which may be esereised at a dis-
tance, and that sight does not operate at a distance.
But this is not all; up to this time 1 have supposed that to
determine the place of an object I have made use only of my
eye and a single finger; hut I could just as well have employed
other means, for example, all my other fiogera.
I suppose that my first finger receives at the instant a a tactile
impression which I attribute to the object A. I make a scries of
movements, corresponding to a series S of muscular sensations.
After these movements, at the instant a', my second finger re-
ceives a tactile impression that I attribute likewise to A. After-
ward, at the instant p, without my having budged, as my mus-
cular sense tells me, this same second finger transmits to me
262 THE VALVE OF SCIENCE
anew a tactile impression which I attribute this time to the
object B ; I then make a series of movements, corresponding to
a series 8' of muscular sensations. I know that this series 8' is
the inverse of the series 8 and corresponds to contrary move-
ments. I know this because many previous experiences have
shown me that if I made successively the two series of movements
corresponding to 8 and to 8', the primitive impressions would be
reestablished, in other words, that the two series mutually com-
pensate. That settled, should I expect that at the instant ^9^,
when the second series of movements is ended, my first finger
would feel a tactile impression attributable to the object B f
To answer this question, those already knowing geometry
would reason as follows : There are chances that the object A has
not budged, between the instants a and a', nor the object B
between the instants p and p'; assume this. At the instant a,
the object A occupied a certain point M of space. Now at this
instant it touched my first finger, and as touch does not operate
at a distance, my first finger was likewise at the point If. I
afterward made the series 8 of movements and at the end of
this series, at the instant a', I ascertained that the object A
touched my second finger. I thence conclude that this second
finger was then at M, that is, that the movements 8 had the result
of bringing the second finger to the place of the first. At the
instant p the object B has come in contact with my second finger :
as I have not budged, this second finger has remained at M\
therefore the object B has come to ilf ; by hypothesis it does not
budge up to the instant p^. But between the instants p and p!
I have made the movements 8'\ as these movements are the in-
verse of the movements 8, they must have for effect bringing the
first finger in the place of the second. At the instant p^ this
first finger will, therefore, be at M ; and as the object B is like-
wise at M, this object B will touch my first finger. To the ques-
tion put, the answer should therefore be yes.
We who do not yet know geometry can not reason thus ; but
we ascertain that this anticipation is ordinarily realized ; and we
can always explain the exceptions by saying that the object A
has moved between the instants a and a', or the object B between
the instants p and ^.
But could not experience have given a contrary results Would
this contrary result have been abaurd in itself! Evidently not.
What should we have done then if experience had given this
contrary resultf Would all geometry thus have become impos-
sible t Not the least in the world. "We should have contented
ourselves with concluding that touch can operate at a distance.
When I say, touch does not operate at a distance, but sight
operates at a distance, this assertion has only one meaning,
which is as follows: To recognize whether B occupies at the
instant j? the point occupied by A at the instant a, I can use
& moltitude of different criteria. In one my eye intervenes,
in another my first finger, in another my second finger, etc.
Well, it is sufficient for the criterion relative to one of my fin-
gers to be satisfied in order that all the others should be satisfied,
but it is not sufBcient that the criterion relative to the eye should
be. This b the sense of my assertion, I content myself with
affirming an experimental fact which is ordinarily verified.
At the end of the preceding chapter we analyzed visual space;
we saw that to engender this space it is necessary to bring in the
retinal sensations, the sensation of convergence and the sensa-
tion of accommodation ; that if these last two were not always
in accord, visual space would have four dimensions in place of
three ; we also saw that if we brought in only the retinal sensa-
tioDS, we should obtain 'simple visual space,' of only two dimen-
sions. On the other hand, consider tactile space, limiting our-
selves to the sensations of a single finger, that is in sum to the
assemblage of positions this finger can occupy. This tactile
space that we shall analyze in the following section and which
consequently I ask permission not to consider further for the
moment, this tactile space, I say, has three dimensions. Why
has space properly so called as many dimensions as tactile space
and more than simple visual space T It is because touch does not
operate at a distance, while vision does operate at a distance.
These two assertions have the same meaning and we have just
seen what this is.
Now I return to a point over which I passed rapidly in order
not to interrupt the discussion. How do we know that the im-
pre^ioDS made on our retina by A at the instant a and B at the
264 THE VALUE OF SCIENCE
instant p are transmitted by the same retinal fiber, although
these impressions are qualitatively different? I have suggested
a simple hypothesis, while adding that other hypotheses, decid-
edly more complex, would seem to me more probably true. Here
then are these hypotheses, of which I have already said a word.
How do we know that the impressions produced by the red object
A at the instant a, and by the blue object B at the instant ^, if
these two objects have been imaged on the same point of the
retina, have something in common? The simple hypothesis
above made may be rejected and we may suppose that these two
impressions, qualitatively different, are transmitted by two dif-
ferent though contiguous nervous fibers. What means have I
then of knowing that these fibers are contiguous ? It is probable
that we should have none, if the eye were immovable. It is the
movements of the eye which have told us that there is the same
relation between the sensation of blue at the point A and the sen-
sation of blue at the point B of the retina as between the sensation
of red at the point A and the sensation of red at the point B.
They have shown us, in fact, that the same movements, corre-
sponding to the same muscular sensations, carry us from the
first to the second, or from the third to the fourth. I do not
emphasize these considerations, which belong, as one sees, to the
question of local signs raised by Lotze.
3. Tactile Space
Thus I know how to recognize the identity of two points, the
point occupied by A at the instant a and the point occupied by
B at the instant p, but only on one condition, namely, that I have
not budged between the instants a and p. That does not suiBSce
for our object. Suppose, therefore, that I have moved in any
manner in the interval between these two instants, how shall I
know whether the point occupied by A at the instant a is identi-
cal with the point occupied by B at the instant pi I suppose
that at the instant a, the object A was in contact with my first
finger and that in the same way, at the instant p, the object B
touches this first finger ; but at the same time, my muscular sense
has told me that in the interval my body has moved. I have
considered above two series of muscular sensations S and 8\ and
SPACE AND IIS THREE DIMENSIONS
266
I have said it sometimes liappetts that we are led to coDsider two
sach series S and S' as inverse one of the other, because we have
often observed that when these two series succeed one another
our primitive impressions are reestablished.
If then my muscular sense tells me that I have moved between
the two instants a and ff, but so as to feel successively the two
series of muscular sensations S and S' that I consider inverses,
I shall still conclude, just as if I had not budged, that the points
occupied by A at the instant a and by B at the instant p are
identical, if I ascertain that my lirst finger touches A at the
instant a, and B at the instant y8.
This solution is not yet completely satisfactory, as one will see.
Let us see, in fact, how many dimensions it would make us at-
tribute to space. I wish to compare the two points occupied by A
and B at the instants a and j3, or (what amounts to the same
thing since I suppose that my finger touches A at the instant a
and B at the instant 0] I wish to compare the two points occu-
pied by my finger at the two instants a and /3. The sole means
1 use for this comparison is the series S of muscular sensations
which have accompanied the movements of my body between
these two instants. The different imaginable series 2 form evi-
dently a physical continuum of which the number of dimensions
is very great. Let us agree, as I have done, not to consider as
distinct the two series 2 and 2 + S + S\ when S and S' are in-
verses one of the other in the sense above given to this word;
in spite of this agreement, the aggregate of distinct series 2 will
still form a physical continuum and the number of dimensions
will be less but still very great.
To each of these series 2 corresponds a point of space ; to two
series 5 and S' thus correspond two points If and M". The means
we have hitherto used enable us to recognize that M and M' are
not distinct in two cases: (1) if 2 is identical with 2'; (2) if 2' =
2 -|- S -f S', S and S' being inverses one of the otlier. If in all
the other cases we should regard M and M' as distinct, the mani-
fold of points would have as many dimensions as the aggregate
of distinct series 2, that is, much more than three.
For those who already know geometry, the following esplana-
lion would be easily comprehensible. Among the imaginable
266 THE VALVE OF SCIENCE
series of muscular sensations, there are those which correspond
to series of movements where the finger does not budge. I say
that if one does not consider as distinct the series S and S -|- <r,
where the series <r corresponds to movements where the fingar
does not budge, the aggregate of series will constitute a con-
tinuum of three dimensions, but that if one regards as distinct
two series S and S' unless S' = S + /S + /S', 8 and 8' being in-
verses, the aggregate of series will constitute a continuum of
more than three dimensions.
In fact, let there be in space a surface A, on this surface a
line By on this line a point M. Let Co be the aggregate of all
series S. Let C^ be the aggregate of all the series S, such that
at the end of corresponding movements the finger is found upon
the surface A, and C, or C^ the aggregate of series 2 such that
at the end the finger is found on B, or at M. It is clear, first that
Ci will constitute a cut which will divide Co, that C, will be a cut
which will divide Ci, and Cj a cut which will divide C,. Thence
it results, in accordance with our definitions, that if C^ is a con-
tinuum of n dimensions, Co will be a physical continuum of
n -j- 3 dimensions.
Therefore, let 5 and S' = 5 + o- be two series forming part
of Cj ; for both, at the end of the movements, the finger is found
at M ; thence results that at the beginning and at the end of the
series o-, the finger is at the same point M. This series a is there-
fore one of those which correspond to movements where the
finger does not budge. If 5 and 5 + o- are not regarded as dis-
tinct, all the series of Cj blend into one ; therefore Cj will have
dimension, and Cq will have 3, as I washed to prove. If, on
the contrary, I do not regard 5 and 5 + o- as blending (unless
(r=8 -\-8', 8 and 8' being inverses), it is clear that C^ will con-
tain a great number of series of distinct sensations; because,
without the finger budging, the body may take a multitude of
different attitudes. Then Cg will form a continuum and Co will
have more than three dimensions, and this also I wished to prove.
We who do not yet know geometry can not reason in this way;
we can only verify. But then a question arises; how, before
knowing geometry, have we been led to distinguish from the
others these series o- where the finger does not budget It is, in
SPACE AND ITS THREE DIMENSIONS
267
Jact, only after having made thia distinction that we could be led
to regard S and S + (r as identical, and it is on this condition
alone, as we have just seen, that we can arrive at apace of three
dimensions.
We are led to distinguish the series <r, because it often happens
that when we have executed the movements which correspond to
these series a of muscular sensations, the tactile sensations which
are transmitted to us by the nerve of the finger that we have
called the first finger, persist and are not altered by these move-
ments. Experience alone tells us that and it alone could tell us.
If we have distinguished the series of muscular sensations
5 + S' formed by the union of two inverse series, it is because
they preserve the totality of our impressions; if now we distin-
guish the series a, it is because they preserve certain of our im-
pressions. (When I say that a series of muscular sensations S
'preserves' one of our impressions A, I mean that we ascertain
that if we feel the impression A, then the muscular sensations 8,
we still feel the impression A after these sensations .S.)
I have said above it often happens that the series a do not
alter the tactile impressions felt by our Brst finger ; I said ofien,
I did not say always. This it is that we express in our ordinary
language by saying that the tactile impressions would not be
altered if the finger has not moved, on the condition that neither
has the object -d, which was in contact with this finger, moved.
Before knowing geometry, we could not give this explanation;
all we could do is to ascertain that the impression often per^
gists, but not always.
But that the impression often continues is enough to make the
series o appear remarkable to us, to lead us to put in the sai
class the series 2 and 2 + 0-, and hence not regard them as dis-
tinct. Under these conditions we have seen that they will en-
gender a physical continuum of three dimensions.
Behold then a space of three dimensions engendered by my
first finger. Each of my fingers will create one like it. It re-
mains to consider how we are led to regard them as identical
with visual space, as identical with geometric space.
Bnt one reflection before going further ; according to the fore-
going, we know the points of space, or more generally the final
268 THE VALUE OF SCIENCE
situation of our body, only by the series of muscular sensations
revealing to us the movements which have carried us from a
certain initial situation to this final situation. But it is clear
that this final situation will depend, on the one hand, upon
these movements and, on the other hand, upon the initial situa-
tion from which we set out. Now these movements are re-
vealed to us by our muscular sensations ; but nothing tells us the
initial situation; nothing can distinguish it for us from all the
other possible situations. This puts well in evidence the essential
relativity of space.
4. Identity of the Different Spaces
We are therefore led to compare the two continua C and C
engendered, for instance, one by my first finger D, the other by
my second finger D\ These two physical continua both have
three dimensions. To each element of the continuum C, or, if
you prefer, to each point of the first tactile space, corresponds a
series of muscular sensations S, which carry me from a certain
initial situation to a certain final situation.^ Moreover, the same
point of this first space will correspond to S and to S + cr, if cr
is a series of which we know that it does not make the finger D
move.
Similarly to each element of the continuum C\ or to each point
of the second tactile space, corresponds a series of sensations S',
and the same point will correspond to 2' and to 2' + <t', if a' is a
series which does not make the finger D' move.
What makes us distinguish the various series designated o- from
those called o-' is that the first do not alter the tactile impressions
felt by the finger D and the second preserve those the finger ly
feels.
Now see what we ascertain : in the beginning my finger D' feels
a sensation A' ; I make movements which produce muscular sen-
sations 8; my finger D feels the impression A; I make move-
ments which produce a series of sensations a; my finger D con-
tinues to feel the impression A, since this is the characteristic
1 In place of saying that we refer space to axes rigidly bound to our
body, perhaps it would be better to say, in conformity to what precedes,
that we refer it to axes rigidly bound to the initial situation of our body.
SPACE AND ITS THREE DIMENSION'S 209
property of the series a ; I then make movements which produce
the series S' of muscular sensations, inverse to S io the sense
above given to this word. I ascertain then that my finger D'
feels anew the impression A'. (It is of course understood that
8 has been suitably chosen.)
This means that the series 8 -\-a-\-S', preserving the tactile
impressions of the finger D', is one of the series I have called t/.
Inversely, if one takes any series a', S' + o' + S will be one of
the series that we call a.
Thas if S is suitably chosen, S -j- o' + S' will be a series a', and
by making a vary in all possible ways, we shall obtain all the
possible series </.
Not yet knowing geometry, we limit ourselves to verifying all
that, but here is how those who know geometrj- would explain the
fact. In the beginning my finger ly is at the point M, in contact
with the object a, which makes it feel the impression A'. I make
the movements corresponding to the series S; I have said that
this series should be suitably chosen, I should so make this
choice tliat these movements carry the finger D to the point
originally occupied by the finger D', that is, to the point M ; this
finger D will thus be in contact with the object a, which will
make it feel the impression A.
I then make the movements corresponding to the series a; in
these movements, by hypothesis, the position of the finger D does
not change, this finger therefore remains in contact with the ob-
ject a and continues to feel the impression A. Finally I make
the movements corresponding to the series S'. As S' is inverse
Io S, these movements carry the finger D' to the point previously
occupied by the finger D, that is, to the point M, If, as may be
supposed, tJie object a has not budged, this finger D' will be
in contact with this object and will feel anew the impression
A'.. . . Q. E. D.
Let us see the consequences. I consider a series of muscular
sensations 2. To this series will correspond a point M of the
first tactile space. Now take again the two series S and 8', in-
verses of one another, of which we have just spoken. To the
series S -}- 1-^-8' will correspond a point N of the second tac-
I tilfi space, since to any series of muscular sensations corresponds,
270 THE VALUE OF^ SCIENCE
as we have said, a point, whether in the first space or in the
second.
I am going to consider the two points N and M, thus defined,
as corresponding. What authorizes me so to do? For this
correspondence to be admissible, it is necessary that if two points
M and M\ corresponding in the first space to two series S and S',
are identical, so also are the two corresponding points of the
second space N and N', that is the two points which correspond
to the two series /S + S + S' and S + S' + 8\ Now we shall see
that this condition is fulfilled.
First a remark. As S and S' are inverses of one another, we
•shall have /S + /S' = 0, and consequently flf + flf' + S = S + 5 +
8' =t S, or again S + flf + /S' + 5' = 5 + S'; but it does not fol-
low that we have 8 -{-1, -{- S' = i; because, though we have used
the addition sign to represent the succession of our sensations,
it is clear that the order of this succession is not indifferent:
we can not, therefore, as in ordinary addition, invert the order
of the terms ; to use abridged language, our operations are asso-
ciative, but not commutative.
That fixed, in order that S and S' should correspond to the
same point M=^M' of the first space, it is necessary and suffi-
cient for us to have 2' = 5 + o-. We shall then have : S -{- S' +
flf' = fif + 5+cr + iS' = /Sf + 5 + iS' + flf + cr + flf'.
But we have just ascertained that S -\- g -\- S' was one of the
series a'. We shall therefore have :/S + S' + iS' = /S-fS +
8' + or', which means that the series /S + 2' + ^' aiid iSf + 2 +
8' correspond to the same point N=N' of the second space.
Q. E. D.
Our two spaces therefore correspond point for point ; they can
be 'transformed' one into the other; they are isomorphic. How
are we led to conclude thence that they are identical ?
Consider the two series o- and 8 -\- <t -\- 8' ==^ <t\ I have said
that often, but not always, the series a preserves the tactile impres-
sion A felt by the finger D ; and similarly it often happens, but
not always, that the series </ preserves the tactile impression A'
felt by the finger D\ Now I ascertain that it happens very often
(that is, much more often than what I have just called 'often')
that when the series o- has preserved the impression A of the
SPACE AND ITS THREE DIMENSIONS 271
finger D, the series </ preserves at the same time the impression
A^ of the finger D' ; and, inversely, that if the first impression is
altered, the second is likewise. That happens very often, but not
always.
We interpret this experimental fact by saying that the un-
known object a which gives the impression A to the finger D is
identical with the unknown object a' which gives the impression
A' to the finger Z>'. And in fact when the first object moves,
which the disappearance of the impression A tells us, the second
likewise moves, since the impression A' disappears likewise.
When the first object remains motionless, the second remains
motionless. If these two objects are identical, as the first is at
the point M of the first space and the second at the point N
of the second space, these two points are identical. This is how
we are led to regard these two spaces as identical ; or better, this
is what we mean when we say that they are identical.
What we have just said of the identity of the two tactile
spaces makes unnecessary our discussing the question of the
identity of tactile space and visual space, which could be treated
in the same way.
5. Space and Empiricism
It seems that I am about to be led to conclusions in conformity
with empiristic ideas. I have, in fact, sought to put in evidence
the role of experience and to analyze the experimental facts
which intervene in the genesis of space of three dimensions. But
whatever may be the importance of these facts, there is one thing
we must not forget and to which besides I have more than once
called attention. These experimental facts are often verified
but not always. That evidently does not mean that space has
often three dimensions, but not always.
I know well that it is easy to save oneself and that, if the
facts do not verify, it will be easily explained by saying that
the exterior objects have moved. If experience succeeds, we say
that it teaches us about space; if it does not succeed, we hie to
exterior objects which we accuse of having moved; in other
words, if it does not succeed, it is given a fillip.
These fillips are legitimate ; I do not refuse to admit them ; but
272 THE VALUE OF SCIENCE
they sofSce to tell us that the properties of space are not experi-
mental truths, properly so called. If we had wished to verify
other laws,, we could have succeeded also, by giving other analo-
gous fillips. Should we not always have been able to justify
these fillips by the same reasons ? One could at most have said to
us: 'Your fillips are doubtless legitimate, but you abuse them;
why move the exterior objects so often t*
To sum up, experience does not prove to us that space has
three dimensions ; it only proves to us that it is convenient to at-
tribute three to it, because thus the number of fillips is reduced
to a minimum.
I will add that experience brings us into contact only with
representative space, which is a physical continuum, never with
geometric space, which is a mathematical continuum. At the
very most it would appear to tell us that it is convenient to give
to geometric space three dimensions, so that it may have as
many as representative space.
The empiric question may be put under another form. Is it
impossible to conceive physical phenomena, the mechanical phe-
nomena, for example, otherwise than in space of three dimen-
sions? We should thus have an objective experimental proof,
so to speak, independent of our physiology, of our modes of
representation.
But it is not so; I shall not here discuss the question com-
pletely, I shall confine myself to recalling the striking example
given us by the mechanics of Hertz. You know that the great
physicist did not believe in the existence of forces, properly so
called ; he supposed that visible material points are subjected to
certain invisible bonds which join them to other invisible points
and that it is the effect of these invisible bonds that we attribute
to forces.
But that is only a part of his ideas. Suppose a system formed
of n material points, visible or not ; that will give in all 3n coor-
dinates ; let us regard them as the coordinates of a single point
in space of 3n dimensions. This single point would be con-
strained to remain upon a surface (of any number of dimensions
< 3n) in virtue of the bonds of which we have just spoken; to
go on this surface from one point to another, it would always
SPACE AND ITS THREE DIMENSIONS
273
tabe the shortest way ; this would be the single principle which
would sum up all mechanics.
Whatever should be thought of this hypothesis, whether we be
allured by its simplicity, or repelled by its artificial character,
the simple fact that Hertz was able to conceive it, and to regard
it as more convenient than our habitual hypotheses, suEBces to
prove that our ordinary ideas, and, in particular, the three di-
mensions of space, are in no wise imposed upon mechanics with
an invincible force.
6. Mind and Space
Experience, therefore, has played only a single role, it has
served as occasion. But this role was none the less very impor-
tant ; and I have thought it necessary to give it prominence.
This role would have been useless if there existed an a priori
form imposing itself upon our sensitivity, and which was space
of three dimensions.
Does this form exist, or, if you choose, can we represent to oar-
selves space of more than three dimensioos ! And first what does
this question mean! In the true sense of the word, it is clear
that we can not represent to ourselves space of four, nor space
of three, dimensions ; we can not first represent them to ourselves
empty, and no more can we represent to ourselves an object
either in space of four, or in space of three, dimensions: (1)
Because these spaces are both infinite and we can not represent
to ourselves a figure in space, that is, the part in the whole, with-
out representing the whole, and that is impossible, because it is
infinite; (2) because these spaces are both mathematical con-
tinna, and we can represent to ourselves only the physical con-
tinnnm; (3) because these spaces are both homogeneous, and
the frames in which we enclose oar sensations, being limited, can
not be homogeneous.
Thus the question put can only be nnderstood in one way;
is it possible to imagine that, the results of the experiences
related above having been different, we might have been led to
attribute to space more than three dimensions; to imagine, for
instance, that the sensation of accommodation might not be con-
stantly in accord with the sensation of convergence of the eyes;
19
274 THE VALUE OF SCIENCE
or indeed that the exi)eriences of which we have spoken in § 2,
and of which we express the result by saying ' that touch does
not operate at a distance/ might have led us to an inverse con-
clusion.
And then yes evidently that is possible ; from the moment one
imagines an experience, one imagines just there by the two con-
trary results it may give. That is possible, but that is diffi-
cult, because we have to overcome a multitude of associations of
ideas, which are the fruit of a long personal experience and of
the still longer experience of the race. Is it these associations
(or at least those of them that we have inherited from our an-
cestors), which constitute this a priori form of which it is said
that we have pure intuition? Then I do not see why one should
declare it refractory to analysis and should deny me the right
of investigating its origin.
When it is said that our sensations are 'extended' only one
thing can be meant, that is that they are always associated with
the idea of certain muscular sensations, corresponding to the
movements which enable us to reach the object which causes
them, which enable us, in other words, to defend ourselves against
it. And it is just because this association is useful for the de-
fense of the organism, that it is so old in the history of the species
and that it seems to us indestructible. Nevertheless, it is only
an association and we can conceive that it may be broken; so
that we may not say that sensation can not enter consciousness
without entering in space, but that in fact it does not enter con-
sciousness without entering in space, which means, without being
entangled in this association.
No more can I understand one's saying that the idea of time
is logically subsequent to space, since we can represent it to our-
selves only under the form of a straight line; as well say that
time is logically subsequent to the cultivation of the prairies,
since it is usually represented armed with a scythe. That one
can not represent to himself simultaneously the different parts of
time, goes without saying, since the essential character of these
parts is precisely not to be simultaneous. That does not mean
that we have not the intuition of time. So far as that goes, no
more should we have that of space, because neither can we rep-
SPACE AND ITS THREE DIMENSIONS
resent it, in the proper sense of the word, for the reasons I have
mentioned. What we represent to ourselves under the name of
straight is a crude image which as ill resembles the geometric
straight as it does time itself.
Why has it been said that every attempt to ^ve a fourth dimen-
sion to space always carries this one back to one of the other
three! It is easy to understand. Consider our muscular sen-
sations and the 'series' they may form. In consequence of nu-
merous experiences, the ideas of these series are associated to-
gether in a very complex woof, our series are classed. Allow
me, for convenience of language, to express my thought in a
way altogether crude and even inexact by saying that our scries
of muscular sensations are classed in three classes correspond-
ing to the three dimensions of space. Of course this classiBca-
tion is much more complicated than that, but that will suffice
to make my reasoning understood. If I wish to imagine a fourth
dimensiou, I shall suppose another series of muscular sensations,
making part of a fourth class. But as all my muscular sensa-
tions have already been classed in one of the three preexistent
classes, I can only represent to myself a series belonging to one
of these three classes, so that my fourth dimension is carried
back to one of the other three.
What does that prove! This; that it woixld have been neces-
sary first to destroy the old classification and replace it by a new
one in which the series of muscular sensations should have been
distributed into four classes. The difBculty would have dis-
appeared.
It is presented sometimes under a more striking form. Sup-
pose I am enclosed in a chamber between the six impassable
boundaries formed by the four walls, the Soor and the ceiling;
it will be impossible for me to get out and to imagine my getting
out. Pardon, can you not imagine that the door opens, or that
two of these walla separate! But of course, you answer, one
must suppose that these walls remain immovable. Yea, but it is
evident that I have the right to move ; and then the walls that we
suppose absolutely at rest will be in motion with regard to me.
Yes, but such a relative motion can not be arbitrary; when ob-
jects are at rest, their relative motion with regard to any axea
276 THE VALUE OF SCIENCE
is that of a rigid solid; now, the apparent motions that yon
imagine are not in conformity with the laws of motion of a rigid
solid. Yes, but it is experience which has taught us the laws
of motion of a rigid solid ; nothing would prevent our imagimng
them different. To sum up, for me to imagine that I get out of
my prison, I have only to imagine that the walls seem to open,
when I move.
I believe, therefore, that if by space is understood a mathemat-
ical continuum of three dimensions, were it otherwise amorphous,
it is the mind which constructs it, but it does not construct it out
of nothing; it needs materials and models. These materials,
like these models, preexist within it. But there is not a single
model which is imposed upon it; it has choice; it may choose,
for instance, between space of four and space of three dimen-
sions. What then is the role of experience? It gives the indi-
cations following which the choice is made.
Another thing: whence does space get its quantitative char-
acter? It comes from the role which the series of muscular sen-
sations play in its genesis. These are series which may repeat
themselves, and it is from their repetition that number comes ; it
is because they can repeat themselves indefinitely that space is
infinite. And finally we have seen, at the end of section 3, that
it is also because of this that space is relative. So it is repeti-
tion which has given to space its essential characteristics; now,
repetition supposes time; this is enough to tell that time is
logically anterior to space.
7. Role of the Semicircular Canals
I have not hitherto spoken of the role of certain organs to
which the physiologists attribute with reason a capital impor-
tance, I mean the semicircular canals. Numerous experiments
have suflSciently shown that these canals are necessary to our
sense of orientation; but the physiologists are not entirely in
accord ; two opposing theories have been proposed, that of Mach-
Delage and that of M. de Cyon.
M. de Cyon is a physiologist who has made his name illustrious
by important discoveries on the innervation of the heart; I can
not, however, agree with his ideas on the question before us. Not
SPACE AND ITS THREE DIMENSIONS 277
being a physiologist, I hesitate to criticize the experiments he has
directed against the adverse theory of Mach-Delage; it seems
to me, however, that they are not convincing, because in many
of them the total pressure was made to vary in one of the canals,
while, physiologically, what varies is the difference between the
pressures on the two extremities of the canal; in others the
organs were subjected to profound lesions, which must alter their
functions
Besides, this is not important; the experiments, if they were
irreproachable, might be convincing against the old theory. They
would not be convincing for the new theory. In fact, if I have
rightly understood the theory, my explaining it wiU be enough
for one to understand that it is impossible to conceive of an experi-
ment confirming it.
The three pairs of canals would have as sole function to tell us
that space has three dimensions. Japanese mice have only two
pairs of canals ; they believe, it would seem, that space has only
two dimensions, and they manifest this opinion in the strangest
way ; they put themselves in a circle, and, so ordered, they spin
rapidly around. The lampreys, having only one pair of canals,
believe that space has only one dimension, but their manifesta-
tions are less turbulent.
It is evident that such a theory is inadmissible. The sense-
organs are designed to tell us of changes which happen in the
exterior world. We could not understand why the Creator should
have given us organs destined to cry without cease : Remember
that space has three dimensions, since the number of these three
dimensions is not subject to change.
We must, therefore, come back to the theory of Mach-Delage.
What the nerves of the canals can tell us is the difference of pres-
sure on the two extremities of the same canal, and thereby: (1)
the direction of the vertical with regard to three axes rigidly
bound to the head; (2) the three components of the acceleration
of translation of the center of gravity of the head; (3) the cen-
trifugal forces developed by the rotation of the head; (4) the
acceleration of the motion of rotation of the head.
It follows from the experiments of M. Delage that it is this
last indication which is much the most important; doubtless be-
27K THE VALUE OF SCIENCE
eaiue the nenres are leas sensible to the difference of pressoie
itself than to the brusque variations of this difference. The first
three indications may thus be neglected.
Knowing the acceleration of the motion of rotation of the head
at each instant, we deduce from it, by an unconscious integrar
tion, the final orientation of the head, referred to a certain initial
orientation taken as origin. The circular canals contribute, there-
fore, to inform us of the movements that we have executed, and
that on the same ground as the muscular sensations. When,
therefore, above we speak of the series S or of the series S, we
should say, not that these were series of muscular sensations
alone, but that they were series at the same time of muscular
sensations and of sensations due to the semicircular canals.
Apart from this additioii, we should have nothing to change in
what precedes.
In the series S and S, these sensations of the semicircular canals
evidently hold a very important place. Yet alone they would
not suffice, because they can tell us only of the movements of the
head ; they tell us nothing of the relative movements of the body
or of the members in regard to the head. And more, it seems thai
they tell us only of the rotations of the head and not of the trans-
lations it may undergo.
PART II
THE PHYSICAL SCIENCES
CHAPTER V
Analysis and Physics
I
You have doubtless often been asked of what good is mathe-
matics and whether these delicate constmctions entirely mind-
made are not artificial and bom of our caprice.
Among those who put this question I should make a distinc-
tion ; practical people ask of us only the means of money-making.
These merit no reply j rather would it be proper to ask of them
what is the good of accumulating so much wealth and whether,
to get time to acquire it, we are to neglect art and science, which
alone give us souls capable of enjoying it, 'and for life's sake to
sacrifice all reasons for living.'
Besides, a science made solely in view of applications is impos-
sible; truths are fecund only if bound together. If we devote
ourselves solely to those truths whence we expect an immediate
result, the intermediary links are wanting and there will no
longer be a chain.
The men most disdainful of theory get from it, without sus-
pecting it, their daily bread; deprived of this food, progress
would quickly cease, and we should soon congeal into the im-
mobility of old China.
But enough of uncompromising practicians! Besides these,
there are those who are only interested in nature and who ask us
if we can enable them to know it better.
To answer these, we have only to show them the two monu-
ments already rough-hewn. Celestial Mechanics and Mathematical
Physics.
279
280 THE VALUE OF SCIENCE
They would doubtless concede that these structures are wdl
worth the trouble they have cost us. But this is not enough.
Mathematics has a triple aim. It must furnish an instrument
for the study of nature. But that is not all : it has a philosophic
aim and, I dare maintain, an esthetic aim. It must aid the
philosopher to fathom the notions of number, of space, of time.
And above all, its adepts find therein delights analogous to those
given by painting and music. They admire the delicate harmony
of numbers and forms ; they marvel when a new discovery opens
to them an unexpected perspective ; and has not the joy they thus
feel the esthetic character, even though the senses take no part
therein t Only a privileged few are called to enjoy it fully, it is
true, but is not this the case for all the noblest arts ?
This is why I do not hesitate to say that mathematics deserves
to be cultivated for its own sake, and the theories inapplicable
to physics as well as the others. Even if the physical aim and
the esthetic aim were not united, we ought not to sacrifice either.
But more : these two aims are inseparable and the best means
of attaining one is to aim at the other, or at least never to lose
sight of it. This is what I am about to try to demonstrate in
setting forth the nature of the relations between the pure sci-
ence and its applications.
The mathematician should not be for the physicist a mere pur-
veyor of formulas ; there should be between them a more intimate
collaboration. Mathematical physics and pure analysis are not
merely adjacent powers, maintaining good neighborly relations;
they mutually interpenetrate and their spirit is the same. This
will be better understood when I have shown what physics gets
from mathematics and what mathematics, in return, borrows
from physics.
II
The physicist can not ask of the analyst to reveal to him a new
truth ; the latter could at most only aid him to foresee it. It is a
long time since one still dreamt of forestalling experiment, or of
constructing the entire world on certain premature hypotheses.
Since all those constructions in which one yet took a naive de-
light it is an age, to-day only their ruins remain.
ANALYSIS AND PHTSJCS
AU laws are therefore deduced from experiment; but to eniin-
ciate tlieni, a special language is needful ; ordinary language is
too poor, it is besides too vague, to express relations so delicate,
BO rich, and ao precise.
This therefore is one reason why the physicist can not do with-
out mathematics ; it furnishes him the only language he can speak.
And a well-made language is no indifferent thing; not to go
beyond physics, the unknown man who invented the word heat
devoted many generations to error. Heat has been treated as a
sabstance, simply because it was designated by a substantive, and
it has been thought indestructible.
On the other hand, he who invented the word electricity had
the unmerited good fortune to implicitly endow physics with a
new law, that of the conservation of electricity, which, by a pure
chance, has been found exact, at least until now.
Well, to continue the simile, the writers who embellish s lan-
guage, who treat it as an object of art, make of it at the same time
a more supple instrument, more apt for rendering shades of
thought.
We understand, then, how the analyst, who pursues a purely
esthetic aim, helps create, just by that, a language more fit to
satisfy the physicist.
But this is not all : law springs from experiment, but not im-
mediately. Experiment is individual, the law deduced from it is
general ; experiment is only approximate, the law is precise, or at
least pretends to be. Experiment is made under conditions
always complex, the enunciation of the luw eliminates these com-
plications. This is what is called ' correcting the systematic errors. *
In a word, to get the law from esperiment, it is necessary to
generalize; this is a necessity imposed upon the most circum-
spect observer. But how generalize T Every particular truth
may evidently be extended in an infinity o£ ways. Among these
thousand routes opening before us, it is necessary to make a
choice, at least provisional ; in this choice, what shall guide us?
It can only be analogy. But how vague is this word 1 Primi-
tive man knew only crude analogies, those which strike the senses,
those of colors or of sounds. He never would have dreamt of
likening light to radiant heat.
282 THE VALUE OF SCIENCE
What has taught us to know the true, profound analogies, those
the eyes do not see but reason divines?
It is the mathematical spirit, which disdains matter to ding
only to pure form. This it is which has taught us to give the same
name to things differing only in material, to call by the same
name, for instance, the multiplication of quaternions and that of
whole numbers.
If quaternions, of which I have just spoken, had not been so
promptly utilized by the English physicists, many persons would
doubtless see in them only a useless fancy, and yet, in teaching us
to liken what appearances separate, they would have already
rendered us more apt to penetrate the secrets of nature.
Such are the services the physicist should expect of analysis;
but for this science to be able to render them, it must be culti-
vated in the broadest fashion without immediate expectation of
utility — the mathematician must have worked as artist.
What we ask of him is to help us to see, to discern our way in
the labyrinth which opens before us. Now, he sees best who
stands highest. Examples abound, and I limit myself to the most
striking.
The first will show us how to change the language suffices to
reveal generalizations not before suspected.
A\Tien Newton's law has been substituted for Kepler's we still
know only elliptic motion. Now, in so far as concerns this motion,
the two laws differ only in form ; we pass from one to the other
by a simple differentiation. And yet from Newton's law may be
deduced by an immediate generalization all the effects of pertur-
bations and the whole of celestial mechanics. If, on the other
hand, Kepler's enunciation had been retained, no one would ever
have regarded the orbits of the perturbed plants, those compli-
cated curves of which no one has ever written the equation, as
the natural generalizations of the ellipse. The progress of obser-
vations would only have served to create belief in chaos.
The second example is equally deserving of consideration.
When Maxwell began his work, the laws of electro-dynamics
admitted up to his time accounted for all the known facts. It was
not a new experiment which came to invalidate them. But in
looking at them under a new bias. Maxwell saw that the equa-
ANALYSIS AND PHYSICS
lions became more symmetrical when a term was added, and
besides, this term was too small to produce effects appreciable
with the old methods.
You know that Maxwell's a priori views awaited for twenty
years an e^cperimental confirmation ; or, if you prefer, Maxwell
was twenty years ahead of experiment. How was this triomph
obtained?
It was because Maxwell was profoundly steeped in the sense of
mathematical symmetry ; would he have been so, if others before
him had not studied this symmetry for its own beauty t
It was because Maxwell was accustomed to 'think in vectors,'
and yet it was through the theory of imaginaries (neomonics)
that vectors were introduced into analysis. And those who in-
vented imaginaries hardly suspected the advantage which would
be obtained from them for the study of the real world, of this the
name given them is proof sufScient.
In a word, Maxwell was perhaps not an able analyst, but this
ability would have been for him only a useless and bothersome
baggage. On the other hand, he had iu the highest degree the
intimate sense of mathematical analogies. Therefore it is that he
made good mathematical physics.
Maxwell's example teaches us still another thing.
How should the equations of mathematical physics be treated !
Should we simply deduce all the consequences, and regard them
as intangible realities t Par from it; what they should teach us
above all is what can and what should be changed. It is thus
that we get from them something useful.
The third example goes to show us bow we may perceive mathe-
matical analogies between phenomena which have physically no
relation either apparent or real, so that the laws of one of these
phenomena aid us to divine those of the other.
The very same equation, that of Laplace, is met in the theory i
of Newtonian attraction, in that of the motion of liquids, in that
of the electric potential, in that of magnetism, in that of the
propagation of heat and in still many others. What is the result*
These theories seem images copied one from the other; they are
mutually illuminating, borrowing their language from each other ;
ask electricians if they do not felicitate themselves on having in-
I
284 THE VALUE OF SCIENCE
vented the phrase flow of force, suggested by hydrodynamics and
the theory of heat.
Thus mathematical analogies not only may make us foresee
physical analogies, but besides do not not cease to be useful when
these latter fail.
To sum up, the aim of mathematical physics is not only to
facilitate for the physicist the numerical calculation of certain
constants or the int^ration of certain differential equations. It
is besides, it is above all, to reveal to him the hidden harmony of
things in making him see them in a new way.
Of all the parts of analysis, the most elevated, the purest, so
to speak, will be the most fruitful in the hands of those who know
how to use them.
Ill
Let us now see what analysis owes to physics.
It would be necessary to have completely forgotten the history
of science not to remember that the desire to understand nature
has had on the development of mathematics the most constant and
happiest influence.
In the flrst place the physicist sets us problems whose solution
he expects of us. But in proposing them to us, he has largely
paid us in advance for the service we shall render him, if we
solve them.
If I may be allowed to continue my comparison with the fine
arts, the pure mathematician who should forget the existence of
the exterior world would be like a painter who knew how to har-
moniously combine colors and forms, but who lacked models. His
creative power would soon be exhausted.
The combinations which numbers and symbols may form are an
infinite multitude. In this multitude how shall we choose those
which are worthy to fix our attention t Shall we let ourselves be
guided solely by our caprice f This caprice, which itself would
besides soon tire, would doubtless carry us very far apart and we
should quickly cease to understand each other.
But this is only the smaller side of the question. Physics will
doubtless prevent our straying, but it will also preserve us from
a danger much more formidable ; it will prevent our ceaselessly
going around in the same circle.
History proves that physics has not only forced us to choose
amoDg problems which came in a crowd ; it has imposed upon as
such as we should without it never have dreamed of. However
varied may be the imagination of man, nature is still a thousand
times richer. To follow her we must take ways we have
neglected, and these patlis lead us often to summits whence we
discover new countries. "What could be more u.%eful !
It is with mathematical symbola as with physical realities; it is
in comparing the different aspects of things that we are able to
comprehend their inner harmony, which alone is beautiful and
consequently worthy of our efforts.
The first example I shall cite is so old we are tempted to foi^et
it; it is nevertheless the most important of all.
The sole natural object of mathematical thought is the whole
number. It is the external world which has imposed the con-
tinuum upon U9, which we doubtless have invented, but which it
has forced us to invent. Without it there would be no infini-
tesimal analysis ; all mathematical science would reduce itself to
antbmetic or to the theory of substitutions.
On the contrary, we have devoted to the study of the con-
tinuum almost all our time and all our strength. Who will regret
it; who will think that this time and this strength have been
wasted! Analysis unfolds before us infinite perspectives that
arithmetic never suspects; it shows us at a glance a majestic
assemblage whose array is simple and symmetric; on the con-
trary, in the theory of numbers, where reigns the unforeseen, the
view is, so to speak, arrested at every step.
Doubtless it will be said that outside of the whole number there
is no rigor, and consequently no mathematical truth; that the
whole number hides everywhere, and that we must strive to render
transparent the screens which cloak it, even if to do so we must
resign ourselves to interminable repetitions. Let us not be such
pnrists and let us be grateful to the continuum, which, if oU
springs from the whole number, was alone capable of making
to much proceed therefrom.
Need I also recall that M. Hermite obtained a surprising ad-
vantage from the introduction of continuous variables into the
theory of numbersi Thus the whole number's own domain is
286 THE VALUE OF SCIENCE
itself invaded, and this invasion has established order where dis-
order reigned.
See what we owe to the continuum and consequently to phys-
ical nature.
Fourier's series is a precious instrument of which analysis
makes continual use, it is by this means that it has been able to
represent discontinuous functions ; Fourier invented it to solve a
problem of physics relative to the propagation of heat. If this
problem had not come up naturally, we should never have dared
to give discontinuity its rights ; we should still long have regarded
continuous functions as the only true functions.
The notion of function has been thereby considerably extended
and has received from some logician-analysts an unforeseen de-
velopment. These analysts have thus adventured into regions
where reigns the purest abstraction and have gone as far away
as possible from the real world. Yet it is a problem of physics
which has furnished them the occasion.
After Fourier's series, other analogous series have entered the
domain of analysis; they have entered by the same door; they
have been imagined in view of applications.
The theory of partial differential equations of the second
order has an analogous history. It has been developed chiefly
by and for physics. But it may take many forms, because such
an equation does not suflBce to determine the unknown function,
it is necessary to adjoin to it complementary conditions which
are called conditions at the limits; whence many different
problems.
If the analysts had abandoned themselves to their natural tend-
encies, they would never have known but one, that which
Madame Kovalevski has treated in her celebrated memoir. But
there are a multitude of others which they would have ignored.
Each of the theories of physics, that of electricity, that of heat,
presents us these equations under a new aspect. It may, there-
fore, be said that without these theories we should not know
partial differential equations.
It is needless to multiply examples. I have given enough to
be able to conclude : when physicists ask of us the solution of a
problem, it is not a duty-service they impose upon us, it is on
the contrary we who owe them thanks.
ANALYSIS AND PHYSICS 287
IV
But this is not all; physics not only gives ns the occasion to
solve problems ; it aids us to find the means thereto, and that in
two ways. It makes us foresee the solution; it suggests argu-
ments to us.
I have spoken above of Laplace's equation which is met in a
multitude of diverse physical theories. It is found again in
geometry, in the theory of conf ormal representation and in pure
analysis, in that of imaginaries.
In this way, in the study of functions of complex variables, the
analyst, alongside of the geometric image, which is his usual in-
strument, finds many physical images which he may make
use of with the same success. Thanks to these images, he can
see at a glance what pure deduction would show him only suc-
cessively. He masses thus the separate elements of the solu-
tion, and by a sort of intuition divines before being able to
demonstrate.
To divine before demonstrating ! Need I recall that thus have
been made all the important discoveries? How many are the
truths that physical analogies permit us to present and that we
are not in condition to establish by rigorous reasoning!
For example, mathematical physics introduces a great number
of developments in series. No one doubts that these develop-
ments converge ; but the mathematical certitude is lacking. These
are so many conquests assured for the investigators who shall
come after us.
On the other hand, physics furnishes us not alone solutions;
it furnishes us besides, in a certain measure, arguments. It will
sufSce to recall how Felix BUein, in a question relative to Rie-
mann surfaces, has had recourse to the properties of electric
currents.
It is true, the arguments of this species are not rigorous, in
the sense the analyst attaches to this word. And here a question
arises: How can a demonstration not suflBciently rigorous for
the analyst sufSce for the physicist! It seems there can not be
two rigors, that rigor is or is not, and that, where it is not there
can not be deduction.
This apparent paradox will be better understood by recalling
288 THE VALUE OF SCIENCE
under what conditions number is applied to natural phenomena.
Whence come in general the difficulties encountered in seeking
rigorf We strike them almost always in seeking to estaUiah
that some quantity tends to some limit, or that some function is
continuous, or that it has a derivative.
Now the numbers the physicist measures by experiment are
never known except approximately; and besides, any function
always differs as littie as you choose from a discontinuous func-
tion, and at the same time it differs as littie as you choose from
a continuous function. The physicist may, therefore, at will
suppose that the function studied is continuous, or that it is dis-
continuous; that it has or has not a derivative; and may do so
without fear of ever being contradicted, either by present ex-
perience or by any future experiment We see that with such
liberty he makes sport of difficulties which stop the analyst. He
may always reason as if all the functions which occur in his
calculations were entire polynomials.
Thus the sketch which suffices for physics is not the deduc-
tion which analysis requires. It does not follow thence that one
can not aid in finding the other. So many physical sketches have
already been transformed into rigorous demonstrations that
to-day this transformation is easy. There would be plenty of
examples did I not fear in citing them to tire the reader.
I hope I have said enough to show that pure analysis and
mathematical physics may serve one another without making any
sacrifice one to the other, and that each of these two sciences
should rejoice in all which elevates its associate.
CHAPTBB VI
Astronomy
Governments and parliaments must find that astronomy is one
of the sciences which cost most dear: the least instrument costs
hundreds of thousands of dollars, the least observatory costs
millions; each eclipse carries with it supplementary appropria-
tions. And all that for stars which are so far away, which are
complete strangers to our electoral contests, and in all probability
will never take any part in them. It must be that our politi-
cians have retained a remnant of idealism, a vague instinct for
what is grand; truly, I think they have been calumniated; they
should be encouraged and shown that this instinct does not de-
ceive them, that they are not dupes of that idealism.
We might indeed speak to them of navigation, of which no
one can underestimate the importance, and which has need of
astronomy. But this would be to take the question by its
smaller side.
Astronomy is useful because it raises us above ourselves ; it is
useful because it is grand ; that is what we should say. It shows
us how small is man's body, how great his mind, since his intel-
ligence can embrace the whole of this dazzling immensity, where
his body is only an obscure point, and enjoy its silent harmony.
Thus we attain the consciousness of our power, and this is some-
thing which can not cost too dear, since this consciousness makes
us mightier.
But what I should wish before all to show is, to what point
astronomy has facilitated the work of the other sciences, more
directly useful, since it has given us a soul capable of compre-
hending nature.
Think how diminished humanity would be if, under heavens
constantly overclouded, as Jupiter's must be, it had forever
remained ignorant of the stars. Do you think that in such a
world we should be what we are! I know well that under this
somber vault we should have been deprived of the light of the
20 289
290 THE VALUE OF SCIENCE
son, necessary to organisms like those which inhabit the eartL
But if you please, we shall assume that these clouds are phos-
phorescent and emit a soft and constant light. Since we are
making hypotheses, another will cost no more. Well ! I repeat
my question: Do you think that in such a world we should be
what we are f
The stars send us not only that visible and gross light which
strikes our bodily eyes, but from them also comes to us a light far
more subtle, which illuminates our minds and whose effects I
shall try to show you. You know what man was on the earth
some thousands of years ago, and what he is to-day. Isolated
amidst a nature where everything was a mystery to him, terrified
at each unexpected manifestation of incomprehensible forces, he
was incapable of seeing in the conduct of the universe anything
but caprice ; he attributed all phenomena to the action of a mul-
titude of little genii, fantastic and exacting, and to act on the
world he sought to conciliate them by means analogous to those
employed to gain the good graces of a minister or a deputy.
Even his failures did not enlighten him, any more than to-day
a beggar refused is discouraged to the point of ceasing to beg.
To-day we no longer beg of nature ; we command her, because
we have discovered certain of her secrets and shall discover
others each day. We command her in the name of laws she can
not challenge, because they are hers ; these laws we do not madly
ask her to change, we are the first to submit to them. Nature
can only be governed by obeying her.
What a change must our souls have undergone to pass from the
one state to the other! Does any one believe that, without the
lessons of the stars, under the heavens perpetually overclouded
that I have just supposed, they would have changed so quickly?
Would the metamorphosis have been possible, or at least would it
not have been much slower?
And first of all, astronomy it is which taught that there are
laws. The Chaldeans, who were the first to observe the heavens
with some attention, saw that this multitude of luminous points
is not a confused crowd wandering at random, but rather a disci-
plined army. Doubtless the rules of this discipline escaped them,
but the harmonious spectacle of the starry night suflSced to give
A.STBONOMT
them the impression of regularity, and that was in itself already
a great thing. Besides, these rules were discerned by Hippar-
chus, Ptolemy, Copernicus, Kepler, one after another, and finally,
it is needless to recall that Newton it was who enunciated the
oldest, the most precise, the most simple, the most general of all
natural laws.
And then, taught by this example, we have seen our little ter-
restrial world better and, under the apparent disorder, there also
we have found again the harmony that the study of the heavens
had revealed to us. It also is regular, it also obeys immutable
laws, but they are more complicated, in apparent conflict one with
another, and an eye untrained by other sights would have seen
there only chaos and the reign of chance or caprice. If we had
not known the stars, some bold spirits might perhaps have
sought to foresee physical phenomena; but their failures would
have been frequent, and they would have escited only the deri-
sion of the vulgar; do we not see, that even in our day the
meteorologists sometimes deceive themselves, and that certain
persons are inclined to laugh at them.
How often would the physicists, disheartened by so many
(^ecks, have fallen into discouragement, if they had not had, to
soBtain their confidence, the brilliant example of the success of
the astronomers! This success showed them that nature obejTB
laws; it only remained to know what laws; for that they only
needed patience, and they had the right to demand that tbo
sceptics should give them credit.
This is not all : astronomy has not only taught us that there are
laws, but that from these laws there is no escape, that with them
there is no possible compromise. How much time should we have
n«eded to comprehend that fact, if we had known only the ter-
restrial world, where each elemental force would always seem to
OS in conflict with other forces T Astronomy has taught us that
the laws are infinitely precise, and that if those we enunciate
are approximative, it is because we do not know them well. Aris-
totle, the most scientific mind of antiquity, still accorded a part
to accident, to chance, and seemed to think that the laws of na-
ture, at least here below, determine only the large features of
phenomena. How much has the ever-increasing precision of
292 THE VALUE OF SCIENCE
astronomical predictions contributed to correct such an error,
which would have rendered nature unintelligible!
But are these laws not local, varying in different places, like
those which men make ; does not that which is truth in one comer
of the universe, on our globe, for instance, or in our little solar
system, become error a little farther away ? And then could it
not be asked whether laws depending on space do not also depend
upon time, whether they are not simple habitudes, transitory,
therefore, and ephemeral f Again it is astronomy that answers
this question. Consider the double stars; all describe conies;
thus, as far as the telescope carries, it does not reach the limits
of the domain which obeys Newton's law.
Even the simplicity of this law is a lesson for us; how many
complicated phenomena are contained in the two lines of its
enunciation ; persons who do not understand celestial mechanics
may form some idea of it at least from the size of the treatises
devoted to this science ; and then it may be hoped that the com-
plication of physical phenomena likevdse hides from us some
simple cause still unknown.
It is therefore astronomy which has shown us what are the
general characteristics of natural laws ; but among these charac-
teristics there is one, the most subtle and the most important of
all, which I shall ask leave to stress.
How was the order of the universe understood by the
ancients; for instance, by Pythagoras, Plato or Aristotle? It
was either an immutable type fixed once for all, or an ideal to
which the world sought to approach. Kepler himself still
thought thus when, for instance, he sought whether the distances
of the planets from the sun had not some relation to the five reg-
ular polyhedrons. This idea contained nothing absurd, but it
was sterile, since nature is not so made. Newton has shown us
that a law is only a necessary relation between the present state
of the world and its immediately subsequent state. All the
other laws since discovered are nothing else; they are in sum,
differential equations; but it is astronomy which furnished the
first model for them, without which we should doubtless long
have erred.
Astronomy has also taught us to set at naught appearances.
ASTBONOMT
298
The day Copemicna proved that what was thought the most stable
was in motion, that what was thought raoviug was fixed, he
showed us how deceptive could be the infantile reasonings which
spring directly from the immediate data of our senses. True,
bis ideas did not easily triumph, but since this triumph there is
no longer a prejudice so inveterate that we can not shake it off.
How can we estimate the value of the new weapon thus wonf
The ancients thought everything was made for man, and this
illusion must be very tenacious, since it must ever be combated.
Yet it is necessary to divest oneself of it ; or else one will be only
an eternal myope, incapable of seeing the truth. To comprehend
nature one must be able to get out of self, so to speak, and to
contemplate her from many different points of view; other^vise
we never shall know more than one side. Now, to get out of
self is what he who refers everj'thing to himself can not do. Who
deUvered us from this illusion J It was those who showed us that
t&e earth is only one of the smallest planets of the solar system,
and that the solar system itself is only an imperceptible point
in the infinite spaces of the stellar universe.
At the same time astronomy taught us not to be afraid of big
numbers. This was needful, not only for knowing the heavens,
but to know the earth itself; and was not so easj- as it seems to
us to-day. Let us try to go back and picture to ourselves what a
Greek would have thought if told that red light vibrates four
hundred millions of millions of times per second. Without any
doubt, such an assertion would have appeared to him pure mad-
ness, and he never would have lowered himself to test it. To-
day a hypothesis will no longer appear absurd to us because it
obligea us to imagine objects much larger or smaller than those
onr senses are capable of showing us, and we no longer com-
prehend those scruples which arrested our predecessors and pre-
vented them from discovering certain truths simply because they
were afraid of them. But whyT It is because we have seen
the heavens enlarging and enlarging without cease; because we
know that the sun is 150 millions of kilometers from the earth
and that the distatiees of the nearest stars are hundreds of
thoosands of times greater yet. Habituated to the contempla-
tioD of the infinitely great, we have become apt to comprehend
294 THE VALUE OF SCIENCE
the infinitely small. Thanks to the education it has reoeived,
our imagination, like the eagle 's eye that the sun does not dazzle,
can look truth in the face.
Was I wrong in saying that it is astronomy which has made
us a soul capable of comprehending nature ; that under heavens
always overcast and starless, the earth itself would have been for
us eternally unintelligible; that we should there have seen only
caprice and disorder; and that, not knowing the world, we should
never have been able to subdue itf What science could have
been more useful t And in thus speaking I put myself at the
point of view of those who only value practical applications.
Certainly, this point of view is not mine ; as for me, on the con-
trary, if I admire the conquests of industry, it is above all be-
cause if they free us from material cares, they will one day give
to all the leisure to contemplate nature. I do not say: Science
is useful, because it teaches us to construct machines. I say:
Machines are useful, because in working for us, they will some
day leave us more time to make science. But finally it is worth
remarking that between the two points of view there is no antag-
onism, and that man having pursued a disinterested aim, all else
has been added unto him.
Auguste Comte has said somewhere, that it would be idle to
seek to know the composition of the sun, since this knowledge
would be of no use to sociology. How could he be so short-
sighted ? Have we not just seen that it is by astronomy that, to
speak his language, humanity has passed from the theological to
the positive state? He found an explanation for that because
it had happened. But how has he not understood that what
remained to do was not less considerable and would be not less
profitable? Physical astronomy, which he seems to condemn,
has already begun to bear fruit, and it will give us much more,
for it only dates from yesterday.
First was discovered the nature of the sun, what the founder of
positivism wished to deny us, and there bodies were found which
exist on the earth, but had here remained undiscovered ; for ex-
ample, helium, that gas almost as light as hydrogen. That al-
ready contradicted Comte. But to the spectroscope we owe a
lesson precious in a quite different way ; in the most distant stars.
A8TB0N0UI 295
it shows us the same substances. It might have been asked
whether the terrestrial elements were not due to some chance
which had brought together more tenuous atoms to construct of
them the more complex edifice that the chemists call atom;
whether, in other regions of the universe, other fortuitous meet-
ings had not engendered edifices entirely different. Now we know
that this is not so, that the laws of our chemistry are the gen-
eral laws of nature, and that they owe nothing to the chance
which caused us to be bom on the earth.
But, it will be said, astronomy has given to the other sciences
all it can give them, and now that the heavens have procured for
us the instruments which enable us to study terrestrial nature,
they could without danger veil themselves forever. After what
we have just said, is there still need to answer this objection t
One could have reasoned the same in Ptolemy's time; then also
men thought they knew everything, and they still had almost
everything to learn.
The stars are majestic laboratories, gigantic crucibles, such as
no chemist could dream. There reign temperatures impossible
for us to realize. Their only defect is being a little far away;
but the telescope will soon bring them near to us, and then we
shall see how matter acts there. What good fortune for the
physicist and the chemist !
Matter will there exhibit itself to us under a thousand different
states, from those rarefied gases which seem to form the nebul®
and which are luminous with I know not what glimmering of
mysterious origin, even to the incandescent stars and to the
planets so near and yet so different.
Perchance even, the stars will some day teach us something
about life ; that seems an insensate dream and I do not at all see
how it can be realized ; but, a hundred years ago, would not the
chemistry of the stars have also appeared a mad dream f
But limiting our views to horizons less distant, there still will
remain to us promises less contingent and yet sufBciently seduc-
tive. If the past has given us much, we may rest assured that
the future will give us still more.
In sum, it is incredible how useful belief in astrology has
been to humanity. If Kepler and Tycho Brahe made a living.
296 THE VALUE OF SCIENCE
it was because they sold to naive kings predictions founded cm
the conjunctions of the stars. If these princes had not been so
credulous, we should perhaps still believe that nature obeys
caprice, and we should still wallow in ignorance.
CHAPTER VII
The History of Mathematical Physics
The Past and the Future of Physics. — What is the present
state of mathematical physics t What are the problems it is led
to set itself t What is its future t Is its orientation about to be
modified t
Ten years hence will the aim and the methods of this science
appear to our immediate successors in the same light as to our-
selves; or, on the contrary, are we about to witness a profound
transformation t Such are the questions we are forced to raise
in entering to-day upon our investigation.
If it is easy to propound them: to answer is difScult. If we
felt tempted to risk a prediction, we should easily resist this
temptation, by thinking of all the stupidities the most eminent
savants of a hundred years ago would have uttered, if some one
had asked them what the science of the nineteenth century
would be. They would have thought themselves bold in their
predictions, and after the event, how very timid we should have
found them. Do not, therefore, expect of me any prophecy.
But if, like all prudent physicians, I shun giving a prognosis,
yet I can not dispense with a little diagnostic ; well, yes, there are
indications of a serious crisis, as if we might expect an approach-
ing transformation. Still, be not too anxious: we are sure the
patient will not die of it, and we may even hope that this crisis
will be salutary, for the history of the past seems to guarantee us
this. This crisis, in fact, is not the first, and to understand it,
it is important to recall those which have preceded. Pardon then
a brief historical sketch.
The Physics of Central Forces. — ^Mathematical physics, as we
know, was bom of celestial mechanics, which gave birth to it at
the end of the eighteenth century, at the moment when it itself
attained its complete development. During its first years espe-
cially, the infant strikingly resembled its mother.
297
298 THE VALUE OF SCIENCE
The astronomic universe is formed of masses, very great, no
doubt, but separated by intervals so immense that they appear
to us only as material points. These points attract each other
inversely as the square of the distance, and this attraction is the
sole force which influences their movements. But if our senses
were sufSciently keen to show us all the details of the bodies
which the physicist studies, the spectacle thus disclosed would
scarcely differ from the one the astronomer contemplates. There
also we should see material points, separated from one another
by intervals, enormous in comparison with their dimensions, and
describing orbits according to regular laws. These infinitesimal
stars are the atoms. Like the stars proper, they attract or repel
each other, and this attraction or this repulsion, following the
straight line which joins them, depends only on the distance.
The law according to which this force varies as function of the
distance is perhaps not the law of Newton, but it is an analogous
law; in place of the exponent — ^2, we have probably a different
exponent, and it is from this change of exponent that arises all
the diversity of physical phenomena, the variety of qualities and
of sensations, all the world, colored and sonorous, which sur-
rounds us; in a word, all nature.
Such is the primitive conception in all its purity. It only
remains to seek in the different cases what value should be given
to this exponent in order to explain all the facts. It is on this
model that Laplace, for example, constructed his beautiful theory
of capillarity; he regards it only as a particular case of attrac-
tion, or, as he says, of universal gravitation, and no one is as-
tonished to find it in the middle of one of the five volumes of the
*Mecanique celeste.' More recently Briot believes he penetrated
the final secret of optics in demonstrating that the atoms of ether
attract each other in the inverse ratio of the sixth power of the
distance; and Maxwell himself, does he not say somewhere that
the atoms of gases repel each other in the inverse ratio of the
fifth power of the distance? We have the exponent — 6, or — 5,
in place of the exponent — 2, but it is always an exponent.
Among the theories of this epoch, one alone is an exception,
that of Fourier ; in it are indeed atoms acting at a distance one
upon the other; they mutually transmit heat, but they do not
THE HISTORY OF MATHEMATICAL PHYSICS 299
attract, they never budge. From this point of view, Fourier's
theory must have appeared to the eyes of his contemporaries, to
those of Fourier himself, as imperfect and provisional.
This conception was not without grandeur; it was seductive,
and many among us have not finally renounced it ; they know that
one will attain the ultimate elements of things only by patiently
disentangling the complicated skein that our senses give us ; that
it is necessary to advance step by step, neglecting no interme-
diary ; that our fathers were wrong in wishing to skip stations ;
but they believe that when one shall have arrived at these ulti-
mate elements, there again will be found the majestic simplicity
of celestial mechanics.
Neither has this conception been useless ; it has rendered us an
inestimable service, since it has contributed to make precise the
fundamental notion of the physical law.
I will explain myself; how did the ancients understand lawt
It was for them an internal harmony, static, so to say, and im-
mutable ; or else it was like a model that nature tried to imitate.
For us a law is something quite different; it is a constant rela-
tion between the phenomenon of to-day and that of to-morrow;
in a word, it is a differential equation.
Behold the ideal form of physical law; well, it is Newton's law
which first clothed it forth. If then one has acclimated this form
in physics, it is precisely by copying as far as possible this law of
Newton, that is by imitating celestial mechanics. This is, more-
over, the idea I have tried to bring out in Chapter VI.
The Physics of the Principles. — ^Nevertheless, a day arrived
when the conception of central forces no longer appeared suffi-
cient, and this is the first of those crises of which I just now
spoke.
What was done then? The attempt to penetrate into the
detail of the structure of the universe, to isolate the pieces of this
vast mechanism, to analyze one by one the forces which put them
in motion, was abandoned, and we were content to take as guides
certain general principles the express object of which is to spare
us this minute study. How so ? Suppose we have before us any
machine; the initial wheel work and the final wheel work alone
SOO THE VALUE OF SCIENCE
are visible, but the transmission, the intermediary machinery 1^
which the movement is communicated from one to the other, is
hidden in the interior and escapes our view; we do not know
whether the communication is made by gearing or by belts^ by
connecting-rods or by other contrivances. Do we say that it
is impossible for us to understand anything about this machine
so long as we are not permitted to take it to pieces t You know
well we do not, and that the principle of the conservation of
energy sufSces to determine for us the most interesting point
We easily ascertain that the final wheel turns ten times less
quickly than the initial wheel, since these two wheels are visible;
we are able thence to conclude that a couple applied to the one
will be balanced by a couple ten times greater applied to the
other. For that there is no need to penetrate the mechanism
of this equilibrium and to know how the forces compensate each
other in the interior of the machine; it sufSces to be assured
that this compensation can not fail to occur.
Well, in regard to the universe, the principle of the conserva-
tion of energy is able to render us the same service. The uni-
verse is also a machine, much more complicated than all those of
industry, of which almost all the parts are profoundly hidden
from us; but in observing the motion of those that we can see,
we are able, by the aid of this principle, to draw conclusions
which remain true whatever may be the details of the invisible
mechanism which animates them.
The principle of the conservation of energy, or Mayer's prin-
ciple, is certainly the most important, but it is not the only one ;
there are others from which we can derive the same advantage.
These are:
Camot's principle, or the principle of the degradation of
energy.
Newton's principle, or the principle of the equality of action
and reaction.
The principle of relativity, according to which the laws of
physical phenomena must be the same for a stationary observer
as for an observer carried along in a uniform motion of trans-
lation ; so that we have not and can not have any means of dis-
cerning whether or not we are carried along in such a motion.
THE HISTORY OF MATHEMATICAL PHYSICS 301
The principle of the conservation of mass, or Lavoisier's
principle.
I will add the principle of least action.
The application of these five or six general principles to the
different physical phenomena is sufficient for our learning of
them all that we could reasonably hope to know of them. The
most remarkable example of this new mathematical physics is,
beyond question, Maxwell's electromagnetic theory of light.
"SVe know nothing as to what the ether is, how its molecules are
disposed, whether they attract or repel each other ; but we know
that this medium transmits at the same time the optical pertur-
bations and the electrical perturbations ; we know that this trans-
mission must take place in conformity with the general princi-
ples of mechanics, and that suffices us for the establishment of
the equations of the electromagnetic field.
These principles are results of experiments boldly generalized ;
but they seem to derive from their very generality a high degree
of certainty. la fact, the more general they are, the more fre-
quent are the opportunities to check them, and the verifications
multiplying, taking the most varied, the most unexpected forms,
end by no longer leaving place for doubt.
Uiility of the Old Physics. — Such is the second phase of the
history of mathematical physics and we have not yet emerged
from it. Shall we say that the first has been uselessT that dur-
ing fifty years science went the wrong way, and that there is
nothing left but to forget so many accumulated efforts that a
vicious conception condemned in advance to failure! Not the
least in the world. Do you think the second phase could have
come into existence without the first T The hypothesis of central
forces contained all the principles ; it involved them as necessary
consequences; it involved both the conservation of energy and
that of masses, and the equality of action and reaction, and the
law of least action, which appeared, it is true, not as experimental
truths, but as theorems; the enunciation of which had at the
same time something more precise and less general than under
their present form.
It is the mathematical physics of our fathers which has famil-
iarized us little by little with these various principles; which has
802 THE VALUE OF SCIENCE
habituated us to recognize them under the different vestments in
which they disguise themselves. They have been compared with
the data of experience, it has been seen how it was necessary to
modify their enunciation to adapt them to these data; thereby
they have been extended and consolidated. Thus th^ came
to be regarded as experimental truths ; the conception of central
forces became then a useless support, or rather an embarrasi-
ment, since it made the principles partake of its hypothetical
character.
The frames then have not broken, because they are elastic ; but
they have enlarged; our fathers, who established them, did not
labor in vain, and we recognize in the science of to-day the gen-
eral traits of the sketch which they traced.
CHAPTER VIII
The Present Crisis of Mathematical Physics
The New Crisis. — ^Are we now about to enter upon a third
period t Are we on the eve of a second crisis t These principles
on which we have built all, are they about to crumble away in
their tumt This has been for some time a pertinent question.
When I speak thus, you no doubt think of radium, that grand
revolutionist of the present time, and in fact I shall come back
to it presently; but there is something else. It is not alone the
conservation of energy which is in question ; all the other princi-
ples are equally in danger, as we shall see in passing them succes-
sively in review.
Camoi's Principle. — ^Let us commence with the principle of
Carnot. This is the only one which does not present itself as an
immediate consequence of the hypothesis of central forces ; more
than that, it seems, if not to directly contradict that hypothesis,
at least not to be reconciled with it without a certain effort. If
physical phenomena were due exclusively to the movements of
atoms whose mutual attraction depended only on the distance, it
seems that all these phenomena should be reversible ; if all the in-
itial velocities were reversed, these atoms, always subjected to
the same forces, ought to go over their trajectories in the contrary
sense, just as the earth would describe in the retrograde sense
this same elliptic orbit which it describes in the direct sense, if
the initial conditions of its motion had been reversed. On this
account, if a physical phenomenon is possible, the inverse phe-
nomenon should be equally so, and one should be able to reascend
the course of time. Now, it is not so in nature, and this is pre-
cisely what the principle of Carnot teaches us; heat can pass
from the warm body to the cold body ; it is impossible afterward
to make it take the inverse route and to reestablish differences
of temperature which have been effaced. Motion can be wholly
dissipated and transformed into heat by friction; the contrary
transformation can never be made except partially.
303
304 THE VALUE OF SCIENCE
We have striven to reconcile this apparent contradiction. If
the world tends toward uniformity, this is not because its ulti-
mate parts, at first unlike, tend to become less and less different;
it is because, shifting at random, they end by blending. For an
eye which should distinguish all the elements, the variety would
remain always as great; each grain of this dust preserves its
originality and does not model itself on its neighbors; but as the
blend becomes more and more intimate, our gross senses perceive
only the uniformity. This is why for example, temperatures
tend to a level, without the possibility of going backwards.
A drop of wine falls into a glass of water; whatever may be
the law of the internal motion of the liquid, we shall soon see it
colored of a uniform rosy tint, and however much from this
moment one may shake it afterwards, the wine and the water
do not seem capable of again separating. Here we have the
type of the irreversible physical phenomenon : to hide a grain of
barley in a heap of wheat, this is easy; afterwards to find it
again and get it out, this is practically impossible. All this
Maxwell and Boltzmann have explained; but the one who has
seen it most clearly, in a book too little read because it is a little
diflBeult to read, is Gibbs, in his ' Elementary Principles of Statis-
tical Mechanics.'
For those who take this point of view, Camot's principle is
only an imperfect principle, a sort of concession to the infirmity
of our senses ; it is because our eyes are too gross that we do not
distinguish the elements of the blend ; it is because our hands are
too gross that we can not force them to separate ; the imaginary
demon of Maxwell, who is able to sort the molecules one by one,
could well constrain the world to return backward. Can it re-
turn of itself? That is not impossible; that is only infinitely
improbable. The chances are that we should wait a long time
for the concourse of circumstances which would permit a retro-
gradation ; but sooner or later they will occur, after years whose
number it would take millions of figures to write. These reser-
vations, however, all remained theoretic ; they were not very dis-
quieting, and Carnot's principle retained all its practical value.
But here the scene changes. The biologist, armed with his micro-
scope, long ago noticed in his preparations irregular movements
THE CBISIS OF MATHEMATICAL PHYSICS
of little particles la suspension ; this is the Brownian movement.
He first thought this was a vital phenomenon, but soon he saw
that the inanimate bodies danced with no less ardor than the
others; then he turned the matter over to the physicists. Un-
happily, the physicists remained long uninterested in this ques-
tion; one concentrates the tight to illuminate the microscopic
preparation, thought they; with light goes heat; thence inequal-
ities of temperature and in the liquid interior currents which
produce the movements referred to. It occurred to M. Oouy to
look more closely, and he saw, or thought he saw, that this ex-
planation is untenable, that the movements become brisker as the
particles are smaller, but that they are not influenced by the
mode of illumination. If then these movements never cease, or
rather are reborn without cease, without borrowii^ anything
from an external source of energy, what ought we to believe t
To be sure, we should not on this account renounce our belief
in the conservation of energy, but we see under our eyes now
motion transformed into heat by friction, now inversely heat
changed into motion, and that without loss since the movement
lasts forever. This is the contrary of Carnot's principle. If
this be so, to see the world return backward, we no longer have
need of the infinitely keen eye of Maxwell's demon; our micro-
scope sufBees, Bodies too large, those, for example, which are
a tenth of a millimeter, are hit from all sides by moving atoma,
but they do not budge, because these shocks are very numerous
and the law of chance makes them compensate each other; but
the smaller particles receive too few shocks for this compensation
to take place with certainty and are incessantly knocked about.
And behold already one of our principles in peril.
The Principle of Relafivity. — Let us pass to the principle of
relativity: this not only is confirmed by daily experience, not
only is it a necessary consequence of the hypothesis of central
forces, hut it is irresistibly imposed upon our good sense, and
yet it also is assailed. Consider two electrified bodies; though
they seem to us at rest, they are both carried along by the mo-
tion of the earth; an electric eharge in motion, Rowland has
taught us, ia equivalent to a current; these two charged bodies
I ate, therefore, equivalent to two parallel currents of the same
306 THE VALUE OF SCIENCE
sense and these two currents should attract each other. In meas-
uring this attraction, we shall measure the velocity of the earth;
not its velocity in relation to the sun or the fixed stars, but its
absolute velocity.
I well know what will be said: It is not its absolute velocity
that is measured, it is its velocity in relation to the ether. How
unsatisfactory that is I Is it not evident that from the principle
so understood we could no longer infer anything f It could no
longer tell us anything just because it would no longer fear any
contradiction. If we succeed in measuring anything, we shall
always be free to say that this is not the absolute velocity, and if
it is not the velocity in relation to the ether, it might always be
the velocity in relation to some new unknown fluid with which
we might fill space.
Indeed, experiment has taken upon itself to ruin this interpre-
tation of the principle of relativity; all attempts to measure the
velocity of the earth in relation to the ether have led to nega-
tive results. This time experimental physics has been more
faithful to the principle than mathematical ph3rsics; the theorists,
to put in accord their other general views, would not have spared
it; but experiment has been stubborn in confirming it. The
means have been varied; finally Michelson pushed precision to
its last limits; nothing came of it. It is precisely to explain
this obstinacy that the mathematicians are forced to-day to em-
ploy all their ingenuity.
Their task was not easy, and if Lorentz has got through it, it is
only by accumulating hypotheses.
The most ingenious idea was that of local time. Imagine twp
observers who wish to adjust their timepieces by optical signals;
they exchange signals, but as they know that the transmission
of light is not instantaneous, they are careful to cross them.
When station B perceives the signal from station A, its clock
should not mark the same hour as that of station A at the
moment of sending the signal, but this hour augmented by a
constant representing the duration of the transmission. Sup-
pose, for example, that station A sends its signal when its clock
marks the hour 0, and that station B perceives it when its clock
marks the hour t. The clocks are adjusted if the slowness equal
THE CRISIS OF MATHEMATICAL PHTSICS
307
to 1 represeDts the duration of the transmission, and to verify
it, station B sends in its turn a signal when its clock marks O;
then station A should perceive it when its clock marks t. The
timepieces are then adjusted.
And in fact they mark the same hour at the same pbTBical
instant, but on the one condition, that the two stations are fixed.
Otherwise the doration of the transmission will not be the same
in the two senses, since the station A, for example, moves for^
ward to meet the optical perturbation emanating from B, whereas
the station B Sees before the perturbation emanating from A.
The watches adjusted in that way will not mark, therefore, the
true time ; they will mark what may be called the local lime, so
that one of them will be alow of the other. It matters little, since
we have do means of perceiving it. All the phenomena which
happen at A, for example, will be late, but all will be equally
80, and the observer will not perceive it, since his watch is slow;
so, as the principle of relativity requires, he will have no means
of knowing whether he is at rest or in absolute motion.
Unhappily, that does not sufSce, and complementary hypoth-
eses are necessary; it is necessary to admit that bodies in mo-
tion undergo a uniform contraction in the sense of the motion.
One of the diameters of the earth, for example, is shrunk by
one two-hundred-millionth in consequence of our planet's motion,
while the other diameter retains its normal length. Thus the last
little differences are compensated. And then, there is still the
hypothesis about forces. Forces, whatever be their origin, grav-
ity as well as elasticity, would be reduced in a certain propor-
tion in a world animated by a uniform translation; or, rather,
this would happen for the components perpendicular to the
translation ; the components parallel would not change. Re-
sume, then, our example of two eleetri6ed bodies; these bodies
repel each other, but at the same time if all is carried along in a
uniform translation, they are equivalent to two parallel currents
of the same sense which attract each other. This eleetrodynamio
attraction diminishes, therefore, the electrostatic repulsion, and
the total repulsion is feebler than if the two bodies were at rest
Bat since to measure this repulsion we must balance it by another
ree, and all these other forces are reduced in the same pro-
308 THE VALUE OF SCIENCE
portion, we perceive nothing. Thus all seems arranged, but are
all the doubts dissipated? What would happen if one could
communicate by non-luminous signals whose velocity of propa-
gation differed from that of light? If, after having adjusted
the watches by the optical procedure, we wished to verify the
adjustment by the aid of these new signals, we should observe
discrepancies which would render evident the common transla-
tion of the two stations. And are such signals inconceivable, if
we admit with Laplace that universal gravitation is transmitted
a million times more rapidly than light?
Thus, the principle of relativity has been valiantly defended
in these latter times, but the very energy of the defense proves
how serious was the attack.
Newton* 8 Principle. — ^Let us speak now of the principle of
Newton, on the equality of action and reaction. This is inti-
mately bound up with the preceding, and it seems indeed that the
fall of the one would involve that of the other. Thus we must
not be astonished to find here the same difficulties.
Electrical phenomena, according to the theory of Lorentz, are
due to the displacements of little charged particles, called elec-
trons, immersed in the medium we call ether. The movements
of these electrons produce perturbations in the neighboring ether;
these perturbations propagate themselves in every direction with
the velocity of light, and in turn other electrons, originally at
rest, are made to vibrate when the perturbation reaches the parts
of the ether which touch them. The electrons, therefore, act on
one another, but this action is not direct, it is accomplished
through the ether as intermediary. Under these conditions can
there be compensation between action and reaction, at least for
an observer who should take account only of the movements
of matter, that is, of the electrons, and who should be ignorant
of those of the ether that he could not see? Evidently not.
Even if the compensation should be exact, it could not be simul-
taneous. The perturbation is propagated with a finite velocity;
it, therefore, reaches the second electron only when the first has
long ago entered upon its rest. This second electron, therefore,
will undergo, after a delay, the action of the first, but will cer-
tainly not at that moment react upon it, since around this first
electron nothing any longer budges.
The analysis of the facta penoits us to be still more precise.
Imagine, for example, a Hertzian oscillator, like those used in
wireless telegraphy ; it sends out energy in every direction ; but
we can provide it with a parabolic mirror, as Hertz did with bis
smallest oscillators, so as to send all the energy produced in a
single direction. "What happens then according to the theoryt
The apparatus recoils, as if it were a cannon and the projected
energy a ball; and that is contrary to the principle of Newton,
since our projectile here has no mass, it is not matter, it is energy.
The ease is still the same, moreover, with a beacon light provided
with a reflector, since light is nothing but a perturbation of the
eleotromagnetic field. This beacon light should recoil as if the
light it sends out were a projectile. What is the force that
should produce this recoil T It is what is called the Alaxwell-
Bartholi pressure. It is very minute, and it has been difficult
to put it in evidence even with the most sensitive radiometers;
but it suffices that it exists.
If all the energy issuing from our oscillator falls on a receiver,
this will act as if it had received a mechanical shock, which will
represent in a sense the compensation of the oscillator's recoil;
the reaction will be equal to the action, but it will not be simul-
taneous; the receiver will move on, but not at the moment when
the oscillator recoils. If tlie energy propagates itself indefihitely
without encountering a receiver, the compensation will never
occur,
SbfiU we say that the space which separates the oscillator from
the receiver and which the perturbation must pass over in going
from the one to the other is not void, that it is full not only of
ether, but of air, or even in the interplanetary spaces of some
fluid subtile but still ponderable ; that this matter undergoes the
shock like the receiver at the moment when the energy reaches
it, and recoils in its turn when the perturbation quits itt That
would save NchIou's principle, but that is not true. If energy
in its diffusion remained always attached to some material sub-
slratam, then matter in motion would carry along light with it,
and Fizeau has demonstrated that it does nothing of the sort,
at least for air. Michelson and Moriey have since confirmed
this. It might be supposed also that the movements of matter
310 THE VALUE OF SCIENCE
proper are exactly compensated by those of the ether; but that
would lead us to the same reflections as before now. The prin-
ciple so understood will explain everything, since, whatever
might be the visible movements, we always could imagine hypo-
thetical movements which compensate them. But if it is able
to explain everything, this is because it does not enable us to
foresee anything; it does not enable us to decide between the
different possible hypotheses, since it explains everything be-
forehand. It therefore becomes useless.
And then the suppositions that it would be necessary to make
on the movements of the ether are not very satisfactory. If the
electric charges double, it would be natural to imagine that the
velocities of the diverse atoms of ether double also ; but, for the
compensation, it would be necessary that the mean velocity of
the ether quadruple.
This is why I have long thought that these consequences of
theory, contrary to Newton's principle, would end some day by
being abandoned, and yet the recent experiments on the move-
ments of the electrons issuing from radium seem rather to con-
firm them.
Lavoisier's Principle, — I arrive at the principle of Lavoisier on
the conservation of mass. Certainly, this is one not to be
touched without unsettling all mechanics. And now certain per-
sons think that it seems true to us only because in mechanics
merely moderate velocities are considered, but that it would cease
to be true for bodies animated by velocities comparable to that
of light. Now these velocities are believed at present to have
been realized; the cathode rays and those of radium may be
formed of very minute particles or of electrons which are dis-
placed with velocities smaller no doubt than that of light, but
which might be its one tenth or one third.
These rays can be deflected, whether by an electric fleld, or
by a magnetic fleld, and we are able, by comparing these deflec-
tions, to measure at the same time the velocity of the electrons
and their mass (or rather the relation of their mass to their
charge). But when it was seen that these velocities approached
that of light, it was decided that a correction was necessary.
These molecules, being electrified, can not be displaced without
THE CRISIS OF MATHEMATICAL PHYSICS
3U
agitating the ether ; to put them in motion it is necessary to over-
come a double inertia, that of the molecule itself and that of the
ether. The total or apparent mass that one measures is com-
posed, therefore, of two parts: the real or mechanical mass of
the molecule and the electrodynamie mass representing the
inertia of the ether.
The calculations of Abraham and the experiments of Kanf-
mann have then shown that the mechanical mass, properly so
called, is null, and that the mass of the electrons, or, at least, of
the negative electrons, is of exclusively electrodynamic origin.
This is what forces us to change the definition of mass; we can
not any longer distinguish mechanical mass and electrodynamic
mass, since then the first would vanish; there is no mass other
than electrodynamic inertia. But in this case the mass can no
longer be constant; it augments with the velocity, and it even
depends on the direction, and a body animated by a notable
velocity will not oppose the same inertia to the forces which tend
to deflect it from its route, as to those which tend to accelerate
or to retard its progress.
There is still a resource; the ultimate elements of bodies are
electrons, some charged negatively, the others charged positively.
The negative electrons have no mass, this is understood; but the
positive electrons, from the little we know of them, seem much
greater. Perhaps they have, besides their electrodynamic mass,
a true mechanical mass. The real mass of a body would, then,
be the sum of the mechanical masses of its positive electrons, the
negative electrons not counting; mass so defined might still be
constant.
Alas! this resource also evades us. Recall what we have said
of the principle of relativity and of the efforts made to save it.
And it is not merely a principle which it is a question of saving,
it is the indubitable results of the experiments of Michelson.
Well, as was above seen, Lorentz, to account for these results,
was obliged to suppose that all forces, whatever their origin,
were reduced in the same proportion in a medium animated by a
uniform translation ; this is not suflScient ; it is not enough that
this take place for the real forces, it must also be the same for
the forces of inertia ; it is therefore necessary, he says, that the
312 THE VALUE OF SCIENCE
masses of all the particles he influenced by a translation to the
same degree as the electromagnetic masses of the electrons.
So the mechanical masses must vary in accordance with the
same laws as the electrodynamic masses ; th^ can not, therefore,
be constant.
Need I point out that the fall of Lavoisier's principle involves
that of Newton's ? This latter signifies that the center of gravity
of an isolated system moves in a straight line ; but if there is no
longer a constant mass, there is no longer a center of gravity,
we no longer know even what this is. This is why I said above
that the experiments on the cathode rays appeared to justify
the doubts of Lorentz concerning Newton's principle.
From all these results, if they were confirmed, would arise an
entirely new mechanics, which would be, above all, characterized
by this fact, that no velocity could surpass that of light,^ any
more than any temperature can fall below absolute zero.
No more for an observer, carried along himself in a transla-
tion he does not suspect, could any apparent velocity surpass
that of light ; and this would be then a contradiction, if we did
not recall that this observer would not use the same clocks as a
fixed observer, but, indeed, clocks marking * local time.'
Here we are then facing a question I content myself with stat-
ing. If there is no longer any mass, what becomes of Newton's
law? Mass has two aspects : it is at the same time a coefiScient of
inertia and an attracting mass entering as factor into Newtonian
attraction. If the coefficient of inertia is not constant, can the
attracting mass be ? That is the question.
Mayer's Principle, — At least, the principle of the conservation
of energy yet remained to us, and this seemed more solid. Shall
I recall to you how it was in its turn thrown into discredit?
This event has made more noise than the preceding, and it is in
all the memoirs. From the first works of Becquerel, and, above
all, when the Curies had discovered radium, it was seen that
every radioactive body was an inexhaustible source of radiation.
Its activity seemed to subsist without alteration throughout the
months and the years. This was in itself a strain on the prin-
1 Because bodies would oppose an increasing inertia to the causes which
would tend to accelerate their motion ; and this inertia would become infinite
when one approached the velocity of light.
THE CBI8I8 OF MATHEMATICAL PHYSICS 313
ciples; these radiations were in fact energy, and from the same
morsel of radium this issued and forever issued. But these
quantities of energy were too slight to be measured ; at least that
was the belief and we were not much disquieted.
The scene changed when Curie bethought himself to put ra-
dium in a calorimeter ; it was then seen that the quantity of heat
incessantly created was very notable.
The explanations proposed were numerous; but in such case
we can not say, the more the better. In so far as no one of them
has prevailed over the others, we can not be sure there is a good
one among them. Since some time, however, one of these ex-
planations seems to be getting the upper hand and we may rea-
sonably hope that we hold the key to the mystery.
Sir W. Bamsay has striven to show that radium is in process
of transformation, that it contains a store of energy enormous
but not inexhaustible. The transformation of radium then
would produce a million times more heat than all known trans-
formations ; radium would wear itself out in 1,250 years ; this is
quite short, and you see that we are at least certain to have this
point settled some hundreds of years from now. While wait-
ing, our doubts remain.
CHAPTER IX
The Future of Mathematical Physics
The Principles and Experiment. — ^In the midst of so much
ruin, what remains standing? The principle of least action is
hitherto intact, and Larmor appears to believe that it will long
survive the others; in reality, it is still more vague and more
general.
In presence of this general collapse of the principles, what at-
titude will mathematical physics take? And first, before too
much excitement, it is proper to ask if all that is really true.
All these derogations to the principles are encountered only
among infinitesimals; the microscope is necessary to see the
Brownian movement; electrons are very light; radium is very
rare, and one never has more than some milligrams of it at a
time. And, then, it may be asked whether, besides the infinites-
imal seen, there was not another infinitesimal unseen counterpoise
to the first.
So there is an interlocutory question, and, as it seems, only
experiment can solve it. We shall, therefore, only have to hand
over the matter to the experimenters, and, while waiting for them
to finally decide the debate, not to preoccupy ourselves with these
disquieting problems, and to tranquilly continue our work as if
the principles were still uncontested. Certes, we have much to
do without leaving the domain where they may be applied in all
security; we have enough to employ our activity during this
period of doubts.
The Bole of the Analy[st. — ^And as to these doubts, is it indeed
true that we can do nothing to disembarrass science of themf
It must indeed be said, it is not alone experimental physics that
has given birth to them ; mathematical physics has well contrib-
uted. It is the experimenters who have seen radium throw out
energy, but it is the theorists who have put in evidence all the
diflSculties raised by the propagation of light across a medium in
motion; but for these it is probable we should not have become
314
THE FUTURE OF MATHEMATICAL PHYSICS
315
conscious of tliem, "Well, then, if they have done their best to
put us into this embarrassment, it is proper also that they help us
to get out of it.
They must subject to critical examination all these new viewa
I have just outlined before you, and abandon the principles only
after Laving made a loyal effort to save them. What can they
do in this sense I That is what I will try to explain.
It is a question before all of endeavoring to obtain a more
satisfactory- theory of the electrodynamics of bodies in motion.
It is there especially, as I have sufficiently shown above, that
difficulties accumulate. It is useless to heap up hypotheses,
we can not satisfy all the principles at once; so far, one has
succeeded in safeguarding some only on condition of sacrificing
the others; hut all hope of obtaining better results ia not yet
lost. Let us take, then, the theory of Lorentz, turn it in all
senses, modify it little by little, and perhaps everything will
arrange itself.
Thus in place of supposing that bodies in motion undergo a
contraction in the sense of the motion, and that this contraction
is the same whatever be the nature of these bodies and the forces
to which they are otherwise subjected, could we not make a more
simple and natural hypothesis f We might imagine, for example,
that it ia the ether which is modified when it is in relative motion
in reference to the material medium which penetrates it, that,
when it is thus modified, it no longer transmits perturbations
with the same velocity in every direction. It might tranamit
more rapidly those which are propagated parallel to the motion
of the medium, whether in the same sense or in the opposite sense,
and less rapidly those which are propagated perpendicularly.
The wave surfaces would no longer be spheres, but ellipsoids,
and we could dispense with that extraordinary contraction of all
bodies.
I cite this only as an example, since the modifications that
might be essayed would be evidently sxisceptible of infinite varia-
tion.
Aberralion and Astronomy. — It ia possible also that astronomy
may some day furnish us data on this point; she it was in the
main who raised the question in making us acquainted with the
316 TEE VALUE OF SCIENCE
phenomenon of the aberration of light. If we make craddy the
theory of aberration, we reach a very curious result. The ap-
parent positions of the stars differ from their real positions be-
cause of the earth's motion, and as this motion is variable, these
apparent positions vary. The real position we can not ascertain,
but we can observe the variations of the apparent position. The
observations of the aberration show us, therefore, not the earth's
motion, but the variations of this motion; they can not, there-
fore, give us information about the absolute motion of the earth.
At least this is true in first approximation, but the case would
be no longer the same if we could appreciate the thousandths of
a second. Then it would be seen that the amplitude of the oscil-
lation depends not alone on the variation of the motion, a varia-
tion which is well known, since it is the motion of our globe on
its elliptic orbit, but on the mean value of this motion, so that
the constant of aberration would not be quite the same for all the
stars, and the differences would tell us the absolute motion of the
earth in space.
This, then, would be, under another form, the ruin of the prin-
ciple of relativity. We are far, it is true, from appreciating the
thousandth of a second, but, after all, say some, the earth's total
absolute velocity is perhaps much greater than its relative veloc-
ity with respect to the sun. If, for example, it were 300 kilo-
meters per second in place of 30, this would suffice to make the
phenomenon observable.
I believe that in reasoning thus one admits a too simple theory
of aberration. Michelson has shown us, I have told you, that the
physical procedures are powerless to put in evidence absolute
motion ; I am persuaded that the same will be true of the astro-
nomic procedures, however far precision be carried.
However that may be, the data astronomy will furnish us in
this regard will some day be precious to the physicist. Mean-
while, I believe that the theorists, recalling the experience of
Michelson, may anticipate a negative result, and that they would
accomplish a useful work in constructing a theory of aberration
which would explain this in advance.
Electrons and Spectra, — This dynamics of electrons can be ap-
proached from many sides, but among the ways leading thither is
THE FUTURE OF MATHEMATICAL PHYSICS
317
one which has been somewhKt neglected, and yet this is one of
those which promise as the most surprises. It is movements of
electrons which produce the lines of the emiGsion spectra ; this is
proved by the Zeeman effect ; in an incandescent body what vi-
brates is sensitive to the magnet, therefore electrified. This is a
very important first point, but no one has gone farther. Why
are the lines of the spectrum distributed in accordance with a
regular law T These laws have been studied hy the experimenters
in their least details; they are very precise and comparatively
simple. A first study of these distributions recalls the harmon-
ics encountered in acoustics; but the difference is great. Not
only are the numbers of vibrations not the successive multiples
of a single number, but we do not even find anything analogous
to the roots of those transcendental equations to which we are
led by 80 many problems of matliematical physics: that of the
vibrations of an elastic body of any form, that of the Hertzian
oscillations in a generator of any form, the problem of Fourier
for the cooling of a solid body.
The laws are simpler, but they are of wholly other nature, and
to cite only one of these differences, for the harmonics of high
order, the number of vibrations tends toward a finite limit,
instead of increasing indefinitely.
That has not yet been accounted for, and I believe that there
we have one of the most important secrets of nature. A Japa-
nese physicist, M. Nagaoka, has recently proposed an explana-
tion; according to him, atoms are composed of a large podtive
electron surrounded by a ring formed of a great number of very
small negative electrons. Such is the planet Saturn with ita
rings. This is a very interesting attempt, but not yet wholly
satisfactory; this attempt should be renewed. We will pene-
trate, 80 to speak, into the inmost recess of matter. And from
the particular point of view which we to-day occupy, when we
know why the vibrations of incandescent bodies differ thus from
ordinary elastic vibrations, why the electrons do not behave like
the matter which is familiar to us, we shall better comprehend the
dynamics of electrons nnd it will be perhaps more easy for us
to reconcile it with the principles.
Conventions Preceding Experiment. — Suppose, now, that all
818 TBE VALVE OF SCIENCE
these efforts fail, and, after all, I do not believe they will, what
most be done? Will it be necessary to seek to mend the broken
principles by giving what we French call a coup de paucef That
evidently is always possible, and I retract nothing of what I have
said above.
Have yon not written, you might say if you wished to seek a
quarrel with me — ^have you not written that the principles,
though of experimental origin, are now unassailable by experi-
ment because they have become conventions f And now you
have just told us that the most recent conquests of experiment
put these principles in danger.
Well, formerly I was right and to-day I am not wrong. For-
merly I was right, and what is now happening is a new proof of
it. Take, for example, the calorimetric experiment of Curie on
radium. Is it possible to reconcile it with the principle of the
conservation of energy ? This has been attempted in many ways.
But there is among them one I should like you to notice; this is
not the explanation which tends to-day to prevail, but it is one
of those which have been proposed. It has been conjectured
that radium was only an intermediary, that it only stored radia-
tions of unknown nature which flashed through space in every
direction, traversing all bodies, save radium, without being al-
tered by this passage and without exercising any action upon
them. Radium alone took from them a little of their energy and
afterward gave it out to us in various forms.
What an advantageous explanation, and how convenient!
First, it is unverifiable and thus irrefutable. Then again it will
serve to account for any derogation whatever to Mayer's prin-
ciple ; it answers in advance not only the objection of Curie, but
all the objections that future experimenters might accumulate.
This new and unknown energy would serve for everything.
This is just what I said, and therewith we are shown that our
principle is unassailable by experiment.
But then, what have we gained by this stroke? The principle
is intact, but thenceforth of what use is it ? It enabled us to fore-
see that in such or such circumstance we could count on such a
total quantity of energy ; it limited us ; but now that this indefi-
nite provision of new energy is placed at our disposal, we are no
TBE FUTUBE OF MATHEMATICAL PHT8IC8 819
longer limited by anything; and, as I have written in 'Science
and Hypothesis,' if a principle ceases to be fecund, experiment
without contradicting it directly will nevertheless have con-
demned it.
Future Mathematical Physics. — This, therefore, is not what
would have to be done; it would be necessary to rebuild anew.
If we were reduced to this necessity, we could moreover console
ourselves. It would not be necessary thence to conclude that
science can weave only a Penelope's web, that it can raise only
ephemeral structures, which it is soon forced to demolish from
top to bottom with its own hands.
As I have said, we have already passed through a like crisis.
I have shown you that in the second mathematical physics, that
of the principles, we find traces of the first, that of central
forces ; it will be just the same if we must know a third. Just so
with the animal that exuviates, that breaks its too narrow cara-
pace and makes itself a fresh one; under the new envelope one
will recognize the essential traits of the organism which have
persisted.
We can not foresee in what way we are about to expand ; per-
haps it is the kinetic theory of gases which is about to undergo
development and serve as model to the others. Then the facts
which first appeared to us as simple thereafter would be merely
resultants of a very great number of elementary facts which only
the laws of chance would make cooperate for a common end.
Physical law would then assume an entirely new aspect ; it would
no longer be solely a differential equation, it would take the char-
acter of a statistical law.
Perhaps, too, we shall have to construct an entirely new me-
chanics that we only succeed in catching a glimpse of, where,
inertia increasing with the velocity, the velocity of light would
become an impassable limit. The ordinary mechanics, more
simple, would remain a first approximation, since it would be
true for velocities not too great, so that the old dynamics would
still be found under the new. We should not have to regret hav-
ing believed in the principles, and even, since velocities too great
for the old formulas would always be only exceptional, the sur-
est way in practise would be still to act as if we continued to
820 TRE VALUE OF SCIENCE
believe in them. They are so useful, it would be neceflsary to
keep a place for them. To determine to exclude them altogether
would be to deprive oneself of a precious weapon. I hasten to
say in conclusion that we are not yet there, and as yet nothing
proves that the principles will not come forth from out the fray
victorious and intact.^
1 These considerations on mathematical physics are borrowed from wj
St Louis address.
PART in
THE OBJECTIVE VALUE
OF SCIENCE
CHAPTER X
Is Science AsTiFicuLf
1. The Philosophy of M. LeBoy
There are many reasons for being sceptics; should we push
this scepticism to the very end or stop on the way? To go to the
end is the most tempting solution, the easiest, and that which
many have adopted, despairing of saving anything from the
shipwreck.
Among the writings inspired by this tendency it is proper to
place in the first rank those of M. LeBoy. This thinker is not
only a philosopher and a writer of the greatest merit, but he has
acquired a deep knowledge of the exact and physical sciences,
and even has shown rare powers of mathematical invention. Let
us recapitulate in a few words his doctrine, which has given rise
to numerous discussions.
Science consists only of conventions, and to this circumstance
solely does it owe its apparent certitude ; the facts of science and,
a fortiori, its laws are the artificial work of the scientist; science
therefore can teach us nothing of the truth; it can only serve
us as rule of action.
Here we recognize the philosophic theory known under the
name of nominalism ; all is not false in this theory ; its legitimate
domain must be left it, but out of this it should not be allowed
to go.
This is not all; M. LeRoy's doctrine is not only nominalistic ;
it has besides another characteristic which it doubtless owes to M.
Bergson, it is anti-intellectualistic. According to M. LeRoy, the
22 321
322 THE VALUE OF SCIENCE
intellect deforms all its touches, and that is still more true of its
necessary instrument 'discourse/ There is reality only in our
fugitive and changing impressions, and even this reality, when
touched, vanishes.
And yet M. LeBoy is not a sceptic ; if he regards the intellect
as incurably powerless, it is only to give more scope to other
sources of knowledge, to the heart, for instance, to sentiment, to
instinct or to faith.
However great my esteem for M. LeBoy 's talent, whatever the
ingenuity of this thesis, I can not wholly accept it. Certes, I
am in accord on many points with M. LeBoy, and he has even
cited, in support of his view, various passages of my writings
which I am by no means disposed to reject. I think myself only
the more bound to explain why I can not go with him all the way.
M. LeBoy often complains of being accused of scepticism.
He could not help being, though this accusation is probably un-
just. Are not appearances against himf Nominalist in doc-
trine, but realist at heart, he seems to escape absolute nominaUsm
only by a desperate act of faith.
The fact is that anti-intellectualistic philosophy in rejecting
analysis and 'discourse,' just by that condemns itself to being
intransmissible ; it is a philosophy essentially internal, or, at the
very least, only its negations can be transmitted; what wonder
then that for an external observer it takes the shape of scepticism t
Therein lies the weak point of this philosophy ; if it strives to
remain faithful to itself, its energy is spent in a negation and a
cry of enthusiasm. Each author may repeat this negation and
this cry, may vary their form, but without adding anything.
And yet, would it not be more logical in remaining silent t
See, you have written long articles; for that, it was necessary
to use words. And therein have you not been much more 'dis-
cursive' and consequently much farther from life and truth than
the animal who simply lives without philosophizing t Would
not this animal be the true philosopher?
However, because no painter has made a perfect portrait,
should we conclude that the best painting is not to paint ? When
a zoologist dissects an animal, certainly he 'alters it.' Yes, in
dissecting it, he condemns himself to never know all of it ; but in
18 SCIENCE ABTIFICIALt 828
not dissecting it, he would condemn himself to never know any-
thing of it and consequently to never see anything of it.
Certes^ in man are other forces besides his intellect; no one
has ever been mad enough to deny that. The first comer makes
these blind forces act or lets them act; the philosopher must
speak of them ; to speak of them, he must know of them the little
that can be known, he should therefore see them act. Howf
With what eyes, if not with his intellect? Heart, instinct, may
guide it, but not render it useless ; they may direct the look, but
not replace the eye. It may be granted that the heart is the
workman, and the intellect only the instrument. Yet is it an
instrument not to be done without, if not for action, at least for
philosophizing f Therefore a philosopher really anti-intellectual«
istic is impossible. Perhaps we shall have to declare for the
supremacy of action; always it is our intellect which will thus
conclude ; in allowing precedence to action it will thus retain the
superiority of the thinking reed. This also is a supremacy not
to be disdained.
Pardon these brief reflections and pardon also their brevity,
scarcely skimming the question. The process of intellectualism
is not the subject I wish to treat : I wish to speak of science, and
about it there is no doubt ; by definition, so to speak, it will be
intellectualistic or it will not be at all. Precisely the question is,
whether it will be.
2. Science, Rule of Action
For M. LeRoy, science is only a rule of action. We are pow-
erless to know anything and yet we are launched, we must act,
and at all hazards we have established rules. It is the aggregate
of these rules that is called science.
It is thus that men, desirous of diversion, have instituted rules
of play, like those of tric-trac for instance, which, better than
science itself, could rely upon the proof by universal consent.
It is thus likewise that, unable to choose, but forced to choose, we
toss up a coin, head or tail to win.
The rule of tric-trac is indeed a rule of action like science,
but does any one think the comparison just and not see
the difference? The rules of the game are arbitrary conven-
824 THE VALUE OF SCIENCE
tions and the contrary convention might have been adopted,
wJUch would have been none the less good. On the contrary,
science is a rule of action which is successful, generally at least,
and I add, while the contrary rule would not have succeeded.
If I say, to make hydrogen cause an acid to act on zinc, I for-
mulate a rule which succeeds; I could have said, make distilled
water act on gold ; that also would have been a rule, only it would
not have succeeded. If, therefore, scientific ^recipes' have a
value, as rule of action, it is because we know they succeed, gener-
ally at least. But to know this is to know something and th^
why tell us we can know nothing f
Science foresees, and it is because it foresees that it can be
useful and serve as rule of action. I well know that its pre-
visions are often contradicted by the event; that shows that
science is imperfect, and if I add that it will always remain so,
I am certain that this is a prevision which, at least, will nev^
be contradicted. Always the scientist is less often mistaken
than a prophet who should predict at random. Besides the
progress though slow is continuous, so that scientists, though
more and more bold, are less and less misled. This is little, but
it is enough.
I well know that M. LeRoy has somewhere said that science
was mistaken of tener than one thought, that comets sometimes
played tricks on astronomers, that scientists, who apparently are
men, did not willingly speak of their failures, and that, if they
should speak of them, they would have to count more defeats
than victories.
That day, M. LeRoy evidently overreached himself. If science
did not succeed, it could not serve as rule of action; whence
would it get its value? Because it is * lived,' that is, because we
love it and believe in it? The alchemists had recipes for making
gold, they loved them and had faith in them, and yet our recipes
are the good ones, although our faith be less lively, because they
succeed.
There is no escape from this dilemma ; either science does not
enable us to foresee, and then it is valueless as rule of action ; or
else it enables us to foresee, in a fashion more or less imperfect,
and then it is not without value as means of knowledge.
It should not even be said that action is the goal of science ;
shoald we condemn studies of the star Sinus, under pretext that
we shall probably never exercise any influence on tliat starl To
my eyes, on the contrary, it is the knowledge which is the end,
and the action which is the means. If I felicitate myself on the
industrial development, it is not alone because it furnishes a
facile argument to the advocates of science ; it is above all because
it gives to the scientist faith in himself and also because it offers
him an immense field of experience where he clashes againat
forces too colossal to be tampered with. Without this ballast,
who knows whether he would not quit solid ground, seduced by
the mirage of some scholastic novelty, or whether he would not
despair, believing he had fashioned only a dream!
3. The Crude Fact arid the Scientific Fact
What was most paradoxical in M. LeRoy's thesis was that
affirmation that the scientist creates th-e fact; this was at the
same time its essential point and it is one of those which have
been most discussed.
Perhaps, says he (I well believe that this was a concession),
it is not the scientist that creates the fact in the rough ; it is at
least he who creates the scientific fact.
This distinction between the fact in the rough and the scien-
tific fact does not by itself appear to me illegitimate. But I
complain first that the boundary has not been traced either
exactly or precisely ; and then that the author has seemed to sup-
pose that the crude fact, not being scientific, is outside of science.
Pinally, I can not admit that the scientist creates without re-
straint the scientific fact, since it is the crude fact which impoaea
it upon him.
The examples given by M. LeRoy have greatly astonished me.
The first is taken from the notion of atom. The atom chosen as
example of fact! I avow that this choice has so disconcerted
me that I prefer to say nothing about it. I have evidently mia-
underatood the author's thought and I could not fruitfully dis-
cuss it.
The second case taken as example is that of an eclipse where
tlie erode phenomenon is a play of light and shadow, but where
826 THE VALUE OF SCIENCE
the astronomer can not intervene without introducing two foreign
elements, to wit, a clock and Newton's law.
Finally, M. heRoy cites the rotation of the earth; it has been
answered : but this is not a fact, and he has replied : it was one
for Galileo, who afSrmed it, as for the inquisitor, who denied it
It always remains that this is not a fact in the same sense as
those just spoken of and that to give them the same name is to
expose one's self to many confusions.
Here then are four degrees :
1°. It grows dark, says the clown.
2°. The eclipse happened at nine o'clock, says the astronomer.
3°. The eclipse happened at the time deducible from the tables
constructed according to Newton's law, says he again.
4^. That results from the earth's turning around the sun, says
Qalileo finally.
Where then is the boundary between the fact in the rough
and the scientific factf To read M. LeBoy one would believe
that it is between the first and the second stage, but who does not
see that there is a greater distance from the second to the third,
and still more from the third to the fourth.
Allow me to cite two examples which perhaps will enlighten us
a little.
I observe the deviation of a galvanometer by the aid of a mov-
able mirror which projects a luminous image or spot on a divided
scale. The crude fact is this : I see the spot displace itself on the
scale, and the scientific fact is this : a current passes in the circuit.
Or again: when I make an experiment I should subject the
result to certain corrections, because I know I must have made
errors. These errors are of two kinds, some are accidental and
these I shall correct by taking the mean ; the others are systematic
and I shall be able to correct those only by a thorough study of
their causes. The first result obtained is then the fact in the
rough, while the scientific fact is the final result after the
finished corrections.
Reflecting on this latter example, we are led to subdivide our
second stage, and in place of saying :
2. The eclipse happened at nine o'clock, we shall say:
2a. The eclipse happened when my clock pointed to nine, and
IS SCIENCE ARTlFICIALt
2b. Mjr clock being ten minutes slow, the eclipse happened at
ten minutes past nine.
And this is not all : the first stage also should be subdivided,
and not between these two subdivisions will be the least distance;
it is necessary to distinguish between the impression of obscur-
ity felt by one witnessing an eclipse, and the affirmation : It grows
dark, which this impression extorts from him. In a sense it is
the first which is the only true fact in the rough, aud Ihe second
is already a sort of scientific fact.
Now then our scale has six stages, and even though there is no
reason for halting at this figure, there we shall stop.
What strikes me at the start is this. At the first of our six
stages, the fact, still completely in the rough, is, so to speak, in-
dividual, it is completely distinct from all other possible facts.
From the second stage, already it is no longer the same. The
entmciation of the fact would suit an infinity of other facts.
So soon as language intervenes, I have at my command only a
finite number of terms to express the shades, in number infinite,
that my impressions might cover. When I say: It grows dark,
that well expresses the impressions I feel in being present at an
eclipse; but even in obscurity a multitude of shades could be
imagined, and if, instead of that actually realized, had happened
a slightly different shade, yet I should still have enunciated this
other fact by saying: It grows dark.
Second remark: even at the second stage, the enunciation of
a fact can only be (rue or false. This is not so of any proposi-
tion; if this proposition is the enunciation of a convention, it
can not be said that this enunciation is true, in the proper sense
of the word, since it could not be true apart from me and is tme
only because T wish it to be.
When, for instance, I say the unit for length is the meter, this
is a decree that I promulgate, it is not something ascertained
which forces itself upon me. It is the same, as I think I have
elsewhere shown, when it is a question, for example, of Euclid's
postulate.
When I am asked: Is it growing darki I always know whether
I ought to reply yes or no. Although an infinity of possible
facts may be susceptible of this same enunciation, it grows dark,
828 THE VALUE OF SCIENCE
I shall always know whether the fact realized belongs or does not
belong among those which answer to this enunciation. Facts are
classed in categories, and if I am asked whether the fact that I
ascertain belongs or does not belong in such a category, I shall
not hesitate.
Doubtless this classification is su£Sciently arbitrary to leave a
large part to man's freedom or caprice. In a word, this classifi-
cation is a convention. This convention being given, if I am
asked : Is such a fact true ? I shall always know what to answer,
and my reply will be imposed upon me by the witness of my
senses.
If therefore, during an eclipse, it is asked : Is it growing darkf
all the world will answer yes. Doubtless those speaking a lan-
guage where bright was called dark, and dark bright, would
answer no. But of what importance is thatf
In the same way, in mathematics, when I have laid doum the
definitions, and the postulates which are conventions, a theorem
henceforth can only be true or false. But to answer the ques-
tion : Is this theorem true ? it is no longer to the witness of my
senses that I shall have recourse, but to reasoning.
A statement of fact is always verifiable, and for the verifica-
tion we have recourse either to the witness of our senses, or to
the memory of this witness. This is properly what characterizes
a fact. If you put the question to me : Is such a fact true 1 I
shall begin by asking you, if there is occasion, to state precisely
the conventions, by asking you, in other words, what language you
have spoken; then once settled on this point, I shall interrogate
my senses and shall answer yes or no. But it will be my senses
that will have made answer, it will not be you when you say to
me: I have spoken to you in English or in French.
Is there something to change in all that when we pass to the
following stages! When I observe a galvanometer, as I have
just said, if I ask an ignorant visitor: Is the current passing?
he looks at the wire to try to see something pass ; but if I put the
same question to my assistant who understands my language, he
will know I mean: Does the spot move? and he will look at the
scale.
What difference is there then between the statement of a fact
IS SCIENCE ARTlFlClALt 329
in the rough and the statement of a scientific fact ! The same
difference as between the statement of the same crude fact in
French and in German, The scientific statement is the transla-
tion of the crude statement into a language which is distinguished
above all from the common German or French, because it is
spoken by a veiy much smaller number of people.
Yet let us not go too fast. To measure a current I may use
a very great number of types of galvaBomet^rs or besides an
eleetrodynamometer. And then when I shall say there is r unni ng
in tbia circuit a current of so many amperes, that will mean:
if I adapt to this circuit such a galvanometer I shall see the
spot come to the division a; but that will mean equally: if I
adapt to this circuit such an eleetrodynamometer, I shall see the
spot go to the division b. And that will mean still many other
things, because the current can manifest itself not only by me-
chanical effects, but by effects chemical, thermal, luminous, etc.
Here then is one same statement which siuts a very great num-
ber of facts absolutely different. WhyT It is because I assuma |
a law according to which, whenever such a mechanical effect shall
happen, such a chemical effect will happen also. Previous experi-
ments, very numerous, have never shown this law to fail, and
then I have understood that I could express by the same state-
ment two facta BO invariably bound one to the other.
When I am asked: Is the current passing 1 I can understand
that that means ; Will such a mechanical effect happen I But I
can understand also: Will such a chemical effect happen? I
shall then verify either the existence of the mechanical effect, or
that of the chemical effect ; that will be indifferent, since in both
caaes the answer must be Uie same.
And if the law should one day be found false T If it was per-
ceived that the concordance of the two effects, mechanical and
chemical, is not constant! That day it would be necessary to
change the scientific language to free it from a grave ambiguity.
And after that 1 Is it thought that ordinary language by aid
of which are expressed the facts of daily life is exempt from
ambiguity 1
Shalt we thence conclude that the fads of daily life are Ih^
work of the grammariansT
830 THE VALUE OF SCIENCE
You ask me : Is there a current f I try whether the mechanical
effect exists, I ascertain it and I answer: Yes, there is a current
You understand at once that that means that the mechanical
effect exists, and that the chemical effect, that I have not investi-
gated, exists likewise. Imagine now, supposing an impossibility,
the law we believe true, not to be, and the chemical effect not to
exist. Under this hypothesis there will be two distinct facts, the
one directly observed and which is true, the other inferred and
which is false. It may strictly be said that we have created the
second. So that error is the part of man's personal collabora-
tion in the creation of the scientific fact.
But if we can say that the fact in question is false, is this not
just because it is not a free and arbitrary creation of our mind, a
disguised convention, in which case it would be neither true nor
false. And in fact it was verifiable ; I had not made the verifica-
tion, but I could have made it If I answered amiss, it was be-
cause I chose to reply too quickly, without having asked nature,
who alone knew the secret.
When, after an experiment, I correct the accidental and sys-
tematic errors to bring out the scientific fact, the case is the same;
the scientific fact will never be anything but the crude fact trans-
lated into another language. When I shall say: It is such an
hour, that will be a short way of saying : There is such a relation
between the hour indicated by my clock, and the hour it marked
at the moment of the passing of such a star and such another
star across the meridian. And this convention of language once
adopted, when I shall be asked: Is it such an hour? it will not
depend upon me to answer yes or no.
Let us pass to the stage before the last : the eclipse happened at
the hour given by the tables deduced from Newton's laws. This
is still a convention of language which is perfectly clear for those
who know celestial mechanics or simply for those who have the
tables calculated by the astronomers. I am asked: Did the
eclipse happen at the hour predicted? I look in the nautical
almanac, I see that the eclipse was announced for nine o'clock
and I understand that the question means: Did the eclipse
happen at nine o'clock? There still we have nothing to change
in our conclusions. The scientific fact is only the crude fact
translated into a convenient language.
It is true that at the last stage things change. Does the
earth rotate t Is this a veriiiaMe fact? Could Galileo and the
Grand Inquisitor, to settle the matter, appeal to the witness of
their senses 1 On the contrary, they were in accord about the
appearances, and whatever had been the accumulated expe-
riences, they would have remained in accord with regard to the
appearances without ever agreeing on their interpretation. It
is just on that account that they were obliged to have recourse
to procedures of discussion so unscientific.
This is why I think they did not disagree about a fact: we
have not the right to give the same name to the rotation of the
earth, which was the object of their discussion, and to the facta
crude or scientific we have hitherto passed in review.
After what precedes, it seems superfluous to investigate
whether the fact in the rough is outside of science, because there
can neither be science without scientific fact, nor scientific fact
without fact in the rough, since the first is only the translation
of the second.
And then, has one the right to say that the scientist creates the
scientific factf First of all, he does not create it from nothing,
since he makes it with the fact in the rough. Consequently he
does not make it freely and as he chooses. However able the
worker may be, his freedom is always limited by the properties of
the raw material on which he works.
After all, what do you mean when you speak of this free
creation of the scientific fact and when you take as example the
astronomer who intervenes actively in the phenomenon of the
eclipse by bringing his clock 1 Do you mean: The eclipse hap-
pened at nine o'clock; but if the astronomer had wished it to
happen at ten, that depended only on him, he had only to
advance his clock an hour!
But the astronomer, in perpetrating that bad joke, would
evidently have been guilty of an equivocation. When he tells
me : The eclipse happened at nine, I understand that nine is the
hour deduced from the crude indication of the pendulum by the
usual series of corrections. If he has given me solely that crude
indication, or if he has made corrections contrary to the babitnal
roles, he has changed the language agreed upon without fore-
832 THE VALUE OF SCIENCE
warning me. If, on the contrary, he took care to forewarn me,
I have nothing to complain of, but then it is always the same
fact expressed in another language.
In sum, aU the scientist creates in a fact is tliie language ivC
which he enunciates it. If he predicts a fact, he will employ this
language, and for all those who can speak and understand it, his
prediction is free from ambiguity. Moreover, this prediction
once made, it evidently does not depend upon him whether it is
fulfilled or not.
What then remains of M. LeRoy's thesis f This remains: the
scientist intervenes actively in choosing the facts worth observ-
ing. An isolated fact has by itself no interest ; it becomes inter-
esting if one has reason to think that it may aid in the prediction
of other facts; or better, if, having been predicted, its verifies^
tion is the confirmation of a law. Who shall choose the facts
which, corresponding to these conditions, are worthy the freedom
of the city in science ? This is the free activity of the scientist
And that is not all. I have said that the scientific fact is the
translation of a crude fact into a certain language ; I should add
that every scientific fact is formed of many crude facts. This is
suflBciently shown by the examples cited above. For instance,
for the hour of the eclipse my clock marked the hour a at the
instant of the eclipse ; it marked the hour p at the moment of the
last transit of the meridian of a certain star that we take as
origin of right ascensions ; it marked the hour y at the moment
of the preceding transit of this same star. There are three dis-
tinct facts (still it will be noticed that each of them results itself
from two simultaneous facts in the rough; but let us pass this
over). In place of that I say: The eclipse happened at the hour
24 {oL-p) / ip-^) , and the three facts are combined in a single
scientific fact. I have concluded that the three readings a, j8, y
made on my clock at three different moments lacked interest and
that the only thing interesting was the combination (oL-p)/(p^)
of the three. In this conclusion is found the free activity of my
mind.
But I have thus used up my power ; I can not make this com-
bination {cL-p)/{p-^) have such a value and not such another,
since I can not influence either the value of a, or that of j8, of
that of y, which are imposed upon me as crude facts.
IS SCIENCE ABTIFICIALt 888
In sum, facts are facts, and if it happens that they satisfy a
prediction, this is not an effect of our free activity. There is no
precise frontier between the fact in the rough and the scientific
fact ; it can only be said that such an enunciation of fact is more
crude or, on the contrary, more scientific than such another.
4. ^Nominalism* and *the Universal Invariant*
If from facts we pass to laws, it is clear that the part of the
free activity of the scientist will become much greater. But
did not M. LeBoy make it still too great f This is what we are
about to examine.
Recall first the examples he has given. When I say: Phos-
phorus melts at 44^, I think I am enunciating a law; in reality
it is just the definition of phosphorus; if one should discover a
body which, possessing otherwise all the properties of phosphorus,
did not melt at 44^, we should give it another name, that is all,
and the law would remain true.
Just so when I say: Heavy bodies falling freely pass over
spaces proportional to the squares of the times, I only give the
definition of free fall. Whenever the condition shall not be
fulfilled, I shall say that the fall is not free, so that the law
wiU never be wrong. It is clear that if laws were reduced to that,
they could not serve in prediction ; then they would be good for
nothing, either as means of knowledge or as principle of action.
When I say: Phosphorus melts at 44°, I mean by that: All
bodies possessing such or such a property (to wit, all the prop-
erties of phosphorus, save fusing-point) fuse at 44**. So under-
stood, my proposition is indeed a law, and this law may be use-
ful to me, because if I meet a body possessing these properties
I shall be able to predict that it will fuse at 44°.
Doubtless the law may be found to be false. Then we shall
read in the treatises on chemistry: ** There are two bodies which
chemists long confounded under the name of phosphorus; these
two bodies differ only by their points of fusion." That would
evidently not be the first time for chemists to attain to the separa-
tion of two bodies they were at first not able to distinguish ; such,
for example, are neodymium and praseodymium, long confounded
under the name of didymium.
834 TRE VALUE OF SCIENCE
I do not think the chemists much fear that a like mischance
will ever happen to phosphorus. And if, to suppose the impos-
sible, it should happen, the two bodies would probably not have
identically the same density, identicaUy the same specific heat»
etc., so that after having determined with care the density, for
instance, one could still foresee the fusion point.
It is, moreover, unimportant; it su£Sces to remark that there
is a law, and that this law, true or false, does not reduce to a
tautology.
Will it be said that if we do not know on the earth a body
which does not fuse at 44° while having all the other properties
of phosphorus, we can not know whether it does not exist on other
planets f Doubtless that may be maintained, and it would then
be inferred that the law in question, which may serve as a rule
of action to us who inhabit the earth, has yet no general value
from the point of view of knowledge, and owes its interest only
to the chance which has placed us on this globe. This is possible,
but, if it were so, the law would be valueless, not because it re-
duced to a convention, but because it would be false.
The same is true in what concerns the fall of bodies. It would
do me no good to have given the name of free fall to falls which
happen in conformity with Galileo 's law, if I did not know that
elsewhere, in such circumstances, the fall will be probably free or
approximately free. That then is a law which may be true or
false, but which does not reduce to a convention.
Suppose the astronomers discover that the stars do not exactly
obey Newton's law. They will have the choice between two
attitudes; they may say that gravitation does not vary exactly
as the inverse of the square of the distance, or else they may say
that gravitation is not the only force which acts on the stars and
that there is in addition a different sort of force.
In the second case, Newton's law will be considered as the
definition of gravitation. This will be the nominalist attitude.
The choice between the two attitudes is free, and is made from
considerations of convenience, though these considerations are
most often so strong that there remains practically little of this
freedom.
We can break up this proposition : (1) The stars obey Newton's
IS SCIENCE ABTIFICIALt 336
law, into two others; (2) gravitation obeys Newton's law; (3)
gravitation is the only force acting on the stars. In this case
proposition (2) is no longer anything but a definition and is
beyond the test of experiment ; but then it will be on proposition
(3) that this check can be exercised. This is indeed necessary,
since the resulting proposition (1) predicts verifiable facts in the
rough.
It is thanks to these artifices that by an unconscious nomi-
nalism the scientists have elevated above the laws what they call
principles. When a law has received a sufficient confirmation
from experiment, we may adopt two attitudes: either we may
leave this law in the fray; it will then remain subjected to an
incessant revision, which without any doubt will end by demon-
strating that it is only approximative. Or else we may elevate
it into a principle by adopting conventions such that the propo-
sition may be certainly true. For that the procedure is always
the same. The primitive law enunciated a relation between two
facts in the rough, A and B; between these two crude facts is
introduced an abstract intermediary C, more or less fictitious
(such was in the preceding example the impalpable entity, gravi-
tation). And then we have a relation between A and C that we
may suppose rigorous and which is the principle; and another
between C and B which remains a law subject to revision.
The principle, henceforth crystallized, so to speak, is no longer
subject to the test of experiment. It is not true or false, it is
convenient.
Great advantages have often been found in proceeding in that
way, but it is clear that if all the laws had been transformed
into principles nothing would be left of science. Every law may
be broken up into a principle and a law, but thereby it is very
clear that, however far this partition be pushed, there will always
remain laws.
Nominalism has therefore limits, and this is what one might
fail to recognize if one took to the very letter M. LeRoy's
assertions.
A rapid review of the sciences will make us comprehend better
what are these limits. The nominalist attitude is justified only
when it is convenient ; when is it so ?
836 THE VALUE OF SCIENCE
Experiment teaches us relations between bodies ; this is the fact
in the rough ; these relations are extremely complicated. Instead
of envisaging directly the relation of the body A and the body B^
we introduce between them an intermediary, which is space, and
we envisage three distinct relations : that of the body A with the
figure A' of space, that of the body B with the figure B' of space,
that of the two figures A' and B' to each other. Why is this
detour advantageous f Because the relation of A and B was com-
plicated, but differed little from that of A' and B'y which is
simple ; so that this complicated relation may be replaced by the
simple relation between A' and £' and by two other relations
which tell us that the differences between A and A', on the one
hand, between B and B\ on the other hand, are very small. For
example, if A and B are two natural solid bodies which are dis-
placed with slight deformation, we envisage two movable rigid
figures A' and B\ The laws of the relative displacement of these
figures A' and B' will be very simple ; they will be those of geom-
etry. And we shall afterward add that the body A^ which always
differs very little from A', dilates from the effect of heat and
bends from the effect of elasticity. These dilatations and flexions,
just because they are very small, will be for our mind relatively
easy to study. Just imagine to what complexities of language
it would have been necessary to be resigned if we had wished to
comprehend in the same enunciation the displacement of the
solid, its dilatation and its flexure ?
The relation between A and B was a rough law, and was broken
up ; we now have two laws which express the relations of A and A\
of B and B\ and a principle which expresses that of A' with B'.
It is the aggregate of these principles that is called geometry.
Two other remarks. We have a relation between two bodies A
and B, which we have replaced by a relation between two figures
A' and B' \ but this same relation between the same two figures
A' and B' could just as well have replaced advantageously a
relation between two other bodies A" and B", entirely different
from A and B. And that in many ways. If the principles and
geometry had not been invented, after having studied the rela-
tion of A and -B, it would be necessary to begin again ab ovo the
study of the relation of A" and B'\ That is why geometry is so
IS SCIENCE ASTlFWIALt
preoiouB. A geometrical relation can advantageously replace a
relation which, considered in the rough state, should be regarded
as mechanical, it can replace another which should be regarded
as optical, etc.
Yet let no one sa; : Bat that proves geometry an experimental
science ; in separating its principles from laws whence they have
been drawn, you artificially separate it itself from the sciences
which have given birth to it. The other sciences have likewise
principles, but that does not preclude our having to call them
experimental.
It must be recognized that it would have been difficult not to
make this separation that is pretended to be artificial. "We know
the role that the kinematics of solid bodies has played in the
genesis of geometry ; should it then he said that geometry is only
a branch of experimental kinematics? But the laws of the recti-
linear propagation of light have also contributed to the forma-
tion of its principles. Must geometry be regarded both as a
branch of kinematics and as a branch of optics t I recall besides
that our Euclidean space which is the proper object of geometry
has been chosen, for reasons of convenience, from among a cer-
tain number of types which preexist in our mind and which are
called groups.
If we pass to mechanics, we still see great principles whose
origin is analogous, and, as their 'radius of action,' so to speak,
is smaller, there is no longer reason to separate them from
mechanics proper and to regard this science as deductive.
In physics, finally, the role of the principles is still more dimin-
ished. And in fact they are only introduced when it is of ad-
vantage. Now they are advantageous precisely because they are
few, since each of them very nearly replaces a great numher of
laws. Therefore it is not of interest to multiply them. Besides
an outcome is necessary, and for that it is needful to end by leav-
ing abstraction to take hold of reality.
Such are the limits of nominalism, and they are narrow.
M. LeRoy has insisted, however, and he has put the question
under another form.
Since the enunciation of our laws may vary with the conven-
tions that we adopt, since these conventions may modify even the
338 THE VALUE OF SCIENCE
natural relations of these laws, is there in the manifold of these
laws something independent of these conventions and which may,
so to speak, play the role of universal invariant t For instance,
the fiction has been introduced of beings who, having been edu-
cated in a world different from ours, would have been led to
create a non-Euclidean geometry. If these beings were after-
ward suddenly transported into our world, they would observe
the same laws as we, but they would enunciate them in an
entirely different way. In truth there would still be something
in common between the two enunciations, but this is because these
beings do not yet differ enough from us. Beings still more strange
may be imagined, and the part common to the two systems of
enunciations will shrink more and more. Will it thus shrink
in convergence toward zero, or will there remain an irreducible
residue which will then be the universal invariant sought f
The question calls for precise statement. Is it desired that
this common part of the enunciations be expressible in words f
It is clear, then, that there are not words common to all languages,
and we can not pretend to construct I know not what universal
invariant which should be understood both by us and by the
fictitious non-Euclidean geometers of whom I have just spoken;
no more than we can construct a phrase which can be understood
both by Germans who do not understand French and by French
who do not understand German. But we have fixed rules which
permit us to translate the French enunciations into German,
and inversely. It is for that that grammars and dictionaries
have been made. There are also fixed rules for translating the
Euclidean language into the non-Euclidean language, or, if there
are not, they could be made.
And even if there were neither interpreter nor dictionary, if
the Germans and the French, after having lived centuries in
separate worlds, found themselves all at once in contact, do you
think there would be nothing in common between the science
of the German books and that of the French books 1 The French
and the Germans would certainly end by understanding each
other, as the American Indians ended by understanding the
language of their conquerors after the arrival of the Spanish.
But, it will be said, doubtless the French would be capable of
18 SCIENCE ABTIFICIALt 339
nnderstanding the Qermans even without having learned Qer-
man, but this is because there remains between the French and
the Germans something in common, since both are men. We
should still attain to an understanding with our hypothetical non-
EuclideanSy though they be not men, because they would still
retain something human. But in any case a minimum of humanity
is necessary.
This is possible, but I shall observe first that this little human-
ness which would remain in the non-Euclideans would suffice not
only to make possible the translation of a Utile of their language,
but to make possible the translation of all their language.
Now, that there must be a minimum is what I concede ; suppose
there exists I know not what fiuid which penetrates between the
molecules of our matter, without having any action on it and
without being subject to any action coming from it. Suppose
beings sensible to the influence of this fluid and insensible to
that of our matter. It is clear that the science of these beings
would differ absolutely from ours and that it would be idle to
seek an 'invariant' common to these two sciences. Or again, if
these beings rejected our logic and did not admit, for instance,
the principle of contradiction.
But truly I think it without interest to examine such
hypotheses.
And then, if we do not push whimsicality so far, if we intro-
duce only fictitious beings having senses analogous to ours and
sensible to the same impressions, and moreover admitting the
principles of our logic, we shall then be able to conclude that
their language, however different from ours it may be, would
always be capable of translation. Now the possibility of trans-
lation implies the existence of an invariant. To translate is
precisely to disengage this invariant. Thus, to decipher a crypto-
gram is to seek what in this document remains invariant, when
the letters are permuted.
What now is the nature of this invariant it is easy to under-
stand, and a word will suffice us. The invariant laws are the
relations between the crude facts, while the relations between the
'scientific facts' remain always dependent on certain conventions.
CHAPTER XI
Science and Beauty
5. Contingence and Detefminism
I DO not intend to treat here the question of the eontingenoe of
the laws of nature, which is evidently insoluble, and on which so
much has already been written. I only wish to call attention to
what different meanings have been given to this word, contm-
gence, and how advantageous it would be to distinguish them.
If we look at any particular law, we may be certain in advance
that it can only be approximate. It is, in fact, deduced from
experimental verifications, and these verifications were and could
be only approximate. We should always expect that more precise
measurements will oblige us to add new terms to our formulas;
this is what has happened, for instance, in the case of Mariotte's
law.
Moreover the statement of any law is necessarily incomplete.
This enunciation should comprise the enumeration of (M the
antecedents in virtue of which a given consequent can happen.
I should first describe all the conditions of the experiment to be
made and the law would then be stated : If all the conditions are
fulfilled, the phenomenon will happen.
But we shall be sure of not having forgotten any of these con-
ditions only when we shall have described the state of the entire
universe at the instant t ; all the parts of this universe may, in
fact, exercise an influence more or less great on the phenomenon
which must happen at the instant t -j- dt.
Now it is clear that such a description could not be found in
the enunciation of the law; besides, if it were made, the law
would become incapable of application ; if one required so many
conditions, there would be very little chance of their ever being
all realized at any moment.
Then as one can never be certain of not having forgotten some
essential condition, it can not be said: If such and such condi-
340
tions are realized, each a pbenomenoD uill occur; it can only be
said; H such and such conditions are realized, it is probable that
aueh a phenomenon will occur, very nearly.
Take the law of gravitation, which ia the least imperfect of all
known laws. It enables us to foresee the motions of the planets.
When I use it, for instance, to calculate the orbit of Saturn, I
neglect the action of the staro, and iu doing so I am certain of
not deceiving myself, because I know that these stars are too fan
away for their action to be sensible,
I announce, then, with a quasi-certitude that the coordinates
of Saturn at such an hour will be comprised between such and
such limits. Yet is that certitude absolute t Could there not
exist in the universe some gigantic mass, much greater than th&t
of all the known stars and whose action could make itself felt
at great distances! That mass might be animated by a colossal
velocity, and after having circulated from all time at such dis-
tances that its influence had remained hitherto insensible to na,
it might come all at once to pass near us. Surely it would pro-
duce in our solar system enormous perturbations that we could
not have foreseen. All that can be said is that such an event ia
wholly improbable, and then, instead of saying: Saturn will be
near sueh a point of the heavens, we must limit ourselves to say-
ing : Saturn will probably be near such a point of the heavens.
Although this probability may be practically equivalent to cer-
tainty, it is only a probability.
For all these reasons, no particular law will ever he more than
approximate and probable. Scientists have never failed to recog-
nize this truth ; only they believe, right or wrong, that every law
may be replaced by another closer and more probable, that this
new law wiU itself be only provisional, but that the same move-
ment can continue indefinitely, so that science in progressing will
possess laws more and more probable, that the approximation will
end by differing as little as you choose from exactitude and the
probability from certitude.
If the scientists who thinfe thus are right, still could it be said
that the laws of nature are contingent, even though each law,
taken in particular, may be quali6ed as contingentt Or must one 1
Teqnire, before concluding the contingence of the natural lawi^J
842 THE VALUE OF SCIENCE
that this process have an end, that the scientist finish some day
by being arrested in his search for a closer and closer approxi-
mation, and that, beyond a cettain limit, he thereafter meet in
nature only caprice f
In the conception of which I have just spoken (and which I
shall call the scientific conception), every law is only a statement
imperfect and provisional, but it must one day be replaced by
another, a superior law, of which it is only a crude image. No
place therefore remains for the intervention of a free will.
It seems to me that the kinetic theory of gases will furnish
us a striking example.
You know that in this theory all the properties of gases are
explained by a simple hypothesis; it is supposed that all the
gaseous molecules move in every direction with great velocities
and that they follow rectilineal paths which are disturbed only
when one molecule passes very near the sides of the vessel or
another molecule. The effects our crude senses enable us to
observe are the mean effects, and in these means, the great devia-
tions compensate, or at least it is very improbable that they do
not compensate ; so that the observable phenomena follow simple
laws such as that of Mariotte or of Gay-Lussac. But this com-
pensation of deviations is only probable. The molecules inces-
santly change place and in these continual displacements the
figures they form pass successively through all possible combina-
tions. Singly these combinations are very numerous ; almost all
are in conformity with Mariotte 's law, only a few deviate from
it. These also will happen, only it would be necessary to wait
a long time for them. If a gas were observed during a sufB-
ciently long time, it would certainly be finally seen to deviate,
for a very short time, from Mariotte 's law. How long would it
be necessary to wait? If it were desired to calculate the prob-
able number of years, it would be found that this number is so
great that to write only the number of places of figures employed
would still require half a score places of figures. No matter;
enough that it may be done.
I do not care to discuss here the value of this theory. It is
evident that if it be adopted, Mariotte 's law will thereafter
appear only as contingent, since a day will come when it will not
SCIENCE AND BEALITT 343
be true. And yet, think you the partisans of the kinetic theory
are adversaries of determinism f Far from it; they are the
most ultra of mechanists. Their molecules follow rigid paths,
from which they depart only under the influence of forces which
vary with the distance, following a perfectly determinate law.
There remains in their system not the smallest place either for
freedom, or for an evolutionary factor, properly so-called, or for
anything whatever that could be called contingence. I add, to
avoid mistake, that neither is there any evolution of Mariotte's
law itself ; it ceases to be true after I know not how many cen-
turies ; but at the end of a fraction of a second it again becomes
true and that for an incalculable number of centuries.
And since I have pronounced the word evolution, let us clear
away another mistake. It is often said: Who knows whether
the laws do not evolve and whether we shall not one day discover
that they were not at the Carboniferous epoch what they are
to-day t What are we to understand by thatt What we think
we know about the past state of our globe, we deduce from its
present state. And how is this deduction madet It is by means
of laws supposed known. The law, being a relation between the
antecedent and the consequent, enables us equally well to deduce
the consequent from the antecedent, that is, to foresee the future,
and to deduce the antecedent from the consequent, that is, to
conclude from the present to the past. The astronomer who
knows the present situation of the stars can from it deduce their
future situation by Newton's law, and this is what he does when
he constructs ephemerides; and he can equally deduce from it
their past situation. The calculations he thus can make can not
teach him that Newton's law will cease to be true in the future,
since this law is precisely his point of departure; not more can
they tell him it was not true in the past. Still, in what concerns
the future, his ephemerides can one day be tested and our de-
scendants will perhaps recognize that they were false. But in
what concerns the past, the geologic past which had no witnesses,
the results of his calculation, like those of all speculations where
we seek to deduce the past from the present, escape by their
very nature every species of test. So that if the laws of nature
were not the same in the Carboniferous age as at the present
S4A THE VALUE OF SCIENCE
epoch, we shall never be able to know it, since we can know
nothing of this age, only what we deduce from the hyx>otheBi8 of
the permanence of these laws.
Perhaps it will be said that this hypothesis might lead to con-
tradictory results and that we shall be obliged to abandon it
Thus, in what concerns the origin of life, we may conclude that
there have always been living beings, since the present world
shows us always life springing from life ; and we may also con-
clude that there have not always been, since the application of
the existent laws of physics to the present state of our globe
teaches us that there was a time when this globe was so warm that
life on it was impossible. But contradictions of this sort can
always be removed in two ways; it may be supposed that the
actual laws of nature are not exactly what we have assumed;
or else it may be supposed that the laws of nature actually are
what we have assumed, but that it has not always been so.
It is evident that the actual laws will never be sufSciently well
known for us not to be able to adopt the first of these two solu-
tions and for us to be constrained to infer the evolution of
natural laws.
On the other hand, suppose such an evolution ; assume, if you
wish, that humanity lasts sufficiently long for this evolution to
have witnesses. The same antecedent shall produce, for instance,
different consequents at the Carboniferous epoch and at the
Quaternary. That evidently means that the antecedents are
closely alike ; if all the circumstances were identical, the Carbon-
iferous epoch would be indistinguishable from the Quaternary.
Evidently this is not what is supposed. What remains is that
such antecedent, accompanied by such accessory circumstance,
produces such consequent ; and that the same antecedent, accom-
panied by such other accessory circumstance, produces such
other consequent. Time does not enter into the affair.
The law, such as ill-informed science would have stated it, and
which would have affirmed that this antecedent always produces
this consequent, without taking account of the accessory circum-
stances, this law, which was only approximate and probable,
must be replaced by another law more approximate and more
probable, which brings in these accessory circumstances. We
SCIENCE AND REALITY
always come back, therefore, to that same process which we have
analyzed above, and if humanity should discover something o£
this sort, it would not say tliat it is the laws which have evolated,
but the circumstances which have changed.
Here, therefore, are several different senses of the word eon-
tingence. M. LeRoy retains them all and he does not sufficiently
distinguish them, but he introduces a new one. Experimental
laws are only approximate, and if some appear to us as exact, it
is because we have artificially transformed them into what I have
above called a principle. We have made this transformation
freely, and as the caprice which has determined ua to make it
is something eminently contingent, we have communicated this
contingence to the law itself. It is in this sense that we have the
right to say that determinism supposes freedom, since it is freely
that we become determinists. Perhaps it will be found that this
is to give large scope to nominalism and that the introduction
of this new sense of the word contingence will not help much to
solve all those questions which naturally arise and of which we
have just been speaking.
I do not at all wish to investigate here the foundations of the
principle of induction; I know very well that I should not suc-
ceed ; it is as difficult to justify this principle as to get on with-
out it. I only wish to show how scientists apply it and are
forced to apply it.
"When the same antecedent recurs, the same consequent must
likewise recur; such is the ordinary statement. But reduced
to these terms this principle could be of no use. For one to be
able to say that the same antecedent recurred, it would be neces-
sary for the circumstances ail to be reproduced, since no one
is absolutely indifferent, and for them to be exacfly reproduced.
And, as that will never happen, the principle can have no
application.
We should therefore modify the enunciation and say: If an
antecedent A has once produced a consequent B, an antecedent
A', slightly different from A, will produce a consequent B',
slightly different from B. But how shall we recognize that the
antecedents A and A' are 'slightly different'I If some one of thu
etremuBtances can be expressed by a number, and this number
346 THE VALUE OF SCIENCE
has in the two cases values very near together, the sense of the
phrase ' slightly different ' is relatively clear; the principle then
signifies that the consequent is a continuous function of the ante-
cedent. And as a practical rule, we reach this conclusion that
we have the right to interpolate. This is in fact what scientists
do every day, and without interpolation all science would be
impossible.
Yet observe one thing. The law sought may be represented by
a curve. Experiment has taught us certain points of this curve.
In virtue of the principle we have just stated, we believe these
points may be connected by a continuous graph. We trace this
graph with the eye. New experiments will furnish us new points
of the curve. If these points are outside of the graph traced in
advance, we shall have to modify our curve, but not to abandon
our principle. Through any points, however numerous they may
be, a continuous curve may always be passed. Doubtless, if this
curve is too capricious, we shall be shocked (and we shall even
suspect errors of experiment), but the principle will not be
directly put at fault.
Furthermore, among the circumstances of a phenomenon, there
are some that we regard as negligible, and we shall consider A
and A' as slightly different if they differ only by these accessory
circumstances. For instance, I have ascertained that hydrogen
unites with oxygen under the influence of the electric spark, and
I am certain that these two gases will unite anew, although the
longitude of Jupiter may have changed considerably in the
interval. We assume, for instance, that the state of distant
bodies can have no sensible influence on terrestrial phenomena,
and that seems in fact requisite, but there are cases where the
choice of these practically indifferent circumstances admits of
more arbitrariness or, if you choose, requires more tact.
One more remark: The principle of induction would be inap-
plicable if there did not exist in nature a great quantity of
bodies like one another, or almost alike, and if we could not
infer, for instance, from one bit of phosphorus to another bit of
phosphorus.
If we reflect on these considerations, the problem of deter-
minism and of contingence will appear to us in a new light.
SCIENCE AND REALITY
Suppose we were able to embrace the series of all phenomena
of the universe in the whole sequence of time. We could envis-
age what might be called the sequences; I mean relations between
antecedent and consequent. I do not wish to speak of constant
relations or laws, I envisage separately (individually, so to
speak) the different sequences realized.
We should then recognize that among these sequences there
are no two altogether alike. But, if the principle of induction,
as we have just stated it, is true, there will be those almost alike
and that can be classed alongside one another. In other words,
^it is possible to make a classification of sequences.
^^■At is to the possibility and the legitimacy of such a classifica-
^^fti that determinism, in the end, reduces. This is all that the
^^^eceding analysis leaves of it. Perhaps under this modest form
it will seem less appalling to the moralist.
It will doubtless be said that this is to come back by a detonr
to M. LeRoy's conclusion which a moment ago we seemed to
reject: we are determinists voluntarily. And in fact all classi-
fication supposes the active intervention of the classifier. I a^tree
that this may be maintained, but it seems to me that this detour
will not have been useless and will have contributed to enlighten
^JU a little.
^^H 6. Objectivity of Science
^^Bk arrive at the question set by the title of this article : What is
^^Be objective value of science T And first what should we under-
stand by objectivity 1
What guarantees the objectivity of the world in which we live
is that this world is common to us with other thinking beings.
Through the communications that we have with other men, we
receive from them ready-raade reasonings; we know that these
reasonings do not come from us and at the same time we recog-
nize in them the work of reasonable beings like ourselves. And
as these reasonings appear to fit the world of our sensations, we
think we may infer that these reasonable beings have seen the
same thing as we; thus it is we know we have not been dreaming.
Such, therefore, is the first condition of objectivity; what is
objective must be common to many minds and consequently trans-
miBgible from one to the other, and as this transmisaioo can only
848 THE VALUE OF SCIENCE
come about by that 'discourse' which inspires so much distrust
in M. LeRoy, we are even forced to conclude : no discourse, no
objectivity.
The sensations of others will be for us a world eternally dosed.
We have no means of verifying that the sensation I call red is
the same as that which my neighbor calls red.
Suppose that a cherry and a red poppy produce on me the
sensation A and on him the sensation B and that, on the con-
trary, a leaf produces on me the sensation B and on him the
sensation A. It is clear we shall never know anything about it;
since I shall call red the sensation A and green the sensation B^
while he will call the first green and the second red. In com-
pensation, what we shall be able to ascertain is that, for him as
for me, the cherry and the red poppy produce the same sensa<
tion, since he gives the same name to the sensations he feels and
I do the same.
Sensations are therefore intransmissible, or rather all that is
pure quality in them is intransmissible and forever impenetrable.
But it is not the same with relations between these sensations.
From this point of view, all that is objective is devoid of all
quality and is only pure relation. Certes, I shall not go so far
as to say that objectivity is only pure quantity (this would be
to particularize too far the nature of the relations in question),
but we understand how some one could have been carried away
into saying that the world is only a differential equation.
With due reserve regarding this paradoxical proposition, we
must nevertheless admit that nothing is objective which is not
transmissible, and consequently that the relations between the
sensations can alone have an objective value.
Perhaps it will be said that the esthetic emotion, which is
common to all mankind, is proof that the qualities of our sensa-
tions are also the same for all men and hence are objective. But
if we think about this, we shall see that the proof is not com-
plete ; what is proved is that this emotion is aroused in John as
in James by the sensations to which James and John give the
same name or by the corresponding combinations of these sensa-
tions; either because this emotion is associated in John with
the sensation A, which John calls red, while parallelly it is asso-
SCIENCE AND BEALITT
elated in Jamea with the sensation B, which James calls red;
or better because this emotion is aroused, not by the qualities
themselves of the sensations, but by the harmonious combina-
tion of their relations of which we undergo the uneonscioos
impression.
Such a sensation is beautiful, not because it possesses such a
quality, but because it occupies such a place in the woof of our
associations of ideas, so that it can not be excited without putting
in motion the 'receiver' which is at the other end of the thread
and which corresponds to the artistic emotion.
Whether we talie the moral, the esthetic or the scientific point
of view, it is always the same thing. Nothing is objective except
what is identical for all ; now we can only speak of such an
identity if a comparison is possible, and can be translated into a
'money of exchange' capable of transmission from one mind to
another. Nothing, therefore, will have objective value except
what is transmissible by 'discourse,' that is, intelligible.
But this is only one side of the question. An absolutely dis-
ordered aggregate could not have objective value since it would
be unintelligible, but no more can a well-ordered assemblage
have it. if it does not correspond to sensations really experienced.
It seems to me superfluous to recall this condition, and I should
not have dreamed of it, if it had not lately been maintained that
physics is not an experimental science. Although this opinion
has no chance of being adopted either by physicists or by phi-
losophers, it is well to be warned so as not to let oneself slip over
the declivity which would lead thither. Two conditions are
therefore to be fulfilled, and if the first separates reality' from
the dream, the second distinguishes it from the romance.
Now what is sciencet I have explained in the preceding
article, it is before all a classification, a manner of bringing
together facts which appearances separate, though they were
bound together by some natural and hidden kinship. Science,
in other words, is a system of relations. Now we have just said,
it is in the relations alone that objectivity must be sought; it
1 1 lt«re OM the nord real as ■ ajmoDj^m. of objective; I tlius eonfonn (o
eomaon nMge; perhaps I am wtong, our dreams are real, but the^r are not
objeetlTA.
860 THE VALUE OF SCIENCE
would be vain to seek it in beings considered as isolated from one
another.
To say that science can not have objective value since it teaches
us only relations, this is to reason backward, since, precisely, it
is relations alone which can be regarded as objective.
External objects, for instance, for which the word abject was
invented, are really objects and not fleeting and fugitive appear-
ances, because they are not only groups of sensations, but groups
cemented by a constant bond. It is this bond, and this bond
alone, which is the object in itself, and this bond is a relation.
Therefore, when we ask what is the objective value of science,
that does not mean: Does science teach us the true nature of
things? but it means: Does it teach us the true relations of
things T
To the first question, no one would hesitate to reply, no ; but I
think we may go farther; not. only science can not teach us the
nature of things ; but nothing is capable of teaching it to us, and
if any god knew it, he could not find words to express it. Not
only can we not divine the response, but if it were given to us
we could understand nothing of it; I ask myself even whether
we really understand the question.
When, therefore, a scientific theory pretends to teach us what
heat is, or what is electricity, or life, it is condemned beforehand;
all it can give us is only a crude image. It is, therefore, pro-
visional and crumbling.
The first question being out of reason, the second remains.
Can science teach us the true relations of things? What it joins
together should that be put asunder, what it puts asunder should
that be joined together?
To understand the meaning of this new question, it is needful
to refer to what was said above on the conditions of objectivity.
Have these relations an objective value? That means: Are
these relations the same for all ? Will they still be the same for
those who shall come after us ?
It is clear that they are not the same for the scientist and the
ignorant person. But that is unimportant, because if the ignorant
person does not see them all at once, the scientist may succeed in
making him see them by a series of experiments and reasonings.
SCIENCE AND REALITY
The thing essential ie that there are points on which all those
acquainted with the experiments made can reach accord.
The question is to know whether this accord will be durable and
whether it will persist for our successors. It may be asked
whether tlie unions that the science of to-day makes will be con-
Srmed by the science of to-morrow. To ftflarm that it will be so
we can not invoke any a priori reason ; but this is a question of
fact, and science has already lived long enough for us to he able
to find out by asking its history whether the edifices it builds
stand the test of time, or whether they are only ephemeral con-
Btructions.
Now what do we see I At the first blush it seems to us that the
theories last only a day and that ruins upon ruins accumulate.
To-day the theories are bom, to-morrow they are the fashion, the
day after to-morrow they are classic, the fourth day they are
superannuated, and the fifth they are forgotten- But if we look
more closely, we see that what thus succumb are the theories
properly so called, those which pretend to teach us what things
are. But there is in them something which usually survives.
If one of them taught us a true relation, this relation is defini-
tively acquired, and it will be found again under a new disguise
in the other theories which will successively come to reign in
place of the old.
Take only a single example: The theory of the undulations of
the ether taught ns that light is a motion ; to-day fashion favors
the electromagnetic theory which teaches us that light is a cur-
rent. We do not consider whether we could reconcile them and
say that light is a current, and that this current is a motion. As
it is probable in any ease that this motion would not be identical
with that which the partisans of the old theory presume, we might
think ourselves justified in saying that this old theory is de-
throned. And yet something of it remains, since between the
hypothetical currents which Maxwell supposes there arc the same
relations as between the hypothetical motions that Presnel sup-
posed. There is, therefore, something which remains over and
this something is the essential. This it is which explains how
we see the present physicists pass without any embarrassment
trom the language of Fresnel to that of Maxwell. Doubtless
862 TEE VALUE OF SCIENCE
many connections that were believed well established have been
abandoned, but the greatest number remain and it would seem
must remain.
And for these, then, what is the measure of their objectivity f
Well, it is precisely the same as for our belief in external objects.
These latter are real in this, that the sensations they make us feel
appear to us as united to each other by I know not what inde-
structible cement and not by the hazard of a day. In the same
way science reveals to us between phenomena other bonds finer
but not less solid; these are threads so slender that they long
remained unperceived, but once noticed there remains no way of
not seeing them ; they are therefore not less real than those which
give their reality to external objects ; small matter that they are
more recently known, since neither can perish before the other.
It may be said, for instance, that the ether is no less real than
any external body; to say this body exists is to say there is
between the color of this body, its taste, its smell, an intimate
bond, solid and persistent; to say the ether exists is to say there
is a natural kinship between all the optical phenomena, and
neither of the two propositions has less value than the other.
And the scientific syntheses have in a sense even more reality
than those of the ordinary senses, since they embrace more terms
and tend to absorb in them the partial syntheses.
It will be said that science is only a classification and that a
classification can not be true, but convenient. But it is true that
it is convenient, it is true that it is so not only for me, but for
all men ; it is true that it will remain convenient for our descend-
ants; it is true finally that this can not be by chance.
In sum, the sole objective reality consists in the relations of
things whence results the universial harmony. Doubtless these
relations, this harmony, could not be conceived outside of a mind
which conceives them. But they are nevertheless objective be-
cause they are, will become, or will remain, conunon to all think-
ing beings.
This will permit us to revert to the question of the rotation of
the earth which will give us at the same time a chance to make
clear what precedes by an example.
SCIENCE AND BEALITT 858
7. The Rotation of the Earth
**. . . Therefore," have I said in Science and Hypothesis,
''this affirmation, the earth turns round, has no meaning ... or
rather these two propositions, the earth turns round, and, it is
more convenient to suppose that the earth turns round, have one
and the same meaning."
These words have given rise to the strangest interpretations.
Some have thought they saw in them the rehabilitation of
Ptolemy's system, and perhaps the justification of Galileo's
condemnation.
Those who had read attentively the whole volume could not,
however, delude themselves. This truth, the earth turns round,
was put on the same footing as Euclid's postulate, for example.
Was that to reject itT But better; in the same language it may
very well be said: These two propositions, the external world
exists, or, it is more convenient to suppose that it exists, have one
and the same meaning. So the hypothesis of the rotation of the
earth would have the same degree of certitude as the very exist-
ence of external objects.
But after what we have just explained in the fourth part, we
may go farther. A physical theory, we have said, is by so much
the more true as it puts in evidence more true relations. In the
light of this new principle, let us examine the question which
occupies us.
No, there is no absolute space ; these two contradictory propo-
sitions: *The earth turns round' and 'The earth does not turn
round' are, therefore, neither of them more true than the other.
To affirm one while denying the other, in the kinematic sense,
would be to admit the existence of absolute space.
But if the one reveals true relations that the other hides from
us, we can nevertheless regard it as physically more true than the
other, since it has a richer content. Now in this regard no doubt
is possible.
Behold the apparent diurnal motion of the stars, and the
diurnal motion of the other heavenly bodies, and besides, the
flattening of the earth, the rotation of Foucault's pendulum, the
gyration of cyclones, the trade- winds, what not else? For the
24
364 THE VALUE OF SCIENCE
Ptolemaist all these phenomena have no bond between them; for
the Copemican they are produced by the one same cause. In
saying, the earth turns round, I afSrm that all these phenomena
have an intimate relation, and that is true, and that remains true,
although there is not and can not be absolute space.
So much for the rotation of the earth upon itself ; what shall we
say of its revolution around the sun T Here again, we have three
phenomena which for the Ptolemaist are absolutely independent
and which for the Copemican are referred back to the same
origin; they are the apparent displacements of the planets on
the celestial sphere, the aberration of the fixed stars, the parallax
of these same stars. Is it by chance that all the planets admit an
inequality whose period is a year, and that this period is precisely
equal to that of aberration, precisely equal besides to that of
parallax? To adopt Ptolemy's system is to answer, yes ; to adopt
that of Copernicus is to answer, no ; this is to affirm that there is
a bond between the three phenomena, and that also is true,
although there is no absolute space.
In Ptolemy's system, the motions of the heavenly bodies can
not be explained by the action of central forces, celestial
mechanics is impossible. The intimate relations that celestial
mechanics reveals to us between all the celestial phenomena are
true relations; to affirm the immobility of the earth would be to
deny these relations, that would be to fool ourselves.
The truth for which Galileo suffered remains, therefore, the
truth, although it has not altogether the same meaning as for
the vulgar, and its true meaning is much more subtle, more pro-
found and more rich.
8. Science for Its Own Sake
Not against M. LeRoy do I wish to defend science for its own
sake ; maybe this is what he condemns, but this is what he culti-
vates, since he loves and seeks truth and could not live without it.
But I have some thoughts to express.
We can not know all facts and it is necessary to choose those
which are worthy of being known. According to Tolstoi, scien-
tists make this choice at random, instead of making it, which
would be reasonable, with a view to practical applications. On
SCIENCE AND BEALITY 866
the contrary, scientists think that certain facts are more interest-
ing than others, because they complete an unfinished harmony,
or because they make one foresee a great number of other facts.
If they are wrong, if this hierarchy of facts that they implicitly
postulate is only an idle illusion, there could be no science for its
own sake, and consequently there could be no science. As for
me, I believe they are right, and, for example, I have shown above
what is the high value of astronomical facts, not because they
are capable of practical applications, but because they are the
most instructive of all.
It is only through science and art that civilization is of value.
Some have wondered at the formula: science for its own sake;
and yet it is as good as life for its own sake, if life is only misery ;
and even as happiness for its own sake, if we do not believe that
all pleasures are of the same quality, if we do not wish to admit
that the goal of civilization is to furnish alcohol to people who
love to drink.
Every act should have an aim. We must suffer, we must work,
we must pay for our place at the game, but this is for seeing 's
sake ; or at the very least that others may one day see.
All that is not thought is pure nothingness ; since we can think
only thoughts and all the words we use to speak of things can
express only thoughts, to say there is something other than
thought, is therefore an affirmation which can have no meaning.
And yet — strange contradiction for those who believe in time —
geologic history shows us that life is only a short episode between
two eternities of death, and that, even in this episode, conscious
thought has lasted and will last only a moment. Thought is only
a gleam in the midst of a long night.
But it is this gleam which is everything.
SCIENCE AND METHOD
4
- w
M
INTRODUCTION
I BRiNa together here different studies relating more or less
directly to questions of scientific methodology. The scientific
method consists in observing and experimenting ; if the scientist
had at his disposal infinite time, it would only be necessary to
say to him : ' Look and notice well ' ; but, as there is not time to
see everything, and as it is better not to see than to see wrongly,
it is necessary for him to make choice. The first question, there-
forC; is how he should make this choice. This question presents
itself as well to the physicist as to the historian; it presents
itself equally to the mathematician, and the principles which
should guide each are not without analogy. The scientist con-
forms to them instinctively, and one can, reflecting on these prin-
ciples, foretell the future of mathematics.
We shall understand them better yet if we observe the scien-
tist at work, and first of all it is necessary to know the xxsycho-
logic mechanism of invention and, in particular, that of mathe-
matical creation. Observation of the processes of the work of
the mathematician is particularly instructive for the psychologist.
In all the sciences of observation account must be taken of the
errors due to the imperfections of our senses and our instru-
ments. Luckily, we may assume that, under certain conditions,
these errors are in part self-compensating, so as to disappear in
the average; this compensation is due to chance. But what is
chance? This idea is difficult to justify or even to define; and
yet what I have just said about the errors of observation, shows
that the scientist can not neglect it. It therefore is necessary to
give a definition as precise as possible of this concept, so indis-
pensable yet so illusive.
These are generalities applicable in sum to all the sciences;
and for example the mechanism of mathematical invention does
not differ sensibly from the mechanism of invention in general.
Later I attack questions relating more particularly to certain
special sciences and first to pure mathematics.
359
8M SCIENCE AND METHOD
In the chapters devoted to these, I have to treat sabjects
a little more abstract I have first to speak of the notion of
space ; every one knows space is relative, or rather every one says
so, but many think still as if they believed it absolute ; it soffices
to reflect a little however to perceive to what contradictions they
are exposed.
The questions of teaching have their importance, first in them-
selves, then because reflecting on the best way to make new
ideas penetrate virgin minds is at the same time reflecting on
how these notions were acquired by our ancestors, and conse-
quently on their true origin, that is to say, in reality on their
true nature. Why do children usually understand nothing of
the definitions which satisfy scientists T Why is it necessary to
give them others f This is the question I set myself in the suc-
ceeding chapter and whose solution should, I think, suggest use-
ful reflections to the philosophers occupied with the logic of
the sciences.
On the other hand, many geometers believe we can reduce
mathematics to the rules of formal logic. Unheard-of efforts
have been made to do this; to accomplish it, some have not
hesitated, for example, to reverse the historic order of the genesis
of our conceptions and to try to explain the finite by the infinite.
I believe I have succeeded in showing, for all those who attack
the problem unprejudiced, that here there is a fallacious illusion.
I hope the reader will understand the importance of the question
and pardon me the aridity of the pages devoted to it.
The concluding chapters relative to mechanics and astronomy
will be easier to read.
Mechanics seems on the point of undergoing a complete revo-
lution. Ideas which appeared best established are assailed by
bold innovators. Certainly it would be premature to decide in
their favor at once simply because they are innovators.
But it is of interest to make known their doctrines, and this
is what I have tried to do. As far as possible I have followed
the historic order; for the new ideas would seem too astonish-
ing unless we saw how they arose.
Astronomy offers us majestic spectacles and raises gigantic
problems. We can not dream of applying to them directly the
INTBODUCTION 361
experimental method; our laboratories are too small. But anid-
ogy with phenomena these laboratories permit us to attain may
nevertheless guide the astronomer. The Milky Way, for ex-
ample, is an assemblage of suns whose movements seem at first
capricious. But may not this assemblage be compared to that of
the molecules of a gas, whose properties the kinetic theory of
gases has made known to usf It is thus by a roundabout way
that the method of the physicist may come to the aid of the
astronomer.
Finally I have endeavored to give in a few lines the history
of the development of French geodesy; I have shown through
what persevering efforts, and often what dangers, the geodesists
have procured for us the knowledge we have of the figure of the
earth. Is this then a question of method f Yes, without doubt,
this history teaches us in fact by what precautions it is necessary
to surround a serious scientific operation and how much time and
pains it costs to conquer one new decimal.
BOOK I
SCIENCE AND THE SCIENTIST
CHAPTEB I
The Choice op Pacts
Tolstoi somewhere explains why 'science for its own sake' is
in his eyes an absurd conception. We can not know all facts,
since their number is practically infinite. It is necessary to
choose; then we may let this choice depend on the pure caprice
of our curiosity ; would it not be better to let ourselves be guided
by utility, by our practical and above all by our moral needs ;
have we nothing better to do than to count the number of lady-
bugs on our planet T
It is clear the word utility has not for him the sense men of
affairs give it, and following them most of our contemporaries.
Little cares he for industrial applications, for the marvels of
electricity or of automobilism, which he regards rather as ob-
stacles to moral progress ; utility for him is solely what can make
man better.
For my part, it need scarce be said, I could never be content
with either the one or the other ideal ; I want neither that plutoc-
racy grasping and mean, nor that democracy goody and mediocre,
occupied solely in turning the other cheek, where would dwell
sages without curiosity, who, shunning excess, would not die of
disease, but would surely die of ennui. But that is a matter of
taste and is not what I wish to discuss.
The question nevertheless remains and should fix our attention ;
if our choice can only be determined by caprice or by immediate
utility, there can be no science for its own sake, and consequently
no science. But is that true? That a choice must be made is
incontestable ; whatever be our activity, facts go quicker than we,
and we can not catch them ; while the scientist discovers one fact,
362
THE CHOICE OF FACTS 363
there happen milliards of milliards in a cubic millimeter of his
body. To wish to comprise nature in science would be to want
to put the whole into the part.
But scientists believe there is a hierarchy of facts and that
among them may be made a judicious choice. They are right,
since otherwise there would be no science, yet science exists. One
need only open the eyes to see that the conquests of industry which
have enriched so many practical men would never have seen the
light, if these practical men alone had existed and if they had not
been preceded by unselfish devotees who died poor, who never
thought of utility, and yet had a guide far other than caprice.
As Mach says, these devotees have spared their successors the
trouble of thinking. Those who might have worked solely in
view of an immediate application would have left nothing behind
them, and, in face of a new need, all must have been begun over
again. Now most men do not love to think, and this is perhaps
fortunate when instinct guides them, for most often, when they
pursue an aim which is immediate and ever the same, instinct
guides them better than reason would guide a pure intelligence.
But instinct is routine, and if thought did not fecundate it, it
would no more progress in man than in the bee or ant. It is
needful then to think for those who love not thinking, and, as
they are numerous, it is needful that each of our thoughts be as
often useful as possible, and this is why a law will be the more
precious the more general it is.
This shows us how we should choose : the most interesting facts
are those which may serve many times ; these are the facts which
have a chance of coming up again. We have been so fortunate as
to be bom in a world where there are such. Suppose that in-
stead of 60 chemical elements there were 60 milliards of them,
that they were not some common, the others rare, but that they
were uniformly distributed. Then, every time we picked up a
new pebble there would be great probability of its being formed
of some unknown substance; all that we knew of other pebbles
would be worthless for it ; before each new object we should be
as the new-bom babe ; like it we could only obey our caprices or
our needs. Biologists would be just as much at a loss if there
were only individuals and no species and if heredity did not
make sons like their fathers.
864 SCIENCE AND METHOD
In such a world there would be no science; perhaps thought
and even life would be impossible, since evolution could not there
develop the preservational instincts. Happily it is not so; like
all good fortune to which we are accustomed, this is not appre-
ciated at its true worth.
Which then are the facts likely to reappear T They are first
the simple facts. It is clear that in a complex fact a thousand
circumstances are united by chance, and that only a chance still
much less probable could reunite them anew. But are there any
simple facts T And if there are, how recognize themT What
assurance is there that a thing we think simple does not hide a
dreadful complexity? All we can say is that we ought to prefer
the facts which seem simple to those where our crude eye discerns
unlike elements. And then one of two things : either this simplic-
ity is real, or else the elements are so intimately mingled as not
to be distinguishable. In the first case there is chance of our
meeting anew this same simple fact, either in all its purity or
entering itself as element in a complex manifold. In the second
case this intimate mixture has likewise more chances of recurring
than a heterogeneous assemblage; chance knows how to mix, it
knows not how to disentangle, and to make with multiple elements
a well-ordered edifice in which something is distinguishable, it
must be made expressly. The facts which appear simple, even
if they are not so, will therefore be more easily revived by chance.
This it is which justifies the method instinctively adopted by the
scientist, and what justifies it still better, perhaps, is that oft-
recurring facts appear to us simple, precisely because we are
used to them.
But where is the simple fact? Scientists have been seeking
it in the two extremes, in the infinitely great and in the infinitely
small. The astronomer has found it because the distances of
the stars are immense, so great that each of them appears but
as a point, so great that the qualitative differences are effaced,
and because a point is simpler than a body which has form and
qualities. The physicist on the other hand has sought the ele-
mentary phenomenon in fictively cutting up bodies into infinites-
imal cubes, because the conditions of the problem, which undergo
slow and continuous variation in passing from one point of the
THE CHOICE OF FACTS 365
body to another, may be regarded as constant in the interior of
each of these little cubes. In the same way the biologist has
been instinctively led to regard the cell as more interesting than
the whole animal, and the outcome has shown his wisdom, since
cells belonging to organisms the most different are more alike,
for the one who can recognize their resemblances, than are these
organisms themselves. The sociologist is more embarrassed ; the
elements, which for him are men, are too unlike, too variable, too
capricious, in a word, too complex ; besides, history never begins
over again. How then choose the interesting fact, which is that
which begins again T Method is precisely the choice of facts; it
is needful then to be occupied first with creating a method, and
many have been imagined, since none imposes itself, so that so-
ciology is the science which has the most methods and the fewest
results.
Therefore it is by the regular facts that it is proper to begin ;
but after the rule is well established, after it is beyond all doubt,
the facts in fidl conformity with it are erelong without interest
since they no longer teach us anything new. It is then the ex-
ception which becomes important. We cease to seek resem-
blances; we devote ourselves above all to the differences, and
among the differences are chosen first the most accentuated, not
only because they are the most striking, but because they will
be the most instructive. A simple example will make my thought
plainer : Suppose one wishes to determine a curve by observing
some of its points. The practician who concerns himself only
with immediate utility would observe only the points he might
need for some special object. These points would be badly dis-
tributed on the curve ; they would be crowded in certain regions,
rare in others, so that it would be impossible to join them by a
continuous line, and they would be unavailable for other applica-
tions. The scientist will proceed differently; as he wishes to
study the curve for itself, he will distribute regularly the points
to be observed, and when enough are known he will join them
by a regular line and then he will have the entire curve. But
for that how does he proceed ? If he has determined an extreme
point of the curve, he does not stay near this extremity, but goes
first to the other end ; after the two extremities the most instruc-
tive point will be the mid-point, and so on.
866 SCIENCE AND METHOD
So when a rule is established we should first seek the cases
where this rule has the greatest chance of failing. Thence,
among other reasons, come the interest of astronomic facts, and
the interest of the geologic past ; by going very far away in space
or very far away in time, we may find our usual rules entirely
overturned, and these grand overtumings aid us the better to see
or the better to understand the little changes which may happen
nearer to us, in the little comer of the world where we are called
to Uve and act. We shall better know this comer for having
traveled in distant countries with which we have nothing to do.
But what we ought to aim at is less the ascertainment of resem-
blances and differences than the recognition of likenesses hidden
under apparent divergences. Particular rules seem at first dis-
cordant, but looking more closely we see in general that they
resemble each other; different as to matter, they are alike as to
form, as to the order of their parts. When we look at them with
this bias, we shall see them enlarge and tend to embrace every-
thing. And this it is which makes the value of certain facts
which come to complete an assemblage and to show that it is the
faithful image of other known assemblages.
I will not further insist, but these few words suffice to show
that the scientist does not choose at random the facte he observes.
He does not, as Tolstoi says, count the lady-bugs, because, how-
ever interesting lady-bugs may be, their number is subject to
capricious variations. He seeks to condense much experience
and much thought into a slender volume ; and that is why a little
book on physics contains so many past experiences and a thou-
sand times as many possible experiences whose result is known
beforehand.
But we have as yet looked at only one side of the question.
The scientist does not study nature because it is useful ; he studies
it because he delights in it, and he delights in it because it is
beautiful. If nature were not beautiful, it would not be worth
knowing, and if nature were not worth knowing, life would not
be worth living. Of course I do not here speak of that beauty
which strikes the senses, the beauty of qualities and of appear-
ances ; not that I undervalue such beauty, far from it, but it has
nothing to do with science ; I mean that prof ounder beauty which
THE CHOICE OF FACTS 367
comes from the harmonious order of the parts and which a pure
intelligence can grasp. This it is which gives body, a structure
so to speak, to the iridescent appearances which flatter our senses,
and without this support the beauty of these fugitive dreams
would be only imperfect, because it would be vague and always
fleeting. On the contrary, intellectual beauty is sufficient unto
itself, and it is for its sake, more perhaps than for the future
good of humanity, that the scientist devotes himself to long and
difficult labors.
It is, therefore, the quest of this especial beauty, the sense of
the harmony of the cosmos, which makes us choose the facts
most fitting to contribute to this harmony, just as the artist
chooses from among the features of his model those which perfect
the picture and give it character and life. And we need not
fear that this instinctive and unavowed prepossession will turn
the scientist aside from the search for the true. One may dream
a harmonious world, but how far the real world will leave it
behind! The greatest artists that ever lived, the Greeks, made
their heavens ; how shabby it is beside the true heavens, ours !
And it is because simplicity, because grandeur, is beautiful,
that we preferably seek simple facts, sublime facts, that we de-
light now to follow the majestic course of the stars, now to ex-
amine with the microscope that prodigious littleness which is
also a grandeur, now to seek in geologic time the traces of a past
which attracts because it is far away.
We see too that the longing for the beautiful leads us to the
same choice as the longing for the useful. And so it is that this
economy of thought, this economy of effort, which is, according
to Mach, the constant tendency of science, is at the same time
a source of beauty and a practical advantage. The edifices that
we admire are those where the architect has known how to pro-
portion the means to the end, where the columns seem to carry
gaily, without effort, the weight placed upon them, like the
gracious caryatids of the Erechtheum.
Whence comes this concordance f Is it simply that the things
which seem to us beautiful are those which best adapt themselves
to our intelligence, and that consequently they are at the same
time the implement this intelligence knows best how to usef
868 SCIENCE AND METHOD
Or 18 there here a play of evolutioii and natural adeetionf Have
the peoples whoee ideal most conformed to their highest interest
exterminated the others and taken their place f All puraaed
their ideals without reference to consequences, but while this
quest led some to destruction, to others it gave empire. One is
tempted to belieye it If the Oreeks triumphed over the bar-
barians and if Europe, heir of Greek thought, dominates the
world, it is because the savages loved loud colors and the clamor-
ous tones of the drum which occupied only their senses, while the
Greeks loved the intellectual beauty which hides beneath sen-
suous beauty, and this intellectual beauty it is which makes in-
telligence sure and strong.
Doubtless such a triumph would horrify Tolstoi, and he would
not like to acknowledge that it might be tmly usef uL But this
disinterested quest of the true for its own beauty is sane also and
able to make man better. I well know that there are mistakes,
that the thinker does not always draw thence the serenity he
should find therein, and even that there are scientists of bad
character. Must we, therefore, abandon science and study only
morals f What ! Do you think the moralists themselves are irre-
proachable when they come down from their pedestal f
CHAPTBE II
The Futube of Mathematics
To foresee the future of mathematics, the true method is to
study its history and its present state.
Is this not for us mathematicians in a way a professional pro-
cedure f We are accustomed to exirapoUrie, which is a means
of deducing the future from the past and present, and as we well
know what this amounts to, we run no risk of deceiving ourselves
about the range of the results it gives us.
We have had hitherto prophets of evil. They blithely reiterate
that all problems capable of solution have already been solved,
and that nothing is left but gleaning. Happily the case of the
past reassures us. Often it was thought all problems were solved
or at least an inventory was made of all admitting solution.
And then the sense of the word solution enlarged, the insoluble
problems became the most interesting of all, and others unfore-
seen presented themselves. For the Greeks a good solution was
one employing only ruler and compasses; then it became one
obtained by the extraction of roots, then one using only algebraic
or logarithmic functions. The pessimists thus found themselves
always outflanked, always forced to retreat, so that at present I
think there are no more.
My intention, therefore, is not to combat them, as they are
dead ; we well know that mathematics will continue to develop,
but the question is how, in what direction? You will answer,
'in every direction,' and that is partly true; but if it were
wholly true it would be a little appalling. Our riches would
soon become encumbering and their accumulation would produce
a medley as impenetrable as the unknown true was for the
ignorant.
The historian, the physicist, even, must make a choice among
facts; the head of the scientist, which is only a comer of the
universe, could never contain the universe entire ; so that among
the innumerable facts nature offers, some will be passed by,
others retained.
25 369
370 SCIENCE AND METHOD
Jujst so, a fortiori, in mathenuitics ; no more can the geometer
hold fast pell-mell all the facts presenting themselyes to him;
all the more becanae he it is, almost I had said his caprice, that
creates these facts. He constructs a wholly new combination by
patting together its elements; nature does not in general give it
to him ready made.
Doubtless it sometimes happens that the mathematician under-
takes a problem to satisfy a need in physics; that the physicist
or engineer asks him to calculate a number for a certain applica-
tion. Shall it be said that we geometers should limit ourselyes
to awaiting orders, and, in place of cultivating our science for
our own delectation, try only to accommodate ourselves to the
wants of our patrons? If mathematics has no other object be-
sides aiding those who study nature, it is from these we should
await orders. Is this way of looking at it legitimate f Certainly
not; if we had not cultivated the exact sciences for themselves,
we should not have created mathematics the instrument^ and the
day the call came from the physicist we should have been
helpless.
Nor do the physicists wait to study a phenomenon until some
urgent need of material life has made it a necessity for them;
and they are right. If the scientists of the eighteenth century
had neglected electricity as being in their eyes only a curiosity
without practical interest, we should have had in the twentieth
century neither telegraphy, nor electro-chemistry, nor electro-
technics. The physicists, compelled to choose, are therefore not
guided in their choice solely by utility. How then do they choose
between the facts of nature ? We have explained it in the pre-
ceding chapter: the facts which interest them are those capable
of leading to the discovery of a law, and so they are analogous
to many other facts which do not seem to us isolated, but closely
grouped with others. The isolated fact attracts all eyes, those of
the layman as well as of the scientist. But what the genuine
physicist alone knows how to see, is the bond which unites many
facts whose analogy is profound but hidden. The story of New-
ton's apple is probably not true, but it is symbolic; let us speak
of it then as if it were true. Well then, we must believe that
before Newton plenty of men had seen apples fall ; not one knew
TEE FUTURE OF MATHEMATICS 371
how to coDclude anything therefrom. Facts would be fiterile
were there not minds capable of choosing among them, discern-
mg those behind which something was hidden, and of recognizing
what is hiding, minds which under the cmde fact perceive the
eool of the fact.
We find just the same thing in mathematics. From the varied
elements at our disposal we can get millions of different com-
binations ; but one of these combinations, in so far as it is isolated,
is absolutely void of value. Often we have taken great pains to
construct it, but it serves no purpose, if not perhaps to furnish a
task in secondary education. Quite otherwise will it be when
thi>t combination shall find place in a class of analogous combina-
tions and we shall have noticed this analogy. We are no longer
in the presence of a fact, but of a law. And upon that day the
real discoverer will not be the workman who shall have patiently
built up certain of these combinations; it will be he who brings
to light their kinship. The first will have seen merely the crude
fact, only the other will have perceived the soul of the fact.
Often to fix this kinship it suffices him to make a new word, and
this word is creative. The history of science furnishes us a
crowd of examples familiar to all.
The celebrated Vienna philosopher Maeh has said that the r61e
of science is to produce economy of thought, just as machines
produce economy of effort. And that is very true. The savage
reckons on his fingers or by heaping pebbles. In teaching chil-
dren the multiplication table we spare them later innumerable
pebble bunchings. Some one has already found out, with pebbles
or otherwise, that 6 times 7 is 42 and lias had the idea of noting
the result, and so we need not do it over again. He did not
waste his time even if he reckoned for pleasure: his operation
took him only two minutes ; it would have taken in all two mil-
liards if a milliard men had had to do it over after him.
The importance of a fact then is measured by its yield, that is
to say, by the amount of thought it permits ua to spare.
In physics the facts of great yield are those entering into a
very general law, since from it they enable us to foresee a great
number of others, and just so it is in mathematics. Suppose I
bare undertaken a complicated calculation and laborioualjp
372 SCIENCE AND METHOD
reached a result: I shall not be compensated for my trouble if
thereby I have not become capable of foreseeing the results of
other analogous calculations and guiding them with a certainty
that avoids the gropings to which one must be resigned in a
fiist attempt. On the other hand, I shall not have wasted my
time if these gropings themselves have ended by revealing to me
the profound analogy of the problem just treated with a much
more extended class of other problems; if they have shown me
at once the resemblances and differences of these, if in a word
they have made me perceive the possibility of a generalization.
Then it is not a new result I have won, it is a new power.
The simple example that comes first to mind is that of an alge-
braic formula which gives us the solution of a type of numeric
problems when finally we replace the letters by numbers. Thanks
to it, a single algebraic calculation saves us the pains of cease-
lessly beginning over again new numeric calculations. But this
is only a crude example; we all know there are analogies inex-
pressible by a formula and all the more precious.
A new result is of value, if at all, when in unifying elements
long known but hitherto separate and seeming strangers one to
another it suddenly introduces order where apparently disorder
reigned. It then permits us to see at a glance each of these
elements and its place in the assemblage. This new fact is not
merely precious by itself, but it alone gives value to all the old
facts it combines. Our mind is weak as are the senses ; it would
lose itself in the world's complexity were this complexity not har-
monious ; like a near-sighted person, it would see only the details
and would be forced to forget each of these details before exam-
ining the following, since it would be incapable of embracing all.
The only facts worthy our attention are those which introduce
order into this complexity and so make it accessible.
Mathematicians attach great importance to the elegance of
their methods and their results. This is not pure dilettantism.
What is it indeed that gives us the feeling of elegance in a solu-
tion, in a demonstration 1 It is the harmony of the diverse parts,
their symmetry, their happy balance; in a word it is all that
introduces order, all that gives unity, that permits us to see
clearly and to comprehend at once both the ensemble and the
details. But this is exactly what yielda great results; in fact the
more we see this aggregate clearly and at a single glance, the
better we perceive its analogies with other neighboring objects,
consequently the more chances we have of divining the possible
generalizations. Elegance may produce the feeling of the unfore-
seen by the unexpected meeting of objects we are not accustomed
to bring together; there again it is fruitful, since it thus unveils
for us kinships before unrecognized. It is fruitful even when it
results only from the contrast between the simplicity of the
means and the complexity of the problem set ; it makes us then
think of the reason for this contrast and very often makes t]s
see that chance ia not the reason ; that it is to be found in some
unexpected law. In a word, the feeling of mathematical ele-
gance is only the satisfaction due to any adaptation of the solu-
tion to the needs of our mind, and it is because of this very
adaptation that this solution can be for us an instrument. Con-
sequently this esthetic satisfaction is bound up with the econ-
omy of thought. Again the comparison of the Erechthcum
comes to my mind, but I must not use it too often.
It is for the same reason that, when a rather long calculation
has led to some simple and striking result, we are not satisfied
until we have shown that we should have been able to foresee,
if not this entire result, at least its most characteristic traits.
Why I "What prevents our beii^ content with a calculation
which has told us, it seems, all we wished to knowT It is be-
cause, in analogous eases, the long calculation might not again
avail, and that this is not so about the reasoning often half in-
tuitive which would have enabled us to foresee. This reasoning
being short, we see at a single glance all its parts, so that we im-
mediately perceive what must be changed to adapt it to all the
problems of the same nature which can occur. And then it
enables as to foresee if the solution of these problems will be
simple, it shows us at least if the calculation is worth under-
taking.
What we have just said suffices to show how vain it would be
to seek to replace by any mechanical procedure the free initiative
of the mathematician. To obtain a result of real value, it is not
enough to grind out calculations, or to have a machine to put
874 SCIENCE AND METHOD
things in order; it is not order alone, it is unexpected order,
which is worth while. The machine may gnaw on the crude fact,
the soul of the fact will always escape it.
Since the middle of the last century, mathematicians are more
and more desirous of attaining absolute rigor; they are right,
and this tendency will be more and more accentuated. In math-
ematics rigor is not everything, but without it there is nothing.
A demonstration which is not rigorous is nothingness. I think
no one will contest this truth. But if it were taken too literally,
we should be led to conclude that before 1820, for example, there
was no mathematics; this would be manifestly excessive; the
geometers of that time understood voluntarily what we explain
by prolix discourse. This does not mean that they did not see it
at all ; but they passed over it too rapidly, and to see it well would
have necessitated taking the pains to say it.
But is it always needful to say it so many times; those who
were the first to emphasize exactness before all else have given
us arguments that we may try to imitate ; but if the demonstra-
tions of the future are to be built on this model, mathematical
treatises will be very long; and if I fear the lengthenings, it is
not solely because I deprecate encumbering libraries, but because
I fear that in being lengthened out, our demonstrations may lose
that appearance of harmony whose usefulness I have just
explained.
The economy of thought is what we should aim at, so it is not
enough to supply models for imitation. It is needful for those
after us to be able to dispense with these models and, in place of
repeating an argument already made, summarize it in a few
words. And this has already been attained at times. For in-
stance, there was a type of reasoning found everywhere, and
everywhere alike. They were perfectly exact but long. Then
all at once the phrase * uniformity of convergence ' was hit upon
and this phrase made those arguments needless; we were no
longer called upon to repeat them, since they could be under-
stood. Those who conquer difficulties then do us a double service :
first they teach us to do as they at need, but above all they
enable us as often as possible to avoid doing as they, yet without
sacrifice of exactness.
THE FUTURE OF MATHEMATICS 375
We have just seen by one example the importance of words in
mathematics, but many others could be cited. It is hard to be-
lieve how much a well-chosen word can economize thought, as
Mach says. Perhaps I have already said somewhere that mathe-
matics is the art of giving the same name to different things. It
is proper that these things, differing in matter, be alike in
form, that they may, so to speak, run in the same mold. When
the language has been well chosen, we are astonished to see that
all the proofs made for a certain object apply immediately to
many new objects ; there is nothing to change, not even the words,
since the names have become the same.
A well-chosen word usually suffices to do away witii the ex-
ceptions from which the rules stated in the old way suffer; this
is why we have created negative quantities, imaginaries, points
at infinity, and what not. And exceptions, we must not forget,
are pernicious because they hide the laws.
Well, this is one of the characteristics by which we recognize
the facts which yield great results. They are those which allow
of these happy innovations of language. The crude fact then
is often of no great interest; we may point it out many times
without having rendered great service to science. It takes value
only when a wiser thinker perceives the relation for which it
stands, and symbolizes it by a word.
Moreover the physicists do just the same. They have in-
vented the word 'energy,* and this word has been prodigiously
fruitful, because it also made the law by eliminating the excep-
tions, since it gave the same name to things differing in matter
and like in form.
Among words that have had the most fortunate influence I
would select * group' and * invariant.' They have made us see
the essence of many mathematical reasonings; they have shown
us in how many cases the old mathematicians considered groups
without knowing it, and how, believing themselves far from one
another, they suddenly found themselves near without knowing
why.
To-day we should say that they had dealt with isomorphic
groups. We now know that in a group the matter is of little
interest, the form alone counts, and that when we know a group
876 SCIENCE AND METHOD
we thus know all the isomorphic groups; and thanks to these
words 'group' and isomorphism,' which condense in a few syl-
lables this subtile rule and quickly make it familiar to all minds,
the transition is immediate and can be done with every econ-
omy of thought effort. The idea of group besides attaches to that
of transformation. Why do we put such a value on the in-
vention of a new transformation f Because from a single the-
orem it enables us to get ten or twenty; it has the same value as
a zero adjoined to the right of a whole number.
This then it is which has hitherto determined the direction of
mathematical advance, and just as certainly will determine it in
the future. But to this end the nature of the problems which
come up contributes equally. We can not forget what must be
our aim. In my opinion this aim is double. Our science borders
upon both philosophy and physics, and we work for our two
neighbors; so we have always seen and shall still see mathema-
ticians advancing in two opposite directions.
On the one hand, mathematical science must reflect upon itself,
and that is useful since reflecting on itself is reflecting on the
human mind which has created it, all the more because it is the
very one of its creations for which it has borrowed least from
without. This is why certain mathematical speculations are
useful, such as those devoted to the study of the postulates, of
unusual geometries, of peculiar functions. The more these spec-
ulations diverge from ordinary conceptions, and consequently
from nature and applications, the better they show us what the
human mind can create when it frees itself more and more from
the tyranny of the external world, the better therefore they let
us know it in itself.
But it is toward the other side, the side of nature, that we must
direct the bulk of our army. There we meet the physicist or
the engineer, who says to us: ** Please integrate this differential
equation for me ; I might need it in a week in view of a construc-
tion which should be finished by that time." **This equation,"
we answer, **does not come under one of the integrable types;
you know there are not many." **Yes, I know; but then what
good are you?" Usually to understand each other is enough;
the engineer in reality does not need the integral in finite terms;
THE FUTURE OF MATHEMATICS 377
he needs to know the general look of the integral function, or he
simply wants a certain nmnher which could readily be deduced
from this integral if it were known. Usually it is not known,
but the number can be calculated without it if we know exactly
what number the engraeer needs and with what approximation.
Formerly an equation was considered solved only when its
solution had been expressed by aid of a finite number of known
functions ; but that is possible scarcely once in a hundred times.
What we always can do, or rather what we should always seek
to do, is to solve the problem qtuilitatively so to speak; that is to
say, seek to know the general form of the curve which represents
the unknown function.
It remains to find the quantitaiive solution of the problem;
but if the unknown can not be determined by a finite calculation,
it may always be represented by a convergent infinite series
vhich enables us to calculate it. Can that be regarded as a true
solution? We are told that Newton sent Leibnitz an anagram
almost like this: aaaaabbbeeeeii, etc. Leibnitz naturally under-
stood nothing at all of it ; but we, who have the key, know that
this anagram meant, translated into modem terms: '^I can inte-
grate all differential equations ' ' ; and we are tempted to say that
Newton had either great luck or strange delusions. He merely
wished to say he could form (by the method of indeterminate
coefiScients) a series of powers formally satisfying the proposed
equation.
Such a solution would not satisfy us to-day, and for two
reasons: because the convergence is too slow and because the
terms follow each other without obeying any law. On the con-
trary, the series seems to us to leave nothing to be desired, first
because it converges very quickly (this is for the practical man
who wishes to get at a number as quickly as possible) and next
because we see at a glance the law of the terms (this is to satisfy
the esthetic need of the theorist).
But then there are no longer solved problems and others
which are not; there are only problems more or less solved,
according as they are solved by a series converging more or less
rapidly, or ruled by a law more or less harmonious. It often
happens however that an imperfect solution guides us toward a
878 SCIENCE AND METHOD
better one. Sometimes the series converges so slowly that the
computation is impracticable and we have only succeeded in
proving the possibility of the problem.
And then the engineer finds this a mockery, and justly, since
it will not aid him to complete his construction by the date fixed.
He little cares to know if it will benefit engineers of the twenty-
second century. But as for us, we think differently and we arc
sometimes happier to have spared our grandchildren a day's
work than to have saved our contemporaries an hour.
Sometimes by groping, empirically, so to speak, we reach a
formula sufficiently convergent. **What more do you wantt"
says the engineer. And yet, in spite of all, we are not satisfied;
we should have liked to foresee that convergence. Why? Be-
cause if we had known how to foresee it once, we woidd know how
to foresee it another time. We have succeeded ; that is a smaU
matter in our eyes if we can not validly expect to do so again.
In proportion as science develops, its total comprehension
becomes more difficult; then we seek to cut it in pieces and to
be satisfied with one of these pieces: in a word, to specialize.
If we went on in this way, it would be a grievous obstacle to the
progress of science. As we have said, it is by unexpected union
between its diverse parts that it progresses. To specialize too
much would be to forbid these drawings together. It is to be
hoped that congresses like those of Heidelberg and Rome, by
putting us in touch with one another, will open for us vistas over
neighboring domains and oblige us to compare them with our
own, to range somewhat abroad from our own little village ; thus
they will be the best remedy for the danger just mentioned.
But I have lingered too long over generalities; it is time to
enter into detail.
Let us pass in review the various special sciences which com-
bined make mathematics ; let us see what each has accomplished,
whither it tends and what we may hope from it. If the pre-
ceding views are correct, we should see that the greatest advances
in the past have happened when two of these sciences have united,
when we have become conscious of the similarity of their form,
despite the difference of their matter, when they have so modeled
themselves upon each other that each could profit by the other's
THE FUTURE OF MATHEMATICS 379
conquests. We should at the same time foresee in combinations
of the same sort the progress of the future.
Abithmetio
Progress in arithmetic has been much slower than in algebra
and analysis, and it is easy to see why. The feeling of continuity
is a precious guide which the arithmetician lacks; each whole
number is separated from the others, — ^it has, so to speak, its own
individuality. Each of them is a sort of exception and this is
why general theorems are rarer in the theory of numbers; this
is also why those which exist are more hidden and longer elude
the searchers.
If arithmetic is behind algebra and analysis, the best thing for
it to do is to seek to model itself upon these sciences so as to
profit by their advance. The arithmetician ought therefore to
take as guide the analogies with algebra. These analogies are
numerous and if, in many cases, they haVe not yet been studied
sufficiently closely to become utilizable, they at least have long
been foreseen, and even the language of the two sciences shows
they have been recognized. Thus we speak of transcendent
numbers and thus we account for the future classification of
these numbers already having as model the classification of tran-
scendent functions, and still we do not as yet very well see how
to pass from one classification to the other; but had it been seen,
it would already have been accomplished and would no longer
be the work of the future.
The first example that comes to my mind is the theory of con-
gruences, where is found a perfect parallelism to the theory of
algebraic equations. Surely we shall succeed in completing this
parallelism, which must hold for instance between the theory of
algebraic curves and that of congruences with two variables.
And when the problems relative to congruences with several
variables shall be solved, this will be a first step toward the solu-
tion of many questions of indeterminate analysis.
Algebra
The theory of algebraic equations will still long hold the atten-
tion of geometers; numerous and very different are the sides
whence it may be attacked.
380 SCIENCE AND METHOD
We need not think algebra is ended because it gives ns roles
to form all possible combinations ; it remains to find the interest-
ing combinations, those which satisfy such and such a condition.
Thus will be formed a sort of indeterminate analysis where the
unknowns will no longer be whole numbers, but polynomials.
This time it is algebra which will model itself upon arithmetic,
following the analogy of the whole number to the integ^ral poly-
nomial with any coefiScients or to the integral polynomial with
integral coefiScients.
Gbombtby
It looks as if geometry could contain nothing which is not
already included in algebra or analysis; that geometric facts are
only algebraic or analytic facts expressed in another language.
It might then be thought that after our review there would
remain nothing more for us to say relating specially to geometry.
This would be to fail to recognize the importance of well-oon-
strueted language, not to comprehend what is added to the things
themselves by the method of expressing these things and conse-
quently of grouping them.
First the geometric considerations lead us to set ourselves new
problems; these may be, if you choose, analytic problems, but
such as we never would have set ourselves in connection with
analysis. Analysis profits by them however, as it profits by those
it has to solve to satisfy the needs of physics.
A great advantage of geometry lies in the fact that in it the
senses can come to the aid of thought, and help find the path to
follow, and many minds prefer to put the problems of analysis
into geometric form. Unhappily our senses can not carry us very
far, and they desert us when we wish to soar beyond the classical
three dimensions. Does this mean that, beyond the restricted
domain wherein they seem to wish to imprison us, we should
rely only on pure analysis and that all geometry of more than
three dimensions is vain and objectless? The greatest masters
of a preceding generation would have answered *yes* ; to-day we
are so familiarized with this notion that we can speak of it, even
in a university course, without arousing too much astonishment
But what good is it T That is easy to see : First it gives us a
TEE FUTURE OF MATHEMATICS 881
very convenient terminology, which expresses concisely what the
ordinary analytic language would say in prolix phrases. More-
over, this language makes us call like things by the same name
and emphasize analogies it will never again let us forget. It
enables us therefore still to find our way in this space which is
too big for us and which we can not see, always recalling visible
space, which is only an imperfect image of it doubtless, but which
is nevertheless an image. Here again, as in all the preceding
examples, it is analogy with the simple which enables us to com-
prehend the complex.
This geometry of more than three dimensions is not a simple
analytic geometry; it is not purely quantitative, but qualitative
also, and it is in this respect above all that it becomes interesting.
There is a science called analysis situs and which has for its
object the study of the positional relations of the different ele-
ments of a figure, apart from their sizes. This geometry is purely
qualitative ; its theorems would remain true if the figures, instead
of being exact, were roughly imitated by a child. We may also
make an analysis situs of more than three dimensions. The
importance of analysis situs is enormous and can not be too much
emphasized ; the advantage obtained from it by Riemann, one of
its chief creators, would suffice to prove this. We must achieve
its complete construction in the higher spaces ; then we shall have
an instrument which will enable us really to see in hyperspace
and supplement our senses.
The problems of analysis situs would perhaps not have sug-
gested themselves if the analytic language alone had been spoken ;
or rather, I am mistaken, they would have occurred surely, since
their solution is essential to a crowd of questions in analysis, but
they would have come singly, one after another, and without our
being able to perceive their common bond.
Cantorism
I have spoken above of our need to go back continually to the
first principles of our science, and of the advantage of this for
the study of the human mind. This need has inspired two en-
deavors which have taken a very prominent place in the most
recent annals of mathematics. The first is Cantorism, which has
382 SCIENCE AND METHOD
rendered our science such conspicuous service. Cantor intro-
duced into science a new way of considering mathematical in-
finity. One of the characteristic traits of Cantorism is that in
place of going up to the general by building up constructions
more and more complicated and defining by construction, it starts
from the genus supremum and defines only, as the scholastics
would have said, per genus proximum et differentiam spedficam.
Thence comes the horror it has sometimes inspired in certain
minds, for instance in Hermite, whose favorite idea was to com-
pare the mathematical to the natural sciences. With most of
us these prejudices have been dissipated, but it has come to
pass that we have encountered certain paradoxes, certain appar-
ent contradictions that would have delighted Zeno the Eleatic
and the school of Megara. And then each must seek the remedy.
For my part, I think, and I am not the only one, that the impor-
tant thing is never to introduce entities not completely definable
in a finite number of words. Whatever be the cure adopted, we
may promise ourselves the joy of the doctor called in to follow
a beautiful pathologic case.
The Investigation of the Postulates
On the other hand, efforts have been made to enumerate the
axioms and postulates, more or less hidden, which serve as foun-
dation to the different theories of mathematics. Professor Hilbert
has obtained the most brilliant results. It seems at first that this
domain would be very restricted and there would be nothing
more to do when the inventory should be ended, which could not
take long. But when we shall have enumerated all, there will be
many ways of classifying all ; a good librarian always finds some-
thing to do, and each new classification will be instructive for
the philosopher.
Here I end this review which I could not dream of making
complete. I think these examples will suffice to show by what
mechanism the mathematical sciences have made their progress
in the past and in what direction they must advance in the future.
CHAPTER III
Mathematioal Cbeation
The genesis of mathematical creation is a problem which
should intensely interest the psychologist. It is the activity in
which the human mind seems to take least from the outside
world, in which it acts or seems to act only of itself and on itself,
so that in studying the procedure of geometric thought we may
hope to reach what is most essential in man's mind.
This has long been appreciated, and some time back the journal
called L^enseignement mathematique, edited by Laisant and
Fehr, began an investigation of the mental habits and methods
of work of different mathematicians. I had finished the main
outlines of this article when the results of that inquiry were
published, so I have hardly been able to utilize them and shall
confine myself to saying that the majority of witnesses confirm
my conclusions; I do not say all, for when the appeal is to uni-
versal suffrage unanimity is not to be hoped.
A first fact should surprise us, or rather would surprise us if
we were not so used to it. How does it happen there are people
who do not understand mathematics? If mathematics invokes
only the rules of logic, such as are accepted by all normal minds;
if its evidence is based on principles common to all men, and that
none could deny without being mad, how does it come about that
so many persons are here refractory t
That not every one can invent is nowise mysterious. That
not every one can retain a demonstration once learned may also
pass. But that not every one can understand mathematical
reasoning when explained appears very surprising when we think
of it. And yet those who can follow this reasoning only with
difficulty are in the majority : that is undeniable, and will surely
not be gainsaid by the experience of secondary-school teachers.
And further: how is error possible in mathematics t A sane
mind should not be guilty of a logical fallacy, and yet there are
383
384 SCIENCE AND METHOD
Yery fine minds who do not trip in brief reasoning such as oocors
in the ordinary doings of life, and who are incapable of follow-
ing or repeating without error the mathematical demonstrations
which are longer, but which after all are only an acoumnlation
of brief reasonings wholly analogous to those they make so easily.
Need we add that mathematicians themselves are not infallible t
The answer seems to me evident. Imagine a long series of
syllogisms, and that the conclusions of the first serve as premises
of the following: we shall be able to catch each of these syllo-
gisms, and it is not in passing from premises to conclusion that
we are in danger of deceiving ourselves. But between the
moment in which we first meet a proposition as conclusion of one
syllogism, and that in which we reencounter it as premise of
another syllogism occasionally some time will elapse, several links
of the chain will have unrolled ; so it may happen that we have
forgotten it, or worse, that we have forgotten its meaning. So
it may happen that we replace it by a slightly different propo-
sition, or that, while retaining the same enunciation, we attribute
to it a slightly different meaning, and thus it is that we are
exposed to error.
Often the mathematician uses a rule. Naturally he begins by
demonstrating this rule ; and at the time when this proof is fresh
in his memory he understands perfectly its meaning and its bear-
ing, and he is in no danger of changing it. But subsequently he
trusts his memory and afterward only applies it in a mechanical
way; and then if his memory fails him, he may apply it all
wrong. Thus it is, to take a simple example, that we sometimes
make slips in calculation because we have forgotten our multi-
plication table.
According to this, the special aptitude for mathematics would
be due only to a very sure memory or to a prodigious force of
attention. It would be a power like that of the whist-player who
remembers the cards played ; or, to go up a step, like that of the
chess-player who can visualize a great number of combinations
and hold them in his memory. Every good mathematician ought
to be a good chess-player, and inversely ; likewise he should be a
good computer. Of course that sometimes happens; thus Gauss
MATHEMATICAL CPEATION 386
was at the same time a geometer of genius and a very precocious
and accurate computer.
But there are exceptions; or rather I err; I can not call them
exceptions without the exceptions being more than the rule.
Gauss it is, on the contrary, who was an exception. As for my-
self, I must confess, I am absolutely incapable even of adding
without mistakes. In the same way I should be but a poor chess-
player; I would perceive that by a certain play I should expose
myself to a certain danger; I would pass in review several other
plays, rejecting them for other reasons, and then finally I should
make the move first examined, having meantime forgotten the
danger I had foreseen.
In a word, my memory is not bad, but it would be insufiScient
to make me a good chess-player. Why then does it not fail me in
a difiScult piece of mathematical reasoning where most chess-
players would lose themselves t Evidently because it is guided
by the general march of the reasoning. A mathematical demon-
stration is not a simple juxtaposition of syllogisms, it is syllo-
gisms placed in a certain order, and the order in which these
elements are placed is much more important than the elements
themselves. If I have the feeling, the intuition, so to speak, of
this order, so as to perceive at a glance the reasoning as a whole,
I need no longer fear lest I forget one of the elements, for each
of them will take its allotted place in the array, and that with-
out any effort of memory on my part.
It seems to me then, in repeating a reasoning learned, that I
could have invented it. This is often only an illusion ; but even
then, even if I am not so gifted as to create it by myself, I my-
self re-invent it in so far as I repeat it.
We know that this feeling, this intuition of mathematical
order, that makes us divine hidden harmonies and relations, can
not be possessed by every one. Some will not have either this
delicate feeling so difficult to define, or a strength of memory
and attention beyond the ordinary, and then they will be abso-
lutely incapable of understanding higher mathematics. Such are
the majority. Others will have this feeling only in a slight
degree, but they will be gifted with an uncommon memory and
a great power of attention. They will learn by heart the details
26
386 SCIENCE AND METHOD
one after another; they can understand mathematics and some-
times make applications, but they cannot create. Others, finally,
will possess in a less or greater degree the special intuition
referred to, and then not only can they understand mathematics
even if their memory is nothing extraordinary, but they may
become creators and try to invent with more or less success
according as this intuition is more or less developed in them.
In fact, what is mathematical creation t It does not consist
in making new combinations with mathematical entities already
known. Any one could do that, but the combinations so made
would be infinite in number and most of them absolutely with-
out interest. To create consists precisely in not making useless
combinations and in making those which are useful and which
are only a small minority. Invention is discernment, choice.
How to make this choice I have before explained; the mathe-
matical facts worthy of being studied are those which, by their
analogy with other facts, are capable of leading us to the knowl-
edge of a mathematical law just as experimental facts lead us to
the knowledge of a physical law. They are those which reveal
to us unsuspected kinship between other facts, long known, but
wrongly believed to be strangers to one another.
Among chosen combinations the most fertile will often be those
formed of elements drawn from domains which are far apart.
Not that I mean as suflBcing for invention the bringing together
of objects as disparate as possible ; most combinations so formed
would be entirely sterile. But certain among them, very rare,
are the most fruitful of all.
To invent, I have said, is to choose ; but the word is perhaps
not wholly exact. It makes one think of a purchaser before whom
are displayed a large number of samples, and who examines
them, one after the other, to make a choice. Here the samples
would be so numerous that a whole lifetime would not suffice to
examine them. This is not the actual state of things. The sterile
combinations do not even present themselves to the mind of the
inventor. Never in the field of his consciousness do combina-
tions appear that are not really useful, except some that he rejects
but which have to some extent the characteristics of useful com-
binations. All goes on as if the inventor were an examiner for
MATHEMATICAL CREATION 387
the second degree who would only have to question the candi-
dates who had passed a previous examination.
But what I have hitherto said is what may be observed or
inferred in reading the writings of the geometers, reading
reflectively.
It is time to penetrate deeper and to see what goes on in the
very soul of the mathematician. For this, I believe, I can do best
by recalling memories of my own. But I shall limit myself to
telling how I wrote my first memoir on Fuchsian functions. I
beg the reader's pardon ; I am about to use some technical expres-
sions, but they need not frighten him, for he is not obliged to
understand them. I shall say, for example, that I have found
the demonstration of such a theorem under such circumstances.
This theorem will have a barbarous name, unfamiliar to many,
but that is unimportant ; what is of interest for the psychologist
is not the theorem but the circumstances.
For fifteen days I strove to prove that there could not be any
functions like those I have since called Fuchsian functions. I
was then very ignorant; every day I seated myself at my work
table, stayed an hour or two, tried a great number of combina-
tions and reached no results. One evening, contrary to my
custom, I drank black coffee and could not sleep. Ideas rose in
crowds; I felt them collide until pairs interlocked, so to speak,
making a stable combination. By the next morning I had estab-
lished the existence of a class of Fuchsian functions, those which
come from the hypergeometric series; I had only to write out
the results, which took but a few hours.
Then I wanted to represent these functions by the quotient of
two series; this idea was perfectly conscious and deliberate, the
analogy with elliptic functions guided me. I asked myself what
properties these series must have if they existed, and I succeeded
without diflBculty in forming the series I have called theta-
Fuchsian.
Just at this time I left Caen, where I was then living, to go on
a geologic excursion under the auspices of the school of mines.
The changes of travel made me forget my mathematical work.
^Having reached Coutances, we entered an omnibus to go some
place or other. At the moment when I put my foot on the step
388 SCIENCE AND METHOD
the idea came to me, without anything in my former thoughts
seeming to have paved the way for it, that the transformations
I had used to define the Fuchsian functions were identical with
those of non-Euclidean geometry. I did not verify the idea; I
should not have had time, as, upon taking my seat in the omni-
bus, I went on with a conversation already commenced^ but I
felt a perfect certainty. On my return to Caen, for conscience'
sake I verified the result at my leisure.
Then I turned my attention to the study of some arithmetical
questions apparently without much success and without a sus-
picion of any connection with my preceding researches. Dis-
gusted with my failure, I went to spend a few days at the sea-
side, and thought of something else. One morning, walking on
the bluff, the idea came to me, with just the same characteristics
of brevity, suddenness and immediate certainty, that the arith-
metic transformations of indeterminate ternary quadratic forms
were identical with those of non-Euclidean geometry.
Returned to Caen, I meditated on this result and deduced the
consequences. The example of quadratic forms showed me that
there were Fuchsian groups other than those corresponding to
the hypergeometric series ; I saw that I could apply to them the
theory of theta-Fuehsian series and that consequently there
existed Fuchsian functions other than those from the hyper-
geometric series, the ones I then knew. Naturally I set my-
self to form all these functions. I made a systematic attack upon
them and carried all the outworks, one after another. There was
one however that still held out, whose fall would involve that of
the whole place. But all my efforts only served at first the better
to show me the diflSculty, which indeed was something. All this
work was perfectly conscious.
Thereupon I left for Mont-Val6rien, where I was to go through
my military service; so I was very differently occupied. One
day, going along the street, the solution of the diflSculty which
had stopped me suddenly appeared to me. I did not try to go
deep into it immediately, and only after my service did I again
take up the question. I had all the elements and had only to
arrange them and put them together. So I wrote out my final
memoir at a single stroke and without diflSculty.
MATHE3IATICAI. CREATION
389
I shall limit myself to this single example; it is useless to
multiply them. In regard to my other researches I would have
to say analogous things, and the observations of other mathe-
maticians given in L'enseignement matkimaHqxie would od17
conUrm them.
Most striking at first is this appearance of sudden illumina-
tion, a manifest sign of long, unconscious prior work. The role
of this unconscious work in mathematical invention appears to
me incontestable, and traces of it would be found in other cases
where it is less evident. Often when one works at a hard ques-
tion, nothing good is accomplished at the first attack. Then
one takes a rest, longer or shorter, and sita down anew to the
work, Durinfj the first half-hour, as before, nothing is found,
and then all of a sudden the decisive idea presents itself to the
mind. It might be said that the conscious work has been more
fruitful because it has been interrupted and the rest has given
back to the mind its force and freshness. But it is more prob-
able that this rest has been filled out with unconscious work and
that the result of this work has afterward revealed itself to the
geometer just as in the eases I have cited ; only the revelation,
instead of coming during a walk or a journey, has happened
during a period of conscious work, but independently of this
work which plays at most a role of excitant, as if it were the goad
stimulating the results already reached during rest, but remain-
ing unconscious, to assume tbe conscious form.
There is another remark to be made about the conditions of
this unconscious work: it is possible, and of a certainty it is only
fruitful, if it is on the one hand preceded and on the other hand
followed by a period of conscious work. These sudden inspira-
tions (and the examples already cited sufficiently prove this)
never happen except after some days of voluntary effort which
has appeared absolutely fruitless and whence nothing good scemn
to have come, where the way taken seems totally astray. These
efforts then have not been as sterile as one thinks; they have set
agoing the unconscious machine and without them it would not
have moved and would have produced nothing.
The need for tlie second period of conscious work, after the
inspiration, is atiU easier to nnderstand. It is neceBBary to put
390 SCIENCE AND METHOD
in shape the results of this inspiration, to deduce from them the
immediate consequences, to arrange them, to word the demonstra-
tions, but above all is verification necessary. I have spoken of
the feeling of absolute certitude accompanying the inspiration;
in the cases cited this feeling was no deceiver, nor is it usually.
But do not think this a rule without exception ; often this feeling
deceives us without being any the less vivid, and we only find it
out when we seek to put on foot the demonstration. I have
especially noticed this fact in regard to ideas coming to me in the
morning or evening in bed while in a semi-hypnagogic state.
Such are the realities; now for the thoughts they force upon
us. The unconscious, or, as we say, the subliminal self plays an
important role in mathematical creation ; this follows from what
we have said. But usually the subliminal self is considered as
purely automatic. Now we have seen that mathematical work is
not simply mechanical, that it could not be done by a machine,
however perfect. It is not merely a question of applying rules,
of making the most combinations possible according to certain
fixed laws. The combinations so obtained would be exceedingly
numerous, useless and cumbersome. The true work of the in-
ventor consists in choosing among these combinations so as to
eliminate the useless ones or rather to avoid the trouble of mak-
ing them, and the rules which must guide this choice are extremely
fine and delicate. It is almost impossible to state them precisely;
they are felt rather than formulated. Under these conditions,
how imagine a sieve capable of applying them mechanically!
A first hypothesis now presents itself : the subliminal self is in
no way inferior to the conscious self; it is not purely automatic;
it is capable of discernment ; it has tact, delicacy ; it knows how
to choose, to divine. What do I say? It knows better how to
divine than the conscious self, since it succeeds where that has
failed. In a word, is not the subliminal self superior to the
conscious self? You recognize the full importance of this ques-
tion. Boutroux in a recent lecture has shown how it came up
on a very different occasion, and what consequences would follow
an affirmative answer. (See also, by the same author. Science
et Religion, pp. 313 ff.)
Is this affirmative answer forced upon us by the facts I have
MATHEMATICAL CREATION 391
just given t I confess that, for my part, I should hate to accept
it Reexamine the facts then and see if they are not compatible
with another explanation.
It is certain that the combinations which present themselves to
the mind in a sort of sadden illumination, after an unconscious
working somewhat prolonged, are generally useful and fertile
combinations, which seem the result of a first impression. Does
it follow that the subliminal self, having divined by a delicate
intuition that these combinations would be useful, has formed
only these, or has it rather formed many others which were
lacking in interest and have remained unconscious t
In this second way of looking at it, all the combinations would
be formed in consequence of the automatism of the subliminal
self, but only the interesting ones would break into the domain
of consciousness. And this is still very mysterious. What is the
cause that, among the thousand products of our unconscious
activity, some are called to pass the threshold, while others remain
below t Is it a simple chance which confers this privilege t Evi-
dently not ; among all the stimuli of our senses, for example, only
the most intense fix our attention, unless it has been drawn to
them by other causes. More generally the privileged uncon-
scious phenomena, those susceptible of becoming conscious, are
those which, directly or indirectly, affect most profoundly our
emotional sensibility.
It may be surprising to see emotional sensibility invoked
d propos of mathematical demonstrations which, it would seem,
can interest only the intellect. This would be to forget the feel-
ing of mathematical beauty, of the harmony of numbers and
forms, of geometric elegance. This is a true esthetic feeling that
all real mathematicians know, and surely it belongs to emo-
tional sensibility.
Now, what are the mathematic entities to which we attribute
this character of beauty and elegance, and which are capable of
developing in us a sort of esthetic emotion t They are thoae
whose elements are harmoniously disposed so that the mind with-
out effort can embrace their totality while realizing the details.
This harmony is at once a satisfaction of our esthetic needs and
an aid to the mind, sustaining and guiding. And at the same
392 SCIENCE AND METHOD
time, in putting under our eyes a well-ordered whole, it makes
us foresee a mathematical law. Now, as we have said above, the
only mathematical facts worthy of fixing our attention and
capable of being useful are those which can teach us a mathe-
matical law. So that we reach the following conclusion: The
useful combinations are precisely the most 'beautiful, I mean
those best able to charm this special sensibility that all mathe-
maticians know, but of which the profane are so ignorant as
often to be tempted to smile at it.
What happens then t Among the great numbers of combina-
tions blindly formed by the subliminal self, almost all are without
interest and without utility; but just for that reason they are
also without effect upon the esthetic sensibility. Consciousness
will never know them; only certain ones are harmonious, and,
consequently, at once useful and beautiful. They will be capable
of touching this special sensibility of the geometer of which I
have just spoken, and which, once aroused, will call our atten-
tion to them, and thus give them occasion to become conscious.
This is only a hypothesis, and yet here is an observation which
may confirm it: when a sudden illumination seizes upon the
mind of the mathematician, it usually happens that it does not
deceive him, but it also sometimes happens, as I have said, that
it does not stand the test of verification ; well, we almost always
notice that this false idea, had it been true, would have gratified
our natural feeling for mathematical elegance.
Thus it is this special esthetic sensibility which plays the role
of the delicate sieve of which I spoke, and that suflSciently ex-
plains why the one lacking it will never be a real creator.
Yet all the diflBculties have not disappeared. The conscious
self is narrowly limited, and as for the subliminal self we know
not its limitations, and this is why we are not too reluctant in
supposing that it has been able in a short time to make more
different combinations than the whole life of a conscious being
could encompass. Yet these limitations exist. Is it likely that
it is able to form all the possible combinations, whose number
would frighten the imagination t Nevertheless that would seem
necessary, because if it produces only a small part of these com-
binations, and if it makes them at random, there would be small {
\
f
MATHEMATICAL CREATION 393
chance that the good, the one we shotQd choose, would be found
among them.
Perhaps we ought to seek the explanation in that preliminary
period of conscious work which always precedes all fruitful
unconscious labor. Permit me a rough comparison. Figure
the future elements of our combinations as something like the
hooked atoms of Epicurus. During the complete repose of the
mind, these atoms are motionless, they are, so to speak, hooked
to the wall ; so this complete rest may be indefinitely prolonged
without the atoms meeting, and consequently without any com-
bination between them.
On the other hand, during a period of apparent rest and
unconscious work, certain of them are detached from the wall and
put in motion. They flash in every direction through the space
(I was about to say the room) where they are enclosed, as would,
for example, a swarm of gnats or, if you prefer a more learned
comparison, like the molecules of gas in the kinematic theory of
gases. Then their mutual impacts may produce new combinations.
What is the role of the preliminary conscious workt It is
evidently to mobilize certain of these atoms, to unhook them from
the wall and put them in swing. We think we have done no
good, because we have moved these elements a thousand different
ways in seeking to assemble them, and have found no satisfactory
aggregate. But, after this shaking up imposed upon them by our
will, these atoms do not return to their primitive rest. They
freely continue their dance.
Now, our will did not choose them at random; it pursued a
perfectly determined aim. The mobilized atoms are therefore
not any atoms whatsoever; they are those from which we might
reasonably expect the desired solution. Then the mobilized atoms
undergo impacts which make them enter into combinations among
themselves or with other atoms at rest which they struck against
in their course. Again I beg pardon, my comparison is very
rough, but I scarcely know how otherwise to make my thought
understood.
However it may be, the only combinations that have a chance
of forming are those where at least one of the elements is one
of those atoms freely chosen by our will. Now, it is evidently
394 SCIENCE AND METHOD
among these that is found what I called the good combination.
Perhaps this is a way of lessening the paradoxical in the original
hypothesis.
Another observation. It never happens that the unconscious
work gives us the result of a somewhat long calculation all made,
where we have only to apply fixed rules. We might think the
wholly automatic subliminal self particularly apt for this sort of
work, which is in a way exclusively mechanical. It seems that
thinking in the evening upon the factors of a multiplication we
might hope to find the product ready made upon our awaken-
ing, or again that an algebraic calculation, for example a veri-
fication, would be made unconsciously. Nothing of the sort, as
observation proves. All one may hope from these inspirations,
fruits of unconscious work, is a point of departure for such calcu-
lations. As for the calculations themselves, they must be made
in the second period of conscious work, that which follows the
inspiration, that in which one verifies the results of this inspira-
tion and deduces their consequences. The rules of these calcu-
lations are strict and complicated. They require discipline, atten-
tion, will, and therefore consciousness. In the subliminal self, on
the contrary, reigns what I should call liberty, if we might give
this name to the simple absence of discipline and to the disorder
bom of chance. Only, this disorder itself permits unexpected
combinations.
I shall make a last remark : when above I made certain personal
observations, I spoke of a night of excitement when I worked in
spite of myself. Such cases are frequent, and it is not necessary
that the abnormal cerebral activity be caused by a physical exci-
tant as in that I mentioned. It seems, in such cases, that one is
present at his own unconscious work, made partially perceptible
to the over-excited consciousness, yet without having changed its
nature. Then we vaguely comprehend what distinguishes the
two mechanisms or, if you wish, the working methods of the two
egos. And the psychologic observations I have been able thus
to make seem to me to confirm in their general outlines the views
I have given.
Surely they have need of it, for they are and remain in spite
of all very hypothetical : the interest of the questions is so great
that I do not repent of having submitted them to the reader.
CHAPTER IV
Change
I
''How dare we speak of the laws of chance t Is not chance
the antithesis of all lawt" So says Bertrand at the beginning of
his Calcul des probabilitSs. Probability is opposed to certitude ;
so it is what we do not know and consequently it seems what we
could not calculate. Here is at least apparently a contradiction,
and about it much has already been written.
And first, what is chance t The ancients distinguished between
phenomena seemingly obeying harmonious laws, established once
for all, and those which they attributed to chance; these were
the ones unpredictable because rebellious to all law. In each
domain the precise laws did not decide everything, they only
drew limits between which chance might act. In this conception
the word chance had a precise and objective meaning : what was
chance for one was also chance for another and even for the gods.
But this conception is not ours to-day. We have become abso-
lute determinists, and even those who want to reserve the rights
of human free will let determinism reign undividedly in the inor-
ganic world at least. Every phenomenon, however minute, has
a cause ; and a mind infinitely powerful, infinitely well-informed
about the laws of nature, could have foreseen it from the begin-
ning of the centuries. If such a mind existed, we could not play
with it at any game of chance ; we should always lose.
In fact for it the word chance would not have any meaning,
or rather there would be no chance. It is because of our weak-
ness and our ignorance that the word has a meaning for us. And,
even without going beyond our feeble humanity, what is chance
for the ignorant is not chance for the scientist. Chance is only
the measure of our ignorance. Fortuitous phenomena are, by
definition, those whose laws we do not know.
But is this definition altogether satisfactory t When the first
395
396 SCIENCE AND METHOD
Chaldean shepherds followed with their eyes the movements of
the stars, they knew not as yet the laws of astronomy ; would ih^
have dreamed of saying that the stars move at random t If a
modem physicist studies a new phenomenon, and if he discovers
its law Tuesday, would he have said Monday that this phenom-
enon was fortuitous t Moreover, do we not often invoke what
Bertrand calls the laws of chance, to predict a phenomenon t
For example, in the kinetic theory of gases we obtain the known
laws of Mariotte and of Gay-Lussac by means of the hypothesis
that the velocities of the molecules of gas vary irregularly, that
is to say at random. All physicists will agree that the observable
laws would be much less simple if the velocities were ruled by
any simple elementary law whatsoever, if the molecules were,
as we say, organized, if they were subject to some discipline. It
is due to chance, that is to say, to our ignorance, that we can draw
our conclusions ; and then if the word chance is simply synony-
mous with ignorance what does that meant Must we therefore
translate as follows!
"You ask me to predict for you the phenomena about to
happen. If, unluckily, I knew the laws of these phenomena I
could make the prediction only by inextricable calculations and
would have to renounce attempting to answer you ; but as I have
the good fortune not to know them, I will answer you at once.
And what is most surprising, my answer will be right.''
So it must well be that chance is something other than the
name we give our ignorance, that among phenomena whose
causes are unknown to us we must distinguish fortuitous phe-
nomena about which the calculus of probabilities will provision-
ally give information, from those which are not fortuitous and of
which we can say nothing so long as we shall not have determined
the laws governing them. For the fortuitous phenomena them-
selves, it is clear that the information given us by the calculus
of probabilities will not cease to be true upon the day when these
phenomena shall be better known.
The director of a life insurance company does not know when
each of the insured will die, but he relies upon the calculus of
probabilities and on the law of great numbers, and he is not
deceived, since he distributes dividends to his stockholders. These
CHANCE 397
dividends would not vanish if a very penetrating and very indis-
crete physician should, after the policies were signed, reveal to
the director the life chances of the insured. This doctor would
dissipate the ignorance of the director, but he would have no
influence on the dividends, which evidently are not an outcome
of this ignorance.
II
To find a better definition of chance we must examine some of
the facts which we agree to regard as fortuitous, and to which
the calculus of probabilities seems to apply ; we then shall investi-
gate what are their common characteristics.
The first example we select is that of unstable equilibrium ; if
a cone rests upon its apex, we know well that it will fall, but we
do not know toward what side ; it seems to us chance alone will
decide. If the cone were perfectly symmetric, if its axis were
perfectly vertical, if it were acted upon by no force other than
gravity, it would not fall at all. But the least defect in symmetry
will make it lean slightly toward one side or the other, and if it
leans, however little, it will fall altogether toward that side.
Even if the symmetry were perfect, a very slight tremor, a breath
of air could make it incline some seconds of arc; this will be
enough to determine its fall and even the sense of its fall which
will be that of the initial inclination.
A very slight cause, which escapes us, determines a consider-
able effect which we can not help seeing, and then we say this
effect is due to chance. If we could know exactly the laws of
nature and the situation of the universe at the initial instant,
we should be able to predict exactly the situation of this same
universe at a subsequent instant. But even when the natural
laws should have no further secret for us, we could know the
initial situation only approxinuUely. If that permits us to fore-
see the subsequent situation tviih the same degree of approxima-
tion, this is all we require, we say the phenomenon has been
predicted, that it is ruled by laws. But this is not always the
case ; it may happen that slight differences in the initial condi-
tions produce very great differences in the final phenomena; a
slight error in the former would make an enormous error in the
398 SCIENCE AND METHOD
latter. Prediction becomes impossible and we have the fortuitous
phenomenon.
Our second example will be very analogous to the first and we
shall take it from meteorology. Why have the meteorologists such
difficulty in predicting the weather with any certainty! Why
do the rains, the tempests themselves seem to us to come by
chance, so that many persons find it quite natural to pray for
rain or shine, when they would think it ridiculous to pray for
an eclipse t We see that great perturbations generally happen in
regions where the atmosphere is in unstable equilibrium. The
meteorologists are aware that this equilibrium is unstable, that a
cyclone is arising somewhere; but where they can not tell; one-
tenth of a degree more or less at any point, and the cyclone
bursts here and not there, and spreads its ravages over countries
it would have spared. This we could have foreseen if we had
known that tenth of a degree, but the observations were neither
sufficiently close nor sufficiently precise, and for this reason all
seems due to the agency of chance. Here again we find the same
contrast between a very slight cause, unappreciable to the ob-
server, and important effects, which are sometimes tremendous
disasters.
Let us pass to another example, the distribution of the minor
planets on the zodiac. Their initial longitudes may have been
any longitudes whatever ; but their mean motions were different
and they have revolved for so long a time that we may say they
are now distributed at random along the zodiac. Very slight
initial differences between their distances from the sun, or, what
comes to the same thing, between their mean motions, have
ended by giving enormous differences between their present
longitudes. An excess of the thousandth of a second in the daily
mean motion will give in fact a second in three years, a degree
in ten thousand years, an entire circumference in three or four
million years, and what is that to the time which has passed since
the minor planets detached themselves from the nebula of
Laplace? Again therefore we see a slight cause and a great
effect ; or better, slight differences in the cause and great differ-
ences in the effect.
The game of roulette does not take us as far as might seem
CHANCE 399
from the preceding example. Assume a needle to be turned on a
pivot over a dial divided into a hundred sectors alternately red
and black. If it stops on a red sector I win ; if not, I lose. Evi-:
dently all depends upon the initial impulse I give the needle.
The needle will make, suppose, ten or twenty turns, but it will
stop sooner or not so soon, according as I shall have pushed it
more or less strongly. It suffices that the impulse vary only by
a thousandth or a two thousandth to make the needle stop over a
black sector or over the following red one. These are differences
the muscular sense can not distinguish and which elude even the
most delicate instruments. So it is impossible for me to foresee
what the needle I have started will do, and this is why my heart
throbs and I hope ever3rthing from luck. The difference in the
cause is imperceptible, and the difference in the effect is for me
of the highest importance, since it means my whole stake.
Ill
Permit me, in this connection, a thought somewhat foreign to
my subject. Some years ago a philosopher said that the future
is determined by the past, but not the past by the future ; or, in
other words, from knowledge of the present we could deduce the
future, but not the past ; because, said he, a cause can have only
one effect, while the same effect might be produced by several
different causes. It is clear no scientist can subscribe to this
conclusion. The laws of nature bind the antecedent to the conse-
quent in such a way that the antecedent is as well determined by
the consequent as the consequent by the antecedent. But whence
came the error of this philosopher t We know that in virtue of
Carnot's principle physical phenomena are irreversible and the
world tends toward uniformity. When two bodies of different
temperature come in contact, the warmer gives up heat to the
colder; so we may foresee that the temperature will equalize.
But once equal, if asked about the anterior state, what can we
answer? We might say that one was warm and the other cold,
but not be able to divine which formerly was the warmer.
And yet in reality the temperatures will never reach perfect
equality. The difference of the temperatures only tends asymp-
totically toward zero. There comes a moment when our ther-
400 SCIENCE AND METHOD
mometers are powerless to make it known. But if we had ther-
mometers a thousand times, a hundred thousand times as sensi-
tive, we should recognize that there still is a slight difference, and
that one of the bodies remains a little warmer than the other, and
so we could say this it is which formerly was much the warmer.
So then there are, contrary to what we found in the former
examples, great differences in cause and slight differences in
effect. Flammarion once imagined an observer going away from
the earth with a velocity greater than that of light; for him time
would have changed sign. History would be turned about, and
Waterloo would precede Austerlitz. Well, for this observer,
effects and causes would be inverted ; unstable equilibrium would
no longer be the exception. Because of the universal irreversi-
bility, all would seem to him to come out of a sort of chaos in
unstable equilibrium. All nature would appear to him delivered
over to chance.
IV
Now for other examples where we shall see somewhat different
characteristics. Take first the kinetic theory of gases. How
should we picture a receptacle filled with gast Innumerable
molecules, moving at high speeds, flash through this receptacle
in every direction. At every instant they strike against its walls
or each other, and these collisions happen under the most diverse
conditions. What above all impresses us here is not the little-
ness of the causes, but their complexity, and yet the former ele-
ment is still found here and plays an important role. If a mole-
cule deviated right or left from its trajectory, by a very small
quantity, comparable to the radius of action of the gaseous mole-
cules, it would avoid a collision or sustain it under different con-
ditions, and that would vary the direction of its velocity after
the impact, perhaps by ninety degrees or by a hundred and
eighty degrees.
And this is not all; we have just seen that it is necessary to
deflect the molecule before the clash by only an infinitesimal, to
produce its deviation after the collision by a finite quantity. If
then the molecule undergoes two successive shocks, it will suflSce
to deflect it before the first by an infinitesimal of the second
order, for it to deviate after the first encounter by an infinites-
CHANCE 401
•
imal of the first order, and after the second hit, by a finite quan-
tity. And the molecule will not undergo merely two shocks; it
will undergo a very great number per second. So that if the
first shock has multiplied the deviation by a very large number
A, after n shocks it will be multiplied by A^. It will therefore
become very great not merely because A is large, that is to say
because little causes produce big effects, but because the exponent
n is large, that is to say because the shocks are very numerous
and the causes very complex.
Take a second example. Why do the drops of rain in a
shower seem to be distributed at random t This is again because
of the complexity of the causes which determine their formation.
Ions are distributed in the atmosphere. For a long while they
have been subjected to air-currents constantly changing, they
have been caught in very small whirlwinds, so that their final
distribution has no longer any relation to their initial distribu-
tion. Suddenly the temperature falls, vapor condenses, and each
of these ions becomes the center of a drop of rain. To know
what will be the distribution of these drops and how many will
fall on each paving-stone, it would not be sufBcient to know the
initial situation of the ions, it would be necessary to compute
the effect of a thousand little capricious air-currents.
And again it is the same if we put grains of powder in sus-
pension in water. The vase is ploughed by currents whose law
we know not, we only know it is very complicated. At the
end of a certain time the grains will be distributed at random,
that is to say uniformly, in the vase ; and this is due precisely to
the complexity of these currents. If they obeyed some simple
law, if, for example the vase revolved and the currents circulated
around the axis of the vase, describing circles, it would no
longer be the same, since each grain ^ ould retain its initial alti-
tude and its initial distance from the axis.
We should reach the same result in considering the mixing of
two liquids or of two fine-grained powders. And to take a
grosser example, this is also what happens when we shuflSe play-
ing-cards. At each stroke the cards undergo a permutation
(analogous to that studied in the theory of substitutions). What
will happen? The probability of a particular permutation (for
27
402 SCIENCE AND METHOD
example, that bringing to the nth place the eard occupying the
^(n}th place before the permutation} dependa upon the player's
habits. But if this player shuflSes the cards long enough, there
will be a great number of successive permutations, and the re-
sulting final order will no longer be governed by aught but
chance; I mean to say that all possible orders will be equally
probable. It is to the great number of successive permutations,
that is to say to the complexity of the phenomenon, that this
result is due.
A final word about the theory of errors. Here it is that the
causes are complex and multiple. To how many snares is not
the observer exposed, even with the best instrument 1 He should
apply himself to finding out the largest and avoiding them.
These are the ones giving birth to gystematic errors. But when
he has eliminated those, admitting that he succeeds, there remain
many small ones which, their effects accumulating, may be-
come dangerous. Thence come the accidental errors; and we at-
tribute them to diance because their causes are too complicated
and too numerous. Here again we have only little causes, but
each of them would produce only a slight effect; it is by their
union and their number that their effects become formidable.
V
We may take still a third point of view, less important than
the first two and upon which I shall lay less stress. When we
seek to foresee an event and examine its antecedents, we strive
to search into the anterior situation. This could not be done for
all parts of the universe and we are content to know what is
passing in the neighborhood of the point where the event should
occur, or what would appear to have some relation to it. An
examination can not be complete and we must know how to choose.
But it may happen that we have passed by circumstances which
at first sight seemed completely foreign to the foreseen happen-
ing, to which one would never have dreamed of attributing any
influence and which nevertheless, contrary to all anticipation,
come to play an important role.
A man passes in the street going to his business; some one
knowing the business could have told why he started at such a
CHANCE 408
time and went by such a street. On the roof works a tiler.
The contractor employing him could in a certain measure fore-
see what he would do. But the passer-by scarcely thinks of the
tiler, nor the tiler of him; they seem to belong to two worlds
completely foreign to one another. And yet the tiler drops a
tile which kills the man, and we do not hesitate to say this is
chance.
Our weakness forbids our considering the entire universe
and makes us cut it up into slices. We try to do this as little
artificially as possible. And yet it happens from time to time
that two of these slices react upon each other. The effects
of this mutual action then seem to us to be due to chance.
Is this a third way of conceiving chance t Not always; in
fact most often we are carried back to the first or the second.
Whenever two worlds usually foreign to one another come thus
to react upon each other, the laws of this reaction must be very
complex. On the other hand, a very slight change in the initial
conditions of these two worlds would have been su£Bcient for the
reaction not to have happened. How little was needed for the
man to pass a second later or the tiler to drop his tile a second
sooner.
VI
All we have said still does not explain why chance obeys laws.
Does the fact that the causes are slight or complex suffice for
our foreseeing, if not their effects in each case, at least what their
effects will be, on the average? To answer this question we had
better take up again some of the examples already cited.
I shall begin with that of the roulette. I have said that the
point where the needle will stop depends upon the initial push
given it. What is the probability of this push having this or
that value? I know nothing about it, but it is difficult for me
not to suppose that this probability is represented by a continuous
analytic function. The probability that the push is comprised
between a and a + « will then be sensibly equal to the probability
of its being comprised between a + e and a + 2€, provided € be
very S7nall. This is a property common to all analytic functions.
Minute variations of the function are proportional to minute
variations of the variable.
404 SCIENCE AND METHOD
But we have assumed that an exceedingly slight variation of
the push suffices to change the color of the sector over which the
needle finally stops. From a to a + e it is red, from a + e to
a -|- 2€ it is black ; the probability of each red sector is therefore
the same as of the following black, and consequently the total
probability of red equals the total probability of black.
The datum of the question is the analytic function representing
the probability of a particular initial push. But the theorem
remains true whatever be this datum, since it depends upon a
property common to all analytic functions. From this it follows
finally that we no longer need the datum.
What we have just said for the case of the roulette applies
also to the example of the minor planets. The zodiac may be
regarded as an immense roulette on which have been tossed many
little balls with different initial impulses varying according to
some law. Their present distribution is uniform and independ-
ent of this law, for the same reason as in the preceding case.
Thus we see why phenomena obey the laws of chance when
slight differences in the causes suffice to bring on great differences
in the effects. The probabilities of these slight differences may
then be regarded as proportional to these differences themselves,
just because these differences are minute, and the infinitesimal
increments of a continuous function are proportional to those of
the variable.
Take an entirely different example, where intervenes especially
the complexity of the causes. Suppose a player shuffles a pack
of cards. At each shuffle he changes the order of the cards, and
he may change them in many ways. To simplify the exposition,
consider only three cards. The cards which before the shuffle
occupied respectively the places 123, may after the shuffle occupy
the places
123, 231, 312, 321, 132, 213.
Each of these six hypotheses is possible and they have respec-
tively for probabilities :
Vu Vzt Ps, P<y Pb, Pa-
The sum of these six numbers equals 1 ; but this is all we know
of them ; these six probabilities depend naturally upon the habits
of the player which we do not know.
CHANCE 406
At the second shuffle and the following, this will recommence,
and under the same conditions ; I mean that p^ for example rep-
resents always the probability that the three cards which occu-
pied after the nth shuffle and before the n -{- 1th the places 123,
occupy the places 321 after the n-|-lth shuffle. And this re-
mains true whatever be the number n, since the habits of the
player and his way of shuffling remain the same.
But if the number of shuffles is very great, the cards which
before the first shuffle occupied the places 123 may, after the
last shuffle, occupy the places
123, 231, 312, 321, 132, 213
and the probability of these six hypotheses will be sensibly the
same and equal to 1/6; and this will be true whatever be the
numbers Pi • • • Pa which we do not know. The gn>^at num-
ber of shuffles, that is to say the complexity of the causes, has
produced uniformity.
This would apply without change if there were more than
three cards, but even with three cards the demonstration would
be complicated ; let it suffice to give it for only two cards. Then
we have only two possibilities 12, 21 with the probabilities p^ and
V2 = 1 — Pi-
Suppose n shuffles and suppose I win one franc if the cards
are finally in the initial order and lose one if they are finally
inverted. Then, my mathematical expectation will be (Pi — P2)*-
The difference pj — pj is certainly less than 1; so that if n
is very great my expectation will be zero; we need not learn p^
and P2 to be aware that the game is equitable.
There would always be an exception if one of the numbers
Pi and P2 was equal to 1 and the other naught. Then it would
not apply because our initial hypotheses would he too simple.
What we have just seen applies not only to the mixing of
cards, but to all mixings, to those of powders and of liquids;
and even to those of the molecules of gases in the kinetic theory
of gases.
To return to this theory, suppose for a moment a gas whose
molecules can not mutually clash, but may be deviated by hitting
the insides of the vase wherein the gas is confined. If the form
406 SCIENCE AND METHOD
of the vase is sufficiently complex the distribution of the mole-
cules and that of the velocities will not be long in becoming uni-
form. But this will not be so if the vase is spherical or if it
has the shape of a cuboid. Whyf Because in the first case the
distance from the center to any trajectory will remain constant;
in the second case this will be the absolute value of the angle of
each trajectory with the faces of the cuboid.
So we see what should be understood by conditions too simple;
they are those which conserve something, which leave an invariant
remaining. Are the differential equations of the problem too
simple for us to apply the laws of chance 1 This question would
seem at first view to lack precise meaning ; now we know what it
means. They are too simple if they conserve something, if they
admit a uniform integral. If something in the initial conditions
remains unchanged, it is clear the final situation can no longer
be independent of the initial situation.
We come finally to the theory of errors. We know not to
what are due the accidental errors, and precisely because we do
not know, we are aware they obey the law of Gauss. Such is the
paradox. The explanation is nearly the same as in the preceding
cases. We need know only one thing: that the errors are very
numerous, that they are very slight, that each may be as well
negative as positive. What is the curve of probability of each
of them? We do not know; we only suppose it is symmetric.
We prove then that the resultant error will follow Gauss's law,
and this resulting law is independent of the particular laws
which we do not know. Here again the simplicity of the result
is born of the very complexity of the data.
VII
But we are not through with paradoxes. I have just recalled
the figment of Flammarion, that of the man going quicker than
light, for whom time changes sign. I said that for him all phe-
nomena would seem due to chance. That is true from a certain
point of view, and yet all these phenomena at a given moment
would not be distributed in conformity with the laws of chance,
since the distribution would be the same as for us, who, seeing
them unfold harmoniously and without coming out of a primal
chaos, do not regard them as ruled by chance.
CHANCE 407
What does that meant For Lumen, Flammarion's man, slight
causes seem to produce great effects ; why do not things go on as
for us when we think we see grand effects due to little causes t
Would not the same reasoning be applicable in his caset
Let us return to the argument. When slight differences in the
causes produce vast differences in the effects, why are these effects
distributed according to the laws of chance f Suppose a differ-
ence of a millimeter in the cause produces a difference of a kilo-
meter in the effect. If I win in case the effect corresponds to a
kilometer bearing an even number, my probability of winning
will be 1/2. Why f Because to make that, the cause must corre-
spond to a millimeter with an even number. Now, according to
all appearance, the probability of the cause varying between
certain limits will be proportional to the distance apart of these
limits, provided this distance be very small. If this hypothesis
were not admitted there would no longer be any way of repre-
senting the probability by a continuous function.
What now will happen when great causes produce small
effects ? This is the case where we should not attribute the phe-
nomenon to chance and where on the contrary Lumen would
attribute it to chance. To a difference of a kilometer in the
cause would correspond a difference of a millimeter in the effect.
Would the probability of the cause being comprised between two
limits n kilometers apart still be proportional to n? We have
no reason to suppose so, since this distance, n kilometers, is
great. But the probability that the effect lies between two
limits n millimeters apart will be precisely the same, so it will not
be proportional to n, even though this distance, n millimeters,
be small. There is no way therefore of representing the law of
probability of effects by a continuous curve. This curve, un-
derstand, may remain continuous in the analytic sense of the
word; to infinitesimal variations of the abscissa will correspond
infinitesimal variations of the ordinate. But practically it will
not be continuous, since very small variations of the ordinate
would not correspond to very small variations of the abscissa. It
would become impossible to trace the curve with an ordinary
pencil ; that is what I mean.
So what must we conclude? Lumen has no right to say that
408 SCIENCE AND METHOD
the probability of the cause {his cause, our effect) ahould be
represented necessarily by a continuous function. But then why
have we this right t It is because this state of unstable equilib-
rium which we have been calling initial is itself only the final
outcome of a long previous history. In the course of this history
complex causes have worked a great while : they have contributed
to produce the mixture of elements and they have tended to make
everything uniform at least within a small region; they have
rounded off the comers, smoothed down the hills and filled up
the valleys. However capricious and irregular may have been the
primitive curve given over to them, they have worked so much
toward making it reg^ular that finally they deliver over to us a
continuous curve. And this is why we may in all confidence
assume its continuity.
Lumen would not have the same reasons for such a conclusion.
For him complex causes would not seem agents of equalization
and regularity, but on the contrary would create only inequality
and differentiation. He would see a world more and more varied
come forth from a sort of primitive chaos. The changes he
could observe would be for him unforeseen and impossible to
foresee. They would seem to him due to some caprice or another;
but this caprice would be quite different from our chance, since
it would be opposed to all law, while our chance still has its laws.
All these points call for lengthy explications, which perhaps
would aid in the better comprehension of the irreversibility of
the universe.
VIII
We have sought to define chance, and now it is proper to put a
question. Has chance thus defined, in so far as this is possible,
objectivity!
It may be questioned. I have spoken of very slight or very
complex causes. But what is very little for one may be very
big for another, and what seems very complex to one may seem
simple to another. In part I have already answered by saying
precisely in what cases differential equations become too simple
for the laws of chance to remain applicable. But it is fitting to
examine the matter a little more closely, because we may take
still other points of view.
CHANCE 409
What means the phrase Wery slight'? To understand it we
need only go back to what has already been said. A difference
is very slight, an interval is very small, when within the limits
of this interval the probability remains sensibly constant. And
why may this probability be regarded as constant within a
small interval? It is because we assume that the law of proba-
bility is represented by a continuous curve, continuous not only
in the analytic sense, but practically continuous, as already ex-
plained. This means that it not only presents no absolute hiatus,
but that it has neither salients nor reentrants too acute or too
accentuated.
And what gives us the right to make this hypothesis? We
have already said it is because, since the beginning of the ages,
there have always been complex causes ceaselessly acting in the
same way and making the world tend toward uniformity without
ever being able to turn back. These are the causes which little
by little have flattened the salients and filled up the reentrants,
and this is why our probability curves now show only gentle un-
dulations. In milliards of milliards of ages another step will
have been made toward uniformity, and these undulations will be
ten times as gentle; the radius of mean curvature of our curve
will have become ten times as great. And then such a length as
seems to us to-day not very small, since on our curve an arc of
this length can not be regarded as rectilineal, should on the con-
trary at that epoch be called very little, since the curvature will
have become ten times less and an arc of this length may be
sensibly identified with a sect.
Thus the phrase *very slight' remains relative; but it is not
relative to such or such a man, it is relative to the actual state of
the world. It will change its meaning when the world shall have
become more uniform, when all things shall have blended still
more. But then doubtless men can no longer live and must give
place to other beings — should I say far smaller or far larger?
So that our criterion, remaining true for all men, retains an
objective sense.
And on the other hand what means the phrase 'very complex'?
I have already given one solution, but there are others. Com-
plex causes we have said produce a blend more and more inti-
410 SCIENCE AND METHOD
mate, but after how long a time will this blend satisfy ns t When
will it have accumulated sufScient complexity t When shall we
have sufSciently shufBed the cards t If we mix two powders, one
blue, the other white, there comes a moment when the tint of the
mixture seems to us uniform because of the feebleness of our
senses; it will be uniform for the presbyte, forced to gaze &<»n
afar, before it will be so for the myope. And when it has become
uniform for all eyes, we still could push back the limit by the use
of instruments. There is no chance for any man ever to discern
the infinite variety which, if the kinetic theory is true, hides
under the uniform appearance of a gas. And yet if we accept
Gk)uy 's ideas on the Brownian movement, does not the microscope
seem on the point of showing us something analogous?
This new criterion is therefore relative like the first ; and if it
retains an objective character, it is because all men have ap-
proximately the same senses, the power of their instruments is
limited, and besides they use them only exceptionally.
IX
It is just the same in the moral sciences and particularly in
history. The historian is obliged to make a choice among the
events of the epoch he studies; he recounts only those which
seem to him the most important. He therefore contents himself
with relating the most momentous events of the sixteenth cen-
tury, for example, as likewise the most remarkable facts of the
seventeenth century. If the first suflBce to explain the second,
we say these conform to the laws of history. But if a great event
of the seventeenth century should have for cause a small fact of
the sixteenth century which no history reports, which all the
world has neglected, then we say this event is due to chance.
This word has therefore the same sense as in the physical sci-
ences ; it means that slight causes have produced great effects.
The greatest bit of chance is the birth of a great man. It is
only by chance that meeting of two germinal cells, of different
sex, containing precisely, each on its side, the mysterious ele-
ments whose mutual reaction must produce the genius. One will
agree that these elements must be rare and that their meeting is
still more rare. How slight a thing it would have required to de-
flect from its route the carrying spermatozoon. It would have
CHANCE 411
sufSced to deflect it a tenth of a millimeter and Napoleon would
not have been bom and the destinies of a continent would have
been changed. No example can better make us understand the
veritable characteristics of chance.
One more word about the paradoxes brought out by the appli-
cation of the calculus of probabilities to the moral sciences. It
has been proved that no Chamber of Deputies will ever fail to
contain a member of the opposition, or at least such an event
would be so improbable that we might without fear wager the
contrary, and bet a million against a sou.
Condorcet has striven to calculate how many jurors it would
require to make a judicial error practically impossible. If we
had used the results of this calculation, we should certainly have
been exposed to the same disappointments as in betting, on the
faith of the calculus, that the opposition would never be without
a representative.
The laws of chance do not apply to these questions. If justice
be not always meted out to accord with the best reasons, it uses
less than we think the method of Bridoye. This is perhaps to
be regretted, for then the system of Condorcet would shield us
from judicial errors.
What is the meaning of thisf We are tempted to attribute
facts of this nature to chance because their causes are obscure;
but this is not true chance. The causes are unknown to us, it is
true, and they are even complex ; but they are not sufficiently so,
since they conserve something. We have seen that this it is which
distinguishes causes 4oo simple.' When men are brought to-
gether they no longer decide at random and independently one
of another; they influence one another. Multiplex causes come
into action. They worry men, dragging them to right or left,
but one thing there is they can not destroy, this is their Panurge
flock-of -sheep habits. And this is an invariant.
X
Difficulties are indeed involved in the application of the
calculus of probabilities to the exact sciences. Why are the
decimals of a table of logarithms, why are those of the number
IT distributed in accordance with the laws of chance t Elsewhere
I have already studied the question in so far as it concerns log-
412 SCIENCE AND METHOD
arithms, and there it is easy. It is clear that a slight difference
of argument will give a slight difference of logarithm, but a great
difference in the sixth decimal of the logarithm. Always we
find again the same criterion.
But as for the number v, that presents more difficulties, and I
have at the moment nothing worth while to say.
There would be many other questions to resolve, had I wished
to attack them before solving that which I more specially set
myself. When we reach a simple result, when we find for ex-
ample a round number, we say that such a result can not be due
to chance, and we seek, for its explanation, a non-fortuitous
cause. And in fact there is only a very slight probability that
among 10,000 numbers chance will give a round number; for
example, the number 10,000. This has only one chance in 10,000.
But there is only one chance in 10,000 for the occurrence of any
other one number; and yet this result will not astonish us, nor
will it be hard for us to attribute it to chance ; and that simply
because it will be less striking.
Is this a simple illusion of ours, or are there cases where this
way of thinking is legitimate t We must hope so, else were all
science impossible. When we wish to check a hypothesis, what
do we do? We can not verify all its consequences, since they
would be infinite in number ; we content ourselves with verifying
certain ones and if we succeed we declare the hypothesis con-
firmed, because so much success could not be due to chance.
And this is always at bottom the same reasoning.
I can not completely justify it here, since it would take too
much time; but I may at least say that we find ourselves con-
fronted by two hypotheses, either a simple cause or that aggre-
gate of complex causes we call chance. We find it natural to
suppose that the first should produce a simple result, and then,
if we find that simple result, the round number for example, it
seems more likely to us to be attributable to the simple cause
which must give it almost certainly, than to chance which could
only give it once in 10,000 times. It will not be the same if we
find a result which is not simple ; chance, it is true, will not give
this more than once in 10,000 times ; but neither has the simple
cause any more chance of producing it.
BOOK II
MATHEMATICAL EEASONING
CHAPTER I
Thb Relativity op Space
I
It is impossible to represent to oneself empty space; all our
efforts to imagine a pure space, whence should be excluded the
changing images of material objects, can result only in a repre-
sentation where vividly colored surfaces, for example, are re-
placed by lines of faint coloration, and we can not go to the very
end in his way without all vanishing and terminating in nothing-
ness. Thence comes the irreducible relativity of space.
Whoever speaks of absolute space uses a meaningless phrase.
This is a truth long proclaimed by all who have reflected upon
the matter, but which we are too often led to forget.
I am at a determinate point in Paris, place du Pantheon for in-
stance, and I say: I shall come back here to-morrow. If I be
asked : Do you mean you will return to the same point of space,
I shall be tempted to answer: yes; and yet I shall be wrong,
since by to-morrow the earth will have journeyed hence, carrying
with it the place du Pantheon, which will have traveled over
more than two million kilometers. And if I tried to speak more
precisely, I should gain nothing, since our globe has run over
these two million kilometers in its motion with relation to the sun,
while the sun in its turn is displaced with reference to the Milky
Way, while the Milky Way itself is doubtless in motion without
our being able to perceive its velocity. So that we are completely
ignorant, and always shall be, of how much the place du Pan-
theon is displaced in a day.
In sum, I meant to say : To-morrow I shall see again the dome
413
414 SCIENCE AND METHOD
and the pediment of the Panth6on, and if there were no Pan-
theon my phrase would be meaningless and space would vanish.
This is one of the most commonplace forms of the principle
of the relativity of space; but there is another, upon which
Delbeuf has particularly insisted. Suppose that in the night
all the dimensions of the universe become a thousand times
greater : the world will have remained similar to itself, giving to
the word similitude the same meaning as in Euclid, Book YL
Only what was a meter long will measure thenceforth a kilometer,
what was a millimeter long will become a meter. The bed where-
on I lie and my body itself will be enlarged in the same pro-
portion.
When I awake to-morrow morning, what sensation shall I feel
in presence of such an astounding transformation t Well, I shall
perceive nothing at all. The most precise measurements will be
incapable of revealing to me anything of this immense convul-
sion, since the measures I use will have varied precisely in the
same proportion as the objects I seek to measure. In reality,
this convulsion exists only for those who reason as if space were
absolute. If I for a moment have reasoned as they do, it is the
better to bring out that their way of seeing implies contradic-
tion. In fact it would be better to say that, space being relative,
nothing at all has happened, which is why we have perceived
nothing.
Has one the right, therefore, to say he knows the distance be-
tween two points? No, since this distance could undergo enor-
mous variations without our being able to perceive them, pro-
vided the other distances have varied in the same proportion.
We have just seen that when I say: I shall be here to-morrow,
this does not mean : To-morrow I shall be at the same point of
space where I am to-day, but rather : To-morrow I shall be at the
same distance from the Pantheon as to-day. And we see that
this statement is no longer suflScient and that I should say : To-
morrow and to-day my distance from the Pantheon will be equal
to the same number of times the height of my body.
But this is not all ; I have supposed the dimensions of the world
to vary, but that at least the world remained always similar to
itself. We might go much further, and one of the most aston-
ishing theories of modern physics furnishes us the occasion.
THE RELATIVITY OF SPACE 415
According to Lorentz and Fitzgerald, all the bodies borne along
in the motion of the earth undergo a deformation.
This deformation is, in reality, very slight, since all dimensions
parallel to the movement of the earth diminish by a hundred
millionth, while the dimensions perpendicular to this movement
are unchanged. But it matters little that it is slight, that it
exists sufiBces for the conclusion I am about to draw. And be-
sides, I have said it was slight, but in reality I know nothing
about it; I have myself been victim of the tenacious illusion
which makes us believe we conceive an absolute space; I have
thought of the motion of the earth in its elliptic orbit around
the sun, and I have allowed thirty kilometers as its velocity.
But its real velocity (I mean, this time, not its absolute velocity,
which is meaningless, but its velocity with relation to the ether) ,
I do not know that, and have no means of knowing it : it is per-
haps 10, 100 times greater, and then the deformation will be 100,
10,000 times more.
Can we show this deformation t Evidently not ; here is a cube
with edge one meter; in consequence of the earth's displacement
it is deformed, one of its edges, that parallel to the motion,
becomes smaller, the others do not change. If I wish to assure
myself of it by aid of a meter measure, I shall measure first
one of the edges perpendicular to the motion and shall find that
my standard meter fits this edge exactly ; and in fact neither of
these two lengths is changed, since both are perpendicular to
the motion. Then I wish to measure the other edge, that parallel
to the motion ; to do this I displace my meter and turn it so aJs to
apply it to the edge. But the meter, having changed orienta-
tion and become parallel to the motion, has undergone, in its
turn, the deformation, so that though the edge be not a meter
long, it will fit exactly, I shall find out nothing.
You ask then of what use is the hypothesis of Lorentz and
of Fitzgerald if no experiment can permit of its verification t
It is my exposition that has been incomplete ; I have spoken only
of measurements that can be made with a meter; but we can
also measure a length by the time it takes light to traverse it, on
condition we suppose the velocity of light constant and inde-
pendent of direction. Lorentz could have accounted for the
416 SCIENCE AND METHOD
facts by supposing the velocity of light greater in the direction
of the earth's motion than in the perpendicular direction.
He preferred to suppose that the velocity is the same in these
different directions, but that the bodies are smaller in the one
than in the other. If the wave surfaces of light had undergone
the same deformations as the material bodies we should never
have perceived the Lorentz-Fitzgerald deformation.
In either case, it is not a question of absolute magnitude, but
of the measure of this magnitude by means of some instrument;
this instrument may be a meter, or the path traversed by light;
it is only the relation of the magnitude to the instrument that
we measure; and if this relation is altered, we have no way of
knowing whether it is the magnitude or the instrument which
has changed.
But what I wish to bring out is, that in this deformation the
world has not remained similar to itself; squares have become
rectangles, circles ellipses, spheres ellipsoids. And yet we have
no way of knowing whether this deformation be real.
Evidently one could go much further : in place of the Lorents-
Fitzgerald deformation, whose laws are particularly simple, we
could imagine any deformation whatsoever. Bodies could be
deformed according to any laws, as complicated as we might wish,
we never should notice it provided all bodies without exception
were deformed according to the same laws. In saying, all bodies
without exception, I include of course our own body and the
light rays emanating from different objects.
If we look at the world in one of those mirrors of complicated
shape which deform objects in a bizarre way, the mutual relations
of the different parts of this world would not be altered; if, in
fact two real objects touch, their images likewise seem to touch.
Of course when we look in such a mirror we see indeed the
deformation, but this is because the real world subsists along-
side of its deformed image ; and then even were this real world
hidden from us, something there is could not be hidden, ourself ;
we could not cease to see, or at least to feel, our body and our
limbs which have not been deformed and which continue to serve
us as instruments of measure.
But if we imagine our body itself deformed in the same way
THE RELATIVITY OF SPACE 417
as if seen in the mirror, these instruments of measure in their
turn will fail us and the deformation will no longer be ascer-
tainable.
Consider in the same way two worlds images of one another;
to each object P of the world A corresponds in the world B an
object P', its image ; the coordinates of this image P* are deter-
minate functions of those of the object P; moreover these func-
tions may be any whatsoever; I only suppose them chosen once
for all. Between the position of P and that of P* there is a
constant relation ; what this relation is, matters not ; enough that
it be constant.
Well, these two worlds will be indistinguishable one from the
other. I mean the first will be for its inhabitants what the
second is for its. And so it will be as long as the two worlds
remain strangers to each other. Suppose we live in world A, we
shall have constructed our science and in particular our geom-
etry ; during this time the inhabitants of world B will have con-
structed a science, and as their world is the image of ours, their
geometry will also be the image of ours or, better, it will be the
same. But if for us some day a window is opened upon world
B, how we shall pity them: **Poor things," we shall say, **they
think they have made a geometry, but what they call so is only
a grotesque image of ours; their straights are all twisted, their
circles are humped, their spheres have capricious inequalities."
And we shall never suspect they say the same of us, and one
never will know who is right.
We see in how broad a sense should be understood the rela-
tivity of space; space is in reality amorphous and the things
which are therein alone give it a form. What then should be
thought of that direct intuition we should have of the straight
or of distance t So little have we intuition of distance in itself
that in the night, as we have said, a distance might become a
thousand times greater without our being able to perceive it, if
all other distances had undergone the same alteration. And even
in a night the world B might be substituted for the world A
without our having any way of knowing it, and then the straight
lines of yesterday would have ceased to be straight and we
should never notice.
28
418 SCIENCE AND METHOD
One part of space is not by itself and in the absolute sense of
the word equal to another part of space ; because if so it is for
us, it would not be for the dwellers in world B ; and these have
just as much right to reject our opinion as we to condemn theirs.
I have elsewhere shown what are the consequences of these
facts from the viewpoint of the idea we should form of non-
Euclidean geometry and other analogous geometries; to that I
do not care to return ; and to-day I shall take a somewhat dif-
ferent point of view.
II
If this intuition of distance, of direction, of the straight line,
if this direct intuition of space in a word does not exist, whence
comes our belief that we have itt If this is only an illusion,
why is this illusion so tenacious t It is proper to examine into
this. We have said there is no direct intuition of size and we
can only arrive at the relation of this magnitude to our instru-
ments of measure. We should therefore not have been able to
construct space if we had not had an instrument to measure it;
well, this instrument to which we relate everything, which we
use instinctively, it is our own body. It is in relation to our
body that we place exterior objects, and the only spatial rela-
tions of these objects that we can represent are their relations
to our body. It is our body which serves us, so to speak, as
system of axes of coordinates.
For example, at an instant a, the presence of the object A is
revealed to me by the sense of sight; at another instant, fi, the
presence of another object, B, is revealed to me by another sense,
that of hearing or of touch, for instance. I judge that this
object B occupies the same place as the object A. What does
that mean? First that does not signify that these two objects
occupy, at two different moments, the same point of an absolute
space, which even if it existed would escape our cognition, since,
between the instants a and p, the solar system has moved and
we can not know its displacement. That means these two objects
occupy the same relative position with reference to our body.
But even this, what does it mean ? The impressions that have
come to us from these objects have followed paths absolutely
TEE RELATIVITY OF SPACE 419
different, the optic nerve for the object A, the acoustic nerve for
the object jB. They have nothing in common from the qualita-
tive point of view. The representations we are able to make of
these two objects are absolutely heterogeneous, irreducible one to
the other. Only I know that to reach the object A I have just
to extend the right arm in a certain way ; even when I abstain
from doing it, I represent to myself the muscular sensations and
other analogous sensations which would accompany this exten-
sion, and this representation is associated with that of the
object A.
Now, I likewise know I can reach the object B by extending my
right arm in the same manner, an extension accompanied by the
same train of muscular sensations. And when I say these two
objects occupy the same place, I mean nothing more.
I also know I could have reached the object A by another
appropriate motion of the left arm and I represent to myself the
muscular sensations which would have accompanied this move-
ment ; and by this same motion of the left arm, accompanied by
the same sensations, I likewise could have reached the object B.
And that is very important, since thus I can defend myself
against dangers menacing me from the object A or the object B.
With each of the blows we can be hit, nature has associated
one or more parries which permit of our guarding ourselves.
The same parry may respond to several strokes ; and so it is, for
instance, that the same motion of the right arm would have
allowed us to guard at the instant a against the object A and at
the instant p against the object B. Just so, the same stroke can
be parried in several ways, and we have said, for instance, the
object A could be reached indifferently either by a certain move-
ment of the right arm or by a certain movement of the left arm.
All these parries have nothing in common except warding off
the same blow, and this it is, and nothing else, which is meant
when we say they are movements terminating at the same point
of space. Just so, these objects, of which we say they occupy
the same point of space, have nothing in common, except that the
same parry guards against them.
Or, if you choose, imagine innumerable telegraph wires, some
centripetal, others centrifugal. The centripetal wires warn us of
420 SCIENCE AND METHOD
accidents happening without; the centrifugal wires carry the
reparation. Connections are so established that when a cen-
tripetal wire is traversed by a current this acts on a relay and so
starts a current in one of the centrifugal wires, and things are
so arranged that several centripetal wires may act on the same
centrifugal wire if the same remedy suits several ills, and that a
centripetal wire may agitate different centrifugal wires, either
simultaneously or in lieu one of the other when the same ill may
be cured by several remedies.
It is this complex system of associations, it is this table of distri-
bution, so to speak, which is all our geometry or, if you wish,
all in our geometry that is instinctive. What we call our intui-
tion of the straight line or of distance is the consciousness we
have of these associations and of their imperious character.
And it is easy to understand whence comes this imperious
character itself. An association will seem to us by so much the
more indestructible as it is more ancient. But these associations
are not, for the most part, conquests of the individual, since their
trace is seen in the new-bom babe : they are conquests of the race.
Natural selection had to bring about these conquests by so much
the more quickly as they were the more necessary.
On this account, those of which we speak must have been of
the earliest in date, since without them the defense of the organ-
ism would have been impossible. From the time when the cell-
ules were no longer merely juxtaposed, but were called upon to
give mutual aid, it was needful that a mechanism organize anal-
ogous to what we have described, so that this aid miss not its
way, but forestall the peril.
When a frog is decapitated, and a drop of acid is placed on a
point of its skin, it seeks to wipe off the acid with the nearest foot,
and, if this foot be amputated, it sweeps it off with the foot of
the opposite side. There we have the double parry of which I
have just spoken, allowing the combating of an ill by a second
remedy, if the first fails. And it is this multiplicity of parries,
and the resulting coordination, which is space.
We see to what depths of the unconscious we must descend
to find the first traces of these spatial associations, since only
the inferior parts of the nervous system are involved. Why be
THE RELATIVITY OF SPACE 421
astonished then at the resistance we oppose to every attempt
made to dissociate what so long has been associated t Now, it is
just this resistance that we call the evidence for the geometric
truths ; this evidence is nothing but the repugnance we feel toward
breaking with very old habits which have always proved good.
Ill
The space so created is only a little space extending no farther
than my arm can reach ; the intervention of the memory is neces-
sary to push back its limits. There are points which will remain
out of my reach, whatever effort I make to stretch forth my hand ;
if I were fastened to the ground like a hydra polyp, for instancei
which can only extend its tentacles, all these points would be
outside of space, since the sensations we could experience from
the action of bodies there situated, would be associated with the
idea of no movement allowing us to reach them, of no appro-
priate parry. These sensations would not seem to us to have
any spatial character and we should not seek to localize them.
But we are not fixed to the ground like the lower animals ; we
can, if the enemy be too far away, advance toward him first and
extend the hand when we are sufiSciently near. This is still a
parry, but a parry at long range. On the other hand, it is a
complex parry, and into the representation we make of it enter
the representation of the muscular sensations caused by the
movements of the legs, that of the muscular sensations caused
by the final movement of the arm, that of the sensations of the
semicircular canals, etc. We must, besides, represent to our-
selves, not a complex of simultaneous sensations, but a complex
of successive sensations, following each other in a determinate
order, and this is why I have just said the intervention of memory
was necessary. Notice moreover that, to reach the same point,
I may approach nearer the mark to be attained, so as to have to
stretch my arm less. What more t It is not one, it is a thousand
parries I can oppose to the same danger. All these parries are
made of sensations which may have nothing in common and yet
we regard them as defining the same point of space, since they
may respond to the same danger and are all associated with the
notion of this danger. It is the potentiality of warding off the
422 SCIENCE AND METHOD
same stroke which makes the unity of these different parries, as
it is the possibility of being parried in the same way which makes
the unity of the strokes so different in kind, which may menace
us from the same point of space. It is this double unity which
makes the individuality of each point of space, and, in the
notion of point, there is nothing else.
The space before considered, which might be called restricted
space, was referred to coordinate axes bound to my body; these
axes were fixed, since my body did not move and only my mem-
bers were displaced. What are the axes to which we naturally
refer the extended space? that is to say the new space just
defined. We define a point by the sequence of movements to be
made to reach it, starting from a certain initial position of the
body. The axes are therefore fixed to this initial position of the
body.
But the position I call initial may be arbitrarily chosen among
all the positions my body has successively occupied ; if the memory
more or less unconscious of these successive positions is necessary
for the genesis of the notion of space, this memory may go back
more or less far into the past. Thence results in the definition
itself of space a certain indetermination, and it is precisely this
indetermination which constitutes its relativity.
There is no absolute space, there is only space relative to a
certain initial position of the body. For a conscious being fixed
to the ground like the lower animals, and consequently knowing
only restricted space, space would still be relative (since it would
have reference to his body), but this being would not be conscious
of this relativity, because the axes of reference for this restricted
space would be unchanging! Doubtless the rock to which this
being would be fettered would not be motionless, since it would
be carried along in the movement of our planet; for us conse-
quently these axes would change at each instant ; but for him they
would be changeless. We have the faculty of referring our
extended space now to the position A of our body, considered as
initial, again to the position B^ which it had some moments
afterward, and which we are free to regard in its turn as initial ;
we make therefore at each instant unconscious transformations
of coordinates. This faculty would be lacking in our imaginary
TBE RELATIVITY OF SPACE 423
being, and from not having traveled, he would think space abso-
lute. At every instant, his system of axes would be imposed
upon him ; this system would have to change greatly in reality,
but for him it would be always the same, since it would be
always the only system. Quite otherwise is it with us, who at
each instant have many systems between which we may choose at
will, on condition of going back by memory more or less far into
the past.
This is not all; restricted space would not be homogeneous;
the different points of this space could not be regarded as equiva-
lent, since some could be reached only at the cost of the greatest
efforts, while others could be easily attained. On the contrary,
our extended space seems to us homogeneous, and we say all its
points are equivalent. What does that meant
If we start from a certain place A, we can, from this position,
make certain movements, M, characterized by a certain complex
of muscular sensations. But, starting from another position, B,
we make movements M' characterized by the same muscular sen-
sations. Let a, then, be the situation of a certain point of the
body, the end of the index finger of the right hand for example,
in the initial position A, and b the situation of this same index
when, starting from this position A, we have made the motions M.
Afterwards, let a' be the situation of this index in the position B,
and b' its situation when, starting from the position B, we have
made the motions 3f^
Well, I am accustomed to say that the points of space a and b
are related to each other just as the points a' and b\ and this
simply means that the two series of movements M and ilf' are
accompanied by the same muscular sensations. And as I am
conscious that, in passing from the position A to the position £,
my body has remained capable of the same movements, I know
there is a point of space related to the point of just as any point
b is to the point a, so that the two points a and a' are equivalent.
This is what is called the homogeneity of space. And, at the same
time, this is why space is relative, since its properties remain the
same whether it be referred to the axes A or to the axes B. So
that the relativity of space and its homogeneity are one sole and
same thing.
424 SCIENCE AND METHOD
r, if I wiih to pav to Ae great wp&ot^ wliidi no longer
aenres onl j for me, bat where I maj lodge the muretae, I get
there bj an act of imaginatioiL. I imagine how a giant would
feel who could reach the planets in a few steps; or, if yoa ehooae,
what I mjself should feel in presence of a miniature world whore
these planets were replaced hj little balls, while on one of these
little balls moved a liliputian I should call mjrself. Bat this aet
of imagination woald be impossible for me had I not previoosij
eonstracted my restricted space and my eztoided space for my
own use.
IV
Why now have all these spaces three dimensions ? Go back
to the "table of distribution" of which we hare spoken. We
have on the one side the list of the different possible dangers;
designate them by Al^ A2^ etc. ; and, on the other side, the list
of the different remedies which I shall call in the same way
Bly jB2, etc. We have then connections between the contact studs
or push buttons of the first list and those of the second, so that
when, for instance, the announcer of danger AZ functions, it
will put or may put in action the relay corresponding to the
parry 54.
As I have spoken above of centripetal or centrifugal wires, I
fear lest one see in all this, not a simple comparison, but a descrip-
tion of the nervous system. Such is not my thought, and that
for several reasons : first I should not permit myself to put forth
an opinion on the structure of the nervous system which I do
not know, while those who have studied it speak only circum-
spectly; again because, despite my incompetence, I well know
this scheme would be too simplistic; and finally because on my
list of parries, some would figure very complex, which might even,
in the case of extended space, as we have seen above, consist of
many steps followed by a movement of the arm. It is not a ques-
tion then of physical connection between two real conductors,
but of psychologic association between two series of sensations.
If Al and A2 for instance are both associated with the parry
51, and if Al is likewise associated with the parry 52, it will
generally happen that A2 and 52 will also themselves be asso-
ciated. If this fundamental law were not generally true, there
THE BELATIVITT OF SPACE 426
would exist only an immense confusion and there would be
nothing resembling a conception of space or a geometry. How
in fact have we defined a point of space. We have done it in two
ways: it is on the one hand the aggregate of announcers A in
connection with the same parry JB; it is on the other hand the
aggregate of parries B in connection with the same announcer A.
If our law was not true, we should say ill and A2 correspond
to the same point since they are both in connection with Bl ; but
we should likewise say they do not correspond to the same point,
since Al would be in connection with B2 and the same would
not be true of A2. This would be a contradiction.
But, from another side, if the law were rigorously and always
true, space would be very diiSEerent from what it is. We should
have categories strongly contrasted between which would be
portioned out on the one hand the announcers A, on the other
hand the parries B; these categories would be excessively nu-
merous, but they would be entirely separated one from another.
Space would be composed of points very numerous, but discrete;
it would be discontinuous. There would be no reason for rang-
ing these points in one order rather than another, nor conse-
quently for attributing to space three dimensions.
But it is not so ; permit me to resume for a moment the lan-
guage of those who already know geometry ; this is quite proper
since this is the language best understood by those I wish to make
understand me.
When I desire to parry the stroke, I seek to attain the point
whence comes this blow, but it suffices that I approach quite near.
Then the parry Bl may answer for Al and for A2, if the point
which corresponds to Bl is sufficiently near both to that corre-
sponding to Al and to that corresponding to A2. But it may
happen that the point corresponding to another parry B2 may be
sufficiently near the point corresponding to ^1 and not suffi-
ciently near the point corresponding to -42 ; so that the parry B2
may answer for Al without answering for A2. For one who
does not yet know geometry, this translates itself simply by a
derogation of the law stated above. And then things will happen
thus:
Two parries JBl and B2 will be associated with the same warn-
426 SCIENCE AND METHOD
ing ill and with a large number of warnings which we ahall
range in the same category as Al and which we shall make corre-
spond to the same point of space. But we may find warnings
A2 which will be associated with B2 without being associated
with Bly and which in compensation will be associated with £3,
which jB3 was not associated with Al, and so forth, so that we
may write the series
Bl, ^1, B2, A2, BZ, A3, B4, A4,
where each term is associated with the following and the preced-
ing, but not with the terms several places away.
Needless to add that each of the terms of these series is not
isolated, but forms part of a very numerous category of other
warnings or of other parries which have the same connections as
it, and which may be regarded as belonging to the same point of
space.
The fundamental law, though admitting of exceptions, remains
therefore almost always true. Only, in consequence of these
exceptions, these categories, in place of being entirely separated,
encroach partially one upon another and mutually penetrate in
a certain measure, so that space becomes continuous.
On the other hand, the order in which these categories are to
be ranged is no longer arbitrary, and if we refer to the preceding
series, we see it is necessary to put 52 between Al and A2 and
consequently between Bl and B3 and that we could not for
instance put it between BS and B4.
There is therefore an order in which are naturally arranged
our categories which correspond to the points of space, and experi-
ence teaches us that this order presents itself under the form
of a table of triple entry, and this is why space has three
dimensions.
V
So the characteristic property of space, that of having three
dimensions, is only a property of our table of distribution, an
internal property of the human intelligence, so to speak. It
would suffice to destroy certain of these connections, that is to
say of the associations of ideas to give a different table of dis-
tribution, and that might be enough for space to acquire a fourth
dimension.
THE BELATIVITY OF SPACE 427
Some persons will be astonished at such a result. The external
world, they will think, should count for something. If the num-
ber of dimensions comes from the way we are made, there might
be thinking beings living in our world, but who might be made
differently from us and who would believe space has more or less
than three dimensions. Has not M. de Cyon said that the Jap-
anese mice, having only two pair of semicircular canals, believe
that space is two-dimensional T And then this thinking being, if
he is capable of constructing a physics, would he not make a phys-
ics of two or of four dimensions, and which in a sense would
still be the same as ours, since it would be the description of the
same world in another language!
It seems in fact that it would be possible to translate our phys-
ics into the language of geometry of four dimensions ; to attempt
this translation would be to take great pains for little profit, and
I shall confine myself to citing the mechanics of Hertz where we
have something analogous. However, it seems that the transla-
tion would always be less simple than the text, and that it would
always have the air of a translation, that the language of three
dimensions seems the better fitted to the description of our world,
although this description can be rigorously made in another
idiom. Besides, our table of distribution was not made at ran-
dom. There is connection between the warning Al and the
parry jBI, this is an internal property of our intelligence; but
why this connection? It is because the parry jBI affords means
effectively to guard against the danger Al; and this is a fact
exterior to us, this is a property of the exterior world. Our
table of distribution is therefore only the translation of an ag-
gregate of exterior facts; if it has three dimensions, this is be-
cause it has adapted itself to a world having certain properties ;
and the chief of these properties is that there exist natural solids
whose displacements follow sensibly the laws we call laws of
motion of rigid solids. If therefore the language of three di-
mensions is that which permits us most easily to describe our
world, we should not be astonished ; this language is copied from
our table of distribution ; and it is in order to be able to live in
this world that this table has been established.
I have said we could conceive, living in our world, thinking
428 SCIENCE AND METHOD
beings whose table of distribution would be four-dimensional
and who consequently would think in hyperspace. It is not
certain however that such beings, admitting they were bom there,
could live there and defend themselves against the thousand
dangers by which they would there be assailed.
VI
A few remarks to end with. There is a striking contrast be-
tween the roughness of this primitive geometry, reducible to
what I call a table of distribution, and the infinite precision of
the geometers' geometry. And yet this is bom of that; but not
of that alone ; it must be made fecund by the faculty we have of
constructing mathematical concepts, such as that of group, for
instance; it was needful to seek among the pure concepts that
which best adapts itself to this rough space whose genesis I have
sought to explain and which is common to us and the higher
animals.
The evidence for certain geometric postulates, we have said, is
only our repugnance to renouncing very old habits. But these
postulates are infinitely precise, while these habits have some-
thing about them essentially pliant. When we wish to think, we
need postulates infinitely precise, since this is the only way to
avoid contradiction ; but among all the possible systems of postu-
lates, there are some we dislike to choose because they are not
suflBciently in accord with our habits; however pliant, however
elastic they may be, these have a limit of elasticity.
We see that if geometry is not an experimental science, it is a
science bom apropos of experience; that we have created the
space it studies, but adapting it to the world wherein we live.
We have selected the most convenient space, but experience has
guided our choice; as this choice has been unconscious, we think
it has been imposed upon us; some say experience imposes it,
others that we are bom with our space ready made ; we see from
the preceding considerations, what in these two opinions is the
part of truth, what of error.
In this progressive education whose outcome has been the con-
struction of space, it is very diflScult to determine what is the
THE BELATIVITY OF SPACE 429
part of the individaal, what the part of the race. How far could
one of us, transported from birth to an entirely different world,
where were dominant, for instance, bodies moving in conformity
to the laws of motion of non-Euclidean solids, renounce the an-
cestral space to build a space completely newt
The part of the race seems indeed preponderant ; yet if to it we
owe rough space, the soft space I have spoken of, the space of
the higher animals, is it not to the unconscious experience of the
individual we owe the infinitely precise space of the geometer!
This is a question not easy to solve. Yet we cite a fact showing
that the space our ancestors have bequeathed us still retains a
certain plasticity. Some hunters learn to shoot fish under water,
though the image of these fish be turned up by refraction. Be-
sides they do it instinctively: they therefore have learned to
modify their old instinct of direction ; or, if you choose, to sub-
stitute for the association Al, j51, another association Al, j52,
because experience showed them the first would not work.
CHAPTER II
Mathematical Definitions and Teachinq
1. I SHOULD speak here of general definitions in mathematicB;
at least that is the title, but it will be impossible to confine my-
self to the subject as strictly as the rule of unity of action would
require ; I shall not be able to treat it without touching upon a
few other related questions, and if thus I am forced from time
to time to walk on the bordering flower-beds on the right or left^
I pray you bear with me.
What is a good definition T For the philosopher or the scientist
it is a definition which applies to all the objects defined, and only
those ; it is the one satisfying the rules of logic. But in teach-
ing it is not that; a good definition is one understood by the
scholars.
How does it happen that so many refuse to understand math-
ematics T Is that not something of a paradox! Lo and behold I
a science appealing only to the fundamental principles of logic,
to the principle of contradiction, for instance, to that which is
the skeleton, so to speak, of our intelligence, to that of which we
can not divest ourselves without ceasing to think, and there are
people who find it obscure! and they are even in the majority!
That they are incapable of inventing may pass, but that they do
not understand the demonstrations shown them, that they re-
main blind when we show them a light which seems to us flash-
ing pure flame, this it is which is altogether prodigious.
And yet there is no need of a wide experience with examina-
tions to know that these blind men are in no wise exceptional
beings. This is a problem not easy to solve, but which should
engage the attention of all those wishing to devote themselves to
teaching.
What is it, to understand? Has this word the same meaning
for all the world ? To understand the demonstration of a theorem,
is that to examine successively each of the syllogisms composing
it and to ascertain its correctness, its conformity to the rules of
430
MATHEMATICAL DEFINITIONS AND TEACHING 431
the garnet Likewise, to understand a definition, is this merely
to recognize that one already knows the meaning of all the terms
employed and to ascertain that it implies no contradiction T
For some, yes ; when they have done this, they will say : I un-
derstand.
For the majority, no. Almost all are much more exacting;
they wish to know not merely whether all the syllogisms of a
demonstration are correct, but why they link together in this
order rather than another. In so far as to them they seem en-
gendered by caprice and not by an intelligence always conscious
of the end to be attained, they do not believe they understand.
Doubtless they are not themselves just conscious of what they
crave and they could not formulate their desire, but if they do
not get satisfaction, they vaguely feel that something is lacking.
Then what happens t In the beginning they still perceive the
proofs one puts under their eyes; but as these are connected
only by too slender a thread to those which precede and those
which follow, they pass without leaving any trace in their head ;
they are soon forgotten ; a moment bright, they quickly vanish in
night eternal. When they are farther on, they will no longer see
even this ephemeral light, since the theorems lean one upon
another and those they would need are forgotten ; thus it is they
become incapable of understanding mathematics.
This is not always the fault of their teacher ; often their mind,
which needs to perceive the guiding thread, is too lazy to seek
and find it. But to come to their aid, we first must know just
what hinders them.
Others will always ask of what use is it; they will not have
understood if they do not find about them, in practise or in
nature, the justification of such and such a mathematical concept.
Under each word they wish to put a sensible image ; the definition
must evoke this image, so that at each stage of the demonstration
they may see it transform and evolve. Only upon this condition
do they comprehend and retain. Often these deceive themselves ;
they do not listen to the reasoning, they look at the figures ; they
think they have understood and they have only seen.
2. How many different tendencies! Must we combat themt
Must we use them ? And if we wish to combat them, which should
432 SCIENCE AND METHOD
be favored t Must we show those content with the pure logic that
they have seen only one side of the matter t Or need we say to
those not so cheaply satisfied that what they demand is not
necessary?
In other words, should we constrain the young people to change
the nature of their minds? Such an attempt would be vain; we
do not possess the philosopher's stone which would enable us to
transmute one into another the metals confided to us; all we
can do is to work with them, adapting ourselves to their
properties.
Many children are incapable of becoming mathematicians, to
whom however it is necessary to teach mathematics; and the
mathematicians themselves are not all cast in the same mold.
To read their works suffices to distinguish among them two
sorts of minds, the logicians like Weierstrass for example, the
intuitives like Biemann. There is the same difference among
our students. The one sort prefer to treat their problems 'by
analysis' as they say, the others *by geometry.'
It is useless to seek to change anything of that, and besides
would it be desirable! It is well that there are logicians and
that there are intuitives; who would dare say whether he pre-
ferred that Weierstrass had never written or that there never
had been a Riemann. We must therefore resign ourselves to the
diversity of minds, or better we must rejoice in it.
3. Since the word understand has many meanings, the defi-
nitions which will be best understood by some will not be best
suited to others. We have those which seek to produce an image,
and those where we confine ourselves to combining empty forms,
perfectly intelligible, but purely intelligible, which abstraction
has deprived of all matter.
I know not whether it be necessary to cite examples. Let us
cite them, anyhow, and first the definition of fractions will furnish
us an extreme case. In the primary schools, to define a frac-
tion, one cuts up an apple or a pie; it is cut up mentally of
course and not in reality, because I do not suppose the budget
of the primary instruction allows of such prodigality. At the
Normal School, on the other hand, or at the college, it is said:
a fraction is the combination of two whole numbers separated by
MATHEMATICAL DEFINITIONS AND TEACHING 4S3
a horizontal bar; vre define by conventions the operations to
which these symbols may be submitted ; it is proved that the rules
ot these operations are the same as in calculating with whole
numbers, sjid we ascertain finally that multiplying the fraction,
according to these rules, by the denominator gives the numerator.
This is all very well because we are addressing young people
long familiarized with the notion of fractions through having cut
up apples or other objects, and whose mind, matured by a hard
mathematical education, has come little by little to desire a purely
logical definition. But the debutant to whom one should try to
give it, how dumf ounded !
Such also are the definitions found in a book justly admired
and greatly honored, the Foundations of Geometry by Hilbert.
See in fact how he begins: We think three systems of thinqs
which we shail call points, straights and planes. What are these
'things' T
We know not, nor need we know ; it would even be a pity to
seek to know ; all we have the right to know of them is what the
assumptions tell us ; this for example : Two distinct points olwoyr
determine a straight, which is followed by this remark: in place
of determine, we may say the two points are on the straight, or
the straight goes through these two points or joins the ttvo points.
Thus 'to be on a straight' is simply defined as synonymouu
with 'determine a straight.' Behold a book of which I think
much good, but which I should not recommend to a school boy.
Yet I could do so without fear, he would not read much of it,
I have taken extreme examples and no teacher would dream ot
going that far. But even stopping short of such models, doea
he not already expose himself to the same dangerl
Suppose we are in a class; the professor dictates; the circle is
the locns of points of the plane equidistant from an interior point
called the center. The good scholar writes this phrase in his
note-booki the bad scholar draws faces; hut neither understands;
then the professor takes the chalk and draws a circle ou the board.
"Ahl" think the scholars, "why did he not say at once: a circle
is a ring, we should have uuderstood." Doubtless the professor
is right. The scholars' definition would have been of no avail,
since it could serve for no demonstration, since besides it would
434 SCIENCE AND METHOD
not give them the salutary habit of analyzing their conceptions.
But one should show them that they do not comprehend what
they think they know, lead them to be conscious of the roughness
of their primitive conception, and of themselves to wish it puri-
fied and made precise.
4. I shall return to these examples ; I only wished to show you
the two opposed conceptions ; they are in violent contrast. This
contrast the history of science explains. If we read a book
written fifty years ago, most of the reasoning we find there seems
lacking in rigor. Then it was assumed a continuous function
can change sign only by vanishing ; to-day we prove it. It was
assumed the ordinary rules of calculation are applicable to
inconmiensurable numbers; to-day we prove it. Many other
things were assumed which sometimes were false.
We trusted to intuition ; but intuition can not give rigor, nor
even certainty ; we see this more and more. It tells us for instance
that every curve has a tangent, that is to say that every con-
tinuous function has a derivative, and that is false. And as we
sought certainty, we had to make less and less the part of
intuition.
What has made necessary this evolution? We have not been
slow to perceive that rigor could not be established in the reason-
ings, if it were not first put into the definitions.
The objects occupying mathematicians were long ill defined;
we thought we knew them because we represented them with the
senses or the imagination; but we had of them only a rough
image and not a precise concept upon which reasoning could take
hold. It is there that the logicians would have done well to direct
their efforts.
So for the incommensurable number, the vague idea of con-
tinuity, which we owe to intuition, has resolved itself into a com-
plicated system of inequalities bearing on whole numbers. Thus
have finally vanished all those difiiculties which frightened our
fathers when they reflected upon the foundations of the infini-
tesimal calculus. To-day only whole numbers are left in analysis,
or systems finite or infinite of whole numbers, bound by a
plexus of equalities and inequalities. Mathematics we say is
arithmetized.
MATHEMATICAL DEFINITIONS AND TEACHING 436
5. But do you tliink mathematics has attained absolute rigor
without making any sacrifice t Not at all; what it has gained in
rigor it has lost in objectivity. It is by separating itself from
reality that it has acquired this perfect purity. We may freely
run over its whole domain, formerly bristling with obstacles, but
these obstacles have not disappeared. They have only been
moved to the frontier, and it would be necessary to vanquish
them anew if we wished to break over this frontier to enter the
realm of the practical.
We had a vague notion, formed of incongruous elements, some
a priori, others coming from experiences more or less digested;
we thought we knew, by intuition, its principal properties. To-
day we reject the empiric elements, retaining only the a priori;
one of the properties serves as definition and all the others are
deduced from it by rigorous reasoning. This is all very well,
but it remains to be proved that this properly, which has become
a definition, pertains to the real objects which experience had
made known to us and whence we drew our vague intuitive
notion. To prove that, it would be necessary to appeal to experi-
ence, or to make an effort of intuition, and if we could not prove
it, our theorems would be perfectly rigorous, but perfectly
useless.
Logic sometimes makes monsters. Since half a century we
have seen arise a crowd of bizarre functions which seem to try
to resemble as little as possible the honest functions which serve
some purpose. No longer continuity, or perhaps continuity, but
no derivatives, etc. Nay more, from the logical point of view,
it is these strange functions which are the most general, those
one meets without seeking no longer appear except as particular
case. There remains for them only a very small comer.
Heretofore when a new function was invented, it was for some
practical end ; to-day they are invented expressly to put at fault
the reasonings of our fathers, and one never will get from them
anything more than that.
If logic were the sole guide of the teacher, it would be neces-
sary to begin with the most general functions, that is to say with
the most bizarre. It is the beginner that would have to be set
436 SCIENCE AND METHOD
grappling with this teratologic museum. If you do not do it,
the logicians might say, you will achieve rigor only by stages.
6. Yes, perhaps, but we can not make so cheap of reality, and
I mean not only the reality of the sensible world, which however
has its worth, since it is to combat against it that nine tenths of
your students ask of you weapons. There is a reality more
subtile, which makes the very life of the mathematical beings,
and which is quite other than logic.
Our body is formed of cells, and the cells of atoms ; are these
cells and these atoms then all the reality of the human bodyt
The way these cells are arranged, whence results the unity of the
individual, is it not also a reality and much more interesting t
A naturalist who never had studied the elephant except in
the microscope, would he think he knew the animal adequately!
It is the same in mathematics. When the logician shall have
broken up each demonstration into a multitude of elementary
operations, all correct, he still will not possess the whole reality ;
this I know not what which makes the unity of the demonstration
will completely escape him.
In the edifices built up by our masters, of what use to admire
the work of the mason if we can not comprehend the plan of
the architect? Now pure logic can not give us this appreciation
of the total effect ; this we must ask of intuition.
Take for instance the idea of continuous function. This is at
first only a sensible image, a mark traced by the chalk on the
blackboard. Little by little it is refined ; we use it to construct
a complicated system of inequalities, which reproduces all the
features of the primitive image; when all is done, we have
removed the centering, as after the construction of an arch;
this rough representation, support thenceforth useless, has dis-
appeared and there remains only the edifice itself, irreproachable
in the eyes of the logician. And yet, if the professor did not
recall the primitive image, if he did not restore momentarily the
centering, how could the student divine by what caprice all these
inequalities have been scaffolded in this fashion one upon another?
The definition would be logically correct, but it would not show
him the veritable reality.
7. So back we must return ; doubtless it is hard for a master
I
MATHEMATICAL DEFINITIONS AND TEACHING 437
to teach what does not entirely satisfy him; but the satisfaction
of the master is not the unique object of teaching; we should first
give attention to what the mind of the pupil is and to what we
wish it to become.
Zoologists maintain that the embryonic development of an
animal recapitulates in brief the whole history of its ancestors
throughout geologic time. It seems it is the same in the develop-
ment of minds. The teacher should make the child go over the
path his fathers trod; more rapidly, but without skipping sta-
tions. For this reason, the history of science should be our first
guide.
Our fathers thought they knew what a fraction was, or con-
tinuity, or the area of a curved surface ; we have found they did
not know it. Just so our scholars think they know it when they
begin the serious study of mathematics. If without warning I
tell them : **No, you do not know it; what you think you under-
stand, you do not understand ; I must prove to you what seems
to you evident," and if in the demonstration I support myself
upon premises which to them seem less evident than the con-
clusion, what shall the unfortunates think? They will think that
the science of mathematics is only an arbitrary mass of useless
subtilities ; either they will be disgusted with it, or they will play
it as a game and will reach a state of mind like that of the Greek
sophists.
Later, on the contrary, when the mind of the scholar, familiar-
ized with mathematical reasoning, has been matured by this long
frequentation, the doubts will arise of themselves and then your
demonstration will be welcome. It will awaken new doubts, and
the questions will arise successively to the child, as they arose suc-
cessively to our fathers, until perfect rigor alone can satisfy him.
To doubt everything does not suflSce, one must know why he
doubts.
8. The principal aim of mathematical teaching is to develop
certain faculties of the mind, and among them intuition is not the
least precious. It is through it that the mathematical world
remains in contact with the real world, and if pure mathematics
could do without it, it would always be necessary to have recoone
to it to fill up the chasm which separates the symbol from reality.
438 SCIENCE AND METHOD
The practician will always have need of it, and for one pure
geometer there should be a hundred practicians.
The engineer should receive a complete mathematical educa-
tion, but for what should it serve him ?
To see the different aspects of things and see them quickly;
he has no time to hunt mice. It is necessary that, in the com-
plex physical objects presented to him, he should promptly recog-
nize the point where the mathematical tools we have put in his
hands can take hold. How could he do it if we should leave
between instruments and objects the deep chasm hollowed out
by the logicians!
9. Besides the engineers, other scholars, less numerous, are in
their turn to become teachers; they therefore must go to the
very bottom; a knowledge deep and rigorous of the firist prin-
ciples is for them before all indispensable. But this is no reason
not to cultivate in them intuition ; for they would get a false idea
of the science if they never looked at it except from a single side,
and besides they could not develop in their students a quality
they did not themselves possess.
For the pure geometer himself, this faculty is necessary; it
is by logic one demonstrates, by intuition one invents. To know
how to criticize is good, to know how to create is better. You
know how to recognize if a combination is correct; what a pre-
dicament if you have not the art of choosing among all the pos-
sible combinations. Logic tells us that on such and such a way
we are sure not to meet any obstacle ; it does not say which way
leads to the end. For that it is necessary to see the end from
afar, and the faculty which teaches us to see is intuition. With-
out it the geometer would be like a writer who should be versed
in grammar but had no ideas. Now how could this faculty
develop if, as soon as it showed itself, we chase it away and pro-
scribe it, if we learn to set it at naught before knowing the
good of it.
And here permit a parenthesis to insist upon the importance of
written exercises. Written compositions are perhaps not suflS-
ciently emphasized in certain examinations, at the polytechnic
school, for instance. I am told they would close the door
MATHEMATICAL DEFINITIONS AND TEACHING 439
against very good scholars who have mastered the course, thor-
oughly understanding it, and who nevertheless are incapable of
making the slightest application. I have just said the word
understand has several meanings: such students only understand
in the first way, and we have seen that suffices neither to make an
engineer nor a geometer. Well, since choice must be made, I pre-
fer those who understand completely.
10. But is the art of sound reasoning not also a precious
thing, which the professor of mathematics ought before all to
cultivate! I take good care not to forget that. It should oc-
cupy our attention and from the very beginning. I should be
distressed to see geometry degenerate into I know not what tach-
ymetry of low grade and I by no means subscribe to the extreme
doctrines of certain German Oberlehrer. But there are occa-
sions enough to exercise the scholars in correct reasoning in the
parts of mathematics where the inconveniences I have pointed
out do not present themselves. There are long chains of the-
orems where absolute logic has reigned from the very first and,
so to speak, quite naturally, where the first geometers have given
us models we should constantly imitate and admire.
It is in the exposition of first principles that it is necessary
to avoid too much subtility ; there it would be most discouraging
and moreover useless. We can not prove everything and we can
not define everything ; and it will always be necessary to borrow
from intuition; what does it matter whether it be done a little
sooner or a little later, provided that in using correctly prem-
ises it has furnished us, we learn to reason soundly.
11. Is it possible to fulfill so many opposing conditions t Is
this possible in particular when it is a question of giving a defi-
nition T How find a concise statement satisfying at once the un-
compromising rules of logic, our desire to grasp the place of the
new notion in the totality of the science, our need of thinking
with images! Usually it will not be found, and this is why it is
not enough to state a definition; it must be prepared for and
justified.
What does that meant You know it has often been said:
every definition implies an assumption, since it affirms the exist-
ence of the object defined. The definition then will not be ju»-
440 SCIENCE AND METHOD
tified, from the purely logical i>oint of view, until one shall have
proved that it involves no contradiction, neither in the terms,
nor with the verities previously admitted.
But this is not enough ; the definition is stated to us as a con-
vention ; but most minds will revolt if we wish to impose it ux>on
them as an arbitrary convention* They will be satisfied only
when you have answered numerous questions.
Usually mathematical definitions, as M. Liard has shown, are
veritable constructions built up wholly of more simple notions.
But why assemble these elements in this way when a thousand
other combinations were possible!
Is it by caprice T If not, why had this combination more right
to exist than all the others! To what need does it respond!
How was it foreseen that it would play an important role in the
development of the science, that it would abridge our reason-
ings and our calculations! Is there in nature some familiar
object which is so to speak the rough and vague image of it!
This is not all; if you answer all these questions in a satis-
factory manner, we shall see indeed that the new-bom had the
right to be baptized; but neither is the choice of a name arbi-
trary; it is needful to explain by what analogies one has been
guided and that if analogous names have been given to different
things, these things at least differ only in material and are allied
in form; that their properties are analogous and so to say
parallel.
At this cost we may satisfy all inclinations. If the statement
is correct enough to please the logician, the justification will
satisfy the intuitive. But there is still a better procedure;
wherever possible, the justification should precede the statement
and prepare for it; one should be led on to the general state-
ment by the study of some particular examples.
Still another thing: each of the parts of the statement of a
definition has as aim to distinguish the thing to be defined from
a class of other neighboring objects. The definition will be un-
derstood only when you have shown, not merely the object de-
fined, but the neighboring objects from which it is proper to dis-
tinguish it, when you have given a grasp of the difference and
when you have added explicitly : this is why in stating the defini-
tion I have said this or that.
MATHEMATICAL DEFINITIONS AND TEACHING 441
But it is time to leave generalities and examine how the some-
what abstract principles I have expounded may be applied in
arithmetic, geometry, analysis and mechanics.
Abithmetio
12. The whole number is not to be defined ; in return, one or-
dinarily defines the operations upon whole numbers; I believe
the scholars learn these definitions by heart and attach no mean-
ing to them. For that there are two reasons : first they are made
to learn them too soon, when their mind as yet feels no need of
them ; then these definitions are not satisfactory from the logical
point of view. A good definition for addition is not to be found
just simply because we must stop and can not define everything.
It is not defining addition to say it consists in adding. All that
can be done is to start from a certain number of concrete exam-
ples and say : the operation we have performed is called addition.
For subtraction it is quite otherwise; it may be logically de-
fined as the operation inverse to addition; but should we begin
in that wayt Here also start with examples, show on these ex-
amples the reciprocity of the two operations ; thus the definition
will be prepared for and justified.
Just so again for multiplication; take a particular problem;
show that it may be solved by adding several equal numbers;
then show that we reach the result more quickly by a multiplica-
tion, an operation the scholars already know how to do by routine
and out of that the logical definition will issue naturally.
Division is defined as the operation inverse to multiplication;
but begin by an example taken from the familiar notion of par-
tition and show on this example that multiplication reproduces
the dividend.
There still remain the operations on fractions. The only
difficulty is for multiplication. It is best to expound first the
theory of proportion ; from it alone can come a logical definition ;
but to make acceptable the definitions met at the beginning of
this theory, it is necessary to prepare for them by numerous ex-
amples taken from classic problems of the rule of three, taking
pains to introduce fractional data.
Neither should we fear to familiarize the scholars with the
442 SCIENCE AND METHOD
notion of proportion by geometric images, either by appealing to
what they remember if they have already studied geometry, or
in having recourse to direct intuition, if they have not studied
it, which besides will prepare them to study it. Finally I shall
add that after defining multiplication of fractions, it is needful
to justify this definition by showing that it is commutative, as-
sociative and distributive, and calling to the attention of the
auditors that this is established to justify the definition.
One sees what a role geometric images play in all this; and
this role is justified by the philosophy and the history of the
science. If arithmetic had remained free from all admixture
of geometry, it would have known only the whole number ; it is
to adapt itself to the needs of geometry that it invented any-
thing else.
Geometry
In geometry we meet forthwith the notion of the straight line.
Can the straight line be defined? The well-known definition,
the shortest path from one point to another, scarcely satisfies
me. I should start simply with the ruler and show at first to
the scholar how one may verify a ruler by turning; this verifi-
cation is the true definition of the straight line; the straight
line is an axis of rotation. Next he should be shown how to
verify the ruler by sliding and he would have one of the most
important properties of the straight line.
As to this other property of being the shortest path from one
point to another, it is a theorem which can be demonstrated
apodictically, but the demonstration is too delicate to find a place
in secondary teaching. It will be worth more to show that a
ruler previously verified fits on a stretched thread. In presence
of difficulties like these one need not dread to multiply assump-
tions, justifying them by rough experiments.
It is needful to grant these assumptions, and if one admits a
few more of them than is strictly necessary, the evil is not very
great; the essential thing is to learn to reason soundly on the
assumptions admitted. Uncle Sarcey, who loved to repeat, often
said that at the theater the spectator accepts willingly all the
postulates imposed upon him at the beginning, but the curtain
MATHEMATICAL DEFINITIONS AND TEACHING 448
once raised, he becomes uncompromising on the logic. Well, it
is just the same in mathematics.
For the circle, we may start with the compasses; the scholars
will recognize at the first glance the curve traced; then make
them observe that the distance of the two points of the instru-
ment remains constant, that one of these points is fixed and the
other movable, and so we shall be led naturally to the logical
definition.
The definition of the plane implies an axiom and this need not
be hidden. Take a drawing board and show that a moving ruler
may be kept constantly in complete contact with this plane and
yet retain three degrees of freedomu Compare with the cylin-
der and the cone, surfaces on which an applied straight retains
only two degrees of freedom; next take three drawing boards;
show first that they will glide while remaining applied to one an-
other and this with three degrees of freedom ; and finally to dis-
tinguish the plane from the sphere, show that two of these boards
which fit a third will fit each other.
Perhaps you are surprised at this incessant employment of
moving things; this is not a rough artifice; it is much more
philosophic than one would at first think. What is geometry
for the philosopher? It is the study of a group. And what
group? That of the motions of solid bodies. How define this
group then without moving some solids f
Should we retain the classic definition of parallels and say
parallels are two coplanar straights which do not meet, however
far they be prolonged? No, since this definition is negative,
since it is unverifiable by experiment, and consequently can not
be regarded as an immediate datum of intuition. No, above all
because it is wholly strange to the notion of group, to the consid-
eration of the motion of solid bodies which is, as I have said, the
true source of geometry. Would it not be better to define first
the rectilinear translation of an invariable figure, as a motion
wherein all the points of this figure have rectilinear trajectories;
to show that such a translation is possible by making a square
glide on a ruler?
From this experimental ascertainment, set up as an assump-
tion, it would be easy to derive the notion of parallel and
Euclid's postulate itself.
444 SCIENCE AND METHOD
Mechanics
I need not return to the definition of velocity, or acceleration,
or other kinematic notions; they may be advantageously con-
nected with that of the derivative.
I shall insist, on the other hand, upon the dynamic notions of
force and mass.
I am struck by one thing : how very far the young people who
have received a high-school education are from applying to the
real world the mechanical laws they have been taught. It is not
only that they are incapable of it ; they do not even think of it
For them the world of science and the world of reality are sepa-
rated by an impervious partition wall.
If we try to analyze the state of mind of our scholars, this will
astonish us less. What is for them the real definition of force f
Not that which they recite, but that which, crouching in a nook
of their mind, from there directs it wholly. Here is the definition :
forces are arrows with which one makes parallelograms. These
arrows are imaginary things which have nothing to do with any-
thing existing in nature. This would not happen if they had been
shown forces in reality before representing them by arrows.
How shall we define force 1
I think I have elsewhere suflSciently shown there is no good
logical definition. There is the anthropomorphic definition, the
sensation of muscular eflfort ; this is really too rough and nothing
useful can be drawn from it.
Here is how we should go: first, to make known the genus
force, we must show one after the other all the species of this
genus; they are very numerous and very different; there is the
pressure of fluids on the insides of the vases wherein they are
contained ; the tension of threads ; the elasticity of a spring ; the
gravity working on all the molecules of a body; friction; the
normal mutual action and reaction of two solids in contact.
This is only a qualitative definition; it is necessary to learn
to measure force. For that begin by showing that one force may
be replaced by another without destroying equilibrium ; we may
find the first example of this substitution in the balance and
Borda's double weighing.
Then show that a weight may be replaced, not only by another
MATHEMATICAL DEFINITIONS AND TEACHING 446
weight, but by force of a different nature: for instance, Prony's
brake permits replacing weight by friction.
From all this arises the notion of the equivalence of two forces.
The direction of a force must be defined. If a force F is equiv-
alent to another force P' applied to the body considered by means
of a stretched string, so that F may be replaced by P' without
affecting the equilibrium, then the point of attachment of the
string will be by definition the point of application of the force
F', and that of the equivalent force F; the direction of the string
will be the direction of the force F' and that of the equivalent
force F.
From that, pass to the comparison of the magnitude of forces.
If a force can replace two others with the same direction, it
equals their sum; show for example that a weight of 20 grams
may replace two 10-gram weights.
Is this enough f Not yet. We now know how to compare the
intensity of two forces which have the same direction and same
point of application ; we must learn to do it when the directions
are different. For that, imagine a string stretched by a weight
and passing over a pulley; we shall say that the tensor of the
two legs of the string is the same and equal to the tension weight.
This definition of ours enables us to compare the tensions of
the two pieces of our string, and, using the preceding defini-
tions, to compare any two forces having the same direction as
these two pieces. It should be justified by showing that the
tension of the last piece of the string remains the same for the
same tensor weight, whatever be the number and the disposition
of the reflecting pulleys. It has still to be completed by showing
this is only true if the pulleys are frictionless.
Once master of these definitions, it is to be shown that the
point of application, the direction and the intensity suffice to
determine a force ; that two forces for which these three elements
are the same are always equivalent and may always be replaced
by one another, whether in equilibrium or in movement, and this
whatever be the other forces acting.
It must be shown that two concurrent forces may always be
replaced by a unique resultant ; and that this resultant remains
446 SCIENCE AND METHOD
the same, whether the body be at rest or in motion and whatever
be the other forces applied to it.
Finally it must be shown that forces thus defined satisfy the
principle of the equality of action and reaction.
Experiment it is, and experiment alone, which can teach us
all that. It will suffice to cite certain common experiments,
which the scholars make daily without suspecting it, and to per-
form before them a few experiments, simple and well chosen.
It is after having passed through all these meanders that one
may represent forces by arrows, and I should even wish that in
the development of the reasonings return were made from time
to time from the symbol to the reality. For instance it would
not be difficult to illustrate the parallelogram of forces by aid
of an apparatus formed of three strings, passing over puUeys,
stretched by weights and in equilibrium while pulling on the
same point.
Knowing force, it is easy to define mass ; this time the defini-
tion should be borrowed from dynamics ; there is no way of doing
otherwise, since the end to be attained is to give understanding
of the distinction between mass and weight. Here again, the
definition should be led up to by experiments ; there is in fact a
machine which seems made expressly to show what mass is,
Atwood's machine; recall also the laws of the fall of bodies, that
the acceleration of gravity is the same for heavy as for light
bodies, and that it varies with the latitude, etc.
Now, if you tell me that all the methods I extol have long been
applied in the schools, I shall rejoice over it more than be sur-
prised at it. I know that on the whole our mathematical teach-
ing is good. I do not wish it overturned ; that would even dis-
tress me. I only desire betterments slowly progressive. This
teaching should not be subjected to brusque oscillations under
the capricious blast of ephemeral fads. In such tempests its
high educative value would soon founder. A good and sound
logic should continue to be its basis. The definition by example
is always necessary, but it should prepare the way for the logical
definition, it should not replace it; it should at least make this
wished for, in the cases where the true logical definition can be
advantageously given only in advanced teaching.
MATHEMATICAL DEFINITIONS AND TEACHING 447
Understand that what I have here said does not imply giving
up what I have written elsewhere. I have often had occasion to
criticize certain definitions I extol to-day. These criticisms hold
good completely. These definitions can only be provisory. But
it is by way of them that we must pass.
CHAPTER III
Mathematics and Logic
Introduction
Can mathematics be reduced to logic without having to appeal
to principles peculiar to mathematics? There is a whole school,
abounding in ardor and full of faith, striving to prove it. They
have their own special language, which is without words, using
only signs. This language is understood only by the initiates,
so that commoners are disposed to bow to the trenchant affirma-
tions of the adepts. It is perhaps not unprofitable to examine
these affirmations somewhat closely, to see if they justify the
peremptory tone with which they are presented.
But to make clear the nature of the question it is necessary to
enter upon certain historical details and in particular to recall
the character of the works of Cantor.
Since long ago the notion of infinity had been introduced
into mathematics; but this infinite was what philosophers call
a becoming. The mathematical infinite was only a quantity
capable of increasing beyond all limit: it was a variable quan-
tity of which it could not be said that it had passed all limits,
but only that it could pass them.
Cantor has undertaken to introduce into mathematics an
actual infinite, that is to say a quantity which not only is capable
of passing all limits, but which is regarded as having already
passed them. He has set himself questions like these : Are there
more points in space than whole numbers? Are there more
points in space than points in a plane? etc.
And then the number of whole numbers, that of the points of
space, etc., constitutes what he calls a transfinite cardinal number,
that is to say a cardinal number greater than all the ordinary
cardinal numbers. And he has occupied himself in comparing
these transfinite cardinal numbers. In arranging in a proper
order the elements of an aggregate containing an infinity of
448
MATHEMATICS AND LOGIC 449
them, he has also imagined what he calls transfinite ordinal
numbers upon which I shall not dwell.
Many mathematicians followed his lead and set a series of
questions of the sort. They so familiarized themselves with
transfinite numbers that they have come to make the theory of
finite numbers depend upon that of Cantor's cardinal numbers.
In their eyes, to teach arithmetic in a way truly logical, one
should begin by establishing the general properties of trans-
finite cardinal numbers, then distinguish among them a very
small class, that of the ordinary whole numbers. Thanks to this
d6tour, one might succeed in proving all the propositions relative
to this little class (that is to say all our arithmetic and our
algebra) without using any principle foreign to logic. This
method is evidently contrary to all sane psychology; it is cer-
tainly not in this way that the human mind proceeded in con-
structing mathematics ; so its authors do not dream, I think, of
introducing it into secondary teaching. But is it at least logic,
or, better, is it correct 1 It may be doubted.
The geometers who have employed it are however very numer-
ous. They have accumulated formulas and they have thought
to free themselves from what was not pure logic by writing
memoirs where the formulas no longer alternate with explana-
tory discourse as in the books of ordinary mathematics, but
where this discourse has completely disappeared.
Unfortunately they have reached contradictory results, what
are called the caniorian aniinamies, to which we shall have
occasion to return. These contradictions have not discouraged
them and they have tried to modify their rules so as to make
those disappear which had already shown themselves, without
being sure, for all that, that new ones would not manifest
themselves.
It is time to administer justice on these exaggerations. I do
not hope to convince them ; for they have lived too long in this
atmosphere. Besides, when one of their demonstrations has
been refuted, we are sure to see it resurrected with insignificant
alterations, and some of them have already risen several times
from their ashes. Such long ago was the Lernaean hydra with its
famous heads which alwa]^s grew again. Hercules got through,
30
450 SCIENCE AND METHOD
since his hydra had only nine heads, or eleven ; bat here there are
too many, some in England, some in Germany, in Italy, in
France, and he would have to give up the straggle. So I appeal
only to men of good judgment unprejudiced.
In these latter years numerous works have been published on
pure mathematics and the philosophy of mathematics, trying to
separate and isolate the logical elements of mathematical reason-
ing. These works have been analyzed and expounded veiy
clearly by M. Couturat in a book entitled: The Principles of
Mathematics.
For M. Couturat, the new works, and in particular those of
Russell and Peano, have finally settled the controversy, so long
pending between Leibnitz and Kant. They have shown that
there are no synthetic judgments a priori (Kant's phrase to
designate judgments which can neither be demonstrated analyti-
cally, nor reduced to identities, nor established experimentally),
they have shown that mathematics is entirely reducible to logic
and that intuition here plays no role.
This is what M. Couturat has set forth in the work just cited ;
this he says still more explicitly in his Kant jubilee discourse,
so that I heard my neighbor whisper: **I well see this is the
centenary of Kant's death,**
Can we subscribe to this conclusive condemnation! I think
not, and I shall try to show why.
II
What strikes us first in the new mathematics is its purely
formal character: **We think," says Hilbert, '* three sorts of
things, which we shall call points, straights and planes. We
convene that a straight shall be determined by two points, and
that in place of saying this straight is determined by these two
points, we may say it passes through these two points, or that
these two points are situated on this straight." What these
things are, not only we do not know, but we should not seek to
know. We have no need to, and one who never had seen either
point or straight or plane could geometrize as well as we. That
MATHEMATICS AND LOGIC 461
the phrase to pctss through, or the phrase to be sitiiated upon
may arouse in us no image, the first is simply a synonym of to
be determined and the second of to determine.
Thus, be it understood, to demonstrate a theorem, it is neither
necessary nor even advantageous to know what it means. The
geometer might be replaced by the logic piano imagined by
Stanley Jevons ; or, if you choose, a machine might be imagined
where the assumptions were put in at one end, while the theorems
came out at the other, like the legendary Chicago machine where
the pigs go in alive and come out transformed into hams and
sausages. No more than these machines need the mathematician
know what he does.
I do not make this formal character of his geometry a reproach
to Hilbert. This is the way he should go, given the problem he
set himself. He wished to reduce to a minimum the number of
the fundamental assumptions of geometry and completely enu-
merate them ; now, in reasonings where our mind remains active,
in those where intuition still plays a part, in living reasonings,
so to speak, it is difficult not to introduce an assumption or a
postulate which passes unperceived. It is therefore only after
having carried back all the geometric reasonings to a form purely
mechanical that he could be sure of having accomplished his
design and finished his work.
What Hilbert did for geometry, others have tried to do for
arithmetic and analysis. Even if they had entirely succeeded,
would the Kantians be finally condemned to silence f Perhaps
not, for in reducing mathematical thought to an empty form,
it is certainly mutilated.
Even admitting it were established that all the theorems could
be deduced by procedures purely analytic, by simple logical
combinations of a finite number of assumptions, and that these
assumptions are only conventions; the philosopher would still
have the right to investigate the origins of these conventions,
to see why they have been judged preferable to the contrary
conventions.
And then the logical correctness of the reasonings leading
from the assumptions to the theorems is not the only thing
which should occupy us. The rules of perfect logic, are th^
462 SCIENCE AND METHOD
the whole of mathematics T As well say the whole art of play-
ing chess reduces to the rules of the moves of the pieces. Among
all the constructs which can be built up of the materials fur-
nished by logic, choice must be made; the true geometer makes
this choice judiciously because he is guided by a sure instinct,
or by some vague consciousness of I know not what more pro-
found and more hidden geometry, which alone gives value to the
edifice constructed.
To seek the origin of this instinct, to study the laws of this
deep geometry, felt, not stated, would also be a fine employment
for the philosophers who do not want logic to be all. But it is
not at this point of view I wish to put myself, it is not thus I
wish to consider the question. The instinct mentioned is neces-
sary for the inventor, but it would seem at first we might do
without it in studying the science once created. Well, what I
wish to investigate is if it be true that, the principles of logic
once admitted, one can, I do not say discover, but demonstrate,
all the mathematical verities without making a new appeal to
intuition.
Ill
I once said no to this question :^ should our reply be modified
by the recent works? My saying no was because '*the principle
of complete induction*' seemed to me at once necessary to the
mathematician and irreducible to logic. The statement of this
principle is: **If a property be true of the number 1, and if we
establish that it is true oi n-\-l provided it be of n, it will be
true of all the whole numbers." Therein I see the mathematical
reasoning par excellence. I did not mean to say, as has been
supposed, that all mathematical reasonings can be reduced to
an application of this principle. Examining these reasonings
<;losely, we there should see applied many other analogous princi-
ples, presenting the same essential characteristics. In this cate-
gory of principles, that of complete induction is only the simplest
of all and this is why I have chosen it as type.
The current name, principle of complete induction, is not
justified. This mode of reasoning is none the less a true mathe-
1 See Science and Hypothesis, chapter I.
MATHEMATICS AND LOGIC 463
matical induction which differs from ordinary induction only by
its certitude.
IV
Definitions and Assumptions
The existence of such principles is a difficulty for the uncom-
promising logicians; how do they pretend to get out of itf The
principle of complete induction, they say, is not an assumption
properly so called or a i^nthetic judgment a priori; it is just
simply the definition of whole number. It is therefore a simple
convention. To discuss this way of looking at it, we must ex-
amine a little closely the relations between definitions and
assumptions.
Let us go back first to an article by M. Couturat on mathe-
matical definitions which appeared in VEnseignement mathe-
matique, a magazine published by Qauthier-Villars and by Georg
at Geneva. We shall see there a distinction between the direct
definition and the definition by postulates.
**The definition by postulates," says M. Couturat, "applies,
not to a single notion, but to a system of notions ; it consists in
enumerating the fundamental relations which unite them and
which enable us to demonstrate all their other properties; these
relations are postulates.''
If previously have been defined all these notions but one, then
this last will be by definition the thing which verifies these pos-
tulates. Thus certain indemonstrable assumptions of mathe-
matics would be only disguised definitions. This point of view
is often legitimate ; and I have myself admitted it in regard for
instance to Euclid's postulate.
The other assumptions of geometry do not suffice to completely
define distance ; the distance then will be, by definition, among all
the magnitudes which satisfy these other assumptions, that
which is such as to make Euclid's postulate true.
Well the logicians suppose true for the principle of complete
induction what I admit for Euclid's postulate; they want to
see in it only a disguised definition.
But to give them this right, two conditions must be fulfilled.
Stuart Mill says every definition implies an assumption, that by
which the existence of the defined object is affirmed. According
454 SCIENCE AND METHOD
to that, it would no longer be the assumption which might be a
disguised definition, it would on the contrary be the definition
which would be a disguised assumption. Stuart Mill meant the
word existence in a material and empirical sense; he meant to
say that in defining the circle we afSrm there are round things in
nature.
Under this form, his opinion is inadmissible. Mathematics is
independent of the existence of material objects ; in mathematics
the word exist can have only one meaning, it means free from
contradiction. Thus rectified, Stuart Mill's thought becomes
exact ; in defining a thing, we affirm that the definition implies no
contradiction.
If therefore we have a system of postulates, and if we can
demonstrate that these postulates imply no contradiction, we
shall have the right to consider them as representing the defini-
tion of one of the notions entering therein. If we can not demon-
strate that, it must be admitted without proof, and that then
will be an assumption; so that, seeking the definition under the
postulate, we should find the assumption under the definition.
Usually, to show that a definition implies no contradiction, we
proceed 61/ example, we try to make an example of a thing satis-
fying the definition. Take the case of a definition by postulates ;
we wish to define a notion A, and we say that, by definition, an
A is anything for which certain postulates are true. If we can
prove directly that all these postulates are true of a certain object
B, the definition will be justified ; the object B will be an example
of an A. We shall be certain that the postulates are not contra-
dictory, since there are cases where they are all true at the same
time.
But such a direct demonstration by example is not always
possible.
To establish that the postulates imply no contradiction, it is
then necessary to consider all the propositions deducible from
these postulates considered as premises, and to show that, among
these propositions, no two are contradictory. If these proposi-
tions are finite in number, a direct verification is possible. This
case is infrequent and uninteresting. If these propositions are
infinite in number, this direct verification can no longer be made;
MATHEMATICS AND LOGIC 455
recourse must be had to procedures where in general it is neces-
sary to invoke just this principle of complete induction which is
precisely the thing to be proved.
This is an explanation of one of the conditions the logicians
should satisfy, and further on we shdU see they have not done it.
V
There is a second. When we give a definition, it is to use it.
We therefore shall find in the sequel of the exposition the
word defined; have we the right to affirm, of the thing repre-
sented by this word, the postulate which has served for definition f
Yes, evidently, if the word has retained its meaning, if we do
not attribute to it implicitly a different meaning. Now this is
what sometimes happens and it is usually difficult to perceive it;
it is needful to see how this word comes into our discourse, and
if the gate by which it has entered does not imply in reality a
definition other than that stated.
This difficulty presents itself in all the applications of math-
ematics. The mathematical notion has been given a definition
very refined and very rigorous ; and for the pure mathematician
all doubt has disappeared; but if one wishes to apply it to the
physical sciences for instance, it is no longer a question of this
pure notion, but of a concrete object which is often only a rough
image of it. To say that this object satisfies, at least approx-
imately, the definition, is to state a new truth, which experience
alone can put beyond doubt, and which no longer has the char-
acter of a conventional postulate.
But without going beyond pure mathematics, we also meet the
same difficulty.
You give a subtile definition of numbers ; then, once this defini-
tion given, you think no more of it ; because, in reality, it is not
it which has taught you what number is; you long ago knew
that, and when the word number further on is found under your
pen, you give it the same sense as the first comer. To know what
is this meaning and whether it is the same in this phrase or that,
it is needful to see how you have been led to speak of number and
to introduce this word into these two phrases. I shall not for
the moment dilate upon this point, because we shall have occasion
to return to it.
456 SCIENCE AND METHOD
Thus consider a word of which we have given explicitly a defi-
nition A ; afterwards in the discourse we make a use of it which
implicitly supposes another definition B, It is possible that
these two definitions designate the same thing. But that this is
so is a new truth which must either be demonstrated or admitted
as an independent assumption.
We shall see farther on that the logicians have not fulfilled the
second condition any better than the first,
VI
The definitions of number are very numerous and very differ-
ent ; I forego the enumeration even of the names of their authors.
We should not be astonished that there are so many. If one
among them was satisfactory, no new one would be given. If
each new philosopher occupying himself with this question has
thought he must invent another one, this was because he was not
satisfied with those of his predecessors, and he was not satisfied
with them because he thought he saw a petitio principii.
I have always felt, in reading the writings devoted to this prob-
lem, a profound feeling of discomfort ; I was always expecting to
run against a petitio principii, and when I did not immediately
perceive it, I feared I had overlooked it.
This is because it is impossible to give a definition without
using a sentence, and diflBcult to make a sentence without using
a number word, or at least the word several, or at least a word
in the plural. And then the declivity is slippery and at each
instant there is risk of a fall into petitio principii.
I shall devote my attention in what follows only to those of
these definitions where the petitio principii is most ably con-
cealed.
VII
Pasigraphy
The symbolic language created by Peano plays a very grand
role in these new researches. It is capable of rendering some
service, but I think M. Couturat attaches to it an exaggerated
importance which must astonish Peano himself.
The essential element of this language is certain algebraic
MATHEMATICS AND LOGIC 457
signs which represent the different conjunctions: if, and, or,
therefore. That these signs may be convenient is possible; but
that they are destined to revolutionize all philosophy is a differ-
ent matter. It is difficult to admit that the word if acquires,
when written 'q, a virtue it had not when written if. This in-
vention of Peano was first called pasigraphy, that is to say the
art of writing a treatise on mathematics without using a single
word of ordinary language. This name defined its range very
exactly. Later, it was raised to a more eminent dignity by con-
ferring on it the title of logistic. This word is, it appears, em-
ployed at the Military Academy, to designate the art of the
quartermaster of cavalry, the art of marching and cantoning
troops; but here no confusion need be feared, and it is at once
seen that this new name implies the design of revolutionizing
logic.
We may see the new method at work in a mathematical memoir
by Burali-Forti, intitled: Una Questione sui numeri transfiniti,
inserted in Volume XI of the Bendiconti del circolo matematico
di Palermo.
I begin by saying this memoir is very interesting, and my tak-
ing it here as example is precisely because it is the most im-
portant of all those written in the new language. Besides, the un-
initiated may read it, thanks to an Italian interlinear translation.
What constitutes the importance of this memoir is that it has
given the first example of those antinomies met in the study of
transfinite numbers and making since some years the despair of
mathematicians. The aim, says Burali-Forti, of this note is to
show there may be two transfinite numbers (ordinals), a and b,
such that a is neither equal to, greater than, nor less than b.
To reassure the reader, to comprehend the considerations which
follow, he has no need of knowing what a transfinite ordinal
number is.
Now, Cantor had precisely proved that between two transfinite
numbers as between two finite, there can be no other relation
than equality, or inequality in one sense or the other. But it is
not of the substance of this memoir that I wish to speak here;
that would carry me much too far from my subject ; I only wish
to consider the form, and just to ask if this form makes it gain
i58 SCIENCE AND METHOD
much in rigor and whether it thus compensates for the efforts it
imposes upon the writer and the reader.
First we see Burali-Forti define the number 1 as follows:
a definition eminently fitted to give an idea of the number 1 to
persons who had never heard speak of it.
I understand Peanian too ill to dare risk a critique, but still I
fear this definition contains a petitio principii, considering that
I see the figure 1 in the first member and Tin in letters in the
second.
However that may be, Burali-Forti starts from this definition
and, after a short calculation, reaches the equation :
(27) leNo,
which tells us that One is a number.
And since we are on these definitions of the first numbers, we
recall that M. Couturat has also defined and 1.
What is zero T It is the number of elements of the null dass.
And what is the null class t It is that containing no element.
To define zero by null, and null by no, is really to abuse the
wealth of language ; so M. Couturat has introduced an improve-
ment in his definition, by writing :
which means : zero is the number of things satisfying a condition
never satisfied.
But as never means in no case I do not see that the progress
is great.
I hasten to add that the definition M. Couturat gives of the
number 1 is more satisfactory.
One, says he in substance, is the number of elements in a class
in which any two elements are identical.
It is more satisfactory, I have said, in this sense that to define
1, he does not use the word one; in compensation, he uses the
word two. But I fear, if asked what is two, M. Couturat would
have to use the word one.
MATHEMATICS AND LOGIC 459
vni
But to return to the memoir of Burali-Forti ; I have said his
conclusions are in direct opposition to those of Cantor. Now, one
day M. Hadamard came to see me and the talk fell upon this
antinomy.
"Burali-Forti's reasoning," I said, **does it not seem to you
irreproachable!" *'No, and on the contrary I find nothing to
object to in that of Cantor. Besides, Burali-Forti had no right
to speak of the aggregate of all the ordinal numbers."
'' Pardon, he had the right, since he could always put
o=r(No,f».
I should like to know who was to prevent him, and can it be
said a thing does not exist, when we have called it OT"
It was in vain, I could not convince him (which besides would
have been sad, since he was right). Was it merely because I do
not speak the Peanian with enough eloquence! Perhaps; but
between ourselves I do not think so.
Thus, despite all this pasigraphic apparatus, the question was
not solved. What does that prove t In so far as it is a question
only of proving one a number, pasigraphy sufSces, but if a diffi-
culty presents itself, if there is an antinomy to solve, pasigraphy
becomes impotent.
CHAPTER IV
The New Logics
The SusseU Logic
To justify its pretensions, logic had to change. We hxve seen
new logics arise of which the most interesting is that of BnasdL
It seems he has nothing new to write about formal logic, as if
Aristotle there had touched bottom. But the domain BusseD
attributes to logic is infinitely more extended than that of the
classic logic, and he has put forth on the subject views which are
original and at times well warranted.
First, Bussell subordinates the logic of classes to that of prop-
ositions, while the logic of Aristotle was above all the logic of
classes and took as its point of departure the relation of subject
to predicate. The classic syllogism, ''Socrates is a man," etc.,
gives place to the hypothetical syllogism: **If A is true, B is
true; now if B is true, C is true," etc. And this is, I think, a
most happy idea, because the classic syllogism is easy to carry
back to the hypothetical syllogism, while the inverse transfor-
mation is not without diflSculty.
And then this is not all. Russeirs logic of propositions is the
study of the laws of combination of the conjunctions if, and, or,
and the negation not.
In adding here two other conjunctions and and or, Russell
opens to logic a new field. The symbols and, or follow the same
laws as the two signs X and +, that is to say the commutative
associative and distributive laws. Thus and represents logical
multiplication, while or represents logical addition. This also
is very interesting.
Russell reaches the conclusion that any false proposition im-
plies all other propositions true or false. M. Couturat says this
conclusion will at first seem paradoxical. It is sufficient how-
ever to have corrected a bad thesis in mathematics to recognize
460
THE NEW LOGICS 461
how right Bussell is. The candidate often is at great pains to
get the first false equation; but that once obtained, it is only
sport then for him to accumulate the most surprising results,
some of which even may be true.
II
We see how much richer the new logic is than the classic logic ;
the symbols are multiplied and allow of varied combinations
which are no longer limited in number. Has one the right to
give this extension to the meaning of the word logic t It would
be useless to examine this question and to seek with Bussell a
mere quarrel about words. Grant him what he demands; but be
not astonished if certain verities declared irreducible to logic
in the old sense of the word find themselves now reducible to
logic in the new sense — something very different.
A great number of new notions have been introduced, and
these are not simply combinations of the old. Bussell knows
this, and not only at the beginning of the first chapter, 'The
Logic of Propositions,' but at the beginning of the second and
third, 'The Logic of Classes' and 'The Logic of Belations,' he
introduces new words that he declares indefinable.
And this is not all; he likewise introduces principles he de-
clares indemonstrable. But these indemonstrable principles are
appeals to intuition, synthetic judgments a priori. We regard
them as intuitive when we meet them more or less explicitly
enunciated in mathematical treatises; have they changed char-
acter because the meaning of the word logic has been enlarged
and we now find them in a book entitled Treatise on Logic?
They have not changed nature; they have only changed place.
Ill
Could these principles be considered as disguised definitions!
It would then be necessary to have some way of proving that
they imply no contradiction. It would be necessary to establish
that, however far one followed the series of deductions, he would
never be exposed to contradicting himself.
We might attempt to reason as follows: We can verify that
462 SCIENCE AND METHOD
the operations of the new logic applied to premises exempt from
contradiction can only give consequences equally exempt from
contradiction* If therefore after n operations we have not met
contradiction, we shall not encounter it after n -{-1. Thus it is
impossible that there should be a moment when contradiction
begins, which shows we shall never meet it. Have we the right to
reason in this way t No, for this would be to make use of com-
plete induction ; and remember, we do not yet know the principU
of complete induction.
We therefore have not the right to regard these assumptions
as disguised definitions and only one resource remains for us, to
admit a new act of intuition for each of them. Moreover I be-
lieve this is indeed the thought of Bussell and M. Coutorat.
Thus each of the nine indefinable notions and of the twenty
indemonstrable propositions (I believe if it were I that did the
counting, I should have found some more) which are the founda-
tion of the new logic, logic in the broad sense, presupposes a new
and independent act of our intuition and (why not say itt) a
veritable synthetic judgment a priori. On this point all seem
agreed, but what Bussell claims, and what seems to me doubtful,
is that after these appeals to intuition, that will be the end of it;
we need make no otlicrs and can build all mathematics without
the intervention of any new element.
IV
M. Couturat often repeats that this new logic is altogether in-
dependent of the idea of number. I shall not amuse myself by
counting how many numeral adjectives his exposition contains,
both cardinal and ordinal, or indefinite adjectives such as several.
We may cite, however, some examples:
**The logical product of two or more propositions is . . .";
**A11 propositions are capable only of two values, true and
false'';
The relative product of two relations is a relation";
A relation exists between two terms,'' etc., etc.
Sometimes this inconvenience would not be unavoidable, but
sometimes also it is essential. A relation is incomprehensible
( (
( {
THE NEW LOGICS 463
without two terms; it is impossible to have the intuition of the
relation, without having at the same time that of its two terms,
and without noticing they are two, because, if the relation is to
be conceivable, it is necessary that there be two and only two.
Arithmetic
I reach what M. Couturat calls the ordinal theory which is
the foundation of arithmetic properly so called. M. Couturat
begins by stating Peano's five assumptions, which are independ-
ent, as has been proved by Peano and Padoa.
1. Zero is an integer.
2. Zero is not the successor of any integer.
3. The successor of an integer is an integer.
To this it would be proper to add,
Every integer has a successor.
4. Two integers are equal if their successors are.
The fifth assumption is the principle of complete induction.
M. Couturat considers these assumptions as disguised defini-
tions; they constitute the definition by postulates of zero, of
successor, and of integer.
But we have seen that for a definition by postulates to be
acceptable we must be able to prove that it implies no contra-
diction.
Is this the case heret Not at all.
The demonstration can not be made by example. We can not
take a part of the integers, for instance the first three, and
prove they satisfy the definition.
If I take the series 0, 1, 2, I see it fulfils the assumptions 1,
2, 4 and 5 ; but to satisfy assumption 3 it still is necessary that
3 be an integer, and consequently that the series 0, 1, 2, 3, fulfil
the assumptions; we might prove that it satisfies assumptions
1, 2, 4, 5, but assumption 3 requires besides that 4 be an integer
and that the series 0, 1, 2, 3, 4 fulfil the assumptions, and so on.
It is therefore impossible to demonstrate the assumptions for
certain integers without proving them for all; we must give up
proof by example.
464 SCIENCE AND METHOD
It is necessary then to take all the consequences of our aasomp-
tions and see if they contain no contradiction*
If these consequences were finite in number, this would be
easy; but they are infinite in number; they are the whole of
mathematics, or at least all arithmetic
What then is to be done! Perhaps strictly we could repeat
the reasoning of number III.
But as we have said, this reasoning is complete induction, and
it is precisely the principle of complete induction whose justifi-
cation would be the point in question.
VI
The Logic of Hilbert
I come now to the capital work of Hilbert which he com-
municated to the Congress of Mathematicians at Heidelberg, and
of which a French translation by M. Pierre Boutroux appeared
in VEnseignement mathematique, while an English translation
due to Halsted appeared in The Monist.^ In this work, which
contains profound thoughts, the author's aim is analogous to
that of Russell, but on many points he diverges from his
predecessor.
**But,'' he says {Monist, p. 340), **on attentive consideration
we become aware that in the usual exposition of the laws of logic
certain fundamental concepts of arithmetic are already employed ;
for example, the concept of the aggregate, in part also the concept
of number.
'* We fall thus into a vicious circle and therefore to avoid para-
doxes a partly simultaneous development of the laws of logic and
arithmetic is requisite."
We have seen above that what Hilbert says of the principles
of logic in the usual exposition applies likewise to the logic of
Russell. So for Russell logic is prior to arithmetic ; for Hilbert
they are * simultaneous.' We shall find further on other differ-
ences still greater, but we shall point them out as we come
to them. I prefer to follow step by step the development
of Hilbert 's thought, quoting textually the most important
passages.
i*The Foundations of Logic and Arithmetic,' Monistf XV., 338-352.
THE NEW LOGICS 4d6
"Let us take as the basis of our consideration first of all a
thought-thing 1 (one) " (p. 341). Notice that in so doing we in
no wise imply the notion of number, because it is understood that
1 is here only a symbol and that we do not at all seek to know
its meaning. ''The taking of this thing together with itself
respectively two, three or more times. ..." Ah ! this time it is
no longer the same; if we introduce the words 'two,' 'three,' and
above all 'more,' 'several,' we introduce the notion of number;
and then the definition of finite whole number which we shall
presently find, will come too late. Our author was too circum-
spect not to perceive this begging of the question. So at the end
of his work he tries to proceed to a truly patching-up process.
Hilbert then introduces two simple objects 1 and =, and con-
siders all the combinations of these two objects, all the combina-
tions of their combinations, etc. It goes without saying that we
must forget the ordinary meaning of these two signs and not
attribute any to them.
Afterwards he separates these combinations into two classes,
the class of the existent and the class of the non-existent, and
till further orders this separation is entirely arbitrary. Every
affirmative statement tells us that a certain combination belongs
to the class of the existent; every negative statement tells us that
a certain combination belongs to the class of the non-existent.
VII
Note now a difference of the highest importance. For Russell
any object whatsoever, which he designates by a;, is an object
absolutely undetermined and about which he supposes nothing;
for Hilbert it is one of the combinations formed with the symbols
1 and = ; he could not conceive of the introduction of anything
other than combinations of objects already defined. Moreover
Hilbert formulates his thought in the neatest way, and I think
I must reproduce in extenso his statement (p. 348) :
"In the assumptions the arbitraries (as equivalent for the
concept 'every' and 'all' in the customary logic) represent only
those thought-things and their combinations with one another,
which at this stage are laid down as fundamental or are to be
31
466 SCIENCE AND METHOD
newly defined. Therefore in the dedaetion of inferences from
the amimptiong, the arbitraries, which oceor in the assump-
tions, can be replaced onlj by sach thonght-things and their
combinations.
"Also we most duly remember, that through the saper-addi-
tion and making fundamental of a new thought-thing the pre-
ceding assumptions undergo an enlargement of their validity,
and where necessary, are to be subjected to a change in con-
formity with the sense."
The contrast with Russell's ^ew-point is complete. For this
philosopher we may substitute for x not only objects already
known, but anything.
Russell is faithful to his point of view, which is that of com-
prehension* He starts from the general idea of being, and
enriches it more and more while restricting it, by adding new
qualities. Hilbert on the contrary recognizes as possible beings
only combinations of objects already known ; so that (looking at
only one side of his thought) we might say he takes the view-
point of extension.
vin
Let us continue with the exposition of Hilbert 's ideas. He
introduces two assumptions which he states in his symbolic
language but which signify, in the language of the uninitiated,
that every quality is equal to itself and that every operation per-
formed upon two identical quantities gives identical results.
So stated, they are evident, but thus to present them would
be to misrepresent Hilbert 's thought. For him mathematics
has to combine only pure symbols, and a true mathematician
should reason upon them without preconceptions as to their
meaning. So his assumptions are not for him what they are for
the common people.
He considers them as representing the definition by postulates
of the symbol (=) heretofore void of all signification. But to
justify this definition we must show that these two assumptions
lead to no contradiction. For this Hilbert used the reasoning of
our number III, without appearing to perceive that he is using
complete induction.
THE NEW LOGICS 467
IX
The end of Hilbert's memoir is altogether enigmatic and I
shall not lay stress upon it. Contradictions accumulate ; we feel
that the author is dimly conscious of the petitio principii he has
committed, and that he seeks vainly to patch up the holes in his
argument.
What does this mean t At the point of proving that the defini-
tion of the whole number by the assumption of complete induc-
tion implies no contradiction, Hilbert withdraws as Russell and
Couturat withdrew, because the difficulty is too great.
X
Oeometry
Qeometry, says M. Couturat, is a vast body of doctrine wherein
the principle of complete induction does not enter. That is true
in a certain measure ; we can not say it is entirely absent, but it
enters very slightly. If we refer to the Bcttional Oeometry of
Dr. Halsted (New York, John Wiley and Sons, 1904) built up
in accordance with the principles of Hilbert, we see the principle
of induction enter for the first time on page 114 (unless I have
made an oversight, which is quite possible).'
So geometry, which only a few years ago seemed the domain
where the reign of intuition was uncontested, is to-day the realm
where the logicians seem to triumph. Nothing could better
measure the importance of the geometric works of Hilbert and
the profound impress they have left on our conceptions.
But be not deceived. What is after all the fundamental
theorem of geometry! It is that the assumptions of geometry
imply no contradiction, and this we can not prove without the
principle of induction.
How does Hilbert demonstrate this essential point t By lean-
ing upon analysis and through it upon arithmetic and through
it upon the principle of induction.
And if ever one invents another demonstration, it will still
be necessary to lean upon this principle, since the possible conse-
quences of the assumptions, of which it is necessary to show
that they are not contradictory, are infinite in number.
2 Second ed., 1907, p. 86; French ed., 1911, p. 97. O. B. H.
468 SCIESCE AMD METHOD
Condusiam
Oar coDclusioD straightwaj is that the prineiple of indnctioD
can not be regarded as the disguised definitiaii of the entire
worid.
Here are three troths: (1) The principle of complete indne-
tion; (2) Euclid's postulate; (3) the physical law aeeording
to which phosphorus melts at 44^ (cited l^ IL Le Boy).
These are said to be three disguised definitions: the first, that
of the whole number; the second, that of the straight Une; the
third, that of phosphorus.
I grant it for the second ; I do not admit it for the other two.
I must explain the reason for this apparent inconsistency.
First, we have seen that a definition is acceptable only on con-
dition that it implies no contradiction. We have shown like-
wise that for the first definition this demonstration is impossible;
on the other hand, we have just recalled that for the second
Hilbert has given a complete proof.
As to the third, evidently it implies no contradiction. Does
this mean that the definition guarantees, as it should, the exist-
ence of the object defined T We are here no longer in the mathe-
matical sciences, but in the physical, and the word existence has
no longer the same meaning. It no longer signifies absence of
contradiction; it means objective existence.
You already see a first reason for the distinction I made between
the three eases; there is a second. In the applications we have
to make of these three concepts, do they present themselves to us
as defined by these three postulates!
The possible applications of the principle of induction are
innumerable ; take, for example, one of those we have expounded
above, and where it is sought to prove that an aggregate of
assumptions can lead to no contradiction. For this we consider
one of the series of syllogisms we may go on with in starting
from these assumptions as premises. When we have finished
the nth syllogism, we see we can make still another and this is
the n -j- 1th. Thus the number n serves to count a series of suc-
cessive operations ; it is a number obtainable by successive addi-
THE NEW LOGICS 469
tions. This therefore is a number from which we may go back
to unity by sv^cessive subtractions. Evidently we could not do
this if we had n==n — 1, since then by subtraction we should
always obtain again the same number. So the way we have been
led to consider this number n implies a definition of the finite
whole number and this definition is the following : A finite whole
number is that which can be obtained by successive additions;
it is such that n is not equal to n — 1.
That granted, what do we dot We show that if there has
been no contradiction up to the nth syllogism, no more will there
be up to the n + 1th, and we conclude there never will be. You
say: I have the right to draw this conclusion, since the whole
numbers are by definition those for which a like reasoning is
legitimate. But that implies another definition of the whole
number, which is as follows : A whole number is that on which we
may reason by recurrence. In the particular case it is that of
which we may say that, if the absence of contradiction up to the
time of a syllogism of which the number is an integer carries
with it the absence of contradiction up to the time of the syllo-
gism whose number is the following integer, we need fear no
contradiction for any of the syllogisms whose number is an
integer.
The two definitions are not identical ; they are doubtless equiva-
lent, but only in virtue of a synthetic judgment a priori; we can
not pass from one to the other by a purely logical procedure.
Consequently we have no right to adopt the second, after having
introduced the whole number by a way that presupposes the first.
On the other hand, what happens with regard to the straight
line! I have already explained this so often that I hesitate to
repeat it again, and shall confine myself to a brief recapitulation
of my thought. We have not, as in the preceding case, two
equivalent definitions logically irreducible one to the other. We
have only one expressible in words. Will it be said there is
another which we feel without being able to word it, since we
have the intuition of the straight line or since we represent to
ourselves the straight line t First of all, we can not represent it
to ourselves in geometric space, but only in representative space,
and then we can represent to ourselves just as well the objects
470 SCIENCE AND METHOD
which possess the other properties of the straight line, save that
of satisfying Euclid's postulate. These objects are 'the non-
Euclidean straights, ' which from a certain point of view are not
entities void of sense, but circles (true circles of true space)
orthogonal to a certain sphere. If, among these objects equally
capable of representation, it is the first (the Euclidean straights)
which we call straights, and not the latter (the non-Euclidean
straights), this is properly by definition.
And arriving finally at the third example, the definition of
phosphorus, we see the true definition would be: Phosphorus is
the bit of matter I see in yonder fiask.
XII
And since I am on this subject, still another word. Of the
phosphorus example I said: **This proposition is a real verifiable
physical law, because it means that all bodies having aU the other
properties of phosphorus, save its point of fusion, melt like it at
44**." And it was answered: '*No, this law is not verifiable,
because if it were shown that two bodies resembling phosphorus
melt one at 44"* and the other at SO"*, it might always be said
that doubtless, besides the point of fusion, there is some other
unkno^^Ti property by which they differ.''
That was not quite what I meant to say. I should have written,
''All bodies possessing such and such properties finite in number
(to wit, the properties of phosphorus stated in the books on
chemistry, the fusion-point excepted) melt at 44**."
And the better to make evident the difference between the case
of the straight and that of phosphorus, one more remark. The
straight has in nature many images more or less imperfect, of
which the chief are the light rays and the rotation axis of the
solid. Suppose we find the ray of light does not satisfy Euclid's
postulate (for example by showing that a star has a negative
parallax), what shall we do? Shall we conclude that the straight
being by definition the trajectory of light does not satisfy the
postulate, or, on the other hand, that the straight by definition
satisfying the postulate, the ray of light is not straight?
Assuredly we are free to adopt the one or the other definition
and consequently the one or the other conclusion; but to adopt
THE NEW LOGICS 471
the first would be stupid, because the ray of light probably
satisfies only imperfectly not merely Euclid's postulate, but the
other properties of the straight line, so that if it deviates from
the Euclidean straight, it deviates no less from the rotation axis
of solids which is another imperfect image of the straight line;
while finally it is doubtless subject to change, so that such a line
which yesterday was straight will cease to be straight to-morrow
if some physical circumstance has changed.
Suppose now we find that phosphorus does not melt at 44^,
but at 43.9''. Shall we conclude that phosphorus being by defini-
tion that which melts at 44^, this body that we did call phos-
phorus is not true phosphorus, or, on the other hand, that phos-
phorous melts at 43.9'' T Here again we are free to adopt the one
or the other definition and consequently the one or the other
conclusion; but to adopt the first would be stupid because we
can not be changing the name of a substance every time we
determine a new decimal of its fusion-point.
XIII
To sum up, Russell and Hilbert have each made a vigorous
effort; they have each written a work full of original views,
profound and often well warranted. These two works give us
much to think about and we have much to learn from them.
Among their results, some, many even, are solid and destined to
live.
But to say that they have finally settled the debate between
Kant and Leibnitz and ruined the Kantian theory of mathe-
matics is evidently incorrect. I do not know whether they really
believed they had done it, but if they believed so, they deceived
themselves.
CHAPTER V
The LiLTEST Efforts of the Logistigianb
I
The logicians have attempted to answer the preceding con-
siderations. For that, a transformation of logistic was necessary,
and Russell in particular has modified on certain points his
original views. Without entering into the details of the debate,
I should like to return to the two questions to my mind most im-
portant : Have the rules of logistic demonstrated their fruitfnl-
ness and infallibility f Is it true they afford means of proving
the principle of complete induction without any appeal to
intuition f
n
The Infallibility of Logistic
On the question of fertility, it seems M. Couturat has naive
illusions. Logistic, according to him, lends invention 'stilts and
wings,' and on the next page: ''Ten years ago, Peano published
the first edition of his Formulaire/' How is that, ten years of
wings and not to have flown !
I have the highest esteem for Peano, who has done very pretty
things (for instance his 'space-filling curve,' a phrase now dis-
carded) ; but after all he has not gone further nor higher nor
quicker than the majority of wingless mathematicians, and would
have done just as well with his legs.
On the contrary I see in logistic only shackles for the inventor.
It is no aid to conciseness — far from it, and if twenty-seven
equations were necessary to establish that 1 is a number, how
many would be needed to prove a real theorem! If we distin-
guish, with Whitehead, the individual x, the class of which the
only member is x and which shall be called t x, then the class of
which the only member is the class of which the only member is x
and which shall be called a a;, do you think these distinctions,
useful as they may be, go far to quicken our pace!
472
THE LATEST EFFORTS OF THE LOGISTICIANS 473
Logistic forces us to say all that is ordinarily left to be under-
stood; it makes us advance step by step; this is perhaps surer
but not quicker.
It is not wings you logisticians give us, but leading-strings.
And then we have the right to require that these leading-strings
prevent our falling. This will be their only excuse. When a
bond does not bear much interest, it should at least be an invest-
ment for a father of a family.
Should your rules be followed blindly f Yes, else only intui-
tion could enable us to distinguish among them ; but then they
must be infallible; for only in an infallible authority can one
have a blind confidence. This, therefore, is for you a necessity.
Infallible you shall be, or not at all.
You have no right to say to us: ''It is true we make mistakes,
but so do you." For us to blunder is a misfortune, a very great
misfortune; for you it is death.
Nor may you ask : Does the infallibility of arithmetic prevent
errors in addition f The rules of calculation are infallible, and
yet we see those blunder who do not apply these rules; but in
checking their calculation it is at once seen where they went
wrong. Here it is not at all the case ; the logicians have applied
their rules, and they have fallen into contradiction ; and so true
is this, that they are preparing to change these rules and to
''sacrifice the notion of class." Why change them if they were
infallible t
"We are not obliged," you say, "to solve hie et nunc all pos-
sible problems." Oh, we do not ask so much of you. If, in face
of a problem, you would give no solution, we should have nothing
to say; but on the contrary you give us two of them and those
contradictory, and consequently at least one false ; this it is which
is failure.
Russell seeks to reconcile these contradictions, which can oidy
be done, according to him, "by restricting or even sacrificing the
notion of class." And M. Couturat, discovering the success of
his attempt, adds: "If the logicians succeed where others have
failed, M. Poincar6 will remember this phrase, and give the honor
of the solution to logistic."
But no I Logistic exists, it has its code which has already had
474 SCIENCE AND METHOD
four editions ; or rather this code is logistic itself. Is Mr. Bos-
sell preparing to show that one at least of the two contradictor/
reasonings has transgressed the codef Not at all; he is pre-
paring to change these laws and to abrogate a certain number of
them. If he succeeds, I shall give the honor of it to Russell's
intuition and not to the Peanian logistic which he will have
destroyed.
Ill
The Liberty of Contradiction
I made two principal objections to the definition of whole
number adopted in logistic. What says M. Couturat to the first
of these objections f
What does the word exist mean in mathematics f It means,
I said, to be free from contradiction. This M. Couturat con-
tests. ''Logical existence," says he, ''is quite another thing
from the absence of contradiction. It consists in the fact that
a class is not empty." To say: a's exist, is, by definition, to
afSrm that the class a is not null.
And doubtless to affirm that the class a is not null, is, by defi-
nition, to affirm that a's exist. But one of the two affirmations
is as denuded of meaning as the other, if they do not both signify,
either that one may see or touch a's which is the meaning physi-
cists or naturalists give them, or that one may conceive an a
without being drawn into contradictions, which is the meaning
given them by logicians and mathematicians.
For M. Couturat, "it is not non-contradiction that proves exist-
ence, but it is existence that proves non-contradiction. ' * To estab-
lish the existence of a class, it is necessary therefore to establish,
by an example, that there is an individual belonging to this class:
"But, it will be said, how is the existence of this individual
proved? Must not this existence be established, in order that
the existence of the class of which it is a part may be deduced?
Well, no; however paradoxical may appear the assertion, we
never demonstrate the existence of an individual. Individuals,
just because they are individuals, are always considered as exist-
ent. . . . We never have to express that an individual exists,
absolutely speaking, but only that it exists in a class." M.
THE LATEST EFFORTS OF THE LOGISTICIANS 475
Couturat finds his own assertion paradoxical, and he will cer-
tainly not be the only one. Yet it must have a meaning. It
doubtless means that the existence of an individual, alone in the
world, and of which nothing is affirmed, can not involve contra-
diction ; in so far as it is all alone it evidently will not embarrass
any one. Well, so let it be ; we shall admit the existence of the
individual, ' absolutely speaking, ' but nothing more. It remains to
prove the existence of the individual 'in a class,' and for that it
will always }}e necessary to prove that the affirmation, ''Such an
individual belongs to such a class," is neither contradictory in
itself, nor to the other postulates adopted.
"It is then," continues M. Couturat, "arbitrary and mis-
leading to maintain that a definition is valid only if we first
prove it is not contradictory." One could not claim in prouder
and more energetic terms the liberty of contradiction. "In any
case, the anus probandi rests upon those who believe that these
principles are contradictory." Postulates are presumed to be
compatible until the contrary is proved, just as the accused
person is presumed innocent. Needless to add that I do not
assent to this claim. But, you say, the demonstration you require
of us is impossible, and you can not ask us to jump over the
moon. Pardon me ; that is impossible for you, but not for us, who
admit the principle of induction as a synthetic judgment a priori.
And that would be necessary for you, as for us.
To demonstrate that a system of postulates implies no contra-
diction, it is necessary to apply the principle of complete induc-
tion; this mode of reasoning not only has nothing 'bizarre' about
it, but it is the only correct one. It is not 'unlikely' that it has
ever been employed; and it is not hard to find 'examples and
precedents' of it. I have cited two such instances borrowed from
Hubert's article. He is not the only one to have used it, and
those who have not done so have been wrong. What I have
blamed Hilbert for is not his having recourse to it (a bom
mathematician such as he could not fail to see a demonstration
was necessary and this the only one possible), but his having
recourse without recognizing the reasoning by recurrence. •
476 SCIENCE AND METHOD
IV
The Second Objection
I pointed out a second error of logistic in Hilbert's article.
To-day Hilbert is excommunicated and M. Coutarat no longer
regards him as of the logistic cult; so he asks if I have found
the same fault among the orthodox. No, I have not seen it in the
pages I have read ; I know not whether I should find it in the
three hundred pages they have written which I have no desire to
read.
Only, they must commit it the day they wish to make any
application of mathematics. This science has not as sole object
the eternal contemplation of its own navel; it has to do with
nature and some day it will touch it. Then it will be necessary
to shake off purely verbal definitions and to stop paying oneself
with words.
To go back to the example of Hilbert: always the point at
issue is reasoning by recurrence and the question of knowing
whether a system of postulates is not contradictory. M. Couturat
will doubtless say that then this does not touch him, but it per-
haps will interest those who do not claim, as he does, the liberty
of contradiction.
We wish to establish, as above, that we shall never encounter
contradiction after any number of deductions whatever, pro-
vided this number be finite. For that, it is necessary to apply the
principle of induction. Should we here understand by finite
number every number to which by definition the principle of
induction applies ? Evidently not, else we should Be led to most
embarrassing consequences. To have the right to lay down a
system of postulates, we must be sure they are not contradictory.
This is a truth admitted by most scientists ; I should have written
by all before reading ^I. Couturat 's last article. But what does
this signify ? Does it mean that we must be sure of not meeting
contradiction after a finite number of propositions, the finite
number being by definition that which has all properties of
recurrent nature, so that if one of these properties fails — ^if, for
instance, we come upon a contradiction — we shall agree to say
that the number in question is not finite? In other words, do
THE LATEST EFFORTS OF THE LOGISTICIANS 477
we mean that we must be sure not to meet contradictions, on
condition of agreeing to stop just when we are about to encounter
one! To state such a proposition is enough to condemn it.
So, Hubert's reasoning not only assumes the principle of in-
duction, but it supposes that this principle is given us not as
a simple definition, but as a synthetic judgment a priori.
To sum up :
A demonstration is necessary.
The only demonstration possible is the proof by recurrence.
This is legitimate only if we admit the principle of induction
and if we regard it not as a definition but as a synthetic judgment.
The Cantor Antinomies
Now to examine Russell's new memoir. This memoir was
written with the view to conquer the difSculties raised by those
Cantor antinomies to which frequent allusion has already been
made. Cantor thought he could construct a science of the
infinite ; others went on in the way he opened, but they soon ran
foul of strange contradictions. These antinomies are already
numerous, but the most celebrated are :
1. The Burali-Porti antinomy;
2. The Zermelo-Konig antinomy;
3. The Richard antinomy.
Cantor proved that the ordinal numbers (the question is of
transfinite ordinal numbers, a new notion introduced by him)
can be ranged in a linear series; that is to say that of two un-
equal ordinals one is always less than the other. Burali-Forti
proves the contrary ; and in fact he says in substance that if one
could range all the ordinals in a linear series, this series would
define an ordinal greater than all the others; we could after-
wards adjoin 1 and would obtain again an ordinal which would
be still greater, and this is contradictory.
We shall return later to the Zermelo-Konig antinomy which is
of a slightly diflFerent nature. The Richard antinomy* is as fol-
lows: Consider all the decimal numbers definable by a finite
1 Bevue g^n^ale de$ sciences, June 30, 1905.
478 SCIENCE AND METHOD
number of words ; these decimal numbers form an aggregate £,
and it is eai^ to see that this aggregate is countable, that is to
say we can number the different decimal numbers of this assem-
blage from 1 to infinity. Suppose the numbering effected, and
define a number N as follows: If the nth decimal of the nth
number of the assemblage E is
0, 1, 2, 3, 4, 6, 6, 7, 8, 9
the nth decimal of N shall be :
1, 2, 3, 4, 5, 6, 7, 8, 1, 1
As we see, N is not equal to the nth number of J?, and as n is
arbitrary, N does not appertain to E and yet N should belong
to this assemblage since we have defined it with a finite number
of words.
We shall later see that M. Richard has himself given with
much sagacity the explanation of his paradox and that this ex-
tends, mutatis mutandis, to the other like paradoxes. Again,
Russell cites another quite amusing paradox: What is the least
whole number which can not be defined by a phrase composed of
less than a hundred English words f
This number exists ; and in fact the numbers capable of being
defined by a like phrase are evidently finite in number since the
words of the English language are not infinite in number. There-
fore among them will be one less than all the others. And, on the
other hand, this number does not exist, because its definition
implies contradiction. This number, in fact, is defined by the
phrase in italics which is composed of less than a hundred Eng-
lish words ; and by definition this number should not be capable
of definition by a like phrase.
VI
Zigzag Theory and No-cla^s Theory
What is Mr. Russell's attitude in presence of these contradic-
tions? After having analyzed those of which we have just
spoken, and cited still others, after having given them a form
recalling Epimenides, he does not hesitate to conclude : "A propo-
THE LATEST EFFORTS OF THE LOGISTICIANS 479
sitional function of one variable does not always determine a
class." A propositional function (that is to say a definition)
does not always determine a class. A 'propositional function'
or 'norm' may be 'non-predicative.' And this does not mean
that these non-predicative propositions determine an empty class,
a null class ; this does not mean that there is no value of x satis-
fying the definition and capable of being one of the elements
of the class. The elements exist, but they have no right to unite
in a syndicate to form a class.
But this is only the beginning and it is needful to know how
to recognize whether a definition is or is not predicative. To solve
this problem Bussell hesitates between three theories which he
calls
A. The zigzag theory;
B. The theory of limitation of size ;
C. The no-class theory.
According to the zigzag theory ' ' definitions (propositional func-
tions) determine a class when they are very simple and cease to
do so only when they are complicated and obscure. ' ' Who, now, is
to decide whether a definition may be regarded as simple enough
to be acceptable f To this question there is no answer, if it be
not the loyal avowal of a complete inability: "The rules which
enable us to recognize whether these definitions are predicative
would be extremely complicated and can not commend them-
selves by any plausible reason. This is a fault which might be
remedied by greater ingenuity or by using distinctions not yet
pointed out. But hitherto in seeking these rules, I have not
been able to find any other directing principle than the absence
of contradiction."
This theory therefore remains very obscure; in this night a
single light — ^the word zigzag. What Russell calls the 'zigzagi-
ness' is doubtless the particular characteristic which distinguishes
the argument of Epimenides.
According to the theory of limitation of size, a class would
cease to have the right to exist if it were too extended. Perhaps
it might be infinite, but it should not be too much so. But we
always meet again the same difSculty; at what precise moment
480 SCIENCE AND METHOD
does it begin to be too much sof Of course this diflSculty is not
solved and Russell passes on to the third theory.
In the no-classes theory it is forbidden to speak the word
'class' and this word must be replaced by various periphruei.
What a change for logistic which talks only of classes and
classes of classes! It becomes necessary to remake the whole
of logistic. Imagine how a page of logistic would look upon sap-
pressing all the propositions where it is a question of class. There
would only be some scattered survivors in the midst of a blank
page. Apparent rari nantes in gurgite v<isto.
Be that as it may, we see how Russell hesitates and the modi-
fications to which he submits the fundamental principles he has
hitherto adopted. Criteria are needed to decide whether a defini-
tion is too complex or too extended, and these criteria can only
be justified by an appeal to intuition.
It is toward the no-classes theory that Russell finally inclines.
Be that as it may, logistic is to be remade and it is not clear
how much of it can be saved. Needless to add that Cantorism
and logistic are alone under consideration; real mathematics,
that which is good for something, may continue to develop in
accordance with its own principles without bothering about the
storms which rage outside it, and go on step by step with its usual
conquests which are final and which it never has to abandon.
VII
The True Solution
Wliat choice ought we to make among these difl^erent theories?
It seems to me that the solution is contained in a letter of "SL
Richard of which I have spoken above, to be found in the Revue
gcncralc dcs sciences of June 30, 1905. After having set forth
the antinomy we have called Richard's antinomy, he gives its
explanation. Recall what has already been said of this antinomy.
E is the aggregate of all the numbers definable by a finite number
of words, without introducing the notion of the aggregate E itself.
Else the definition of E would contain a vicious circle ; we must
not define E by the aggregate E itself.
Now we have defined N with a finite number of words, it is
THE LATEST EFFORTS OF THE L0GISTICIAN8 481
true, but with the aid of the notion of the aggregate E. And
this is why N is not part of E. In the example selected by M.
Bichard, the conclusion presents itself with complete evidence
and the evidence will appear still stronger on consulting the
text of the letter itself. But the same explanation holds good
for the other antinomies, as is easily verified. Thus the defini-
tions which should be regarded as not predicative are those
which contain a vidoiLS circle. And the preceding examples suJBS-
ciently show what I mean by that. Is it this which Russell calls
the 'zigzaginess'f I put the question without answering it
vm
The Demonstrations of the Principle of Induction
Let us now examine the pretended demonstrations of the
principle of induction and in particular those of Whitehead and
of Burali-Porti.
We shall speak of Whitehead's first, and take advantage of
certain new terms happily introduced by Russell in his recent
memoir. Call recurrent class every class containing zero, and
containing n -|- 1 if it contains n. Call inductive number every
number which is a part of aU the recurrent classes. Upon what
condition will this latter definition, which plays an essential
role in Whitehead's proof, be 'predicative' and consequently
acceptable f
In accordance with what has been said, it is necessary to
understand by all the recurrent classes, all those in whose defini-
tion the notion of inductive number does not enter. Else we fall
again upon the vicious circle which has engendered the antinomies.
Now Whitehead has not taken this precaution. Whitehead's
reasoning is therefore fallacious ; it is the same which led to the
antinomies. It was illegitimate when it gave false results; it
remains illegitimate when by chance it leads to a true result.
A definition containing a vicious circle defines nothing. It is
of no use to say, we are sure, whatever meaning we may give to
our definition, zero at least belongs to the class of inductive
numbers; it is not a question of knowing whether this class is
void, but whether it can be rigorously deliminated. A 'non-
32
482 SCIENCE AND METHOD
predicative' class is not an empty class, it is a class whose
boundary is undetermined. Needless to add that this particular
objection leaves in force the general objections applicable to all
the demonstrations.
IX
Burali-Forti has given another demonstration.' But he is
obliged to assume two postulates: First, there alw