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SCIENCE AND EDUCATION
A SERIES OF VOLUMES FOR THE PROMOTION OF
SCIENTIFIC RESEARCH AND EDUCATIONAL PROGRESS
EDITED BY J. MCKEEN CATTELL
VOLUME I--THE FOUNDATIONS OF SCIENCE
UNDER THE SAME EDITORSHIP
SCIENCE AND EDUCATION. A series of volumes for the promotion of
scientific research and educational progress.
Volume I. The Foundations of Science. By H. POINCARÉ. Containing
the authorised English translation by George Bruce Halsted of
"Science and Hypothesis," "The Value of Science," and "Science
and Method."
Volume II. 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.
Councilman Theobald Smith, G. N. Stewart, C. M. Jackson,
E. P. Lyon, James B. Herrick, John M. Dodson, C. R. Bardeen,
W. Ophuls, S. J. Meltzer, James Ewing, W. W. Keen, Henry H.
Donaldson, Christian A. Herter, and Henry P. Bowditch.
Volume III. University Control. By J. MCKEEN CATTELL and other
authors.
AMERICAN MEN OF SCIENCE. A Biographical Directory.
SCIENCE. A weekly journal devoted to the advancement of science.
The official organ of the American Association for the
Advancement of Science.
THE POPULAR SCIENCE MONTHLY. A monthly magazine devoted to the
diffusion of science.
THE AMERICAN NATURALIST. A monthly journal devoted to the
biological sciences, with special reference to the factors
of evolution.
THE SCIENCE PRESS
NEW YORK GARRISON, N. Y.
THE FOUNDATIONS
OF SCIENCE
SCIENCE AND HYPOTHESIS
THE VALUE OF SCIENCE
SCIENCE AND METHOD
BY
H. POINCARÉ
AUTHORIZED TRANSLATION BY
GEORGE BRUCE HALSTED
WITH A SPECIAL PREFACE BY POINCARÉ, AND AN INTRODUCTION
BY JOSIAH ROYCE, HARVARD UNIVERSITY
THE SCIENCE PRESS
NEW YORK AND GARRISON, N. Y.
1913
Copyright, 1913
BY THE SCIENCE PRESS
PRESS OF
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.
CONTENTS
PAGE
Henri Poincaré ix
Author's Preface to the Translation 3
SCIENCE AND HYPOTHESIS
Introduction by Royce 9
Introduction 27
PART I. _Number and Magnitude_
CHAPTER I.--On the Nature of Mathematical Reasoning 31
Syllogistic Deduction 31
Verification and Proof 32
Elements of Arithmetic 33
Reasoning by Recurrence 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 Remarks (Curves without Tangents) 50
The Physical Continuum of Several Dimensions 52
The Mathematical Continuum of Several Dimensions 53
PART II. _Space_
CHAPTER III.--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
Riemann'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
Solid Bodies and Geometry 72
Law of Homogeneity 74
The Non-Euclidean World 75
The World of Four Dimensions 78
Conclusions 79
CHAPTER V.--Experience and Geometry 81
Geometry and Astronomy 81
The Law of Relativity 83
Bearing of Experiments 86
Supplement (What is a Point?) 89
Ancestral Experience 91
PART III. _Force_
CHAPTER VI.--The Classic Mechanics 92
The Principle of Inertia 93
The Law of Acceleration 97
Anthropomorphic Mechanics 103
The School of the Thread 104
CHAPTER VII.--Relative Motion and Absolute Motion 107
The Principle of Relative Motion 107
Newton's Argument 108
CHAPTER VIII.--Energy and Thermodynamics 115
Energetics 115
Thermodynamics 119
General Conclusions on Part III 123
PART IV. _Nature_
CHAPTER IX.--Hypotheses in Physics 127
The Rôle of Experiment and Generalization 127
The Unity of Nature 130
The Rôle of Hypothesis 133
Origin of Mathematical Physics 136
CHAPTER X.--The Theories of Modern 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
Rouge 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
CHAPTER XIII.--Electrodynamics 184
Ampère's Theory 184
Closed Currents 185
Action of a Closed Current on a Portion of Current 186
Continuous Rotations 187
Mutual Action of Two Open Currents 189
Induction 190
Theory of Helmholtz 191
Difficulties Raised by these Theories 193
Maxwell's Theory 193
Rowland's Experiment 194
The Theory of Lorentz 196
THE VALUE OF SCIENCE
Translator's Introduction 201
Does the Scientist Create Science? 201
The Mind Dispelling Optical Illusions 202
Euclid not Necessary 202
Without Hypotheses, no Science 203
What Outcome? 203
Introduction 205
PART I. _The Mathematical Sciences_
CHAPTER I.--Intuition and Logic in Mathematics 210
CHAPTER 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
CHAPTER 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
Rôle of the Semicircular Canals 276
PART II. _The Physical Sciences_
CHAPTER 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 VIII.--The Present Crisis in Physics 303
The New Crisis 303
Carnot's Principle 303
The Principle of Relativity 305
Newton's Principle 308
Lavoisier's Principle 310
Mayer's Principle 312
CHAPTER IX.--The Future of Mathematical Physics 314
The Principles and Experiment 314
The Rôle of the Analyst 314
Aberration and Astronomy 315
Electrons and Spectra 316
Conventions preceding Experiment 317
Future Mathematical Physics 319
PART III. _The Objective Value of Science_
CHAPTER X.--Is Science Artificial? 321
The Philosophy of LeRoy 321
Science, Rule 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 Rotation 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 II. _Mathematical Reasoning_
CHAPTER I.--The Relativity 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 Radium 486
CHAPTER II.--Mechanics and Optics 496
CHAPTER III.--The New Mechanics and Astronomy 512
BOOK IV. _Astronomic Science_
CHAPTER I.--The Milky Way and the Theory of Gases 523
CHAPTER II.--French Geodesy 535
General Conclusions 544
Index 547
HENRI POINCARÉ
SIR GEORGE DARWIN, worthy son of an immortal father, said, referring to
what Poincaré was to him and to his work: "He must be regarded as the
presiding genius--or, shall I say, my patron saint?"
Henri Poincaré 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 École Polytechnique, where, like John Bolyai at
Maros Vásárhely, 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 _Nommé_, March 26, 1879. But he won his doctorate in the
University of Paris, August 1, 1879, and was appointed to teach in the
Faculté des Sciences de Caen, December 1, 1879, whence he was quickly
called to the University 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
l'Académie des Sciences, and, March 5, 1908, was chosen Membre de
l'Académie Française. 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 Poincaré tells us the very
minute it happened. Mounting 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 perspective new and
immense. Moreover, the magic wand of his whole 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 commenced 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
possibility of alternative laws of nature, which in the Introduction to
the _Électricité 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_, convenient; but perhaps another
may be vastly more advantageous. The old conception of _true_ has been
revised. The first expression of the new idea occurs on the title page
of John Bolyai's marvelous _Science Absolute of Space_, in the phrase
"haud unquam 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,
Poincaré proved the existence of a previously unknown pear-shaped
figure, with the possibility that the progressive 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
Méthodes nouvelles de la Méchanique céleste_, Sir George Darwin says:
"It is probable that for half a century to come it will be the mine from
which humbler investigators will excavate their materials." Brilliant
was his appreciation of Poincaré 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 Academy of Sciences. His work must be
reckoned with the greatest mathematical achievements of mankind.
The kernel of Poincaré's power lies in an oracle Sylvester often quoted
to me as from Hesiod: The whole is less than its part.
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 immense 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."
GEORGE BRUCE HALSTED.
* * * * *
SCIENCE AND HYPOTHESIS
* * * * *
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 understanding physics; that this
way seems to them superior to all others; that they feel no need of
changing it, nor even of knowing 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 conquests they
alone could have made; they will make still more of which others would
be incapable. It would therefore be deplorable if there were no longer
Anglo-Saxons.
But continentals have on their part done things an Englishman 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 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 savoring 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 appreciating certain theories more interesting than utilitarian, such
as the non-Euclidean geometries. According to that, the first two parts
of this book, on number and space, should seem to them void of all
substance and would only baffle them.
But that is not true. And first of all, are they such uncompromising
realists as has been said? Are they absolutely refractory, I do not say
to metaphysic, but at least to everything metaphysical?
Recall the name of Berkeley, born 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 ascensions
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 groups of substitutions, and under its
most abstract form, the farthest removed from the practical.
Moreover, Dr. Halsted gives regularly each year a review of all
productions 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 ideas 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 burn
all bridges of reliance upon sensory intuition, and this is, I confess,
a boldness which seems to me almost rashness.
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 sacrifice 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 also? What things do they hide? 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.
In 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 mathematical 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 foreknowledge as only
provisional, and they hasten to ask its definitive 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 ordinarily choose is not that preferred by the
Anglo-Saxons.
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 engrossed 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 therefore 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 studying the work of
minds so unlike their own, they will immediately 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 geometric 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 perceived 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 perceiving them,
have the advantage of showing them the hypothetical 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.
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 formulas 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 construct the ether
by material models, or in France 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.
POINCARÉ.
INTRODUCTION
BY PROFESSOR JOSIAH ROYCE
HARVARD 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.
Poincaré'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. Poincaré has so notably contributed.
I
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. Poincaré 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
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. Poincaré refers in his
opening paragraphs, or whether, like Spencer 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
Modern Physics' first appeared, this sense of scientific orthodoxy was
shocked amongst many of our American readers 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 presumption 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 modern '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 details more assured. Phenomena men
know and predict better than ever. But regarding the most general
theories, and the most fundamental, of science, there is no longer any
notable scientific 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 regarding 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 circle-squarers 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 inconceivable' and the doctrine of energy beyond the reach of
novel formulations. 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 interpretations. Certain properties of
the irrational number [pi] are known, in sufficient 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 Dedekind, of Cantor, of Weierstrass and of various others,
the general 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 several very
notable respects, unconquered territory, despite the work of Hilbert and
of Pieri, 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 radioactive bodies will occupy in that theory. The
alchemists would not be welcome workers in modern 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 invariant, than the theoretical
orthodoxy of a recent period supposed. Again, regarding the foundations
of biology, a theoretical orthodoxy grows less possible, less definable,
less conceivable (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 objects, 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, you work as if you were a partisan of 'mechanics.' In
any case, your special science prospers by reason of the empirical
discoveries that you 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 dream, and that the
results of science are coextensive with the range of actual empirical
observation and of successful prediction? 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 man 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
reasonable. 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 observed; 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 resignation? Why not substitute, for the
old scientific orthodoxy, simply a confession of ignorance, and a
resolution to devote ourselves to the business of enlarging the bounds
of actual empirical knowledge?
Such comments upon the situation just characterized are frequently 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
these very comments. Never has there been better reason than there is
to-day to be content, if rational man could be content, with a pure
positivism. The splendid triumphs of special research in the most
various fields, the constant increase in our practical control over
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'? Why
pursue the elusive theoretical 'unification' any further, when what we
daily get from our sciences is an increasing wealth of detailed
information and of practical guidance?
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 orthodoxy 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 regarding 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. Poincaré, in his
book, sketches with the hand of the master, is another illustration of
the consciousness 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 oppression 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 away? 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, namely, man's knowledge of the phenomenal
order of nature, be constantly united in men's minds with that which is
certain to decay, namely, the theoretical formulation of special
knowledge in more or less completely unified systems of doctrine?
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 different 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 construction--an
abandonment of dogma, but a vindication of the rights of the
constructive reason.
III
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 suggests 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 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 frequently 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 explicitly
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 distinct 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. Poincaré'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 concepts 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.
Poincaré's logical theory of this second class of hypotheses undertakes
to accomplish, with modern 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
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. Poincaré'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 the data of sense must be.
This, of course, is not M. Poincaré'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. Poincaré, on the contrary, all this
adjustment of our interpretations 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 necessity. Unverifiable
and irrefutable hypotheses in science are indeed, in general,
indispensable aids to the organization and to the guidance of our
interpretation of experience. But it is experience 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. Poincaré'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
organization is due to hypotheses of the kind here in question, thus
resembles that of a constitutional government--neither absolutely
necessary, nor yet determined apart from the will of the subjects, nor
yet accidental--a free, yet not a capricious establishment of good
order, in conformity with empirical needs.
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 described 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
confirmed 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 existence of continuous
extensive _quanta_ in nature; (2) The principles 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. Hypotheses 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 intelligent beings whose experience could be best
interpreted without using these hypotheses. Yet these hypotheses are
_not_ subject 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 hypotheses 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 appreciation of the significance
of empirical investigation. But the central problem of the logic of
science thus becomes the problem of the relation between the two
fundamentally distinct types of hypotheses, _i. e._, between those which
can not be verified or refuted through experience, and those which can
be empirically tested.
IV
The detailed treatment which M. Poincaré 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 observing that, while M.
Poincaré 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. Poincaré'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 briefly 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, undertake theoretical
constructions which relate to past time. Hypotheses 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 predictions of coming experience. These latter will be subject to
control. 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 predictions proved
verifiable. The result has been a considerable 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, regarding 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 hypotheses 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
anthropology, or Old Testament history, the principal theoretical
constructions 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 confrères of the sciences 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 criticism'--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 importance 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
verifiable 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 theoretical
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 constructions in fact resemble
in a measure those which M. Poincaré 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 we investigate, in the popular phrase, 'talk 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, internally rational, as well as (when that is possible)
subject to external 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. Poincaré speaks when he characterizes the principles of mechanics and
of the theory of energy. I venture to call attention here to this
conception, which seems to me to illustrate M. Poincaré's view of the
functions of hypothesis in scientific work.
The modern science of pathology is usually regarded as dating from the
earlier researches of Virchow, whose 'Cellular Pathology' was the
outcome of a very careful and elaborate induction. Virchow, himself,
felt a strong aversion to mere speculation. 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 researches 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 diseases 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 us the course of the vital
processes under altered conditions_" ('dasz sie nur Ablauf der
Lebenserscheinungen unter veränderten Bedingungen darstellen').
The enormous importance of this theoretical presupposition for all the
early successes of modern pathological investigation is generally
recognized by the experts. I do not doubt this opinion. It appears to be
a commonplace of the history of this science. But in Virchow's later
years this very presupposition seemed to some of his contemporaries to
be called in question by the successes of recent bacteriology. The
question arose whether the theoretical foundations of Virchow's
pathology had not been set aside. And in fact the theory 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 untouched by
the newer discoveries. 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 themselves, 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 _veränderte 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
involves 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 understood 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
_veränderte 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 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
supposition, 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 interpretation 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 processes are linked. Now without some such leading idea, the
cellular 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 refutable hypothesis. One value of
unverifiable and irrefutable hypotheses 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
control, 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 irrefutable character,
suggested, but not to be finally tested, by experience, the hypotheses
in the narrower sense would lack that guidance which, as M. Poincaré 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 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 philosophical 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.
Poincaré 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 problems 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 students 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 Poincaré was born at Nancy, in 1854, the son of a professor
in the Faculty of Medicine at Nancy. He studied at the École
Polytechnique and at the École des Mines, and later received his
doctorate in mathematics in 1879. In 1883 he began courses of
instruction in mathematics at the École Polytechnique; in 1886 received
a professorship of mathematical 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 various branches of his
chosen sciences is long; and his original 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 Poincaré 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
modern 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. Poincaré has long been active. When, in 1893,
the admirable _Revue de Métaphysique et de Morale_ began to appear, M.
Poincaré 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
together in the present volume are in large part the outcome of M.
Poincaré's contributions to the _Revue de Métaphysique et de Morale_.
The reader of M. Poincaré's book is in presence, then, of a great
special investigator who is also a philosopher.
SCIENCE AND HYPOTHESIS
INTRODUCTION
For a superficial observer, scientific truth is beyond the possibility
of doubt; the logic of science is infallible, and if the scientists 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 corner of the universe."
Behold what is for many people in the world, for scholars getting their
first notions of physics, the origin of scientific certitude. This is
what they suppose to be the rôle of experimentation 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 rôle of hypothesis; we shall then recognize, not
only that it is necessary, but that usually it is legitimate. We shall
also see that there are several sorts of hypotheses; that some are
verifiable, and once confirmed by experiment 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 understand 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. Experiment 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 convention
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 definitions and if the world he thinks he discovers
is not simply created by his own caprice.[1] Under these conditions
science would be certain, but deprived of significance.
[1] See Le Roy, 'Science et Philosophie,' _Revue de Métaphysique
et de Morale_, 1901.
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
naïve 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.
What is the nature of mathematical reasoning? Is is really deductive, 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 mathematics puts into
the hands of the investigator, we had to analyze another fundamental
notion, that of mathematical magnitude. Do we find it in nature, or do
we ourselves introduce it there? And, in this latter case, do we not
risk marring everything? Comparing 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 imposed 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 geometry? A deeper discussion will show us it is not. We
therefore conclude that the first principles of geometry are only
conventions; 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 directly 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 hypotheses 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,
and of each of them something remains. It is this something we must seek
to disentangle, since there and there alone is the veritable reality.
The method of the physical sciences rests on the induction which makes
us expect the repetition of a phenomenon when the circumstances under
which it first happened are reproduced. If _all_ these circumstances
could be reproduced at once, this principle 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 rôle 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 Maxwell, and what unconscious hypotheses
were made by Ampère and the other founders of electrodynamics.
PART I
NUMBER AND MAGNITUDE
CHAPTER I
ON THE NATURE OF MATHEMATICAL REASONING
I
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 so 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 mathematical necessity, we have yet the
resource of classing them among synthetic _a priori_ judgments. This is
not to solve the difficulty, 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
reasoning 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 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 intention of
generalizing some proposition already known. Does the mathematical
method proceed from the particular to the general, and, if so, how then
can it be called deductive?
If finally the science of number were purely analytic, or could be
analytically derived from a small number of synthetic 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 sufficiently 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 _x_ + 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:
(4) _x_ + 2 = (_x_ + 1) + 1.
That presupposed, we have
2 + 1 + 1 = 3 + 1 (Definition 2),
3 + 1 = 4 (Definition 3),
2 + 2 = (2 + 1) + 1 (Definition 4),
whence
2 + 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 properly 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 precisely because it is purely analytic and because it is
sterile. It is sterile because the conclusion is nothing but the
premises translated 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.
III
Let us, therefore, see the geometer at work and seek to catch his
process.
The task is not without difficulty; it does not suffice to open a work
at random and analyze any demonstration in it.
We must first exclude geometry, where the question is complicated by
arduous problems relative to the rôle of the postulates, to the nature
and the origin of the notion of space. For analogous reasons we can not
turn to the infinitesimal analysis. 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 undergone 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 precisely 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 mathematical rigor; they would see in it only
useless and irksome subtleties; 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 impress itself
upon all good minds? This is a logical and psychological 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 OF ADDITION.--I suppose already defined the operation
_x_ + 1, which consists in adding the number 1 to a given number _x_.
This definition, whatever it be, does not enter into our subsequent
reasoning.
We now have to define the operation _x_ + _a_, which consists in adding
the number _a_ to a given number _x_.
Supposing we have defined the operation
_x_ + (_a_ - 1),
the operation _x_ + _a_ will be defined by the equality
(1) _x_ + _a_ = [_x_ + (_a_ - 1)] + 1.
We shall know then what _x + a_ is when we know what _x_ + (_a_ - 1)
is, and as I have supposed that to start with we knew what _x_ + 1 is,
we can define successively and 'by recurrence' the operations _x_ + 2,
_x_ + 3, etc.
This definition deserves a moment's attention; it is of a particular
nature which already distinguishes it from the purely logical
definition; the equality (1) contains an infinity of distinct
definitions, each having a meaning only when one knows the preceding.
PROPERTIES OF ADDITION.--_Associativity._--I say that
_a_ + (_b_ + _c_) = (_a_ + _b_) + _c_.
In fact the theorem is true for _c_ = 1; it is then written
_a_ + (_b_ + 1) = (_a_ + _b_) + 1,
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_ = [gamma], I say it will be true for
_c_ = [gamma] + 1.
In fact, supposing
(_a_ + _b_) + [gamma] = _a_ + (_b_ + [gamma]),
it follows that
[(_a_ + _b_) + [gamma]] + 1 = [_a_ + (_b_ + [gamma])] + 1
or by definition (1)
(_a_ + _b_) + ([gamma] + 1) = _a_ + (_b_ + [gamma] + 1)
= _a_ + [_b_ + ([gamma] + 1)],
which shows, by a series of purely analytic deductions, that the
theorem is true for [gamma] + 1.
Being true for _c_ = 1, we thus see successively that so it is for
_c_ = 2, for _c_ = 3, etc.
_Commutativity._--1º I say that
_a_ + 1 = 1 + _a_.
The theorem is evidently true for _a_ = 1; we can _verify_ by purely
analytic reasoning that if it is true for _a_ = [gamma] it will be true
for _a_ = [gamma] + 1; for then
([gamma] + 1) + 1 = (1 + [gamma]) + 1 = 1 + ([gamma] + 1);
now it is true for _a_ = 1, 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.
2º I say that
_a_ + _b_ = _b_ + _a_.
The theorem has just been demonstrated for _b_ = 1; it can be verified
analytically that if it is true for _b_ = [beta], it will be true for
_b_ = [beta] + 1.
The proposition is therefore established by recurrence.
DEFINITION OF MULTIPLICATION.--We shall define multiplication by the
equalities.
(1) _a_ × 1 = _a_.
(2) _a_ × _b_ = [_a_ × (_b_ - 1)] + _a_.
Like equality (1), equality (2) contains an infinity of definitions;
having defined a × 1, it enables us to define successively: _a_ × 2,
_a_ × 3, etc.
PROPERTIES OF MULTIPLICATION.--_Distributivity._--I say that
(_a_ + _b_) × _c_ = (_a_ × _c_) + (_b_ × _c_).
We verify analytically that the equality is true for _c_ = 1; then
that if the theorem is true for _c_ = [gamma], it will be true
for _c_ = [gamma] + 1.
The proposition is, therefore, demonstrated by recurrence.
_Commutativity._--1º I say that
_a_ × 1 = 1 × _a_.
The theorem is evident for _a_ = 1.
We verify analytically that if it is true for _a_ = [alpha], it will be
true for _a_ = [alpha] + 1.
2º I say that
_a_ × _b_ = _b_ × _a_.
The theorem has just been proven for _b_ = 1. We could verify
analytically that if it is true for _b_ = [beta], it will be true
for _b_ = [beta] + 1.
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 _n_ - 1, it is
true of _n_, and thence conclude that it is true for all the whole
numbers.
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 transformation, which lends
itself to many more differing combinations than does the simple
syllogism; but it is still an instrument purely analytic, and incapable
of teaching us anything new. If mathematics had no other instrument, it
would therefore be forthwith 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.
V
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 ourselves 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.
It is now easy to comprehend why every particular consequence 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 numbers, we only
wish to show it true of the number 6, for example, it will suffice for
us to establish the first 5 syllogisms of our cascade; 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 syllogisms 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 sufficiently
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 impracticable. 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 preponderant rôle, and without it
there would be no science, because there would be nothing general.
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 reasoning 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 contradiction.
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 contradiction
would suffice; it would always allow of our developing as many
syllogisms 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, experience becomes powerless.
This rule, inaccessible to analytic demonstration and to experience, is
the veritable type of the synthetic _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 induction? 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, demonstration 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_ + 1 = 1 + _a_
and afterwards used it to establish the equality
_a_ + _b_ = _b_ + _a_
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 mysterious to us, since we
have ascertained the analogies between demonstration by recurrence and
ordinary induction.
Without doubt recurrent reasoning in mathematics and inductive 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
_a_ + 2 = 2 + _a_
it suffices to twice apply the rule
(1) _a_ + 1 = 1 + _a_
and write
(2) _a_ + 2 = _a_ + 1 + 1 = 1 + _a_ + 1 = 1 + 1 + _a_ = 2 + _a_.
The equality (2) thus deduced in purely analytic way from the equality
(1) is, however, not simply a particular ease 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 'construct'
combinations more and more complicated. Coming back then by the analysis
of these combinations, of these aggregates, so to speak, to their
primitive elements, they perceive the relations 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 evidently the
aggregates can not be regarded as more particular than their elements.
Great importance, and justly, has been attached to this procedure 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?
Why reason on a polygon, for instance, which is always decomposable 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 prolonged
exertions that they could be found by studying directly the relations of
the elementary triangles. The knowledge of the general theorem spares us
these efforts.
A construction, therefore, becomes interesting only when it can be
ranged beside other analogous constructions, forming species 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.
CHAPTER II
MATHEMATICAL MAGNITUDE AND EXPERIENCE
To learn what mathematicians understand by a continuum, 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 importance than the whiteness of the chalk.
The pure analyst has not this rock to fear. He has disengaged the
science of mathematics from all foreign elements, and can answer our
question: 'What exactly is this continuum about which mathematicians
reason?' Many analysts who reflect on their art have answered already;
Monsieur Tannery, for example, in his _Introduction à la théorie 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
conception, wherein is supposed between the elements of the continuum 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 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 necessary 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 property of these steps which is used
in their reasonings[2] is that of being before or after such and such
steps; therefore also this alone should occur in the definition.
[2] With those contained in the special conventions which serve to
define addition and of which we shall speak later.
So how the intermediary terms should be intercalated need not concern
us; on the other hand, no one will doubt the possibility 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 themselves to constructing
this continuous scale of fractional and irrational 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
difficulty, they have chiefly striven to define the incommensurable
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 objects; 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 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 class 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 case, 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 all the others, nor in the second class a number
greater than all the others. Suppose, for example, 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. Evidently 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
sqrt(2) or (2)^{1/2}
is nothing but the symbol of this particular mode of partition of
commensurable numbers; and to each mode of partition corresponds 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, does not the
difficulty begin even for the fractional numbers themselves? 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 example, that a weight
_A_ of 10 grams and a weight _B_ of 11 grams produce identical
sensations, that the weight _B_ is just as indistinguishable 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_ = _C_, _A_ < _C_,
which may be regarded as the formula of the physical continuum.
But here is an intolerable discord with the principle of contradiction,
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 OF THE MATHEMATICAL CONTINUUM.--_First Stage._ So far it would
suffice, 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 supplement the feebleness of our senses,
if, for example, we make use of a microscope? Terms such as _A_ and _B_,
before indistinguishable, appear now distinct; but between _A_ and _B_,
now become distinct, will be intercalated a new term, _D_, that we can
distinguish neither from _A_ nor from _B_. Despite the employment 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 sufficiently powerful to decompose the
physical continuum into discrete elements, 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 aggregate of
whole numbers as similar to the aggregate of even numbers, 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 indefinitely, though we have never had to
count more than a finite number of objects.
Just so, as soon as we have been led to intercalate means between two
consecutive terms of a series, we feel that this operation 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 formation 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 coordinates 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
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 incommensurable.
That would not yet be sufficient, 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 certain breadth;
these bands moreover will encroach one on the other, since there must be
no interval between them. The common 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 rays 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.
_Résumé._--In recapitulation, the mind has the faculty of creating
symbols, and it is thus that it has constructed the mathematical
continuum, which is only a particular system of symbols. Its power is
limited only by the necessity of avoiding all contradiction; 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 intuitive, which are derived from empirical notions
more or less elaborated.
MEASURABLE MAGNITUDE.--The magnitudes we have studied hitherto are not
_measurable_; we can indeed say whether a given 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 terms 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 applicable.
This can only be done by the aid of a new and special _convention_. 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 our 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 convention regarded as equidistant.
This is a way of defining the addition of two magnitudes, because 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:
1º Is the creative power of the mind exhausted by the creation of the
mathematical continuum?
No: the works of Du Bois-Reymond demonstrate it in a striking way.
We know that mathematicians distinguish between infinitesimals of
different orders and that those of the second order are infinitesimal,
not only in an absolute way, but also in relation to those of the first
order. It is not difficult to imagine infinitesimals of fractional or
even of irrational order, and thus we find again that scale of the
mathematical continuum which has been dealt with in the preceding
pages.
Further, 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 + [epsilon], and that however small
[epsilon] may be. Here, then, are new terms intercalated in our series,
and if I may be permitted to revert to the phraseology lately employed
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 infinitesimals 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.
2º Once in possession of the concept of the mathematical continuum, 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 defined as an analytic
continuum of the second order.
Without doubt some artifice analogous to those we have discussed above
would have sufficed to remove the contradiction; but, as this is met
with only in very exceptional cases, it has received no further
attention.
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 OF 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 results of Fechner's experiments; I have
shown that these results are summed up in the contradictory formulas
_A_ = _B_, _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 mathematicians; 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.
That being granted, a system of elements will form a _continuum_ if we
can pass from any one of them to any other, by a series of consecutive
elements such that each is indistinguishable from the preceding. This
_linear_ series is to the _line_ of the mathematician what an isolated
_element_ was to the point.
Before going farther, I must explain what is meant by a _cut_. Consider
a continuum _C_ and remove from it certain of its elements which for an
instant we shall regard as no longer belonging 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 aggregate of the remaining elements ceasing to form a
unique continuum.
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
recognized because it will be impossible to find a linear series of
consecutive elements of _C_, each of these elements indistinguishable
from the preceding, the first being _A_ and the last _B_, _without one
of the elements of this series being indistinguishable from one of the
elements of the cut_.
On the contrary, it may happen that the cut made is insufficient 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 reducing 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 dimensions. If cuts
which are continua of one dimension suffice, we shall say _C_ has two
dimensions; if cuts of two dimensions suffice, 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 OF SEVERAL DIMENSIONS.--Thence the notion of
the mathematical continuum of _n_ dimensions 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 measurement of these magnitudes,
in which it is only a question of knowing, for example, whether on a
curve _ABC_, the point _B_ is between 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 attention of
the greatest geometers and where we see flow one from another a series
of remarkable theorems. What distinguishes 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 continuum 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
Every 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 undemonstrable 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 propositions
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:
1º Through two points can pass only one straight;
2º The straight line is the shortest path from one point to another;
3º 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 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 Académie 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 entitled: _Ueber die
Hypothesen welche der Geometrie zu Grunde 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 GEOMETRY.--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 be 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 difference 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 propositions
have no relation to those of Euclid, but they are not less logically
bound one to another.
RIEMANN'S GEOMETRY.--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. Admit 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 certainly attribute to space only two dimensions.
But suppose now that these imaginary animals, while remaining 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 rôle 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 _unbounded_ 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 rôle 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.
There is a sort of opposition between Riemann's geometry and that of
Lobachevski.
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 Lobachevski;
Greater than two right angles in that of Riemann.
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 Lobachevski.
Add that Riemann's space is finite, although unbounded, in the sense
given above to these two words.
THE SURFACES OF CONSTANT CURVATURE.--One objection still remained
possible. The theorems of Lobachevski and of Riemann present no
contradiction; but however numerous the consequences these two geometers
have drawn from their hypotheses, they must have stopped before
exhausting them, since their number would be infinite; who can say then
that if they had pushed their deductions farther they would not have
eventually reached some contradiction?
This difficulty does not exist for Riemann's geometry, provided it is
limited to two dimensions; in fact, as we have seen, two-dimensional
Riemannian geometry does not differ from spherical geometry, which is
only a branch of ordinary geometry, and consequently is beyond all
discussion.
Beltrami, in correlating likewise Lobachevski's two-dimensional 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 general, this flexible and inextensible figure
can not be displaced without leaving the surface; but there are certain
particular surfaces 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 constant in length. On
the contrary, such a movement would appear absurd to animals without
thickness living on a surface of variable 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 OF NON-EUCLIDEAN GEOMETRIES.--So vanishes the objection
so far as two-dimensional geometries are concerned.
It would be easy to extend Beltrami's reasoning to three-dimensional
geometries. The minds that space of four dimensions does not repel will
see no difficulty 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 languages 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.
Now take Lobachevski's theorems and translate them with the aid of this
dictionary as we translate a German text with the aid of a
German-English dictionary. _We shall thus obtain theorems 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 triangle 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
translations 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 question 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 capable of applications; I have not
the time to speak here of these applications, 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 dictionaries
analogous to the preceding could be constructed, which would enable us
by a simple 'translation' to transform Lobachevski'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 successively abandoned there
are still left over some propositions common 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 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 mathematical entity exists, provided its definition implies no
contradiction, 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 meaning 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 properties 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.
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 enunciated, for they
require the admission that it is possible to transport 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 Riemann.
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 rotation 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 Lobachevski 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 introduced 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.
A theorem of Sophus Lie dominates this whole discussion. It may be thus
enunciated:
Suppose the following premises are admitted:
1º Space has _n_ dimensions;
2º The motion of a rigid figure is possible;
3º It requires _p_ conditions to determine the position of this figure
in space.
_The number of geometries compatible with these premises will be
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.
RIEMANN'S GEOMETRIES.--Yet this result seems contradicted by Riemann,
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. Now,
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 incompatible
with the motion of a rigid figure, which in the theorem of Lie is
supposed possible. These geometries of Riemann, 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 OF 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 proportional to the surface of the triangle; will it not
perhaps become sensible when we shall operate on larger triangles or
when our measurements shall become more precise? The Euclidean geometry
would thus be only a provisional geometry.
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 rôle
in the first chapter:
_If a theorem is true for the number 1, and if it has been proved that
it is true of n + 1 provided it is true of n, it will 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 contradicted 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 geometric figures
behaved like solids. What geometry would borrow 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 geometry 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 _axioms of geometry therefore are neither synthetic_ a priori
_judgments nor experimental facts_.
They are _conventions_; our choice among all possible conventions 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 axioms 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
true?
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 another; it can only be
_more convenient_.
Now, Euclidean geometry is, and will remain, the most convenient:
1º Because it is the simplest; and it is so not only in consequence of
our mental habits, or of I know not what direct intuition 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 signification.
2º Because it accords sufficiently well with the properties of natural
solids, those bodies which our hands and our eyes compare and with which
we make our instruments of measure.
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 transported 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 statement 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:
1º It is continuous;
2º It is infinite;
3º It has three dimensions;
4º It is homogeneous, that is to say, all its points are identical one
with another;
5º 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
visual space is not homogeneous_. All the points of the retina, aside
from the images which may there be formed, do not play the same rôle.
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 impressions, 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 consequently 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
rôle as the other two. What may be called _complete visual space_ is
therefore not an isotropic space.
It has, it is true, precisely three dimensions, which means that the
elements of our visual sensations (those at least which combine to form
the notion of extension) will be completely defined 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'_, which respectively accompany them, will be
equally indistinguishable.
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 independent 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 indications 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 discussion I have given for
sight.
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 muscular sensations.
The corresponding frame constitutes what may be called _motor space_.
Each muscle gives rise to a special sensation capable of augmenting 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 muscles_.
I know it will be said that if the muscular sensations contribute 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 sense of direction, geometric space
would indeed be a form imposed upon our sensibility.
But I perceive nothing at all of this when I analyze my sensations.
What I do see is that the sensations which correspond to movements 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, contrary habits would have
arisen and our muscular sensations would have been associated according
to other laws.
CHARACTERISTICS OF PERCEPTUAL SPACE.--Thus perceptual space, under its
triple form, visual, tactile and motor, is essentially different from
geometric space.
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 what?
Does it mean that we _represent_ to ourselves external objects in
geometric space?
Our representations are only the reproduction of our sensations; 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 perspective, and we can represent 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 mean?
_It simply means that we represent to ourselves the movements it would
be necessary to make to reach that object_; and one may not say that to
represent to oneself these movements, it is necessary 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 sensations which accompany
them and which have no geometric character whatever, which consequently
do not at all imply the preexistence of the notion of space.
CHANGE OF STATE AND CHANGE OF 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 explanation that I am about
to develop.
_None of our sensations, isolated, could have conducted us to the 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 distinction.
At one time we say that the objects which cause these impressions 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 modification in an
aggregate of impressions_.
How then could we have been led to distinguish between the two? 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 voluntary 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:
1º Involuntarily and without experiencing muscular sensations; 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.
It follows from this that sight and touch could not have given us the
notion of space without the aid of the 'muscular sense.'
Not only could this notion not be derived from a single sensation 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 whatever 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 OF COMPENSATION.--How is a like compensation possible, of
such sort that two changes, otherwise independent of each other,
reciprocally correct each other?
A mind already familiar with geometry would reason as follows:
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_ position 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 position, 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, and so must it be with the whole of our
body in the second change which corrects the first.
Under these conditions, compensation may take place.
But we who as yet know nothing of geometry, since for us the notion of
space is not yet formed, we can not reason thus, we can not foresee _a
priori_ whether compensation is possible. But experience teaches us that
it sometimes happens, and it is from this experimental fact that we
start to distinguish changes of state from changes of position.
SOLID BODIES AND GEOMETRY.--Among surrounding objects there are some
which frequently undergo displacements susceptible of being thus
corrected by a correlative movement of our own body; these are the
_solid bodies_. The other objects, whose form is variable, only
exceptionally undergo like displacements (change of position without
change of form). When a body changes its place _and its shape_, we can
no longer, by appropriate movements, bring back our sense-organs into
the same _relative_ situation with regard to this body; consequently we
can no 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 is displaced almost in accordance with the same laws as solid
bodies. Thus we distinguish 'deformations' from other changes of state;
in these deformations, each element undergoes a mere change of position,
which can be corrected, but the modification undergone by the aggregate
is more profound and is no longer susceptible of correction by a
correlative movement.
Such 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 had not already taught us to distinguish changes of
position.
_Therefore, if there were no solid bodies in nature, there would be no
geometry._
Another remark also deserves a moment's attention. Suppose a solid body
to occupy successively the positions [alpha] and [beta]; in its first
position, it will produce on us the totality of impressions _A_, and in
its second position the totality of impressions _B_. Let there be now a
second solid body, having qualities entirely different from the first,
for example, a different color. Suppose it to pass from the position
[alpha], where it gives us the totality of impressions _A'_, to the
position [beta], where it gives the totality of impressions _B'_.
In general, the totality _A_ will have nothing in common with the
totality _A'_, nor the totality _B_ with the totality _B'_. The
transition from the totality _A_ to the totality _B_ and that from the
totality _A'_ to the totality _B'_ are therefore two changes which _in
themselves_ have in general nothing in common.
And yet we regard these two changes both as displacements and,
furthermore, we consider them as the _same_ displacement. How can that
be?
It is simply because they can both be corrected by the _same_
correlative movement of our body.
'Correlative movement' therefore constitutes the _sole connection_
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:
1º We are led at first to distinguish two categories of phenomena:
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;
2º We notice that certain changes of each of these categories may be
corrected by a correlative change of the other category;
3º We distinguish among external changes those which have thus a
correlative in the other category; these we call displacements; 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 OF HOMOGENEITY.--The first of these laws is the law of homogeneity.
Suppose that, by an external change [alpha], we pass from the totality
of impressions _A_ to the totality _B_, then that this change [alpha] is
corrected by a correlative voluntary movement [beta], so that we are
brought back to the totality _A_.
Suppose now that another external change [alpha]' makes us pass anew
from the totality _A_ to the totality _B_.
Experience teaches us that this change [alpha]' is, like [alpha],
susceptible of being corrected by a correlative voluntary movement
[beta]' and that this movement [beta]' corresponds to the same muscular
sensations as the movement [beta] which corrected [alpha].
This fact is usually enunciated by saying that _space is homogeneous and
isotropic_.
It may also be said that a movement which has once been produced 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 mathematical
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 power;
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-EUCLIDEAN WORLD.--If geometric space were a frame imposed on
_each_ of our representations, considered individually, 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 case; geometry is only the résumé of the laws
according to which these images succeed each other. Nothing then
prevents us from imagining a series of representations, similar in all
points to our ordinary representations, but succeeding one another
according to laws different from those to which we are accustomed.
We can conceive then that beings who received their education 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.
To specify still more precisely the law in accordance with which this
temperature varies: Let _R_ be the radius of the limiting sphere;
let _r_ be the distance of the point considered from the center
of this sphere. The absolute temperature shall be proportional
to _R_^{2} - _r_^{2}.
I shall further suppose that, in this world, all bodies have the same
coefficient 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 proportion 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 by the differences of
temperature_ just spoken of.
No doubt, in our world, natural solids likewise undergo variations 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 variations 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 _R_^{2} - _r_^{2}. It is easy to 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 _corrected_ by correlative
movements of the sentient beings inhabiting 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 deformation, 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 _non-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 correlative voluntary movement.
If they construct a geometry, it will not be, as ours is, the study of
the movements of our rigid solids; it will be the study of the changes
of position which they will thus have distinguished and which are none
other than the 'non-Euclidean displacements'; _it will be non-Euclidean
geometry_.
Thus beings like ourselves, educated in such a world, would not have the
same geometry as ours.
THE WORLD OF FOUR DIMENSIONS.--We can represent to ourselves 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 suffice 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 various
perspectives of the same body, taken from different points of view.
At the same time, we find that the transition from one perspective 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 dimensions 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.
We can easily represent to ourselves these perspectives, since they are
of only three dimensions.
Imagine that the various perspectives of the same object succeed 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 operations combine
according to any law we choose, for example, so as to form a group with
the same structure as that of the movements 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 dimensions. In
this sense we may say the fourth dimension is imaginable.
CONCLUSIONS.--We see that experience plays an indispensable rôle 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 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 ordinary 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 GEOMETRY
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
[pi], what shall we have done? We shall have made an experiment on the
properties of the matter with which we constructed this _round thing_,
and of that of which the measure used was made.
3. GEOMETRY AND ASTRONOMY.--The question has also been put in another
way. If Lobachevski's geometry is true, the parallax 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 observations 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 suppose 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, so that
experience, in establishing these phenomena, would directly contradict
the non-Euclidean hypothesis? 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 experience, in ascertaining the existence of these
lengths, would directly contradict the hypothesis that there are fathoms
divided into six feet?
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 suppose 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 Lobachevski. It would be ascertained
that a definite concrete object, accessible to experiment, for example,
a pencil of rays of light, possesses the property _A_; we should
conclude that it is rectilinear, and then investigate whether or not it
has the property _B_.
But _this is not so_; no property exists which, like this property _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 property: the
straight line is a line such that a figure of which this line forms a
part can be moved without the mutual distances of its points varying and
so that all points of this line remain fixed"?
This, in fact, is a property which, in Euclidean or non-Euclidean space,
belongs to the straight and belongs only to it. But how shall we
ascertain experimentally whether it belongs to this or that concrete
object? It will be necessary to measure distances, and how shall one
know that any concrete magnitude which I have measured with my material
instrument really represents the abstract distance?
We have only pushed back the difficulty.
In reality the property just enunciated is not a property of the
straight line alone, it is a property of the straight line and
distance. For it to serve as absolute criterion, we should have to be
able to establish not only that it does 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 magnitude other than
distance. Now this 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 postulate; nor,
on the other hand, will any experience ever contradict 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?
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 shall, 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 orientation in space.
The laws of the phenomena which will happen in this system will depend
on the state of these bodies and their mutual distances; but, because of
the relativity and passivity of space, they will not depend on the
absolute position and orientation of the system.
In other words, the state of the bodies and their mutual distances at
any instant will depend solely on the state of these same bodies and on
their mutual distances at the initial instant, but will not at all
depend on the absolute initial position of the system or on its absolute
initial orientation. This is what for brevity I shall call the _law of
relativity_.
Hitherto I have spoken as a Euclidean geometer. As I have said, an
experience, whatever it be, admits of an interpretation on the Euclidean
hypothesis; but it admits of one equally on the non-Euclidean
hypothesis. Well, we have made a series of experiments; we have
interpreted them on the Euclidean hypothesis, and we have recognized
that these experiments thus interpreted 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 experience had 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 consequently augment or diminish, which
might modify the laws of the phenomena happening there.
But if our system is the entire universe, experience is powerless to
give information about its absolute position and orientation in space.
All that our instruments, however perfected they may be, can tell us
will be the state of the various parts of the universe 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 interpretation of
experimental facts. If the law is true in the Euclidean interpretation,
it will also be true in the non-Euclidean interpretation.
Allow me here a short digression. I have spoken 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; 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 and 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 experiments, 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 Foucault's pendulum. The absolute rotation
of this planet 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
difficulty.
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 myself with it, and
have only mentioned it incidentally.
What is important is the conclusion: experiment can not decide 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 insufficient, because 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 suffice for calculating 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 bearing only on your bits of wood can reveal to you
nothing except concerning these bits of wood. Just so your experiments,
however 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.
Still 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.
I suppose you will grant me these are not geometric properties; at least
I am sure you will grant me these properties are independent 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_, _OG_, _OH_,
united at one of their extremities _O_. Let us besides have a second
solid body, for example a bit of wood, to be marked with three little
flecks of ink which I shall call [alpha], [beta], [gamma]. I further
suppose it ascertained that [alpha][beta][gamma] may be brought into
contact with _AGO_ (I mean [alpha] with _A_, and at the same time [beta]
with _G_ and [gamma] with _O_), then that we may bring successively into
contact [alpha][beta][gamma] with _BGO_, _CGO_, _DGO_, _EGO_, _FGO_,
then with _AHO_, _BHO_, _CHO_, _DHO_, _EHO_, _FHO_, then [alpha][gamma]
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 Lobachevski's geometry). 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 [alpha][beta][gamma] was a rigid solid of
our ordinary geometry presenting the form of a right-angled triangle,
and if the points _ABCDEFGH_ 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 _G_ and the other _H_.
Suppose now that in place of the preceding determination it is observed
that as above [alpha][beta][gamma] can be successively applied to _AGO_,
_BGO_, _CGO_, _DGO_, _EGO_, _AHO_, _BHO_, _CHO_, _DHO_, _EHO_, _FHO_,
then that [alpha][beta] (and no longer [alpha][gamma]) can be
successively applied to _AB_, _BC_, _CD_, _DE_, _EF_ and _FA_.
These are determinations which could be made if non-Euclidean geometry
were true, if the bodies [alpha][beta][gamma] and _OABCDEFGH_ were rigid
solids, and if the first were a right-angled triangle and the second a
double regular hexagonal pyramid of suitable 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 Lobachevskian group. They would
suffice, therefore (if one made them), to prove that the bodies in
question do not move according to the Euclidean group.
Thus, without making any hypothesis about form, about the nature of
space, about the relations of bodies to space, and without 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 _R_ 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 which 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 _OABCDEFGH_ and _O'A'B'C'D'E'F'G'H'_ and two
triangles [alpha][beta][gamma] and [alpha]'[beta]'[gamma]'. The first
double pyramid might be rectilinear and the second curvilinear; the
triangle [alpha][beta][gamma] 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 [alpha][beta][gamma], and the second with
the double pyramid _O'A'H'_ and the triangle [alpha]'[beta]'[gamma]'.
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, but 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 Métaphysique et
de Morale_ and in _The Monist_. When we say space has three dimensions,
what do we mean?
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 muscular sensations, and this series
_S_ will define _B_. Observe, however, that we shall often regard two
series _S_ and _S'_ as defining the same attitude _B_ (since the initial
and final attitudes _A_ and _B_ remaining the same, the intermediary
attitudes and the corresponding 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 compensate an external change, and
which we have called 'displacements.' 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 that is an
illusion. What we see when we try to represent to ourselves 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? Or further, what criterion will enable me
to apprehend this?
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 enunciating 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 determines 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? 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. 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 corresponding 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 characteristics 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 compare natural
phenomena? And, this group chosen, which of its sub-groups shall we take
to characterize a point of space? Experience has guided us by showing us
which choice best adapts itself to the properties of our body. But its
rôle 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 mean? Is it meant that we could not experimentally demonstrate
Euclid's postulate, but that our ancestors have been able to do it? 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 advantageous_ 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 long? 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
difficulty 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 convention, what is
hypothesis.
That is not all:
1º 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.
2º 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.
3º Not only have we no direct intuition of the equality of two
durations, but we have not even direct intuition of the simultaneity of
two events occurring in different places: this I have explained in an
article entitled _La mesure du temps_.[3]
[3] _Revue de Métaphysique et de Morale_, t. VI., pp. 1-13
(January, 1898).
4º 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 _Leçons de
mécanique 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 Euclidean
geometry.
THE PRINCIPLE OF 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 mind? If it were so, 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 curvature of its
trajectory can not change, if no external cause intervenes to modify
them?
Is the principle of inertia, which is not an _a priori_ truth, therefore
an experimental fact? But has any one ever experimented on bodies
withdrawn from the action of every force? and, if so, how was it known
that these bodies were subjected to no force? The example ordinarily
cited is that of a ball rolling a very long time on a marble table; but
why do we say it is subjected to no force? Is this because it is too
remote from all other bodies to experience any appreciable action from
them? 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 enunciation:
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 principle of sufficient
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.
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 accelerations.
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 inhabiting 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 determine 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 constellations. 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
cataclysm, 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
experiment, or can it be? When Newton wrote the _Principia_ he quite
regarded this truth as experimentally acquired and demonstrated. It was
so in his eyes, not only through the anthropomorphism 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 probability 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 verified 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? 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. Why? 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 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 derogation 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 observe _n_ molecules and
ascertain that their 3_n_ coordinates satisfy a system of 3_n_
differential equations of the fourth order (and not of the second order
as the law of inertia would require). We know that by introducing 3_n_
auxiliary variables, a system of 3_n_ equations of the fourth order can
be reduced to a system of 6_n_ equations of the second order. If then we
suppose these 3_n_ auxiliary variables represent the coordinates of _n_
invisible molecules, 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 OF 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 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
difficulty arising from the measurement of time. But how measure force,
or mass? 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_?
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?
These difficulties 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 absolutely sterile.
For a definition to be of any use, it must teach us to _measure_ force;
moreover that suffices; 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 produce 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 suppose 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 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? 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 evidently apply to the force of the resiliency of a
dynamometer, which the temperature and a multitude of circumstances may
cause to vary.
This is not all; we can not say the weight of the body _P_ may be
applied to the body _C_ and directly balance the force _F_. 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 _R_ 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 _R_, in virtue of the principle of the
equality of action and reaction; lastly, the force _R_ is equal to the
weight of _P_, since it balances it. It is from these three equalities
we deduce as consequence the equality of _F_ 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
insufficient; 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 definition_.
We are therefore reduced to Kirchhoff's definition; _force is equal to
the mass multiplied 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, for we do not
know what mass is. It enables us doubtless to calculate the relation of
two forces applied to the same body at different 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 reaction, 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 _A_; 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_^{2}.
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 theoretically
indispensable. It might very well have been that attraction was
inversely proportional to the square of the distance without being
proportional to the product of the masses, that it was equal
to _f_/_r_^{2}, but without our having _f_ = _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? Is this
hypothesis rigorously exact? Is it certain it will never be contradicted
by experiment? Who would dare affirm that? 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 inapplicable.
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 system isolated
from all external action will be null. Or, in other words, _the motion
of the center of gravity of this system will be 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 attributed 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.
Therefore nothing remains and our efforts have been fruitless; we are
driven to the following definition, which is only an avowal of
powerlessness: _masses are 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 principles 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 gravity, 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 mechanics are
rigorously true. "In the opinion of many physicists," 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 experimental
truths; but we have been obliged to use them as definitions. It is _by
definition_ that force is equal to the product of mass by acceleration;
here, then, is a principle which is henceforth 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 absolutely
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 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 uniform, 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 us? 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 will never be able to contradict
them._
ANTHROPOMORPHIC MECHANICS.--"Kirchhoff," it will be said, "has only
acted in obedience to the general tendency of mathematicians 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, irreducible, indefinable; we all know what 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 sensations of cold or sensations of warmth,
could read a thermometer just as well as any one else, and that would
suffice him for constructing the whole theory of heat.
Now this immediate notion of effort is of no use to us for measuring
force; it is clear, for instance, that I should feel more fatigue in
lifting a weight of fifty kilos than a man accustomed 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 remembrance 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 rôle 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 OF THE THREAD.'--M. Andrade, in his _Leçons de mécanique
physique_, has rejuvenated anthropomorphic mechanics. 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 [alpha]; 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 [alpha]; 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
[beta]. The four observed accelerations should be proportional. We have
thus an experimental verification of the law of acceleration above
enunciated.
Or still better, a body is submitted to the simultaneous action 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 experimental 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 undergone by this thread,
which is reasonable enough; we have further assumed that if a body is
attached to this thread, the effort transmitted 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 identical
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 détour? 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 question, 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?
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.
Andrade's ideas are nevertheless very interesting; if they 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 naïve
anthropomorphism to the present conceptions of science.
We see at the start a very particular and in sum rather crude
experiment; at the finish, a law perfectly general, perfectly precise,
the certainty of which we regard as absolute. This certainty 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? Conventions, yes; arbitrary, no; they would
be if we lost sight of the experiments which led the creators of the
science to adopt them, and which, imperfect 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
RELATIVE MOTION AND ABSOLUTE MOTION
THE PRINCIPLE OF RELATIVE MOTION.--The attempt has sometimes 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 regulations for the
degree B. ès 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 furnished us the definition we lacked.
The principle of relative motion is none the less highly interesting 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 velocities and
positions, provided the movable axes to which the relative motion is
referred move uniformly in a straight line. Or, if we prefer, their
accelerations depend only on the differences of 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 mathematical 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?
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 question 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 principle of relative motion was
for us not solely a result of experiment and that _a priori_ every
contrary hypothesis would be repugnant 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 without being uniform; the paradox does not bear a
moment's examination. If I am on board, and if the train, striking any
obstacle, 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 relative 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 pendulum 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?
But it does not suffice 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 intelligently. 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 regarding 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 the various parts of the system, as real
attractions do, the other on their relative velocities, as real
frictions do.
Many difficulties, 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 difficulty would seem to them already sufficiently 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 symmetry; they would have to distinguish 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 difficulty doubtless, they would
invent something which would be no more extraordinary than the glass
spheres of Ptolemy, and so it would go on, complications 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 astronomy 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 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
affirmation: 'the earth turns round' has no meaning, 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 contradiction; 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 these 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 likewise by equations
of the second order, it seems that the mind ought to be entirely
satisfied. In what measure 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
system analogous to our solar system, but where one can not perceive
fixed stars foreign to this system, so that astronomers 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 of 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 derivatives with respect to the time, that would not suffice to
determine the values of these same distances at a subsequent instant.
There would still be lacking one datum, and this datum might be for
instance what astronomers call the area-constant.
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 hand, 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 constant, 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 beginning was that their system should appear as isolated, and
that they should observe no star foreign to it. Under these conditions,
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.
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 certain 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 calculate 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 satisfied? 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 polyhedron or of the symmetric polyhedron, and one would escape
from this consequence only by regarding the area-constant as accidental.
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.
The values of the distances at any instant depend upon their initial
values, upon those of their first derivatives and also upon something
else. What is this _something else_?
If we will not admit that this may be simply one of the second
derivatives, we have only the choice of hypotheses. Either it may be
supposed, as is ordinarily done, that this something else is the
absolute orientation of the universe in space, or the rapidity with
which this orientation varies; and this supposition may be correct; it
is certainly the most convenient solution for geometry; it is not the
most satisfactory for the philosopher, because this orientation does not
exist.
Or it may be supposed that this something else is the position or the
velocity of some invisible body; this has been done by certain persons
who have even called it the body alpha, although we are doomed never to
know anything of this body but its name. This is an artifice entirely
analogous to that of which I spoke at the end of the paragraph devoted
to my reflections on the principle of inertia.
But, after all, the difficulty is artificial. Provided the future
indications of our instruments can depend only on the indications they
have given us or would have given us formerly, this is all that is
necessary. Now as to this we may rest easy.
CHAPTER VIII
ENERGY AND THERMODYNAMICS
ENERGETICS.--The difficulties inherent in the classic mechanics 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 fundamental rôle in
this theory. They are _kinetic energy_, or _vis viva_, and _potential
energy_.
All the changes which bodies in nature can undergo are regulated by two
experimental laws:
1º The sum of kinetic energy and potential energy is constant. This is
the principle of the conservation of energy.
2º If a system of bodies is at _A_ at the time t_{0} and at _B_ at the
time t_{1}, 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 t_{0}
and t_{1}, may be as 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:
1º 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:
2º 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 difficulties
almost as great as those of force and mass in the first system. 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 depending 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 quantity, accessible to
experiment, must remain constant. This quantity is the sum of two terms;
the first depends only on the position 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 if _T_ + _U_ is a constant, so is any function of _T_ +
_U_,
{Phi}(_T_ + _U_).
But this function {Phi}(_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_ + _U_ (or a linear
function of _T_ + _U_, which comes to the same thing, since this linear
function may always be reduced to _T_ + _U_ 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 coefficients of this expression will give us
the masses.
Thus, in this simple case, the fundamental ideas may be defined without
difficulty. 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 velocity and their acceleration. If material
points should attract each other according to an analogous law, _U_
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 _U_? Consequently, how
distinguish the two parts of energy?
But still more; how define energy itself? We no longer have any reason
to take as definition _T_ + _U_ rather than any other function of _T_ +
_U_, when the property which characterized _T_ + _U_ 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 principle of the
conservation of energy should be written:
_T_ + _U_ + _Q_ = const.
where _T_ would represent the sensible kinetic energy, _U_ the potential
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, _U_ independent of
these velocities and of the state of the bodies, _Q_ independent 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 electrostatic
energy due to their mutual action will evidently depend upon their
charge, that is to say, on their state; but it will equally depend upon
their position. If these bodies are in motion, they will act one upon
another electrodynamically and the electrodynamic 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 separation 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 [phi](_T_ + _U_ +
_Q_).
If _T_ + _U_ + _Q_ were of the particular form I have above considered,
no ambiguity would result; among the functions [phi](_T_ + _U_ + _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? 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 energy: _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 insufficient.
Another objection seems to me still more grave: the principle 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 principle 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.
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 statement presents the molecule to
us, so to speak, as a living and free being. Clearly it would be better
to replace it by an enunciation less objectionable, and where, as the
philosophers would say, final causes would not seem to be substituted
for efficient causes.
THERMODYNAMICS.[4]--The rôle of the two fundamental principles of
thermodynamics in all branches of natural philosophy becomes daily more
important. Abandoning the ambitious theories 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 time? No one doubts it; but
whence comes this confidence?
[4] The following lines are a partial reproduction of the preface
of my book _Thermodynamique_.
An eminent physicist said to me one day _à propos_ of the law of errors:
"All the world believes it firmly, because the mathematicians imagine
that it is a fact of observation, and the observers 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 question, 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 generalization 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 Newton'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 laws? 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 perpetual
motion; of course we are on our guard at such a prospect, and we think
ourselves less rash in affirming Mayer's principle than in denying it.
That is perhaps not wholly accurate; the impossibility of perpetual
motion implies the conservation of energy only for reversible phenomena.
The imposing simplicity of Mayer's principle likewise contributes to
strengthen our faith. In a law deduced immediately from experiment, like
Mariotte's, this simplicity would rather seem to us a reason for
distrust; but here this is no longer the case; we see elements, at first
sight disparate, arrange themselves 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 impossible 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? In the determinist hypothesis, the state
of the universe is determined by an extremely great number _n_ of
parameters which I shall call x_{1}, x_{2},... x_{_n_}. 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_{1}, x_{2},... x_{_n_}, 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.
To simplify the enunciation, suppose that the sum of the work of the
external forces is null, as well as that of the quantities 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 vanishing 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? 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 influence 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 restriction, 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 conservation of
energy can have only one signification, which is that there is a
property common 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, even 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 approximations, and it is hoped to replace them little
by little by laws more and more precise. If, on the other hand, the
principle of Clausius reduces to an inequality, this is not caused by
the imperfection of our means of observation, but by the very nature of
the question.
GENERAL CONCLUSIONS ON PART THIRD
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 concerns almost isolated
systems. On the other hand, they are postulates applicable to the
totality of the universe and regarded as rigorously true.
If these postulates possess a generality and a certainty which are
lacking to the experimental verities whence they are drawn, this is
because they reduce in the last analysis to a mere convention which we
have the right to make, because we are certain beforehand that no
experiment can ever contradict it.
This convention, however, is not absolutely arbitrary; it does not
spring from our caprice; we adopt it because certain experiments have
shown us that it would be convenient.
Thus is explained how experiment could make the principles of mechanics,
and yet why it can not overturn them.
Compare with geometry: The fundamental propositions of geometry, as for
instance Euclid's postulate, are nothing more than conventions, and it
is just as unreasonable to inquire whether they are true or false as to
ask whether the metric system is true or false.
Only, these conventions are convenient, and it is certain experiments
which have taught us that.
At first blush, the analogy is complete; the rôle of experiment seems
the same. One will therefore be tempted to say: Either mechanics must be
regarded as an experimental science, and then the same must hold for
geometry; or else, on the contrary, geometry is a deductive science, and
then one may say as much of mechanics.
Such a conclusion would be illegitimate. The experiments which have led
us to adopt as more convenient the fundamental conventions of geometry
bear on objects which have nothing in common with those geometry
studies; they bear on the properties of solid bodies, on the rectilinear
propagation of light. They are experiments of mechanics, experiments of
optics; they can not in any way be regarded as experiments of geometry.
And even the principal reason why our geometry seems convenient to us is
that the different parts of our body, our eye, our limbs, have the
properties of solid bodies. On this account, our fundamental experiments
are preeminently physiological experiments, which bear, not on space
which is the object the geometer must study, 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 mechanics 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 compared to that superb body of doctrine called
geometry?
One sees now why the teaching of mechanics should remain experimental.
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 objectivity. 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 attributes an absolute value.
Some philosophers have generalized too far; they believed the principles
were the whole science and consequently that the whole science was
conventional.
This paradoxical doctrine, called nominalism, will not bear
examination.
How can a law become a principle? It expressed a relation between two
real terms _A_ and _B_. But it was not rigorously true, it was only
approximate. We introduce arbitrarily an intermediary 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 principle which
expresses the relation of _A_ to _C_ and an experimental law,
approximate and subject to revision, which expresses the relation of _C_
to _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