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gift of Mrs. Clarence I. Lewis
STANFORD UNIVERSITY LIBRARIES
SCIENCE AND EDUCATION
A SERIES OF VOLUBIES FOR THE PROMOTION OF SCIENTIFIC RESEARCH AND EDUCATIONAL PROGRESS
Edited bt J. McEEEN CATTELL
VOLUME I— THE FOUNDATIONS OF SCIENCE
UNDER THE SAME EDITORSHIP
SCISNCE AND EDUCATION. A series of volumes for the promotion of scientific research and educational progress.
Volume I. The FonndationB of Science. By H. PoincarA. Ck>ntaining the authorised English translation by George Bruce Halsted of "Science and Hypothesis," "The Value of Science," and "Science and Method."
Volume n. Medical Research and Education. By Richard Mills Pearce, William H. Welch, W. H. Howell, Franklin P. Mall, Lewellys F. Barker, Charles S. Minot, W. B. Cannon, W. T. Council- man, Theobald Smith, G. N. Stewart, C. M. Jack- son, E. P. Lyon, James B. Herrick, John M. Dod- son, C. R. Bardeen, W. Ophtds, S. J. Meltier, James Ewing, W. W. Keen, Henry H. Donaldson, Christ- ian A. Herter, and Henry P. Bowditch.
Volume m. UniTenity Control. By J. McKbbn Cattbll and other authors.
AMERICAN MEN OF SCIENCE. A Biographical Directory.
SCISNCE. A weekly journal devoted to the advancement of science. The official organ of the American Asso- ciation for the Advancement of Science.
THE POPULAR SCISNCE MONTHLY. A monthly magasine devoted to the diffusion of science.
THE AMERICAN NATURALIST. A monthly journal devoted to the biological sciences, with spedid refer- ence to the factors of evolution.
THE SCIENCE PRESS
HBW TORK OARRISOIT, IT. T.
THE FOUNDATIONS OF SCIENCE
SCIENCE AND HYPOTHESIS THE VALUE OF SCIENCE SCIENCE AND METHOD
BT
H. POINCARE
AUTHOBIZED TBANSLA.TION BT
GEORGE BRUCE HALSTED
WITH A SPECIAL PBEFACB BT POINCAB^, AND AN INTRODUCTION
BT JOSIAH BOTCE, HABTABD TTNITEItSITT
THE SCIENCE PRESS
NEW YORK AND GARRISON, N. Y.
1913
Ck)pyright, 1913 Bt The Sgebngob Pbbsb
MKSOF
TNI NEW IRA PRINTINQ OOMMNY
LANCAtTtR« PA.
■H'
CONTENTS
PAOX
Henri Poincard zi
Author 'b Preface to the Translation 3
SCIENCE AND HYPOTHESIS
Introduction hj Bojee 9
Introduction 27
Past I. Number and Magnitude
Chapter I. — On the Nature of Mathematical Beasoning 31
Sjllogistic Deduction 31
Verification and Proof 32
Elements of Arithmetic 33
Reasoning hj Becurrence 37
Induction ... * 40
Mathematical Construction 41
Chapter II. — ^Mathematical Magnitude and Experience 43
Definition of Incommensurables 44
The Physical Continuum 46
Creation of the Mathematical Continuum 46
Measurable Magnitude 49
Various Bemarks (Curves without Tangents) 50
The Physical Continuum of Several Dimensions 52
The Mathematical Continuum of Several Dimensions 53
Part II. Space
Chapter HE. — The Non-Euclidean Geometries 55
The Bolyai-Lobachevski Geometry 56
Riemann 's Geometry 57
The Surfaces of Constant Curvature 58
Interpretation of Non-Euclidean Geometries 59
The Implicit Axioms 60
The Fourth Geometry 62
Lie's Theorem 62
Biemann 's Geometries 63
On the Nature of Axioms 63
Chapter IV. — Space and Geometry 66
Geometric Space and Perceptual Space 66
Visual Space 67
Tactile Space and Motor Space 68
Characteristics of Perceptual Space 69
Change of State and Change of Position 70
Conditions of Compensation 72
V
vi CONTENTS
Solid Bodies and (Geometry 72
Law of Homogeneity 74
The Non-Euclidean World 75
The World of Pour Dimenaiona 78
Conclusions 79
Chaptbb V. — ^Experience and Geometry 81
Geometry and Aatronomy 81
The Law of Belativity 83
Bearing of Experiments 86
Supplement (What is a Pointf ) 89
Ancestral Experience 91
Pabt m. Force
CHAPm VI. — The Classic Mechanics 92
The Principle of Inertia 93
The Law of Acceleration 97
Anthropomorphic Mechanics 103
The School of the Thread 104
Ohaptbr YII. — ^Belatiye Motion and Absolute Motion 107
The Principle of Belative Motion 107
Newton 's Argument 108
Chapter VIII. — ^Energy and Thermodynamics 115
Energetics 115
Thermodynamics 119
General Conclusions on Part HI 123
Past IV. Natwre
Chapthi IX.— Hypotheses in Physics 127
The Bdle of Experiment and (Generalization 127
The Unity of Nature 130
The Bdle of Hypothesis 133
Origin of Mathematical Physics 136
Chapter X. — The Theories of Modem Physics 140
Meaning of Physical Theories 140
Physics and Mechanism 144
Present State of the Science 148
Chapter XI. — The Calculus of Probabilities 155
Classification of the Problems of Probability 158
Probability in Mathematics 161
Probability in the Physical Sciences 164
Bouge et noir 167
The Probability of Causes 169
The Theory of Errors 170
Conclusions 172
Chapter XII. — Optics and Electricity 174
Fresnel 's Theory 174
Maxwell's Theory 175
The Mechanical Explanation of Physical Phenomena 177
CONTENTS vii
Xm.— Electrodynamics 184
Ampere's Theory 184
Closed Currents 185
Action of a Closed Current on a Portion of Current 186
Continuous Botations 187
Mutual Action of Two Open Currents 189
Induction 190
Theory of Helmholtz 191
Difficulties Baised by these Theories 193
Maxwell's Theory 193
Bowland 's Experiment 194
The Theory of Lorentz 196
THE VALUE OP SCIENCE
Translator 's Introduction 201
Does the Scientist Create Sciencef 201
The Mind Dispelling Optical Illusions 202
Euclid not Necessary 202
Without Hypotheses, no Science 203
What Outcomef 203
Introduction 205
Past I. The Mathematical Sciences
Chaptkb I. — ^Intuition and Logic in Mathematics 210
Crafteb, II.— -The Measure of Time 223
Chapter III.— The Notion of Space 235
Qualitative Geometry 238
The Physical Continuum of Several Dimensions 240
The Notion of Point 244
The Notion of Displacement 247
Visual Space 252
Chaptib IV. — Space and its Three Dimensions 256
The Group of Displacements 256
Identity of Two Points 259
Tactile Space 264
Identity of the Different Spaces 268
Space and Empiricism 271
B6le of the Semicircular Canals 276
Paet II. TTw Physical Sciences
Chaptee. V.^Analysis and Physics 279
Chapter VI. — ^Astronomy 289
Chapter VII. — The History of Mathematical Physics 297
The Physics of Central Forces 297
The Physics of the Principles 299
Chapter Vin. — ^The Present Crisis in Physics 303
The New Crisis 303
Camot's Principle 303
viii CONTENTS
The Principle of Eelativity 305
Newton's Principle 308
Lavoisier 's Principle 310
Majer 's Principle 312
Chapter IX. — The Future of Mathematical Physics 314
The Principles and Experiment 314
The BMe of the Analyst 314
Aberration and Astronomy 315
Electrons and Spectra 316
Conventions preceding Experiment 317
Futare Mathematical Physics 319
Part III. The Objective Value of Science
Chapter X. — Is Science Artificialf 321
The Philosophy of LeBoy 321
Science, Bule of Action 323
The Crude Fact and the Scientific Fact 325
Nominalism and the Universal Invariant 333
Chapter XI. — Science and Reality 340
Contingence and Determinism 340
Objectivity of Science 347
The notation of the Earth 353
Science for Its Own Sake 354
SCIENCE AND METHOD
Introduction 359
Book I. Science and the Scientist
Chapter I. — The Choice of Facts 362
Chapter II. — The Future of Mathematics 369
Chapter III. — ^Mathematical Creation 383
Chapter IV. — Chance 395
Book n. Maihematicdl Seasoning
Chapter I.— The Belativity of Space 413
Chapter II. — ^Mathematical Definitions and Teaching 430
Chapter III. — ^Mathematics and Logic 448
Chapter IV. — The New Logics 460
Chapter V. — The Latest Efforts of the Logisticians 472
Book III. The New Mechanics
Chapter I. — Mechanics and Badium 486
Chapter II. — ^Mechanics and Optics 496
Chapter HJ. — The New Mechanics and Astronomy 515
Book IV. Astronomic Science
Chapter I. — The Milky Way and the Theory of Gases 522
Chapter I. — ^French Geodesy 535
General Conclusions 544
Index 547
HENRI POINCARE
Sm George Darwin, worthy son of an immortal father, said, referring to what Poincar^ was to him and to his work: **He must be regarded as the presiding genius — or, shall I say, my patron saint t"
Henri Poincar6 was born April 29, 1854, at Nancy, where his father was a physician highly respected. His schooling was broken into by the war of 1870-71, to get news of which he learned to read the German newspapers. He outclassed the other boys of his age in all subjects and in 1873 passed highest into the Ecole Polytechnique, where, like John Bolyai at Maros Y&s4rhely, he followed the courses in mathematics without taking a note and without the syllabus. He proceeded in 1875 to the School of Mines, and was Nomme, March 26, 1879. But he won his doctorate in the University of Paris, August 1, 1879, and was appointed to teach in the Faculty des Sciences de Caen, December 1, 1879, whence he was quickly called to the Uni- versity of Paris, teaching there from October 21, 1881, until his death, July 17, 1912. So it is an error to say he started as an engineer. At the early age of thirty-two he became a member of TAcad^mie des Sciences, and, March 5, 1908, was chosen Membre de TAcademie Frangaise. July 1, 1909, the number of his writings was 436.
His earliest publication was in 1878, and was not important. Afterward came an essay submitted in competition for the Grand Prix offered in 1880, but it did not win. Suddenly there came a change, a striking fire, a bursting forth, in February, 1881, and Poincare tells us the very minute it happened. Mount- ing an omnibus, **at the moment when I put my foot upon the step, the idea came to me, without anything in my previous thoughts seeming to foreshadow it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.'' Thereby was opened a perspec- tive new and immense. Moreover, the magic wand of his whole
ix
X THE FOUNDATIONS OF SCIENCE
life-work had been grasped, the Aladdin's lamp had been rubbed, non-Euclidean geometry, whose necromancy was to open up a new theory of our universe, whose brilliant exposition was com- menced in his book Science and Hypothesis, which has been translated into six languages and has already had a circulation of over 20,000. The non-Euclidean notion is that of the possi- bility of alternative laws of nature, which in the Introduction to the Electridte et Optique, 1901, is thus put: ''If therefore a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of others which will account equally well for all the peculiarities disclosed by experiment."
The scheme of laws of nature so largely due to Newton is merely one of an infinite number of conceivable rational schemes for helping us master and make experience; it is commode, con- venient; but perhaps another may be vastly more advantageous. The old conception of true has been revised. The first expres- sion of the new idea occurs on the title page of John Bolyai's marvelous Science Absolute of Space, in the phrase **haud un- quam a priori decidenda."
With bearing on the history of the earth and moon system and the origin of double stars, in formulating the geometric criterion of stability, Poincar^ proved the existence of a previously un- known pear-shaped figure, with the possibility that the progres- sive deformation of this figure with increasing angular velocity might result in the breaking up of the rotating body into two detached masses. Of his treatise Les Methodes nouvelles de la Mechanique celeste. Sir George Darwin says: **It is probable that for half a century to come it will be the mine from wh^ch humbler investigators will excavate their materials." Brilliant was his appreciation of Poincar6 in presenting the gold medal of the Royal Astronomical Society. The three others most akin in genius are linked with him by the Sylvester medal of the Royal Society, the Lobachevski medal of the Physico-Mathematical Society of Kazan, and the Bolyai prize of the Hungarian Acad- emy of Sciences. His work must be reckoned with the greatest mathematical achievements of mankind.
The kernel of Poincar6's power lies in an oracle Sylvester often quoted to me as from Hesiod : The whole is less than its part.
HENBI POINCABE xi
He penetrates at once the divine simplicity of the perfectly general case, and thence descends, as from Olympus, to the special concrete earthly particulars.
A combination of seemingly extremely simple analytic and geometric concepts gave necessary general conclusions of im- mense scope from which sprang a disconcerting wilderness of possible deductions. And so he leaves a noble, fruitful heritage.
Says Love: ''His right is recognized now, and it is not likely that future generations will revise the judgment, to rank among the greatest mathematicians of all time."
Geobgb Bruce Halsted.
SCIENCE AND HYPOTHESIS
I
AUTHOR'S PREFACE TO THE
TRANSLATION
I AM exceedingly grateful to Dr. Halsted, who has been so good as to present my book to American readers in a translation, clear and faithful.
Every one knows that this savant has already taken the trouble to translate many European treatises and thus has powerfully contributed to make the new continent understand the thought of the old.
Some people love to repeat that Anglo-Saxons have not the same way of thinking as the Latins or as the Germans ; that they have quite another way of understanding mathematics or of un- derstanding physics ; that this way seems to them superior to all others ; that they feel no need of changing it, nor even of know- ing the ways of other peoples.
In that they would beyond question be wrong, but I do not believe that is true, or, at least, that is true no longer. For some time the English and Americans have been devoting themselves much more than formerly to the better understanding of what is thought and said on the continent of Europe.
To be sure, each people will preserve its characteristic genius, and it would be a pity if it were otherwise, supposing such a thing possible. If the Anglo-Saxons wished to become Latins, they would never be more than bad Latins; just as the French, in seeking to imitate them, could turn out only pretty poor Anglo-Saxons.
And then the English and Americans have made scientific eonquests they alone could have made ; they will make still more of which others would be incapable. It would therefore be de- plorable if there were no longer Anglo-Saxons.
But continentals have on their part done things an English- man could not have done, so that there is no need either for wishing all the world Anglo-Saxon.
Each has his characteristic aptitudes, and these aptitudes
3
4 SCIENCE AND HTP0THESI8
should be diverse, else would the scientific concert resemble a quartet where every one wanted to play the violin.
And yet it is not bad for the violin to know what the violon- cello is playing, and vice versa.
This it is that the English and Americans are comprehending more and more; and from this point of view the translations undertaken by Dr. Halsted are most opportune and timely.
Consider first what concerns the mathematical sciences. It is frequently said the English cultivate them only in view of their applications and even that they despise those who have other aims; that speculations too abstract repel them as savor- ing of metaphysic.
The English, even in mathematics, are to proceed always from the particular to the general, so that they would never have an idea of entering mathematics, as do many Germans, by the gate of the theory of aggregates. They are always to hold, so to speak, one foot in the world of the senses, and never burn the bridges keeping them in communication with reality. They thus are to be incapable of comprehending or at least of appreciat- ing certain theories more interesting than utilitarian, such as the non-Euclidean geometries. According to that, the first twK) parts of this book, on number and space, should seem to them void of all substance and would only baflBe them.
But that is not true. And first of all, are they such uncom- promising realists as has been said? Are they absolutely refrac- tory, I do not say to metaphysic, but at least to everything metaphysical ?
Recall the name of Berkeley, bom in Ireland doubtless, but immediately adopted by the English, who marked a natural and necessary stage in the development of English philosophy.
Is this not enough to show they are capable of making ascen- sions otherwise than in a captive balloon?
And to return to America, is not the Monist published at Chicago, that review which even to us seems bold and yet which finds readers?
And in mathematics? Do you think American geometers are concerned only about applications? Far from it. The part of the science they cultivate most devotedly is the theory of
AUTHOE'3 PREFACE TO TRANSLATION 6
groups of snbstitations, and under its most abstract form, the farthest removed from the practical.
Moreover, Dr. Halsted gives regularly each year a review of all productioDS relative to the non-Euclidean geometry, and he has about him a public deeply interested in his work. He has initiated this public into the ideaa of Hilbert, and he has even written an elementary treatise on 'Rational Geometry,' based on the principles of the renowned German savant.
To introduce this principle into teaching is surely this time to bum all bridges of reliance upon sensory intuition, and this is, I confess, a boldness which seems to me almost rash&ess.
The American public is therefore much better prepared than has been thought for investigating the origin of the notion of space.
Moreover, to analyze this concept is not to sactifiee reality to I know not what phantom. The geometric language is after all only a language. Space is only a word that we have believed a thing. "What is the origin of this word and of other words alsol What things do they hidet To ask this is permissible; to forbid it would be, on the contrary, to be a dupe of words ; it would be to adore a metaphysical idol, like savage peoples who prostrate themselves before a statue of wood without daring to take a look at what is within.
Iq the study of nature, the contrast between the Anglo-Saxon spirit and the Latin spirit is still greater.
The Latins seek in general to put their thought in mathe- matical form; the English prefer to express it by a material representation.
Both doubtless rely only on experience for knowing the world; when they happen to go beyond this, they consider their fore- knowledge as only provisional, and they hasten to ask its defini- tive confirmation from nature herself.
But experience is not all, and the savant is not passive; he does not wait for the truth to come and find him, or for a chance meeting to bring him face to face with it. He must go to meet it, and it is for his thinking to reveal to him the way leading thither. For that there is need of an instrument ; well, just there begins the difference — the instrument the Latins ordi- narily choose is not that preferred by the Anglo-Saxons.
6 SCIENCE AND HYPOTHESIS
For a Latin, truth can be expressed only by equations; it must obey laws simple, logical, symmetric and fitted to satisfy minds in love with mathematical elegance.
The Anglo-Saxon to depict a phenomenon will first be en- grossed in making a model, and he will make it with common materials, such as our crude, unaided senses show us them. He also makes a hypothesis, he assumes implicitly that nature, in her finest elements, is the same as in the complicated aggregates which alone are within the reach of our senses. He concludes from the body to the atom.
Both thelrefore make hypotheses, and this indeed is necessary, since no scientist has ever been able to get on without them. The essential thing is never to make them unconsciously.
From this point of view again, it would be well for these two sorts of physicists to know something of each other; in study- ing the work of minds so unlike their own, they will immedi- ately recognize that in this work there has been an accumulation of hypotheses.
Doubtless this will not suffice to make them comprehend that they on their part have made just as many; each sees the mote without seeing the beam ; but by their criticisms they will warn their rivals, and it may be supposed these will not fail to render them the same service.
The English procedure often seems to us crude, the analogies they think they discover to us seem at times superficial ; they are not sufficiently interlocked, not precise enough; they sometimes permit incoherences, contradictions in terms, which shock a geo- metric spirit and which the employment of the mathematical method would immediately have put in evidence. But most often it is, on the other hand, very fortunate that they have not per- ceived these contradictions; else would they have rejected their model and could not have deduced from it the brilliant results they have often made to come out of it.
And then these very contradictions, when they end by per- ceiving them, have the advantage of showing them the hypothet- ical character of their conceptions, whereas the mathematical method, by its apparent rigor and inflexible course, often inspires in us a confidence nothing warrants, and prevents our looking about us.
AUTHOR'S PREFACE TO TRANSLATION 7
From another point of view, however, the two conceptions are very unlike, and if all must be said, they are very unlike because of a common fault.
The English wish to make the world out of what we see. I mean what we see with the unaided eye, not the microscope, nor that still more subtile microscope, the human head guided by scientific induction.
The Latin wants to make it out of formulas, but these for- mulas are still the quintessenced expression of what we see. In a word, both would make the unknown out of the known, and their excuse is that there is no way of doing otherwise.
And yet is this legitimate, if the unknown be the simple and the known the complex?
Shall we not get of the simple a false idea, if we think it like the complex, or worse yet if we strive to make it out of elements which are themselves compounds?
Is not each great advance accomplished precisely the day some one has discovered under the complex aggregate shown by our senses something far more simple, not even resembling it — as when Newton replaced Kepler's three laws by the single law of gravitation, which was something simpler, equivalent, yet unlike ?
One is justified in asking if we are not on the eve of just such a revolution or one even more important. Matter seems on the point of losing its mass, its solidest attribute, and resolving itself into electrons. Mechanics must then give place to a broader conception which will explain it, but which it will not explain.
So it was in vain the attempt was made in England to con- struct the ether by material models, or in Prance to apply to it the laws of dynamic.
The ether it is, the unknown, which explains matter, the known; matter is incapable of explaining the ether.
POINCARfi.
INTRODUCTION
BY PEOFE880B JOSIAH EOYCE Habvasd University
The treatise of a master needs no commendation through the words of a mere learner. But, since my friend and former fellow student, the translator of this volume, has joined with another of my colleagues. Professor Cattell, in asking me to undertake the task of calling the attention of my fellow students to the importance and to the scope of M. Poincare's volume, I accept the office, not as one competent to pass judgment upon the book, but simply as a learner, desirous to increase the number of those amongst us who are already interested in the type of researches to which M. Poincare has so notably contributed.
The branches of inquiry collectively known as the Philosophy of Science have undergone great changes since the appearance of Herbert Spencer's First Principles, that volume which a large part of the general public in this country used to regard as the representative compend of all modern wisdom relating to the foundations of scientific knowledge. The summary which M. Poincare gives, at the outset of his own introduction to the present work, where he states the view which the 'superficial observer' takes of scientific truth, suggests, not indeed Spencer's own most characteristic theories, but something of the spirit in which many disciples of Spencer interpreting their master's formulas used to conceive the position which science occupies in dealing with experience. It was well known to them, indeed, that experience is a constant guide, and an inexhaustible source both of novel scientific results and of unsolved problems; but the fundamental Spencerian principles of science, such as *the persistence of force,' the 'rhythm of motion' and the rest, were treated by Spencer himself as demonstrably objective, although
9
10 SCIENCE AND HYPOTHESIS
indeed 'relative' truths, capable of being tested once for all by the 'inconceivability of the opposite,' and certain to hold true for the whole 'knowable' universe. Thus, whether one dwelt upon the results of such a mathematical procedure as that to which M. Poincar6 refers in his opening paragraphs, or whether, like Spen- cer himself, one applied the 'first principles' to regions of less exact science, this confidence that a certain orthodoxy regarding the principles of science was established forever was characteristic of the followers of the movement in question. Experience, lighted up by reason, seemed to them to have predetermined for all future time certain great theoretical results regarding the real constitution of the 'knowable' cosmos. Whoever doubted this doubted 'the verdict of science.'
Some of us well remember how, when Stallo's 'Principles and Theories of Modem Physics' first appeared, this sense of scien- tific orthodoxy was shocked amongst many of our American read- ers and teachers of science. I myself can recall to mind some highly authoritative reviews of that work in which the author was more or less sharply taken to task for his ignorant presump- tion in speaking with the freedom that he there used regarding such sacred possessions of humanity as the fundamental concepts of physics. That very book, however, has quite lately been translated into German as a valuable contribution to some of the most recent efforts to reconstitute a modem 'philosophy of nature.' And whatever may be otherwise thought of Stallo's critical methods, or of his results, there can be no doubt that, at the present moment, if his book were to appear for the first time, nobody would attempt to discredit the work merely on account of its disposition to be agnostic regarding the objective reality of the concepts of the kinetic theory of gases, or on account of its call for a logical rearrangement of the fundamental concepts of the theory of energy. We are no longer able so easily to know heretics at first sight.
For we now appear to stand in this position: The control of natural phenomena, which through the sciences men have attained, grows daily vaster and more detailed, and in its de- tails more assured. Phenomena men know and predict better than ever. But regarding the most general theories, and the
INTRODUCTION 11
most fundamental, of science, there ia no longer an? notablt Kieatific orthodoxy. Thus, as knowledge grows firmer and wider, conceptual construction becomes less rigid. The field of the theoretical philosophy of nature — ^yes, the field of the logic of science — this whole region is to-day an open one. "Whoever will work there must indeed accept the verdict of experience regard- ing what happens in the natural world. So far he is indeed bound. But he may undertake without hindrance from mere tradition the task of trying afresh to reduce what happens to conceptual unity. The cdrele-squares and the inventors of devices for perpetual motion are indeed still as unwelcome in scientific company as they were in the days when scientific orthodoxy was more rigidly defined ; but that is not because the foundations of geometry are now viewed as completely settled, beyond controversy, nor yet because the 'persistence of force' has been finally so defined>as to make the 'opposite ineonceiT- able ' and the doctrine of energy beyond the reach of novel formu- lations. No, the circle-squarers and the inventors of devices for perpetual motion are to-day discredited, not because of any unorthodoxy of their general philosophy of nature, but because their views regarding special facts and processes stand in conflict with certain equally special results of science which themselves admit of very various general theoretical interpre- tations. Certain properties of the irrational number ir are known, in suificient multitude to justify the mathematician in declining to listen to the arguments of the circle-squarer ; but, despite great advances, and despite the assured results of Dede- kind, of Cantor, of "Weierstrass and of various others, the gen- eral theory of the logic of the numbers, rational and irrational, still presents several important features of great obscurity ; and the philosophy of the concepts of geometry yet remains, in sev- eral very notable respects, unconquered territory, despite the work of Hilbert and of Fieri, and of our author himself. The ordinary inventors of the perpetual motion machines still stand in conflict with accepted generalizations; but nobody knows as yet what the final form of the theory of energy will be, nor can any one say precisely what place the phenomena of the radioac- tive bodies will occupy in that theory. The alchemists would not
12 SCIENCE AND BYP0TBESI8
be welcome workers in modem laboratories; yet some sorts of transformation and of evolution of the elements are to-day matters which theory can find it convenient, upon occasion, to treat as more or less exactly definable possibilities; while some newly observed phenomena tend to indicate, not indeed that the ancient hopes of the alchemists were well founded, but that the ultimate constitution of matter is something more fluent, less in- variant, than the theoretical orthodoxy of a recent period sap- posed. Again, regarding the foundations of biology, a theoret- ical orthodoxy grows less possible, less definable, less conceiv- able (even as a hope) the more knowledge advances. Once 'mechanism' and 'vitalism' were mutually contradictory theories regarding the ultimate constitution of living bodies. Now they are obviously becoming more and more 'points of view,' diverse but not necessarily conflicting. So far as you find it convenient to limit your study of vital processes to those phenomena which distinguish living matter from all other natural obects, you may assume, in the modern 'pragmatic' sense, the attitude of a 'neo- vitalist. ' So far, however, as you are able to lay stress, with good results, upon the many ways in which the life processes can be assimilated to those studied in physics and in chemistry, yon work as if you were a partisan of 'mechanics.' In any case, your special science prospers by reason of the empirical discov- eries that jou make. And your theories, whatever they are, must not run counter to any positive empirical results. But otherwise, scientific orthodoxy no longer predetermines what alone it is respectable for you to think about the nature of living substance.
This gain in the freedom of theory, coming, as it does, side by side with a constant increase of a positive knowledge of nature, lends itself to various interpretations, and raises various obvious questions.
II
One of the most natural of these interpretations, one of the most obvious of these questions, may be readily stated. Is not the lesson of all these recent discussions simply this, that general theories are simply vain, that a philosophy of nature is an idle
INTRODUCTION 13
dream, and that the results of science are coextensive with the range of actual empirical observation and of successful predic- tion? If this is indeed the lesson, then the decline of theoretical orthodoxy in science is — ^like the eclipse of dogma in religion — merely a further lesson in pure positivism, another proof that nttn does best when he limits himself to thinking about what can be found in human experience, and in trying to plan what can be done to make human life more controllable and more reason- able. What we are free to do as we please — ^is it any longer a serious business? What we are free to think as we please — ^is it of any further interest to one who is in search of truth? If certain general theories are mere conceptual constructions, which to-day are, and to-morrow are cast into the oven, why dignify them by the name of philosophy? Has science any place for such theories? Why be a *neo-vitalist,' or an 'evolutionist,' or an * atomist, ' or an ' Energetiker ' ? Why not say, plainly : * * Such and such phenomena, thus and thus described, have been ob- served; such and such experiences are to be expected, since the hypotheses by the terms of which we are required to expect them have been verified too often to let us regard the agreement with experience as due merely to chance; so much then with reasonable assurance we know; all else is silence— or else is some matter to be tested by another experiment?" Why not limit our philosophy of science strictly to such a counsel of resig- nation? Why not substitute, for the old scientific orthodoxy, simply a confession of ignorance, and a resolution to devote our- selves to the business of enlarging the bounds of actual em- pirical knowledge?
Such comments upon the situation just characterized are fre- quently made. Unfortunately, they seem not to content the very age whose revolt from the orthodoxy of traditional theory, whose uncertainty about all theoretical formulations, and whose vast wealth of empirical discoveries and of rapidly advancing special researches, would seem most to justify tliese very com- ments. Never has there been better reason than there is to-day to be content, if rational man could be content, with a pure pos- itivism. The splendid triumphs of special research in the most various fields, the constant increase in our practical control over
14 SCIENCE AND ETPOTHESIS
nature — ^these, our positive and growing possessions, stand in glaring contrast to the failure of the scientific orthodoxy of a former period to fix the outlines of an ultimate creed about the nature of the knowable universe. Why not 'take the cash and let the credit go'f Why pursue the elusive theoretical 'unifica- tion' any further, when what we daily get from our sciences is an increasing wealth of detailed information and of practical guidance T
As a fact, however, the known answer of our own age to these very obvious comments is a constant multiplication of new efforts towards large and unifying theories. If theoretical ortho- doxy is no longer clearly definable, theoretical construction was never more rife. The history of the doctrine of evolution, even in its most recent phases, when the theoretical uncertainties re- garding the 'factors of evolution' are most insisted upon, is full of illustrations of this remarkable union of scepticism in critical work with courage regarding the use of the scientific imagination. The history of those controversies regarding theoretical physics, some of whose principal phases M. Poincare, in his book, sketches with the hand of the master, is another illustration of the con- sciousness of the time. Men have their freedom of thought in these regions; and they feel the need of making constant and constructive use of this freedom. And the men who most feel this need are by no means in the majority of cases professional metaphysicians — or students who, like myself, have to view all these controversies amongst the scientific theoreticians from without as learners. These large theoretical constructions are due, on the contrary, in a great many cases to special workers, who have been driven to the freedom of philosophy by the oppres- sion of experience, and who have learned in the conflict with special problems the lesson that they now teach in the form of general ideas regarding the philosophical aspects of science.
Why, then, does science actually need general theories, despite the fact that these theories inevitably alter and pass awayf What is the service of a philosophy of science, when it is certain that the philosophy of science which is best suited to the needs of one generation must be superseded by the advancing insight of the next generation? Why must that which endlessly grows^
INTRODUCTION 15
namdy, man's knowledge of the phenomenal order of natnre^ be constantly united in men's minds with that which is certain to decay, namely, the theoretical formulation of special knowl- edge in more or less completely unified systems of doctrine T
I understand our author's volume to be in the main an answer to this question. To be sure, the compact and manifold teachings which this text contains relate to a great many dif- ferent special issues. A student interested in the problems of the philosophy of mathematics, or in the theory of probabilities, or in the nature and office of mathematical physics, or in still other problems belonging to the wide field here discussed, may find what he wants here and there in the text, even in case the general issues which give the volume its unity mean little to him, or even if he differs from the author's views regarding the principal issues of the book. But in the main, this volume must be regarded as what its title indicates — a critique of the nature and place of hypothesis in the work of science and a study of the logical relations of theory and fact. The result of the book is a substantial justification of the scientific utility of theoretical con- struction— an abandonment of dogma, but a vindication of the rights of the constructive reason.
Ill
The most notable of the results of our author's investigation of the logic of scientific theories relates, as I understand his work, to a topic which the present state of logical investigation, just summarized, makes especially important, but which has thus far been very inadequately treated in the text-books of inductive logic. The useful hypotheses of science are of two kinds :
1. The hypotheses which are valuable precisely because they are either verifiable or else refutable through a definite appeal to the tests furnished by experience ; and
2. The hypotheses which, despite the fact that experience sug- gests them, are valuable despite, or even because, of the fact that experience can neither confirm nor refute them. The contrast between these two kinds of hypotheses is a prominent topic of our author's discussion.
Hypotheses of the general type which I have here placed first
16 SCIENCE AND HYPOTHESIS
in order are the ones which the text-books of inductive logic and those summaries of scientific method which are customary in the course of the elementary treatises upon physical science are already accustomed to recognize and to characterize. The value of such hypotheses is indeed undoubted. But hypotheses of the type which I have here named in the second place are far less frequentiy recognized in a perfectly explicit way as useful aids in the work of special science. One usually either fails to admit their presence in scientific work, or else remains silent as to the reasons of their usefulness. Our author's treatment of the work of science is therefore especially marked by the fact that he ex- plicitiy makes prominent both the existence and the scientific importance of hypotheses of this second type. They occupy in his discussion a place somewhat analogous to each of the two dis- tinct positions occupied by the 'categories' and the 'forms of sensibility/ on the one hand, and by the 'regulative principles of the reason,' on the other hand, in the Kantian theory of our knowledge of nature. That is, these hypotheses which can neither be confirmed nor refuted by experience appear, in M. Poincar6's account, partly (like the conception of * continuous quantity') as devices of the understanding whereby we give conceptual unity and an invisible connectedness to certain types of phenomenal facts which come to us in a discrete form and in a confused variety; and partly (like the larger organizing con- cepts of science) as principles regarding the structure of the world in its wholeness ; i. e., as principles in the light of which we try to interpret our experience, so as to give to it a totality and an inclusive unity such as Euclidean space, or such as the world of the theory of energy is conceived to possess. Thus viewed, M. Poincare's logical theory of this second class of hypotheses under- takes to accomplish, with modem means and in the light of to-day's issues, a part of what Kant endeavored to accomplish in his theory of scientific knowledge with the limited means which were at his disposal. Those aspects of science which are determined by the use of the hypotheses of this second kind appear in our author's account as constituting an essential human way of viewing nature, an interpretation rather than a portrayal or a prediction of the objective facts of nature, an
INTRODUCTION 17
adjustment of our conceptions of things to the internal needs of our intelligence, rather than a grasping of things as they are in themselves.
To be sure, M. Poincare's view, in this portion of his work, obviously differs, meanwhile, from that of Kant, as well as this agrees, in a measure, with the spirit of the Kantian epistemology. I do not mean therefore to class our author as a Kantian. For Kant, the interpretations imposed by the * forms of sensibility,' and by the 'categories of the understanding,' upon our doctrine of nature are rigidly predetermined by the unalterable 'form' of our intellectual powers. We 'must' thus view facts, whatever tiie data of sense must be. This, of course, is not M. Poincar^'s view. A similarly rigid predetermination also limits the Kantian 'ideas of the reason' to a certain set of principles whose guidance of the course of our theoretical investigations is indeed only 'regulative,' but is 'a priori,' and so unchangeable. For M. Poincar^, on the contrary, all this adjustment of our interpre- tations of experience to the needs of our intellect is something far less rigid and unalterable, and is constantly subject to the suggestions of experience. We must indeed interpret in our own way; but our way is itself only relatively determinate; it is essentially more or less plastic ; other interpretations of experience are conceivable. Those that we use are merely the ones found to be most convenient. But this convenience is not absolute neces- sity. Unverifiable and irrefutable hypotheses in science are in- deed, in general, indispensable aids to the organization and to the guidance of our interpretation of experience. But it is expe- rience itself which points out to us what lines of interpretation will prove most convenient. Instead of Kant's rigid list of a priori 'forms,' we consequently have in M. Poincare's account a set of conventions, neither wholly subjective and arbitrary, nor yet imposed upon us unambiguously by the external compulsion of experience. The organization of science, so far as this organ- ization is due to hypotheses of the kind here in question, thus resembles that of a constitutional government — neither abso- lutely necessary, nor yet determined apart from the will of the subjects, nor yet accidental — a free, yet not a capricious estab- lishment of good order, in conformity with empirical needs.
3
18 SCIENCE AND HYPOTHESIS
Characteristic remains, however, for our author, as, in his decidedly contrasting way, for Kant, the thought that without principles which at every stage transcend precise confirmation through such experience as is then accessible the organization of experience is impossible. Whether one views these principles as conventions or as a priori 'forms,' they may therefore be de- scribed as hypotheses, but as hypotheses that, while lying at the basis of our actual physical sciences, at once refer to experience and help us in dealing with experience, and are yet neither con- firmed nor refuted by the experiences which we possess or which we can hope to attain.
Three special instances or classes of instances, according to our author's account, may be used as illustrations of this general type of hypotheses. They are: (1) The hypothesis of the exist- ence of continuous extensive quanta in nature; (2) The prin- ciples of geometry; (3) The principles of mechanics and of the general theory of energy. In case of each of these special types of hypotheses we are at first disposed, apart from reflection, to say that we find the world to be thus or thus, so that, for instance, we can confirm the thesis according to which nature contains continuous magnitudes; or can prove or disprove the physical truth of the postulates of Euclidean geometry ; or can confirm by definite experience the objective validity of the principles of mechanics. A closer examination reveals, according to our author, the incorrectness of all such opinions. H3rpotheses of these various special types are needed ; and their usefulness can be empirically shown. They are in touch with experience; and that they are not merely arbitrary conventions is also verifiable. They are not a priori necessities ; and we can easily conceive in- telligent beings whose experience could be best interpreted with- out using these hypotheses. Yet these hypotheses are not sub- ject to direct confirmation or refutation by experience. They stand then in sharp contrast to the scientific hypotheses of the other, and more frequently recognized, type, i. e., to the hy- potheses which can be tested by a definite appeal to experience. To these other hypotheses our author attaches, of course, great importance. His treatment of them is full of a living apprecia- tion of the significance of empirical investigation. But the cen-
INTRODUCTION 19
tral problem of the logic of science thus becomes the problem of the relation between the two fundamentally distinct types of hypotheses, ♦. e., between those which can not be verified or re- futed through experience, and those which can be empirically tested.
IV
The detailed treatment which M. Poincar6 gives to the problem thus defined must be learned from his text. It is no part of my purpose to expound, to defend or to traverse any of his special conclusions regarding this matter. Yet I can not avoid observ- ing that, while M. Poincar^ strictly confines his illustrations and his expressions of opinion to those regions of science wherein, as special investigator, he is himself most at home, the issues which he thus raises regarding the logic of science are of even more critical importance and of more impressive interest when one applies M. Poincare's methods to the study of the concepts and presuppositions of the organic and of the historical and social sciences, than when one confines one's attention, as our author here does, to the physical sciences. It belongs to the province of an introduction like the present to point out, however briefiy and inadequately, that the significance of our author's ideas extends far beyond the scope to which he chooses to confine their discussion.
The historical sciences, and in fact all those sciences such as geology, and such as the evolutionary sciences in general, un- dertake theoretical constructions which relate to past time. Hy- potheses relating to the more or less remote past stand, however, in a position which is very interesting from the point of view of the logic of science. Directly speaking, no such hypothesis is capable of confirmation or of refutation, because we can not return into the past to verify by our own experience what then happened. Yet indirectly, such hypotheses may lead to predic- tions of coming experience. These latter will be subject to con- troL Thus, Schliemann's confidence that the legend of Troy had a definite historical foundation led to predictions regarding what certain excavations would reveal. In a sense somewhat different from that which filled Schliemann's enthusiastic mind, these pre- dictions proved verifiable. The result has been a considerable
20 SCIENCE AND HYPOTHESIS
change in the attitude of historians toward the legend of Troy. Geological investigation leads to predictions regarding the order of the strata or the course of mineral veins in a district, regard- ing the fossils which may be discovered in given formations, and so on. These hypotheses are subject to the control of experience. The various theories of evolutionary doctrine include many hy- potheses capable of confirmation and of refutation by empirical tests. Yet, despite all such empirical control, it still remains true that whenever a science is mainly concerned with the remote past, whether this science be archeology, or geology, or anthro- pology, or Old Testament history, the principal theoretical con- structions always include features which no appeal to present or to accessible future experience can ever definitely test. Hence the suspicion with which students of experimental science often regard the theoretical constructions of their confreres of the sci- ences that deal with the past. The origin of the races of men, of man himself, of life, of species, of the planet ; the hypotheses of anthropologists, of archeologists, of students of 'higher criti- cism'— ^all these are matters which the men of the laboratory often regard with a general incredulity as belonging not at all to the domain of true science. Yet no one can doubt the im- portance and the inevitableness of endeavoring to apply scientific method to these regions also. Science needs theories regarding the past history of the world. And no one who looks closer into the methods of these sciences of past time can doubt that verifi- able and unverifiable hypotheses are in all these regions inevitably interwoven; so that, while experience is always the guide, the attitude of the investigator towards experience is determined by interests which have to be partially due to what I should call that 'internal meaning,' that human interest in rational theoret- ical construction which inspires the scientific inquiry; and the theoretical constructions which prevail in such sciences are neither unbiased reports of the actual constitution of an external reality, nor yet arbitrary constructions of fancy. These con- structions in fact resemble in a measure those which M. Poincarfi in this book has analyzed in the case of geometry. They are constructions molded, but not predetermined in their details, by experience. We report facts ; we let the facts speak ; but we, as
INTRODUCTION 21
we inyestigate, in the popular phrase, Halk back' to the facts. We interpret as well as report Man is not merely made for science, but science is made for man. It expresses his deepest intellectual needs, as well as his careful observations. It is an effort to bring internal meanings into harmony with external verifications. It attempts therefore to control, as well as to submit, to conceive with rational unity, as well as to accept data. Its arts are those directed towards self-possession as well as towards an imitation of the outer reality which we find. It seeks therefore a disciplined freedom of thought. The discipline is as essential as the freedom; but the latter has also its place. The theories of science are human, as well as objective, inter- nally rational, as well as (when that is possible) subject to ex- ternal tests.
In a field very different from that of the historical sciences, namely, in a science of observation and of experiment, which is at the same time an organic science, I have been led in the course of some study of the history of certain researches to notice the existence of a theoretical conception which has proved extremely fruitful in guiding research, but which apparently resembles in a measure the type of hypotheses of which M. Poincar4 speaks when he characterizes the principles of mechanics and of the theory of energy. I venture to call attention here to this con- ception, which seems to me to illustrate M. Poincare's view of the functions of hypothesis in scientific work.
The modem science of pathology is usually regarded as dating from the earlier researches of Virchow, whose * Cellular Path- ology' was the outcome of a very careful and elaborate induc- tion. Virchow, himself, felt a strong aversion to mere specula- tion. He endeavored to keep close to observation, and to relieve medical science from the control of fantastic theories, such as those of the Naturphilosophen had been. Yet Virchow 's re- searches were, as early as 1847, or still earlier, already under the guidance of a theoretical presupposition which he himself states as follows: **We have learned to recognize," he says, **that dis- eases are not autonomous organisms, that they are no entities that have entered into the body, that they are no parasites which take root in the body, but that they merely show tis the course of
22 8CIESCE AXD HYPOTHESIS
the vital proeeM$es under mltered ccmdUions" Cdaas sie nnr AMauf der Lebensendieiiiiiiigeii anter Teiudcrten Bedingnn-
gen dAntdkn')-
The enoTiDoiis importmiiee of this theoredcal presupposition for all the earljr socccsscs of modem pmtiiologieal inresligation k generalljr recognized by the experts. I do not doubt this opinion. It spi>ear8 to be a eommonplsee of tiie history of this aeienee. Bnt in Yirchow's later jrears this Tery presupposition seemed to some of his contemporaries to be ealled in qaestion by the soccesses of recent bacteriology. The qaestion arose whether the theoretical foundations of Virchow's pathology had not been set aside. And in fact the theoiy of the parasitical origin of a vast number of diseased conditions has indeed come upon an empirical basis to be generally recognized. Yet to the end of his own career Virchow stoutly maintained that in all its essential significance his own fundamental principle remained quite un- touched by the newer discoreries. And, as a fact, this view could indeed be maintained. For if diseases proved to be the consequences of the presence of parasites, the diseases them- selves, so far as they belonged to the diseased organism, were still not the parasites, but were, as before, the reaction of the organism to the verdnderie Bedingungen which the presence of the parasites entailed. So Virchow could well insist And if the famous principle in question is only stated with sufficient generality, it amounts simply to saying that if a disease in- volves a change in an organism, and if this change is subject to law at all, then the nature of the organism and the reaction of the organism to whatever it is which causes the disease must be underHtood in case the disease is to be understood.
For this very reason, however, Virchow's theoretical principle in its most general form could be neither confirmed nor refuted by experience. It would remain empirically irrefutable, so far as I can see, even if we should learn that the devil was the true cause of all diseases. For the devil himself would then simply predetermine the verdnderte Bedingungen to which the diseased organism would be reacting. Let bullets or bacteria, poisons or compressed air, or the devil be the Bedingungen to which a diseased organism reacts, the postulate that Virchow
INTRODUCTION 23
states in the passage just quoted will remain irrefutable, if only this postulate be interpreted to meet the case. For the principle in question merely says that whatever entity it may be, bullet, or poison, or devil, that affects the organism, the disease is not that entity, but is the resulting alteration in the process of the organism.
I insist, then, that this principle of Virchow's is no trial sup- position, no scientific hypothesis in the narrower sense — capable of being submitted to precise empirical tests. It is, on the contrary, a very precious leading idea, a theoretical interpre- tation of phenomena, in the light of which observations are to be made — *a regulative principle' of research. It is equivalent to a resolution to search for those detailed connections which link the processes of disease to the normal process of the organism. Such a search undertakes to find the true unity, whatever that may prove to be, wherein the pathological and the normal proc- esses are linked. Now without some such leading idea, the cellu- lar pathology itself could never have been reached ; because the empirical facts in question would never have been observed. Hence this principle of Virchow's was indispensable to the growth of his science. Yet it was not a verifiable and not a re- futable hypothesis. One value of unverifiable and irrefutable hyx)otheses of this type lies, then, in the sort of empirical inquiries which they initiate, inspire, organize and guide. In these inquiries hypotheses in the narrower sense, that is, trial propositions which are to be submitted to definite empirical con- trol, are indeed everywhere present. And the use of the other sort of principles lies wholly in their application to experience. Yet without what I have just proposed to call the 'leading ideas' of a science, that is, its principles of an unverifiable and irre- futable character, suggested, but not to be finally tested, by experience, the hypotheses in the narrower sense would lack that guidance which, as M. Poincare has shown, the larger ideas of science give to empirical investigation.
V
I have dwelt, no doubt, at too great length upon one aspect only of our author's varied and well-balanced discussion of the
24 SCIENCE AND HYPOTHESIS
problems and concepts of scientific theory. Of the hypotheses in the narrower sense and of the value of direct empirical control, he has also spoken with the authority and the originality which belong to his position. And in dealing with the foundations of mathematics he has raised one or two questions of great philo- sophical import into which I have no time, even if I had the right, to enter here. In particular, in speaking of the essence of mathematical reasoning, and of the difficult problem of what makes possible novel results in the field of pure mathematics, M. Poincar6 defends a thesis regarding the office of 'demonstration by recurrence' — ^a thesis which is indeed disputable, which has been disputed and which I myself should be disposed, so far as I at present understand the matter, to modify in some respects, even in accepting the spirit of our author's assertion. Yet there can be no doubt of the importance of this thesis, and of the fact that it defines a characteristic that is indeed fundamental in a wide range of mathematical research. The philosophical prob- lems that lie at the basis of recurrent proofs and processes are, as I have elsewhere argued, of the most fundamental importance.
These, then, are a few hints relating to the significance of our author's discussion, and a few reasons for hoping that our own students will profit by the reading of the book as those of other nations have already done.
Of the person and of the life-work of our author a few words are here, in conclusion, still in place, addressed, not to the stu- dents of his own science, to whom his position is well known, but to the general reader who may seek guidance in these pages.
Jules Henri Poincar6 was born at Nancy, in 1854, the son of a professor in the Faculty of Medicine at Nancy. He studied at the i^cole Polytechnique and at the i^cole des Mines, and later received his doctorate in mathematics in 1879. In 1883 he began courses of instruction in mathematics at the £cole Polytechnique ; in 1886 received a professorship of mathe- matical physics in the Faculty of Sciences at Paris; then became member of the Academy of Sciences at Paris, in 1887, and devoted his life to instruction and investigation in the regions of pure mathematics, of mathematical physics and of celestial mechanics. His list of published treatises relating to
INTRODUCTION 25
yarious branches of his chosen sciences is long; and his ori- ginal memoirs have included several momentous investigations, which have gone far to transform more than one branch of research. His presence at the International Congress of Arts and Science in St. Louis was one of the most noticeable features of that remarkable gathering of distinguished foreign guests. In Poincar6 the reader meets, then, not one who is primarily a speculative student of general problems for their own sake, but an original investigator of the highest rank in several distinct, although interrelated, branches of modem research. The theory of functions — ^a highly recondite region of pure mathematics — owes to him advances of the first importance, for instance, the definition of a new type of functions. The 'problem of the three bodies, ' a famous and fundamental problem of celestial mechanics, has received from his studies a treatment whose significance has been recognized by the highest authorities. His international reputation has been confirmed by the conferring of more than one important prize for his researches. His membership in the most eminent learned societies of various nations is widely extended; his volumes bearing upon various branches of mathematics and of mathematical physics are used by special students in all parts of the learned world ; in brief, he is, as geometer, as analyst and as a theoretical physicist, a leader of his age.
Meanwhile, as contributor to the philosophical discussion of the bases and methods of science, M. Poincar^ has long been active. When, in 1893, the admirable Revue de Meiaphysique et de Morale began to appear, M. Poincar^ was soon found amongst the most satisfactory of the contributors to the work of that journal, whose office it has especially been to bring philosophy and the various special sciences (both natural and moral) into a closer mutual understanding. The discussions brought to- gether in the present volume are in large part the outcome of M. Poincar^'s contributions to the Revue de Meiaphysique et de Morale. The reader of M. Poincar^'s book is in presence, then, of a great special investigator who is also a philosopher.
SCIENCE AND HYPOTHESIS
INTRODUCTION
Fob a superficial observer, scientific truth is beyond the possi- bility of doubt ; the logic of science is infallible, and if the scien- tists are sometimes mistaken, this is only from their mistaking its rules.
''The mathematical verities flow from a small number of self- evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a crowd of consequences will follow by a series of mathematical deductions, and thus each experiment will make known to us a comer of the universe."
Behold what is for many people in the world, for scholars get- ting their first notions of physics, the origin of scientific certi- tude. This is what they suppose to be the role of experimenta- tion and mathematics. This same conception, a hundred years ago, was held by many savants who dreamed of constructing the world with as little as possible taken from experiment.
On a little more reflection it was perceived how great a place hypothesis occupies; that the mathematician can not do without it, still less the experimenter. And then it was doubted if all these constructions were really solid, and believed that a breath would overthrow them. To be skeptical in this fashion is still to be superficial. To doubt everything and to believe everything are two equally convenient solutions; each saves us from thinking.
Instead of pronouncing a summary condemnation, we ought therefore to examine with care the role of hypothesis; we shall then recognize, not only that it is necessary, but that usually it is
27
28 SCIENCE AND HYPOTHESIS
le^timate. We shall also see that there are several sorts of hy- potheses ; that some are verifiabley and once confirmed by experi- ment become fruitful truths; that others, powerless to lead us astray, may be useful to us in fixing our ideas; that others, finally, are hypotheses only in appearance and are reducible to disguised definitions or conventions.
These last are met with above all in mathematics and the related sciences. Thence precisely it is that these sciences get their rigor; these conventions are the work of the free activity of our mind, which, in this domain, recognizes no obstacle. Here our mind can affirm, since it decrees ; but let us understctnd that while these decrees are imposed upon our science, which, without them, would be impossible, they are not imposed upon nature. Are they then arbitrary! No, else were they sterile. Experi- ment leaves us our freedom of choice, but it guides us by aiding us to discern the easiest way. Our decrees are therefore like those of a prince, absolute but wise, who consults his council of state.
Some people have been struck by this character of free conven- tion recognizable in certain fundamental principles of the sciences. They have wished to generalize beyond measure, and, at the same time, they have forgotten that liberty is not license. Thus they have reached what is called nominalism, and have asked themselves if the savant is not the dupe of his own defi- nitions and if the world he thinks he discovers is not simply created by his own caprice.^ Under these conditions science would be certain, but deprived of significance.
If this were so, science would be powerless. Now every day we see it work under our very eyes. That could not be if it taught us nothing of reality. Still, the things themselves are not what it can reach, as the naive dogmatists think, but only the relations between things. Outside of these relations there is no knowable reality.
Such is the conclusion to which we shall come, but for that we must review the series of sciences from arithmetic and geometry to mechanics and experimental physics.
i-See Le B07, 'Science et Philosophie, ' Bevue de M^aphysique et de Morale, 1901.
INTRODUCTION 29
What is the nature of mathematical reasoning f Is is really deductivey as is commonly supposed? A deeper analysis shows us that it is not, that it partakes in a certain measure of the nature of inductive reasoning, and just because of this is it so fruitful. None the less does it retain its character of rigor absolute; this is the first thing that had to be shown.
Knowing better now one of the instruments which mathemat- ics puts into the hands of the investigator, we had to analyze an- other fundamental notion, that of mathematical magnitude. Do we find it in nature, or do we ourselves introduce it there f And, in this latter case, do we not risk marring everything! Com- paring the rough data of our senses with that extremely complex and subtile concept which mathematicians call magnitude, we are forced to recognize a difference ; this frame into which we wish to force everything is of our own construction; but we have not made it at random. We have made it, so to speak, by measure and therefore we can make the facts fit into it without changing what is essential in them.
Another frame which we impose on the world is space. Whence come the first principles of geometry! Are they im- posed on us by logic ! Lobachevski has proved not, by creating non-Euclidean geometry. Is space revealed to us by our senses ! Still no, for the space our senses could show us differs absolutely from that of the geometer. Is experience the source of geom- etry ? A deeper discussion will show us it is not. We therefore conclude that the first principles of geometry are only conven- tions ; but these conventions are not arbitrary and if transported into another world (that I call the non-Euclidean world and seek to imagine), then we should have been led to adopt others.
In mechanics we should be led to analogous conclusions, and should see that the principles of this science, though more di- rectly based on experiment, still partake of the conventional character of the geometric postulates. Thus far nominalism triumphs ; but now we arrive at the physical sciences, properly so called. Here the scene changes; we meet another sort of hy- potheses and we see their fertility. Without doubt, at first blush, the theories seem to us fragile, and the history of science proves to us how ephemeral they are; yet they do not entirely perish,
30 SCIENCE AND HYPOTHESIS
and of each of them something remains. It is this something we most seek to disentangle, since there and there alone is the veritable reality.
The method of the physical sciences rests on the induction which makes ns expect the repetition of a phenomenon when the circumstances under which it first happened are reproduced* U all these circumstances could be reproduced at once, this prin- ciple could be applied without fear; but that will never happen; some of these circumstances will always be lacking. Are we absolutely sure they are unimportant! Evidently not. That may be probable, it can not be rigorously certain. Hence the important role the notion of probability plays in the physical sciences. The calculus of probabilities is therefore not merely a recreation or a guide to players of baccarat, and we must seek to go deeper with its foundations. Under this head I have been able to give only very incomplete results, so strongly does this vague instinct which lets us discern probability defy analysis.
After a study of the conditions under which the physicist works, I have thought proper to show him at work. For that I have taken instances from the history of optics and of electricity. We shall see whence have sprung the ideas of Fresnel, of Max- well, and what unconscious hypotheses were made by Ampere and the other founders of electrodynamics.
PARTI
NUMBER AND MAGNITUDE
CHAPTER I On the Nature of Mathematical BsASONiNa
The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill 80 many volumes are nothing but devious ways of saying A is A ?
Without doubt, we can go back to the axioms, which are at the source of all these reasonings. If we decide that these can not be reduced to the principle of contradiction, if still less we see in them experimental facts which could not partake of mathe- matical necessity, we have yet the resource of classing them among synthetic a priori judgments. This is not to solve the diflS- culty, but only to baptize it ; and even if the nature of synthetic judgments were for us no mystery, the contradiction would not have disappeared, it would only have moved back ; syllogistic rea- soning remains incapable of adding anything to the data given it ; these data reduce themselves to a few axioms, and we should find nothing else in the conclusions.
No theorem could be new if no new axiom intervened in its demonstration; reasoning could give us only the immediately
31
32 SCIENCE AND HYPOTHESIS
evident verities borrowed from direct intuition ; it would be only an intermediary parasite, and therefore should we not have good reason to ask whether the whole syllogistic apparatus did not serve solely to disguise our borrowing?
The contradiction will strike us the more if we open any book on mathematics ; on every page the author will announce his in- tention of generalizing some proposition already known. Does the mathematical method proceed from the particular to the gen- eral, and, if so, how then can it be called deductive f
If finally the science of number were purely analytic, or could be analytically derived from a small number of Gfynthetic judgments, it seems that a mind sufficiently powerful could at a glance perceive all its truths; nay more, we might even hope that some day one would invent to express them a language suffi- ciently simple to have them appear self-evident to an ordinary intelligence.
If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from the syllogism.
The difference must even be profound. We shall not, for example, find the key to the mystery in the frequent use of that rule according to which one and the same uniform operation applied to two equal numbers will give identical results.
All these modes of reasoning, whether or not they be reducible to the syllogism properly so called, retain the analytic character, and just because of that are powerless.
II
The discussion is old; Leibnitz tried to prove 2 and 2 make 4; let us look a moment at his demonstration.
I will suppose the number 1 defined and also the operation a? + 1 which consists in adding unity to a given number x.
These definitions, whatever they be, do not enter into the course of the reasoning.
I define then the numbers 2, 3 and 4 by the equalities
(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4.
In the same way, I define the operation x + 2 by the relation:
MATHEMATICAL REASONING 33
(4) a? + 2= (« 4-1)4-1. That presupposed, we have
2 4-1 4-1 = 3 4- 1 (Definition 2),
3 4-1 = 4 (Definition 3), 24-2= (2 4- 1)4-1 (Definition 4),
whence
24-2 = 4 Q.E.D.
It can not be denied that this reasoning is purely analytic. But ask any mathematician: 'That is not a demonstration prop- erly so called,' he will say to you: 'that is a verification.' We have confined ourselves to comparing two purely conventional definitions and have ascertained their identity ; we have learned nothing new. Verification differs from true demonstration pre- cisely because it is purely analytic and because it is sterile. It is sterile because the conclusion is nothing but the premises trans- lated into another language. On the contrary, true demonstration is fruitful because the conclusion here is in a sense more general than the premises.
The equality 2 + 2 = 4 is thus susceptible of a verification only because it is particular. Every particular enunciation in mathematics can always be verified in this same way. But if mathematics could be reduced to a series of such verifications, it would not be a science. So a chess-player, for example, does not create a science in winning a game. There is no science apart from the general.
It may even be said the very object of the exact sciences is to spare us these direct verifications.
Ill
Let us, therefore, see the geometer at work and seek to catch his 'process.
The task is not without diflSculty; it does not suflSce to open a work at random and analyze any demonstration in it.
We must first exclude geometry, where the question is com- plicated by arduous problems relative to the role of the postu- lates, to the nature and the origin of the notion of space. For analogous reasons we can not turn to the infinitesimal analysis.
34 SCIENCE AND HYPOTHESIS
We must seek mathematical thought where it has remained pure^ that is, in arithmetic.
A choice still is necessary; in the higher parts of the theory of numbers, the primitive mathematical notions have already un- dergone an elaboration so profound that it becomes difficult to analyze them.
It is, therefore, at the beginning of arithmetic that we must expect to find the explanation we seek, but it happens that pre- cisely in the demonstration of the most elementary theorems the authors of the classic treatises have shown the least precision and rigor. We must not impute this to them as a crime; they have yielded to a necessity ; beginners are not prepared for real mathe- matical rigor ; they would see in it only useless and irksome sub- tleties; it would be a waste of time to try prematurely to make them more exacting; they must pass over rapidly, but without skipping stations, the road traversed slowly by the founders of the science.
Why is so long a preparation necessary to become habituated to this perfect rigor, which, it would seem, should naturally im- press itself upon all good minds? This is a logical and psy- chological problem well worthy of study.
But we shall not take it up; it is foreign to our purpose; all I wish to insist on is that, not to fail of our purpose, we must recast the demonstrations of the most elementary theorems and give them, not the crude form in which they are left, so as not to harass beginners, but the form that will satisfy a skilled geometer.
Definition op Addition. — I suppose already defined the operation a; + 1, which consists in adding the number 1 to a given number x.
This definition, whatever it be, does not enter into our sub- sequent reasoning.
We now have to define the operation « -f a, which consists in adding the number a to a given number x.
Supposing we have defined the operation
a?+(a — 1),
the operation a; + a will be defined by the equality (1) x + a=lx-\' (a — 1)]+1.
MATHEMATICAL EEASONINQ 86
We shall know then what x-\-a \a when we know what «-|- (<* — 1) is, and as I have supposed that to start with we knew what a?-|-l ^ w® <^*^ define successively and *by recur- rence ' the operations a? + 8, a; + 3, etc.
This definition deserves a moment's attention; it is of a par- ticular nature which already distinguishes it from the purely logical definition; the equality (1) contains an infinity of dis- tinct definitions, each having a meaning only when one knows the preceding.
Pbopebtibs op ADDrnoN. — Assodaiivity. — I say that
a+(& + c) = (a + &)+c. In fact the theorem is true for c = l; it is then written
o+(& + l) = (o+b)+l,
which, apart from the difference of notation, is nothing but the equality (1), by which I have just defined addition. Supposing the theorem true for c=y, I say it will be true for
C=3y4-1.
In fact, supposing
(a + &)+7 = a+(& + 7),
it follows that
[(a + b)4-7]+l = [a+(& + 7)]+l
or by definition (1)
(a+ &) + (7 + 1) =a + (& + 7 4- 1) =a + [6 4- (7 + 1)],
which shows, by a series of purely analytic deductions, that the theorem is true for y + 1.
Being true for c = 1, we thus see successively that so it is for c=2, for c = 3, etc.
Commutaiivity. — 1° I say that
a + 1 = 1 + a.
The theorem is evidently true for a=il; we can verify by purely analytic reasoning that if it is true for a=y it will be true for a =y + 1 ; for then
(7 + 1)4-1= (1 + 7) +1 = 1 + (7 + 1);
now it is true for a = l, therefore it will be true for a = 2, for a =3, etc., which is expressed by saying that the enunciated proposition is demonstrated by recurrence.
36 SCIENCE AND HYPOTHESIS
2** I say that
The theorem has just been demonstrated for & =: 1 ; it can be verified analytically that if it is true for b=fi,it will be true for
The proposition is therefore established by recurrence. Definition op Multiplication. — ^We shall define multiplica- tion by the equalities.
(1) axi = a.
(2) aXh = [aX (6 — l)] + o.
Like equality (1), equality (2) contains an infinity of defini- tions ; having defined a X !> it enables us to define successively : a X 2, a X 3, etc.
Properties op Multiplication. — Distributivity. — ^I say that
(a + 6) Xc=(oXc) + (bxc).
We verify analytically that the equality is true for c = l ; then that if the theorem is true for c = y, it will be true for c =y + 1. The proposition is, therefore, demonstrated by recurrence. Commutativity, — 1° I say that
a X 1 = 1 X a.
The theorem is evident for a=l.
We verify analytically that if it is true for o = o, it will be
true for o =s o + 1.
2M say that
a X ft = & X o.
The theorem has just been proven for 6 = 1. We could verify analytically that if it is true for b=py it will be true for b = p + l.
IV
Here I stop this monotonous series of reasonings. But this very monotony has the better brought out the procedure which is uniform and is met again at each step.
This procedure is the demonstration by recurrence. We first establish a theorem for n = 1 ; then we show that if it is true of w — 1, it is true of n, and thence conclude that it is true for all the whole numbers.
MATHEMATICAL REASONING 37
We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus ; this calculus is an instrument of transforma- tion, which lends itself to many more differing combinations than joes the simple syllogism; but it is still an instrument purely analytic, and incapable of teaching us anything new. If mathe- matics had no other instrument, it would therefore be forth- with arrested in its development; but it has recourse anew to the same procedure, that is, to reasoning by recurrence, and it is able to continue its forward march.
If we look closely, at every step we meet again this mode of reasoning, either in the simple form we have just given it, or under a form more or less modified.
Here then we have the mathematical reasoning par excellence, and we must examine it more closely.
The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
That this may the better be seen, I will state one after another these syllogisms which are, if you will allow me the expression, arranged in 'cascade.'
These are of course hypothetical syllogisms. The theorem is true of the number 1.
Now, if it is true of 1, it is true of 2.
Therefore it is true of 2.
Now, if it is true of 2, it is true of 3.
Therefore it is true of 3, and so on.
We see that the conclusion of each syllogism serves as minor to the following.
Furthermore the majors of all our syllogisms can be reduced to a single formula.
If the theorem is true of n — 1, so it is of n.
We see, then, that in reasoning by recurrence we confine our- selves to stating the minor of the first syllogism, and the general formula which contains as particular cases all the majors.
This never-ending series of syllogisms is thus reduced to a phrase of a few lines.
38 SCIENCE AND HTP0THE8I8
It is now easy to comprehend why every particular conse- quence of a theorem can, as I have explained above, be verified by purely analytic procedures.
If instead of showing that our theorem is true of all num- bers, we only wish to show it true of the number 6, for example, it will sufSce for us to establish the first 5 syllogisms of our cas- cade ; 9 would be necessary if we wished to prove the theorem for the number 10; more would be needed for a larger number; but, however great this number might be, we should always end by reaching it, and the analytic verification would be possible.
And yet, however far we thus might go, we could never rise to the general theorem, applicable to all numbers, which alone can be the object of science. To reach this, an infinity of syl- logisms would be necessary ; it would be necessary to overleap an abyss that the patience of the analyst, restricted to the resources of formal logic alone, never could fill up.
I asked at the outset why one could not conceive of a mind sufSciently powerful to perceive at a glance the whole body of mathematical truths.
The answer is now easy; a chess-player is able to combine four moves, five moves, in advance, but, however extraordinary he may be, he will never prepare more than a finite number of them; if he applies his faculties to arithmetic, he will not be able to perceive its general truths by a single direct intuition ; to arrive at the smallest theorem he can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become im- practicable. But it becomes indispensable as soon as we aim at the general theorem, to which analytic verification would bring us continually nearer without ever enabling us to reach it.
In this domain of arithmetic, we may think ourselves very far from the infinitesimal analysis, and yet, as we have just seen, the idea of the mathematical infinite already plays a preponder- ant role, and without it there would be no science, because there would be nothing general.
MATHEMATICAL BEA80NIN0 39
VI
The judgment on which reasoning by recurrence rests can be put under other forms; we may say, for example, that in an infinite collection of different whole numbers there is always one which is less than all the others.
We can easily pass from one enunciation to the other and thus get the illusion of having demonstrated the legitimacy of reason- ing by recurrence. But we shall always be arrested, we shall always arrive at an undemonstrable axiom which will be in reality only the proposition to be proved translated into another language*
We can not therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of con- tradiction.
Neither can this rule come to us from experience; experience could teach us that the rule is true for the first ten or hundred numbers; for example, it can not attain to the indefinite series of numbers, but only to a portion of this series, more or less long but always limited.
Now if it were only a question of that, the principle of con- tradiction would sufiSce ; it would always allow of our developing as many i^llogisms as we wished ; it is only when it is a question of including an infinity of them in a single formula, it is only before the infinite that this principle fails, and there too, experi- ence becomes powerless. This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the S3mthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry.
Why then does this judgment force itself upon us with an irresistible evidence? It is because it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.
But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by indue-
40 SCIENCE AND HYPOTHESIS
tion f We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on ; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited.
Here is, it must be admitted, a striking analogy with the usual procedures of induction. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demon- stration by recurrence, on the contrary, imposes itself necessarily because it is only the affirmation of a property of the mind itself.
VII
Mathematicians, as I have said before, always endeavor to generalize the propositions they have obtained, and, to seek no other example, we have just proved the equality :
a + l = l + a and afterwards used it to establish the equality
which is manifestly more general.
Mathematics can, therefore, like the other sciences, proceed from the particular to the general.
This is a fact which would have appeared incomprehensible to us at the outset of this study, but which is no longer mys^ terious to us, since we have ascertained the analogies between demonstration by recurrence and ordinary induction.
Without doubt recurrent reasoning in mathematics and in- ductive reasoning in physics rest on different foundations, but their march is parallel, they advance in the same sense, that is to say, from the particular to the general.
Let us examine the case a little more closely.
To demonstrate the equality
it suffices to twice apply the rule
(1) a+l = l + a and write
(2) a-h 2 = a + 1 -h 1 = 1 -ha + 1 = 1 + 1+0 = 2 + a.
MATHEMATICAL BEASONING 41
The equality (2) thus deduced in purely analytic way from the equality (1) is, however, not simply a particular case of it; it is something quite different.
We can not therefore even say that in the really analytic and deductive part of mathematical reasoning we proceed from the general to the particular in the ordinary sense of the word.
The two members of the equality (2) are simply combinations more complicated than the two members of the equality (1), and analysis only serves to separate the elements which enter into these combinations and to study their relations.
Mathematicians proceed therefore *by construction,' they 'con- struct' combinations more and more complicated. Coming back then by the analysis of these combinations, of these aggregates, 80 to speak, to their primitive elements, they perceive the rela- tions of these elements and from them deduce the relations of the aggregates themselves.
This is a purely analytical proceeding, but it is not, however, a proceeding from the general to the particular, because evi- dently the aggregates can not be regarded as more particular than their elements.
Oreat importance, and justly, has been attached to this pro- cedure of 'construction,' and some have tried to see in it the necessary and sufficient condition for the progress of the exact sciences.
Necessary, without doubt ; but sufficient, no.
For a construction to be useful and not a vain toil for the mind, that it may serve as stepping-stone to one wishing to mount, it must first of all possess a sort of unity enabling us to see in it something besides the juxtaposition of its elements.
Or, more exactly, there must be some advantage in considering the construction rather than its elements themselves.
What can this advantage be t
Why reason on a polygon, for instance, which is always de- composable into triangles, and not on the elementary triangles?
It is because there are properties appertaining to polygons of any number of sides and that may be immediately applied to any particular polygon.
Usually, on the contrary, it is only at the cost of the most
f
42 SCIENCE AND HYPOTHESIS
prolonged exertions that they could be fonnd by studying directly the relations of the elementary triangles. The knowl- edge of the general theorem spares us these efforts.
A construction, therefore, becomes interesting only when it can be ranged beside other analogous constructions, forming spe- cies of the same genus.
If the quadrilateral is something besides the juxtaposition of two triangles, this is because it belongs to the genus polygon.
Moreover, one must be able to demonstrate the properties of the genus without being forced to establish them successively for each of the species.
To attain that, we must necessarily mount from the particular to the general, ascending one or more steps.
The analytic procedure 'by construction' does not oblige us to descend, but it leaves us at the same level.
We can ascend only by mathematical induction, which alone can teach us something new. Without the aid of this induction, different in certain respects from physical induction, but quite as fertile, construction would be powerless to create science.
Observe finally that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science, for the different moves of the same game do not resemble one another.
CHAPTBE II Mathematical Maqnttude and Expebienge
To learn what mathematicians understand by a continunmy one should not inquire of geometry. The geometer always seeks to represent to himself more or less the figures he studies, but his representations are for him only instruments; in making geometry he uses space just as he does chalk; so too much weight should not be attached to non-essentials, often of no more im- portance than the whiteness of the chalk.
The pure analyst has not this rock to fear. He has disen- gaged the science of mathematics from all foreign elements, and can answer our question: 'What exactly is this continuum about which mathematicians reason T Many analysts who reflect on their art have answered already; Monsieur Tannery, for example, in his Introduction d la thSorie des fonctions d^une variable.
Let us start from the scale of whole numbers; between two consecutive steps, intercalate one or more intermediary steps, then between these new steps still others, and so on indefinitely. Thus we shall have an unlimited number of terms; these will be the numbers called fractional, rational or commensurable. But this is not yet enough ; between these terms, which, however, are already infinite in number, it is still necessary to intercalate others called irrational or incommensurable. A remark before going further. The continuum so conceived is only a collection of individuals ranged in a certain order, infinite in number, it is true, but exterior to one another. This is not the ordinary con- ception, wherein is supposed between the elements of the con- tinuum a sort of intimate bond which makes of them a whole, where the point does not exist before the line, but the line before the point Of the celebrated formula, *the continuum is unity in multiplicity,' only the multiplicity remains, the unity has disappeared. The analysts are none the less right in defining their continuum as they do, for they always reason on just this as soon as they pique themselves on their rigor. But this is
43
44 SCIENCE AND HYPOTHESIS
enough to apprise us that the veritable mathematical continuum is a very different thing from that of the physicists and that of the metaphysicians.
It may also be said perhaps that the mathematicians who are content with this definition are dupes of words, that it is neces- sary to say precisely what each of these intermediary steps is, to explain how they are to be intercalated and to demonstrate that it is possible to do it. But that would be wrong ; the only prop- erty of these steps which is used in their reasonings^ is that of being before or after such and such steps; therefore also this alone should occur in the definition.
So how the intermediary terms should be intercalated need not concern us ; on the other hand, no one will doubt the possi- bility of this operation, unless from forgetting that possible, in the language of geometers, simply means free from contradiction.
Our definition, however, is not yet complete, and I return to it after this over-long digression.
Definition of Incommensurables. — The mathematicians of the Berlin school, Kronecker in particular, have devoted them- selves to constructing this continuous scale of fractional and irra- tional numbers without using any material other than the whole number. The mathematical continuum would be, in this view, a pure creation of the mind, where experience would have no part.
The notion of the rational number seeming to them to present no diflSculty, they have chiefly striven to define the incommen- surable number. But before producing here their definition, I must make a remark to forestall the astonishment it is sure to arouse in readers unfamiliar with the customs of geometers.
Mathematicians study not objects, but relations between ob- jects; the replacement of these objects by others is therefore indifferent to them, provided the relations do not change. The matter is for them unimportant, the form alone interests them.
Without recalling this, it would scarcely be comprehensible that Dedekind should designate by the name incommensurable number a mere symbol, that is to say, something very different
iWith those contained in the special conventions which serve to define addition and of which we shaU speak later.
MATHEMATICAL MAGNITUDE AND EXPERIENCE 45
from the ordinary idea of a quantity, which should be measurable and almost tangible.
Let us see now what Dedekind's definition is:
The commensurable numbers can in an infinity of ways be partitioned into two classes, such that any number of the first dass is greater than any number of the second class.
It may happen that among the numbers of the first class there is one smaller than all the others ; if, for example, we range in the first class all numbers greater than 2, and 2 itself, and in the second class all numbers less than 2, it is clear that 2 will be the least of all numbers of the first class. The number 2 may be chosen as symbol of this partition.
It may happen, on the contrary, that among the numbers of the second class is one greater than all the others; this is the ease, for example, if the first class comprehends all numbers greater than 2, and the second all numbers less than 2, and 2 itself. Here again the number 2 may be chosen as symbol of this partition.
But it may equally well happen that neither is there in the first class a number less than aU the others, nor in the second class a number greater than all the others. Suppose, for ex- ample, we put in the first class all commensurable numbers whose squares are greater than 2 and in the second all whose squares are less than 2. There is none whose square is precisely 2. Evi- dently there is not in the first class a number less than all the others, for, however near the square of a number may be to 2, we can always find a commensurable number whose square is still closer to 2.
In Dedekind's view, the incommensurable number
V2 or (2)*
is nothing but the symbol of this particular mode of partition of commensurable numbers; and to each mode of partition cor- responds thus a number, commensurable or not, which serves as its symbol.
But to be content with this would be to forget too far the origin of these symbols ; it remains to explain how we have been led to attribute to them a sort of concrete existence, and, besides,
46 SCIENCE AND HYPOTHESIS
does not the difiScolty begin even for the fractional numbers themselves f Should we have the notion of these numbers if we had not previously known a matter that we conceive as infinitely divisible, that is to say, a continuum?
The Physical Continuum. — ^We ask ourselves then if the notion of the mathematical continuum is not simply drawn from experience. If it were, the raw data of experience, which are our sensations, would be susceptible of measurement. We might be tempted to believe they really are so, since in these latter days the attempt has been made to measure them and a law has even been formulated, known as Fechner's law, according to which sensation is proportional to the logarithm of the stimulus.
But if we examine more closely the experiments by which it has been sought to establish this law, we shall be led to a diametrically opposite conclusion. It has been observed, for ex- ample, that a weight A of 10 grams and a weight B of 11 grams produce identical sensations, that the weight B is just as indis- tinguishable from a weight C of 12 grams, but that the weight A is easily distinguished from the weight C. Thus the raw results of experience may be expressed by the following relations :
A = B, B=zC, A<C,
which may be regarded as the formula of the physical continuum.
But here is an intolerable discord with the principle of con- tradiction, and the need of stopping this has compelled us to invent the mathematical continuum.
We are, therefore, forced to conclude that this notion has been created entirely by the mind, but that experience has given the occasion.
We can not believe that two quantities equal to a third are not equal to one another, and so we are led to suppose that A is different from B and B from C, but that the imperfection of our senses has not permitted of our distinguishing them.
Creation op the Mathematical Continuum. — First Stage. So far it would sufSce, in accounting for the facts, to intercalate between A and B a few terms, which would remain discrete. What happens now if we have recourse to some instrument to
MATHEMATICAL MAGNITUDE AND EXPERIENCE 47
sapplement the feebleness of our senses, if, for example, we make use of a microscope f Terms such as A and B, before indis- tingnishable, appear now distinct ; but between A and B, now be- come distinct, will be intercalated a new term, D, that we can distingmsh neither from A nor from B. Despite the employ-^ ment of the most highly perfected methods, the raw results of our experience will always present the characteristics of the physical continuum with the contradiction which is inherent in it.
We shall escape it only by incessantly intercalating new terms between the terms already distinguished, and this operation must be continued indefinitely. We might conceive the stopping of this operation if we could imagine some instrument sufSciently powerful to decompose the physical continuum into discrete ele- ments, as the telescope resolves the milky way into stars. But this we can not imagine ; in fact, it is with the eye we observe the image magnified by the microscope, and consequently this image must always retain the characteristics of visual sensation and consequently those of the physical continuum.
Nothing distinguishes a length observed directly from the half of this length doubled by the microscope. The whole is homogeneous with the part; this is a new contradiction, or rather it would be if the number of terms were supposed finite ; in fact, it is clear that the part containing fewer terms than the whole could not be similar to the whole.
The contradiction ceases when the number of terms is regarded as infinite ; nothing hinders, for example, considering the aggre- gate of whole numbers as similar to the aggregate of even num- bers, which, however, is only a part of it ; and, in fact, to each whole number corresponds an even number, its double.
But it is not only to escape this contradiction contained in the empirical data that the mind is led to create the concept of a continuum, formed of an indefinite number of terms.
All happens as in the sequence of whole numbers. We have the faculty of conceiving that a unit can be added to a collection of units ; thanks to experience, we have occasion to exercise this faculty and we become conscious of it; but from this moment we feel that our power has no limit and that we can count in- definitely, though we have never had to count more than a finite number of objects.
48 SCIENCE AND HTPOTHESIS
Just so, as soon as we have been led to intercalate means between two consecutive terms of a series, we feel that this opera- tion can be continued beyond all limit, and that there is, so to speak, no intrinsic reason for stopping.
As an abbreviation, let me call a mathematical continuum of the first order every aggregate of terms formed according to the same law as the scale of commensurable numbers. If we afterwards intercalate new steps according to the law of for- mation of incommensurable numbers, we shall obtain what we will call a continuum of the second order.
Second Stage. — ^We have made hitherto only the first stride; we have explained the origin of continua of the first order ; but it is necessary to see why even they are not sufficient and why the incommensurable numbers had to be invented.
If we try to imagine a line, it must have the characteristics of the physical continuum, that is to say, we shall not be able to represent it except with a certain breadth. Two lines then will appear to us under the form of two narrow bands, and, if we are content with this rough image, it is evident that if the two lines cross they will have a common part.
But the pure geometer makes a further effort ; without entirely renouncing the aid of the senses, he tries to reach the concept of the line without breadth, of the point without extension. This he can only attain to by regarding the line as the limit toward which tends an ever narrowing band, and the point as the limit toward which tends an ever lessening area. And then, our two bands, however narrow they may be, will always have a common area, the smaller as they are the narrower, and whose limit will be what the pure geometer calls a point.
This is why it is said two lines which cross have a point in common, and this truth seems intuitive.
But it would imply contradiction if lines were conceived as continua of the first order, that is to say, if on the lines traced by the geometer should be found only points having for coordi- nates rational numbers. The contradiction would be manifest as soon as one affirmed, for example, the existence of straights and circles.
It is clear, in fact, that if the points whose coordinates are
MATHEMATICAL MAGNITUDE AND EXPERIENCE 49
commensurable were alone regarded as real, the circle inscribed in a square and the diagonal of this square would not intersect, since the coordinates of the point of intersection are incom- mensurable.
That would not yet be sufiScient, because we should get in this way only certain incommensurable numbers and not all those numbers.
But conceive of a straight line divided into two rays. Each of these rays will appear to our imagination as a band of a cer- tain breadth; these bands moreover will encroach one on the other, since there must be no interval between them. The com- mon part will appear to us as a point which will always remain when we try to imagine our bands narrower and narrower, so that we admit as an intuitive truth that if a straight is cut into two raya their common frontier is a point ; we recognize here the conception of Dedekind, in which an incommensurable number was regarded as the common frontier of two classes of rational numbers.
Such is the origin of the continuum of the second order, which is the mathematical continuum properly so called.
Resume. — ^In recapitulation, the mind has the faculty of cre- ating symbols, and it is thus that it has constructed the mathe- matical continuum, which is only a particular system of symbols. Its power is limited only by the necessity of avoiding all contra- diction ; but the mind only makes use of this faculty if experience furnishes it a stimulus thereto.
In the case considered, this stimulus was the notion of the physical continuum, drawn from the rough data of the senses. But this notion leads to a series of contradictions from which it is necessary successively to free ourselves. So we are forced to imagine a more and more complicated system of symbols. That at which we stop is not only exempt from internal contradiction (it was so already at all the stages we have traversed), but neither is it in contradiction with various propositions called in- tuitive, which are derived from empirical notions more or less elaborated.
Measubable Maonttude. — The magnitudes we have studied hitherto are not measurable; we can indeed say whether a given
60 SCIENCE AND HTP0THESI8
one of these magnitudes is greater than another, but not whether it is twice or thrice as great.
So far, I have only considered the order in which our temu are ranged. But for most applications that does not suffice. We must learn to compare the interval which separates any two terms. Only on this condition does the continuum become a measurable magnitude and the operations of arithmetic ap- plicable.
This can only be done by the aid of a new and special con- vention. We will agree that in such and such a case the interval comprised between the terms A and B is equal to the interval which separates C and D. For example, at the beginning of oiu work we have set out from the scale of the whole numbers and we have supposed intercalated between two consecutive steps n intermediary steps ; well, these new steps will be by conventios regarded as equidistant.
This is a way of defining the addition of two magnitudes, be- cause if the interval AB is by definition equal to the interval CD^ the interval AD will be by definition the sum of the intervals AB and AC.
This definition is arbitrary in a very large measure. It is not completely so, however. It is subjected to certain conditions and, for example, to the rules of commutativity and associativity of addition. But provided the definition chosen satisfies these rules, the choice is indifferent, and it is useless to particularize it.
Various Remarks. — ^We can now discuss several important questions :
1** Is the creative power of the mind exhausted by the creation of the mathematical continuiunf
No : the works of Du Bois-Beymond demonstrate it in a striking way.
We know that mathematicians distinguish between infinitesi- mals of different orders and that those of the second order are infinitesimal, not only in an absolute way, but also in relatioi to those of the first order. It is not difficult to imagine infinites- imals of fractional or even of irrational order, and thus we find again that scale of the mathematical continuum which has beei dealt with in the preceding pages.
MATHEMATICAL MAGNITUDE AND EXPERIENCE 51
Farther, there are infinitesimals which are infinitely small in relation to those of the first order, and, on the contrary, infinitely great in relation to those of order 1 + c, and that however small c may be. Here, then, are new terms intercalated in our series, and if I may be permitted to revert to the phraseology lately em- ployed which is very convenient though not consecrated by usage, I shall say that thus has been created a sort of continuum of the third order.
It would be easy to go further, but that would be idle; one would only be imagining symbols without possible application, and no one will think of doing that. The continuum of the third order, to which the consideration of the different orders of infini- tesimals leads, is itself not useful enough to have won citizenship, and geometers regard it only as a mere curiosity. The mind uses its creative faculty only when experience requires it.
2^ Once in possession of the concept of the mathematical con- tinuum, is one safe from contradictions analogous to those which gave birth to it?
No, and I will give an example.
One must be very wise not to regard it as evident that every curve has a tangent ; and in fact if we picture this curve and a straight as two narrow bands we can always so dispose them that they have a part in common without crossing. If we imagine then the breadth of these two bands to diminish indefinitely, this common part will always subsist and, at the limit, so to speak, the two lines will have a point in common without crossing, that is to say, they will be tangent.
The geometer who reasons in this way, consciously or not, is only doing what we have done above to prove two lines which cut have a point in common, and his intuition might seem just as legitimate.
It would deceive him however. We can demonstrate that there are curves which have no tangent, if such a curve is de- fined as an analytic continuum of the second order.
Without doubt some artifice analogous to those we have dis- enssed above would have suflBced to remove the contradiction; but, as this is met with only in very exceptional cases, it has received no further attention.
52 SCIENCE AND HYPOTHESIS
Instead of seeking to reconcile intuition with analysis, we have been content to sacrifice one of the two, and as analysis must remain impeccable, we have decided against intuition.
The Physical Continuum op Several Dimensions. — ^We have discussed above the physical continuum as derived from the immediate data of our senses, or, if you wish, from the rough re- sults of Fechner's experiments; I have shown that these results are summed up in the contradictory formulas
A=zB, B = C, A<C.
Let us now see how this notion has been generalized and how from it has come the concept of many-dimensional continua.
Consider any two aggregates of sensations. Either we can discriminate them one from another, or we can not, just as in Fechner's experiments a weight of 10 grams can be distinguished from a weight of 12 grams, but not from a weight of 11 grams. This is all that is required to construct the continuum of several dimensions.
Let us call one of these aggregates of sensations an element. That will be something analogous to the point of the mathe- maticians; it will not be altogether the same thing however. We can not say our element is without extension, since we can not distinguish it from neighboring elements and it is thus surrounded by a sort of haze. If the astronomical comparison may be allowed, our * elements' would be like nebulae, whereas the mathematical points would be like stars.
^p That being granted, a system of elements will form a con- Hinuum if we can pass from any one of them to any other, by a V,*- if series of consecutive elements such that each is indistinguish- j able from the preceding. This linear series is to the line of the ^i" ■ mathematician what an isolated element was to the point.
/ Before going farther, I must explain what is meant by a I cut. Consider a continuum C and remove from it certain of its 1 elements which for an instant we shall regard as no longer be- \ longing to this continuum. The aggregate of the elements so removed will be called a cut. It may happen that, thanks to this cut, C may be subdivided into several distinct continua, the ag- t gregate of the remaining elements ceasing to form a unique con- / tinuum.
V
V
MATHEMATICAL MAGNITUDE AND EXPERIENCE 63
There will then be on C two elements, A and B, that must be \ regarded as belonging to two distinct continua, and this will be J v ^ recognized because it will be impossible to find a linear series j ^ of consecutive elements of C, each of these elements indistin- \^ i^ '^ guishable from the preceding, the first being A and the last B, I without one of the elements of this series being indistinguishable \ . /^ from one of the elements of the cut. j ^ a-'
On the contrary, it may happen that the cut made is insuffi-j cient to subdivide the continuum C. To classify the physical! continua, we will examine precisely what are the cuts which must \ be made to subdivide them.
If a physical continuum C can be subdivided by a cut reduc- ing to a finite number of elements all distinguishable from one another (and consequently forming neither a continuum, nor several continua), we shall say C is a one-dimensional continuum.
If, on the contrary, C can be subdivided only by cuts which are themselves continua, we shall say C has several dimen- sions. If cuts which are continua of one dimension sufiSce, we shall say C has two dimensions ; if cuts of two dimensions sufSce, we shall say C has three dimensions, and so on.
Thus is defined the notion of the physical continuum of several dimensions, thanks to this very simple fact that two aggregates of sensations are distinguishable or indistinguishable.
The Mathematical Continuum op Several Dimensions. — Thence the notion of the mathematical continuum of n dimen- sions has sprung quite naturally by a process very like that we discussed at the beginning of this chapter. A point of such a continuum, you know, appears to us as defined by a system of n distinct magnitudes called its coordinates.
These magnitudes need not always be measurable; there is, for instance, a branch of geometry independent of the measure- ment of these magnitudes, in which it is only a question of know- ing, for example, whether on a curve ABC, the point B is be- tween the points A and C, and not of knowing whether the arc AB is equal to the arc BC or twice as great. This is what is called Analysis Situs.
This is a whole body of doctrine which has attracted the
64 SCIENCE AND HYPOTHESIS
attention of the greatest geometers and where we see flow one fram another a series of remarkable theorems. What distin- guishes these theorems from those of ordinary geometry is that they are purely qualitative and that they would remain true if the figures were copied by a draughtsman so awkward as to grossly distort the proportions and replace straights by strokes more or less curved.
Through the wish to introduce measure next into the contin- uum just defined this continuum becomes space, and geometry is born. But the discussion of this is reserved for Part Second.
PART II
SPACE
CHAPTER III The Non-Euclidean Geometries
Evert conclusion supposes premises ; these premises themselves either are self-evident and need no demonstration, or can be established only by relying upon other propositions, and since we can not go back thus to infinity, every deductive science, and in particular geometry, must rest on a certain number of unde- monstrable axioms. All treatises on geometry begin, therefore, by the enunciation of these axioms. But among these there is a distinction to be made: Some, for example, * Things which are equal to the same thing are equal to one another, ' are not propo- sitions of geometry, but propositions of analysis. I regard them as analytic judgments a priori, and shall not concern myself with them.
But I must lay stress upon other axioms which are peculiar to geometry. Most treatises enunciate three of these explicitly :
1® Through two points can pass only one straight;
2** The straight line is the shortest path from one point to another ;
3** Through a given point there is not more than one parallel to a given straight.
Although generally a proof of the second of these axioms is omitted, it would be possible to deduce it from the other two and from those, much more numerous, which are implicitly admitted without enunciating them, as I shall explain further on.
It was long sought in vain to demonstrate likewise the third axiom, known as Euclid^ s Postulate. What vast effort has been wasted in this chimeric hope is truly unimaginable. Finally, in
55
56 SCIENCE AND HYPOTHESIS
the first quarter of the nineteenth century, and almost at the same time, a Hungarian and a Russian, Bolyai and Lobachevski, established irrefutably that this demonstration is impossible ; they have almost rid us of inventors of geometries *sans postulatum'; since then the Academic des Sciences receives only about one or two new demonstrations a year.
The question was not exhausted; it soon made a great stride by the publication of Riemann's celebrated memoir en- titled: Ueber die Hypothesen welche der Oeometrie zu Orunde liegen. This paper has inspired most of the recent works of which I shall speak further on, and among which it is proper to cite those of Beltrami and of Helmholtz.
The Bolyai-Lobachevski Geometby. — If it were possible to deduce Euclid's postulate from the other axioms, it is evident that in denying the postulate and admitting the other axioms, we should be led to contradictory consequences; it would therefore be impossible to base on such premises a coherent geometry.
Now this is precisely what Lobachevski did.
He assumes at the start that: Through a given point can he drawn two parallels to a given straight.
And he retains besides all Euclid's other axioms. From these hypotheses he deduces a series of theorems among which it is impossible to find any contradiction, and he constructs a geometry whose faultless logic is inferior in nothing to that of the Euclidean geometry.
The theorems are, of course, very different from those to which we are accustomed, and they can not fail to be at first a little disconcerting.
Thus the sum of the angles of a triangle is always less than two right angles, and the diflference between this sum and two right angles is proportional to the surface of the triangle.
It is impossible to construct a figure similar to a given figure but of different dimensions.
If we divide a circumference into n equal parts, and draw tangents at the points of division, these n tangents will form a polygon if the radius of the circle is small enough; but if this radius is sufficiently great they will not meet.
It is useless to multiply these examples; Lobachevski 's propo-
THE NON'EUCLIDEAN GEOMETRIES 67
sitions have no relation to those of Euclid, but they are not less logically bound one to another.
Biemann's Geometby. — Imagine a world uniquely peopled by beings of no thickness (height) ; and suppose these infinitely flat' animals are all in the same plane and can not get out. Ad- mit besides that this world is sufficiently far from others to be free from their influence. While we are making hypotheses, it costs us no more to endow these beings with reason and believe them capable of creating a geometry. In that case, they will cer- tainly attribute to space only two dimensions.
But suppose now that these imaginary animals, while remain- ing without thickness, have the form of a spherical, and not of a plane, figure, and are all on the same sphere without power to get off. What geometry will they construct? First it is clear they will attribute to space only two dimensions; what will play for them the role of the straight line will be the shortest path from one point to another on the sphere, that is to say, an arc of a great circle ; in a word, their geometry will be the spherical geometry.
What they will call space will be this sphere on which they must stay, and on which happen all the phenomena they can know. Their space will therefore be unbou^ided since on a sphere one can always go forward without ever being stopped, and yet it will be finite; one can never find the end of it, but one can make a tour of it.
Well, Riemann's geometry is spherical geometry extended to three dimensions. To construct it, the German mathematician had to throw overboard, not only Euclid's postulate, but also the first axiom : Only one straight can pass through two points.
On a sphere, through two given points we can draw in general only one great circle (which, as we have just seen, would play the role of the straight for our imaginary beings) ; but there is an exception : if the two given points are diametrically opposite, an infinity of great circles can be drawn through them.
In the same way, in Riemann's geometry (at least in one of its forms) , through two points will pass in general only a single straight; but there are exceptional cases where through two points an infinity of straights can pass.
68 SCIENCE AND HYPOTHESIS
There is a sort of opposition between Riemann's geometry and that of Lobaehevski.
Thus the sum of the angles of a triangle is :
Equal to two right angles in Euclid's geometry;
Less than two right angles in that of Lobaehevski ;
Greater than two right angles in that of Biemann.
The number of straights through a given point that can be drawn coplanar to a given straight, but nowhere meeting it, is equal :
To one in Euclid's geometry;
To zero in that of Riemann ;
To infinity in that of Lobaehevski.
Add that Biemann 's space is finite, although unbounded, in the sense given above to these two words.
The Surfaces op Constant Cubvatube. — One objection still remained possible. The theorems of Lobaehevski and of Bie- mann present no contradiction ; but however numerous the con- sequences these two geometers have drawn from their hypotheses, they must have stopped before exhausting them, since their num- ber would be infinite ; who can say then that if they had pushed their deductions farther they would not have eventually reached some contradiction f
This diflSculty does not exist for Biemann 's geometry, pro- vided it is limited to two dimensions; in fact, as we have seen, two-dimensional Riemannian geometry does not differ from spher- ical geometry, which is only a branch of ordinary geometry, and consequently is beyond all discussion.
Beltrami, in correlating likewise Lobaehevski 's two-dimen- sional geometry with a branch of ordinary geometry, has equally refuted the objection so far as it is concerned.
Here is how he accomplished it. Consider any figure on a surface. Imagine this figure traced on a flexible and inextensible canvas applied over this surface in such a way that when the canvas is displaced and deformed, the various lines of this figure can change their form without changing their length. In gen- eral, this flexible and inextensible figure can not be displaced without leaving the surface ; but there are certain particular sur-
TBE NON-EUCLIDEAN GEOMETRIES 69
faces for which such a movement would be possible ; these are the surfaces of constant curvature.
If we resume the comparison made above and imagine beings without thickness living on one of these surfaces, they will regard as possible the motion of a figure all of whose lines remain con- stant in length. On the contrary, such a movement would appear absurd to animals without thickness living on a surface of vari- able curvature.
These surfaces of constant curvature are of two sorts: Some are of positive curvature, and can be deformed so as to be applied over a sphere. The geometry of these surfaces reduces itself therefore to the spherical geometry, which is that of Riemann.
The others are of negative curvature. Beltrami has shown that the geometry of these surfaces is none other than that of Lobachevski. The two-dimensional geometries of Riemann and Lobachevski are thus correlated to the Euclidean geometry.
Interpretation op Non-Euclidean Geometries. — So van- ishes the objection so far as two-dimensional geometries are con- cerned.
It would be easy to extend Beltrami's reasoning to three- dimensional geometries. The minds that space of four dimen- sions does not repel will see no diflSculty in it, but they are few. I prefer therefore to proceed otherwise.
Consider a certain plane, which I shall call the fundamental plane, and construct a sort of dictionary, by making correspond each to each a double series of terms written in two columns, just as correspond in the ordinary dictionaries the words of two lan- guages whose significance is the same :
Space: Portion of space situated above the fundamental plane.
Plane: Sphere cutting the fundamental plane orthogonally.
Straight: Circle cutting the fundamental plane orthogonally.
Sphere: Sphere.
Circle: Circle.
Angle: Angle.
Distance between two points: Logarithm of the cross ratio of these two points and the intersections of the fundamental plane with a circle passing through these two points and cutting it orthogonally. Etc., Etc.
60 SCIENCE AND HYPOTHESIS
Now take Lobachevski's theorems and translate them with the aid of this dictionary as we transate a German text with the aid of a German-English dictionary. We shall thus, obtain the- orems of the ordinary geometry. For example, that theorem of Lobachevski : ^the sum of the angles of a triangle is less than two right angles' is translated thus: ^'If a curvilinear triangle has for sides circle-arcs which prolonged would cut orthogonally the fundamental plane, the sum of the angles of this curvilinear tri- angle will be less than two right angles." Thus, however far the consequences of Lobachevski's hypotheses are pushed, they will never lead to «a contradiction. In fact, if two of Lobachevski's theorems were contradictory, it would be the same with the trans- lations of these two theorems, made by the aid of our dictionary, but these translations are theorems of ordinary geometry and no one doubts that the ordinary geometry is free from contradiction. Whence comes this certainty and is it justified? That is a ques- tion I can not treat here because it would require to be enlarged upon, but which is very interesting and I think not insoluble.
Nothing remains then of the objection above formulated. This is not all. Lobachevski's geometry, susceptible of a concrete interpretation, ceases to be a vain logical exercise and is capal)le of applications ; I have not the time to speak here of these appli- cations, nor of the aid that Klein and I have gotten from them for the integration of linear differential equations.
This interpretation moreover is not unique, and several dic- tionaries analogous to the preceding could be constructed, which would enable us by a simple 'translation' to transform Loba- chevski's theorems into theorems of ordinary geometry.
The Implicit Axioms. — Are the axioms explicitly enunciated in our treatises the sole foundations of geometry? We may be assured of the contrary by noticing that after they are succes- sively abandoned there are still left over some propositions com- mon to the theories of Euclid, Lobachevski and Riemann. These propositions must rest on premises the geometers admit without enunciation. It is interesting to try to disentangle them from the classic demonstrations.
Stuart Mill has claimed that every definition contains an
THE NON'EUCLIDEAN GEOMETRIES 61
axiom, because in defining one affirms implicitly the existence of the object defined. This is going much too far ; it is rare that in mathematics a definition is given without its being followed by the demonstration of the existence of the object defined, and when this is dispensed with it is generally because the reader can easily supply it. It must not be forgotten that the word existence has not the same sense when it refers to a mathematical entity and when it is a question of a material object. A mathe- matical entity exists, provided its definition implies no contradic- tioUy either in itself, or with the propositions already admitted.
But if Stuart Mill's observation can not be applied to all definitions, it is none the less just for some of them. The plane is sometimes defined as follows :
The plane is a surface such that the straight which joins any two of its points is wholly on this surface.
This definition manifestly hides a new axiom; it is true we might change it, and that would be preferable, but then we should have to enunciate the axiom explicitly.
Other definitions would suggest reflections not less important.
Such, for example, is that of the equality of two figures ; two figures are equal when they can be superposed; to superpose them one must be displaced until it coincides with the other ; but how shall it be displaced? If we should ask this, no doubt we should be told that it must be done without altering the shape and as a rigid solid. The vicious circle would then be evident.
In fact this definition defines nothing; it would have no mean- ing for a being living in a world where there were only fluids. If it seems clear to us, that is because we are used to the proper- ties of natural solids which do not differ much from those of the ideal solids, all of whose dimensions are invariable.
Yet, imperfect as it may be, this definition implies an axiom.
The possibility of the motion of a rigid figure is not a self- evident truth, or at least it is so only in the fashion of Euclid's postulate and not as an analytic judgment a priori would be.
Moreover, in studying the definitions and the demonstrations of geometry, we see that one is obliged to admit without proof not only the possibility of this motion, but some of its properties besides.
62 SCIENCE AND HYPOTHESIS
This is at once seen from the definition of the straight line. Many defective definitions have been given, but the true one is that which is implied in all the demonstrations where the straight line enters:
''It may happen that the motion of a rigid figure is such that all the points of a line belonging to this figure remain motionless while all the points situated outside of this line move. Such a line will be called a straight line." We have designedly, in this enunciation, separated the definition from the axiom it implies.
Many demonstrations, such as those of the cases of the equality of triangles, of the possibility of dropping a perpendicular from a point to a straight, presume propositions which are not enun- ciated, for they require the admission that it is possible to trans- port a figure in a certain way in space.
The Fourth Geometry. — Among these implicit axioms, there is one which seems to me to merit some attention, because when it is abandoned a fourth geometry can be constructed as coherent as those of Euclid, Lobachevski and Biemann.
To prove that a perpendicular may always be erected at a point A to a straight AB, we consider a straight AC movable around the point A and initially coincident with the fixed straight AB; and we make it turn about the point A until it comes into the prolongation of AB.
Thus two propositions are presupposed : First, that such a ro- tation is possible, and next that it may be continued until the two straights come into the prolongation one of the other.
If the first point is admitted and the second rejected, we are led to a series of theorems even stranger than those of Loba- chevski and Riemann, but equally exempt from contradiction.
I shall cite only one of these theorems and that not the most singular: A real straight may be perpendicular to itself.
LiE^s Theorem. — The number of axioms implicitly intro- duced in the classic demonstrations is greater than necessary, and it would be interesting to reduce it to a minimum. It may first be asked whether this reduction is possible, whether the number of necessary axioms and that of imaginable geometries are not infinite.
THE NON^-EUCLIDEAN GEOMETBIES 63
A theorem of Sophus Lie dominates this whole discussion. It may be thus enunciated:
Suppose the following premises are admitted:
1^ Space has n dimensions;
2^ The motion of a rigid figure is possible;
3^ It requires p conditions to determine the position of this figure in space.
The number of geometries compatible with these premises luill he limited.
I may even add that if n is given, a superior limit can be assigned to p.
If therefore the possibility of motion is admitted, there can be invented only a finite (and even a rather small) number of three-dimensional geometries.
Biemann's Geometries. — ^Tet this result seems contradicted by Biemann, for this savant constructs an infinity of different geometries, and that to which his name is ordinarily given is only a particular case.
All depends, he says, on how the length of a curve is defined. NW, there is an infinity of ways of defining this length, and each of them may be the starting point of a new geometry.
That is perfectly true, but most of these definitions are incom- patible with the motion of a rigid figure, which in the theorem of Lie is supposed possible. These geometries of Rieraann, in many ways so interesting, could never therefore be other than purely analytic and would not lend themselves to demonstrations analogous to those of Euclid.
On the Nature op Axioms. — Most mathematicians regard Lobachevski's geometry only as a mere logical curiosity; some of them, however, have gone farther. Since several geometries are possible, is it certain ours is the true one ? Experience no doubt teaches us that the sum of the angles of a triangle is equal to two right angles ; but this is because the triangles we deal with are too little; the difference, according to Lobachevski, is propor- tiozud to the surface of the triangle ; will it not perhaps become KQgible when we shall operate on larger triangles or when our nteasurements shall become more precise ? The Euclidean geom- etry would thus be only a provisional geometry.
64 SCIENCE AND HYPOTHESIS
To discuss this opinion, we should first ask ourselves what is the nature of the geometric axioms.
Are they synthetic a priori judgments, as Kant said!
They would then impose themselves upon us with such force that we could not conceive the contrary proposition, nor build upon it a theoretic edifice. There would be no non-Euclidean geometry.
To be convinced of it take a veritable synthetic a priori judgment, the following, for instance, of which we have seen the preponderant role in the first chapter :
// a theorem is true for the number 1, and if it has been proved that it is true of n-\-l provided it is true of n, it u)iU be true of all the positive whole numbers.
Then try to escape from that and, denying this proposition, try to found a false arithmetic analogous to non-Euclidean geometry — ^it can not be done ; one would even be tempted at first blush to regard these judgments as analytic.
Moreover, resuming our fiction of animals without thickness, we can hardly admit that these beings, if their minds are like ours, would adopt the Euclidean geometry which would be con- tradicted by all their experience.
Should we therefore conclude that the axioms of geometry are experimental verities? But we do not experiment on ideal straights or circles; it can only be done on material objects. On what then could be based experiments which should serve as foundation for geometry? The answer is easy.
We have seen above that we constantly reason as if the geo- metric figures behaved like solids. What geometry would bor- row from experience would therefore be the properties of these bodies. The properties of light and its rectilinear propagation have also given rise to some of the propositions of geometry, and in particular those of projective geometry, so that from this point of view one would be tempted to say that metric geometry is the study of solids, and projective, that of light.
But a difficulty remains, and it is insurmountable. If geom- etry were an experimental science, it would not be an exact science, it would be subject to a continual revision. Nay, it would from this very day be convicted of error, since we know that there is no rigorously rigid solid.
THE NON-^EUCLIDEAN GEOMETRIES 66
The axioms of geometry therefore are neither synthetic a priori judgments nor experimental facts.
They are conventions; our choice amon^ all possible conven- tions is guided by experimental facts ; but it remains free and is limited only by the necessity of avoiding all contradiction. Thus it is that the postulates can remain rigorously true even though the experimental laws which have determined their adoption are only approximative.
In other words, the ctxioms of geometry (I do not speak of those of arithmetic) are merely disguised definitions.
Then what are we to think of that question : Is the Euclidean geometry truet
It has no meaning.
As well ask whether the metric system is true and the old measures false ; whether Cartesian coordinates are true and polar coordinates false. One geometry can not be more true than an- other; it can only be more convenient.
Now, Euclidean geometry is, and will remain, the most con- venient :
1^ Because it is the simplest ; and it is so not only in conse- quence of our mental habits, or of I know not what direct in- tuition that we may have of Euclidean space ; it is the simplest in itself, just as a polynomial of the first degree is simpler than one of the second; the formulas of spherical trigonometry are more complicated than those of plane trigonometry, and they would still appear so to an analyst ignorant of their geometric signifi- cation.
2® Because it accords sufficiently well with the properties of natural solids, those bodies which our hands and our eyes com- pare and with which we make our instruments of measure.
6
CHAPTER IV Space and Geometry
Let us begin by a little paradox.
Beings with minds like ours, and having the same senses as we, but without previous education, would receive from a suitably chosen external world impressions such that they would be led to construct a geometry other than that of Euclid and to localize the phenomena of that external world in a non-Euclidean space, or even in a space of four dimensions.
As for us, whose education has been accomplished by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring its phenomena to our Euclidean space. Conversely, if these beings were trans- ported into our environment, they would be led to relate our phenomena to non-Euclidean space.
Nay more; with a little effort we likewise could do it. A person who should devote his existence to it might perhaps attain to a realization of the fourth dimension.
Geometric Space and Perceptual Space. — It is often said the images of external objects are localized in space, even that they can not be formed except on this condition. It is also said that this space, which serves thus as a ready prepared frame for our sensations and our representations, is identical with that of the geometers, of which it possesses all the properties.
To all the good minds who think thus, the preceding state- ment must have appeared quite extraordinary. But let us see whether they are not subject to an illusion that a more profound analysis would dissipate.
What, first of all, are the properties of space, properly so called! I mean of that space which is the object of geometry and which I shall call geometric space.
The following are some of the most essential:
1** It is continuous;
66
SPACE AND GEOMETBT 67
2° It is infinite;
3° It has three dimensions;
4° It is homogeneous, that is to say, all its i>oints are identical one with another;
5^ It is isotropic, that is to say, all the straights which pass through the same point are identical one with another.
Compare it now to the frame of our representations and our sensations, which I may call perceptual space.
Visual Space. — Consider first a purely visual impression, due to an image formed on the bottom of the retina.
A cursory analysis shows us this image as continuous, but as possessing only two dimensions; this already distinguishes from geometric space what we may call pure visual space.
Besides, this image is enclosed in a limited frame.
Finally, there is another difference not less important: this pure visvM space is not homogeneous. All the points of the retina, aside from the images which may there be formed, do not play the same role. The yellow spot can in no way be regarded as identical with a point on the border of the retina. In fact, not only does the same object produce there much more vivid im- pressions, but in every limited frame the point occupying the center of the frame will never appear as equivalent to a point near one of the borders.
No doubt a more profound analysis would show us that this continuity of visual space and its two dimensions are only an illusion ; it would separate it therefore still more from geometric space, but we shall not dwell on this remark.
Sight, however, enables us to judge of distances and conse- quently to perceive a third dimension. But every one knows that this perception of the third dimension reduces itself to the sensation of the effort at accommodation it is necessary to make, and to that of the convergence which must be given to the two eyes, to perceive an object distinctly.
These are muscular sensations altogether different from the visual sensations which have given us the notion of the first two dimensions. The third dimension therefore will not appear to us as playing the same role as the other two. What may be called complete visual space is therefore not an isotropic space.
68 SCIENCE AND HYPOTHESIS
It has, it is true, precisely three dimensions, which means that the elements of our visual sensations (those at least which com- bine to form the notion of extension) will be completely de- fined when three of them are known; to use the language of mathematics, they will be functions of three independent variables.
But examine the matter a little more closely. The third dimension is revealed to us in two different ways: by the effort of accommodation and by the convergence of the eyes.
No doubt these two indications are always concordant, there is a constant relation between them, or, in mathematical terms, the two variables which measure these two muscular sensations do not appear to us as independent ; or again, to avoid an appeal to mathematical notions already rather refined, we may go back to the language of the preceding chapter and enunciate the same fact as follows : If two sensations of convergence, A and B, are indistinguishable, the two sensations of accommodation, A' and B'f which respectively accompany them, will be equally indistin- guishable.
But here we have, so to speak, an experimental fact; a priori nothing prevents our supposing the contrary, and if the contrary takes place, if these two muscular sensations vary independently of one another, we shall have to take account of one more inde- pendent variable, and 'complete visual space' will appear to us as a physical continuum of four dimensions.
We have here even, I will add, a fact of external experience. Nothing prevents our supposing that a being with a mind like ours, having the same sense organs that we have, may be placed in a world where light would only reach him after having traversed reflecting media of complicated form. The two indi- cations which serve us in judging distances would cease to be connected by a constant relation. A being who should achieve in such a world the education of his senses would no doubt attribute four dimensions to complete visual space.
Tactile Space and Motor Space. — ^'Tactile space' is still more complicated than visual space and farther removed from geometric space. It is superfluous to repeat for touch the discus- sion I have given for sight.
SPACE AND GEOMETBT 69
But apart from the data of sight and touch, there are other sensations which contribute as much and more than they to the genesis of the notion of space. These are known to every one; they accompany all our movements, and are usually called mus- cular sensations.
The corresponding frame constitutes what may be called motor space.
Each muscle gives rise to a special sensation capable of aug- menting or of diminishing, so that the totality of our muscular sensations will depend upon as many variables as we have muscles. From this point of view, motor space would have as many dimensions as we have mtiscles.
I know it will be said that if the muscular sensations con- tribute to form the notion of space, it is because we have the sense of the direction of each movement and that it makes an integrant part of the sensation. If this were so, if a muscular sensation could not arise except accompanied by this geometric 9ense of direction, geometric space would indeed be a form im- posed upon our sensibility.
But I perceive nothing at all of this when I analyze my sen- sations.
What I do see is that the sensations which correspond to move- ments in the same direction are connected in my mind by a mere association of ideas. It is to this association that what we call 'the sense of direction' is reducible. This feeling therefore can not be found in a single sensation.
This association is extremely complex, for the contraction of the same muscle may correspond, according to the position of the limbs, to movements of very different direction.
Besides, it is evidently acquired; it is, like all associations of ideas, the result of a habit; this habit itself results from very numerous experiences; without any doubt, if the education of our senses had been accomplished in a different environment, where we should have been subjected to different impressions, con- trary habits would have arisen and our muscular sensations would have been associated according to other laws.
Chabacteristics op Perceptual Space. — Thus perceptual space, under its triple form, visual, tactile and motor, is essen- tially different from geometric space.
70 SCIENCE AND HYPOTHESIS
It is neither homogeneous, nor isotropic ; one can not even say that it has three dimensions.
It is often said that we 'project' into geometric space the objects of our external perception; that we 'localize' them.
Has this a meaning, and if so whatf
Does it mean that we represent to ourselves external objects in geometric space f
Our representations are only the reproduction of our sensa- tions; they can therefore be ranged only in the same frame as these, that is to say, in perceptual space.
It is as impossible for us to represent to ourselves external bodies in geometric space, as it is for a painter to paint on a plane canvas objects with their three dimensions.
Perceptual space is only an image of geometric space, an image altered in shape by a sort of perspe<Hive, and we can repre- sent to ourselves objects only by bringing them under the laws of this perspective.
Therefore we do not represent to ourselves external bodies in geometric space, but we reason on these bodies as if they were situated in geometric space.
When it is said then that we 'localize' such and such an object at such and such a point of space, what does it meant
It simply means that we represent to ourselves the movements it would be necessary to m>ake to reach that object; and one may not say that to represent to oneself these movements, it is neces- sary to project the movements themselves in space and that the notion of space must, consequently, pre-exist.
When I say that we represent to ourselves these movements, I mean only that we represent to ourselves the muscular sensa- tions which accompany them and which have no geometric char- acter whatever, which consequently do not at all imply the pre- existence of the notion of space.
Change op State and Change op Position. — ^But, it will be said, if the idea of geometric space is not imposed upon our mind, and if, on the other hand, none of our sensations can furnish it, how could it have come into existence?
This is what we have now to examine, and it will take some time, but I can summarize in a few words the attempt at explana- tion that I am about to develop.
SPACE AND GEOMETRY 71
None of our sensations, isolated, could have conducted us to ike idea of space; we are led to it only in studying the laws, according to which these sensations succeed each other.
We see first that our impressions are subject to change; but among the changes we ascertain we are soon led to make a dis- tinction.
At one time we say that the objects which cause these im- pressions have changed state, at another time that they have changed position, that they have only been displaced.
Whether an object changes its state or merely its position, this is always translated for us in the same manner: by a modifi- cation in an aggregate of impressions.
How then could we have been led to distinguish between the twof It is easy to account for. If there has only been a change of position, we can restore the primitive aggregate of impressions by making movements which replace us opposite the mobile object in the same relative situation. We thus correct the modification that happened and we reestablish the initial state by an inverse modification.
If it is a question of sight, for example, and if an object changes its place before our eye, we can * follow it with the eye' and maintain its image on the same point of the retina by appropriate movements of the eyeball.
These movements we are conscious of because they are volun- tary and because they are accompanied by muscular sensations, but that does not mean that we represent them to ourselves in geometric space.
So what characterizes change of position, what distinguishes it from change of state, is that it can always be corrected in this way.
It may therefore happen that we pass from the totality of impressions A to the totality B in two different ways :
1** Involuntarily and without experiencing muscular sensa- tions ; this happens when it is the object which changes place ;
2*" Voluntarily and with muscular sensations; this happens when the object is motionless, but we move so that the object has relative motion with reference to us.
If this be so, the passage from the totality A to the totality B is only a change of position.
72 SCIENCE AND STPOTHESIS
It follows from this that sight and toach could not have given UB the notion of space without the aid of the 'muscular sense.'
Not only could this notion not be derived from a single sen' sation or even from a series of sensations, but what is more, an immobile being could never have acquired it, since, not being able to correct by his movements the effects of the changes of position of exterior objects, he would have had no reason what- ever to distinguish them from changes of state. Just as little could he have acquired it if his motions had not been voluntary or were unaccompanied by any sensations.
Conditions op Compensation. — How is a like compensation possible, of such sort that two changes, otherwise independent of each other, reciprocally correct each othert
A mind already familiar with geometry would reason as fol- lows: Evidently, if there is to be compensation, the various parts of the external object, on the one hand, and the various sense organs, on the other hand, must be in the same relative poffltioD after the double change. And, for that to be the case, the various parts of the external object must likewise have retained in reference to each other the same relative pontion, and the same must be true of the various parts of our body in regard to each other.
In other words, the external object, in the first change, must be displaced as is a rigid solid, nnd so must it be with the whole of our body in the second change which corrects the first.
Under these conditions, oompunsation may take place.
But we who as yet know nothing of geometry, since for ia"fl| notion of space is not yet formed, we can not reason i can not foresee a priori whether compensatiou is pOssibl experience teaches us that it sometimes happens, and it j this experimental fact that we star: to distingoiili t state from changes of position.
Solid Bodies and Gbouets there are some which ^tqv ceptible of being thus eon our own body; these are
SPACE AND GEOMETRY
V3
whose form is variable, only exceptionally undergo like displace- ments (change of position without change of form). When « body changes ita place and its shape, we can no longer, by appro- priate movements, bring back our sense-organs into the same relative situation with regard to this body; consequently we can DO longer reestablish the primitive totality of impressions.
It is only later, and as a consequence of new experiences, that we learn how to decompose the bodies of variable form into smaller elements, such that each ia displaced almost in accord- ance with the same laws as solid bodies. Thus we distinguish 'deformations" from other changes of state; in these deforma- tions, each element undergoes a mere change of position, which can he corrected, but the modification undergone by the aggre- gate is more profound and is no longer susceptible of correction by a correlative movement.
8uch a notion is already very complex and must have been relatively late in appearing ; moreover it could not have arisen if the observation of solid bodies f d not already taught us to dis- tiDguish changes of portion.
Therefore, if there were no solid bodies in nature, there wcndd ht no geometry.
Another remark also deserves a moment's attention. Suppose a solid body to occupy successively the positions a. and p; in its first position, it wrill produce on us the totality of impressions A, lity of impressions B. Let viag qualities entirely djffer- ilerent color. Suppose it to us the totality of im- the totality of irn-
eommon with
;y B'. The trau-
that from the
:e8 which in
74 SCIENCE AND HYPOTHESIS
It is simply because they can both be corrected by the same correlative movement of our body.
'Correlative movement' therefore constitutes the sole connec- tion between two phenomena which otherwise we never should have dreamt of likening.
On the other hand, our body, thanks to the number of its articulations and muscles, may make a multitude of different movements; but all are not capable of 'correcting' a modification of external objects ; only those will be capable of it in which our whole body, or at least all those of our sense-organs which come into play, are displaced as a whole, that is, without their relative positions varying, or in the fashion of a solid body.
To summarize:
1^ We are led at first to distinguish two categories of phe- nomena :
Some, involuntary, unaccompanied by muscular sensations, are attributed by us to external objects ; these are external changes ;
Others, opposite in character and attributed by us to the movements of our own body, are internal changes ;
2** We notice that certain changes of each of these categories may be corrected by a correlative change of the other category;
3** We distinguish among external changes those which have thus a correlative in the other category; these we call displace- ments; and just so among the internal changes, we distinguish those which have a correlative in the first category.
Thus are defined, thanks to this reciprocity, a particular class of phenomena which we call displacements.
The laws of these phenomena constitute the object of geometry.
Law op Homogeneity. — The first of these laws is the law of homogeneity.
Suppose that, by an external change a, we pass from the total- ity of impressions A to the totality B, then that this change a is corrected by a correlative voluntary movement j8, so that we are brought back to the totality A.
Suppose now that another external change a makes us pass anew from the totality A to the totality B.
Experience teaches us that this change a is, like a, sus- ceptible of being corrected by a correlative voluntary movement
SPACE AND GEOMETRY 75
fif and that this moyement p' corresponds to the same mnscnlar sensations as the movement p which corrected a.
This fact is usually enunciated by saying that space is homo- geneous and isotropic.
It may also be said that a movement which has once been pro- duced may be repeated a second and a third time, and so on, without its properties varying.
In the first chapter, where we discussed the nature of mathe- matical reasoning, we saw the importance which must be attributed to the possibility of repeating indefinitely the same operation.
It is from this repetition that mathematical reasoning gets its I>ower; it is, therefore, thanks to the law of homogeneity, that it has a hold on the geometric facts.
For completeness, to the law of homogeneity should be added a multitude of other analogous laws, into the details of which I do not wish to enter, but which mathematicians sum up in a word by saying that displacements form 'a group.'
The Non-Eucudean World. — If geometric space were a frame imposed on each of our representations, considered indi- vidually, it would be impossible to represent to ourselves an image stripped of this frame, and we could change nothing of our geometry.
But this is not the ease ; geometry is only the resume of the laws according to which these images succeed each other. Noth- ing then prevents us from imagining a series of representations, similar in all points to our ordinary representations, but suc- ceeding one another according to laws different from those to which we are accustomed.
We can conceive then that beings who received their educa- tion in an environment where these laws were thus upset might have a geometry very different from ours.
Suppose, for example, a world enclosed in a great sphere and subject to the following laws:
The temperature is not uniform; it is greatest at the center, and diminishes in proportion to the distance from the center, to sink to absolute zero when the sphere is reached in which this world is enclosed.
76 SCIENCE AND HYPOTHESIS
To specify still more precisely the law in accordance with which this temperature varies: Let B be the radius of the lim- iting sphere; let r be the distance of the point considered from the center of this sphere. The absolute temperature shall be proportional to J2* — r*.
I shall further suppose that, in this world, all bodies have the same coefScient of dilatation, so that the length of any rule is proportional to its absolute temperature.
Finally, I shall suppose that a body transported from one point to another of different temperature is put immediately into thermal equilibrium with its new environment.
Nothing in these hypotheses is contradictory or unimaginable.
A movable object will then become smaller and smaller in pro- portion as it approaches the limit-sphere.
Note first that, though this world is limited from the point of view of our ordinary geometry, it will appear infinite to its inhabitants.
In fact, when these try to approach the limit-sphere, they cool off and become smaller and smaller. Therefore the steps they take are also smaller and smaller, so that they can never reach the limiting sphere.
If, for us, geometry is only the study of the laws according to which rigid solids move, for these imaginary beings it will be the study of the laws of motion of solids distorted hy the differ- ences of temperature just spoken of.
No doubt, in our world, natural solids likewise undergo varia- tions of form and volume due to warming or cooling. But we neglect these variations in laying the foundations of geometry, because, besides their being very slight, they are irregular and consequently seem to us accidental.
In our hypothetical world, this would no longer be the case, and these variations would follow regular and very simple laws.
Moreover, the various solid pieces of which the bodies of its inhabitants would be composed would undergo the same, varia- tions of form and volume.
I will make still another hypothesis; I will suppose light traverses media diversely refractive and such that the index of refraction is inversely proportional to J2* — r*. It is easy to
SPACE AND GEOMETRY 77
see that, under these conditions, the rays of light would not be rectilinear, but circular.
To justify what precedes, it remains for me to show that certain changes in the position of external objects can be cor- reded by correlative movements of the sentient beings inhabit- ing this imaginary world, and that in such a way as to restore the primitive aggregate of impressions experienced by these sentient beings.
Suppose in fact that an object is displaced, undergoing de- formation, not as a rigid solid, but as a solid subjected to unequal dilatations in exact conformity to the law of temperature above supposed. Permit me for brevity to call such a movement a nan-Euclidean displacement.
If a sentient being happens to be in the neighborhood, his impressions will be modified by the displacement of the object, but he can reestablish them by moving in a suitable manner. It suffices if finally the aggregate of the object and the sentient being, considered as forming a single body, has undergone one of those particular displacements I have just called non-Euclidean. This is possible if it be supposed that the limbs of these beings dilate according to the same law as the other bodies of the world they inhabit.
Although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement and their various parts are no longer in the same relative position, nevertheless we shall see that the impressions of the sentient being have once more become the same.
In fact, though the mutual distances of the various parts may have varied, yet the parts originally in contact are again in contact. Therefore the tactile impressions have not changed.
On the other hand, taking into account the hypothesis made above in regard to the refraction and the curvature of the rays of light, the visual impressions will also have remained the same.
These imaginary beings will therefore like ourselves be led to classify the phenomena they witness and to distinguish among them the * changes of position' susceptible of correction by a cor- relative voluntary movement.
If they construct a geometry, it will not be, as ours is, the
78 SCIENCE AND HYPOTHESIS
study of the moyements of our rigid solids ; it will be the study of the changes of position which they will thus have distin- guished and which are none other than the 'non-Euclidean dis- placements'; t^ tvUl he non-EucUdean geometry.
Thus beings like ourselves, educated in such a world, would not have the same geometry as ours.
The World op Four Dimensions. — ^We can represent to our- selves a four-dimensional world just as well as a non-Euclidean.
The sense of sight, even with a single eye, together with the muscular sensations relative to the movements of the eyeball, would sufSce to teach us space of three dimensions.
The images of external objects are painted on the retina, which is a two-dimensional canvas; they are perspectives.
But, as eye and objects are movable, we see in succession vari- ous perspectives of the same body, taken from different points of view.
At the same time, we find that the transition from one per- spective to another is often accompanied by muscular sensations.
If the transition from the perspective A to the perspective B, and that from the perspective A' to the perspective B' are accompanied by the same muscular sensations, we liken them one to the other as operations of the same nature.
Studying then the laws according to which these operations combine, we recognize that they form a group, which has the same structure as that of the movements of rigid solids.
Now, we have seen that it is from the properties of this group we have derived the notion of geometric space and that of three dimensions.
We understand thus how the idea of a space of three dimen- sions could take birth from the pageant of these perspectives, though each of them is of only two dimensions, since they follow one another according to certain laws.
Well, just as the perspective of a three-dimensional figure can be made on a plane, we can make that of a four-dimensional figure on a picture of three (or of two) dimensions. To a geometer this is only child's play.
We can even take of the same figure several perspectives from several different points of view.
SPACE AND GEOMETRY 79
We can easily represent to ourselves these perspectives, since thej are of only three dimensions.
Imagine that the various perspectives of the same object suc- ceed one another, and that the transition from one to the other is accompanied by muscular sensations.
We shall of course consider two of these transitions as two operations of the same nature when they are associated with the same muscular sensations.
Nothing then prevents us from imagining that these opera- tions combine according to any law we choose, for example, so as to form a group with the same structure as that of the move- ments of a rigid solid of four dimensions.
Here there is nothing unpicturable, and yet these sensations are precisely those which would be felt by a being possessed of a two-dimensional retina who could move in space of four dimen- sions. In this sense we may say the fourth dimension is imaginable.
CONCLUSIONS. — ^We see that experience plays an indispensable role in the genesis of geometry ; but it would be an error thence to conclude that geometry is, even in part, an experimental science.
If it were experimental, it would be only approximative and provisional. And what rough approximation!
Geometry would be only the study of the movements of solids ; but in reality it is not occupied with natural solids, it has for object certain ideal solids, absolutely rigid, which are only a simplified and very remote image of natural solids.
The notion of these ideal solids is drawn from all parts of our mind, and experience is only an occasion which induces us to bring it forth from them.
The object of geometry is the study of a particular * group'; but the general group concept pre-exists, at least potentially, in our minds. It is imposed on us, not as form of our sense, but as form of our understanding.
Only, from among all the possible groups, that must be chosen which will be, so to speak, the standard to which we shall refer natural phenomena.
Experience guides us in this choice without forcing it upon
80 SCIENCE AND HYPOTHESIS
us; it tells us not which is the truest geometry, but which is the most convenient.
Notice that I have been able to describe the fantastic worlds above imagined without ceasing to employ the language of ordi- nary geometry.
And, in fact, we should not have to change it if transported thither.
Beings educated there would doubtless find it more convenient to create a geometry different from ours, and better adapted to their impressions. As for us, in face of the same impressions, it is certain we should find it more convenient not to change our habits.
CHAPTER V Experience and Qeometby
1. Already in the preceding pages I have several times tried to show that the principles of geometry are not experimental facts and that in particular Euclid's postulate can not be proven experimentally.
However decisive appear to me the reasons already given, I believe I should emphasize this point because here a false idea is profoundly rooted in many minds.
2. If we construct a material circle, measure its radius and circumference, and see if the ratio of these two lengths is equal to ir, what shall we have done T We shall have made an experi- ment on the properties of the matter with which we constructed this round thing, and of that of which the measure used was made.
3. Qeometry and Astronomy. — The question has also been put in another way. If Lobachevski's geometry is true, the paral- lax of a very distant star will be finite; if Riemann's is true, it will be negative. These are results which seem within the reach of experiment, and there have been hopes that astronomical obser- vations might enable us to decide between the three geometries.
But in astronomy * straight line' means simply *path of a ray of light. '
If therefore negative parallaxes were found, or if it were demonstrated that all parallaxes are superior to a certain limit, two courses would be open to us; we might either renounce Euclidean geometry, or else modify the laws of optics and sup- pose that light does not travel rigorously in a straight line.
It is needless to add that all the world would regard the latter solution as the more advantageous.
The Euclidean geometry has, therefore, nothing to fear from fresh experiments.
4. Is the position tenable, that certain phenomena, possible in Euclidean space, would be impossible in non-Euclidean space,
7 81
82 SCIENCE AND ETPOTBESIS
SO that ezperienee, in eetsbliBhing these phenomena, woald di- rectly contradict the non-Euclidean hypothesis t For my part I think no such question can be put. To my mind it is precisely equivalent to the following, whose absurdity is patent to all eyes: are there lengths expressible in meters and centimeters, but which can not be measured in fathoms, feet and inches, so that experi- ence, in ascertaining the existence of these lengths, would directly contradict the hypothesis that there are fathoms divided into six feet I
Examine the question more closely. I suppose that the straight line possesses in Euclidean space any two properties which I shall call A and B ; that in non-Euclidean space it still possesses the property A, but no longer has the property B ; finally I sup* pose that in both Euclidean and non-Euclidean space the straight line is the only line having the property A.
If this were so, experience would be capable of deciding between the hypothesis of Euclid and that of Lobacbevski. It woold be ascertained that a definite concrete object, accessible to experi- ment, for example, a pencil of rays of light, possesBea the proper^ A ; we should conclude that it is rectilinear, and then inrectitgatt whether or not it has the property B. W^M
But this is not so; no property exists which, like this propeHI^^ A, can be an absolute criterion enabling us to recognize the straight line and to distinguish it from every other line.
Shall we say, for instance: "the following is such a pre the straight line is a line such that a figure of which 1 forma a part can be moved without the mutual d points varying and so that all points oF this line remain t
This, in fact, is a property which, in Euclidean or D can space, belongs to the straight and belongs only t how shall we ascertain experimentally whether it h or that concrete object? It will be necessary i tances, and how slmll one know i which I have measured 'with my m sents the abstract distance T
We have only pushed twfl^
In reality the prop«t7 } the straight line al<ni^ i*
EXFESIENCB AND GEOMETRY
distance. For it to serve as absolute criterion, we should have to be able to establish not only that it doea not also belong to a line other than the straight and to distance, but in addition that it does not belong to a line other than the straight and to a ma^tude other than distance. Now thia is not true.
It is therefore impossible to imagine a concrete experiment which can be interpreted in the Euclidean system and not in the Lobachevskian system, so that I may conclude :
No experience will ever be in contradiction to Euclid's pos- tulate; nor, on the other hand, will any experience ever contra- dict the postulate of Lobachevski.
5. But it is not enough that the Euclidean (or non-Euclidean) geometry can never be directly contradicted by experience. Might it not happen that it can accord with experience only by violating the principle of sufficient reason or that of the relativity of space T
I will explain myself: consider any material system; we shall have to regard, on the one hand, 'the state' of the various bodies of this system (for instance, their temperature, their electric potential, etc.), and, on the other hand, their position in space; and among the data which enable us to define this position we aball, moreover, distinguish the mutual distances of these bodies, which define their relative positions, from the conditions which define the absolute position of the system and its absolute orien- tation in sjwce.
Til.' ■ i ■ I'i i ■; ■:; i :! i '. 'ippea in this system
will ' : their mutual dis-
lani!' - ' . ■ •■■. ity of space, they
H-ill not Jtj- ' tionof the
In ..i»>-r- '• . iiitiml dis-
tail''-' .r iiL'se
84 SCIENCE AND HYPOTHESIS
the non-Euclidean hypothesis. Well, we have made a series of experiments ; we have interpreted them on the Euclidean hyjxoth- esis, and we have recognized that these experiments thus inter- preted do not violate this 'law of relativity.'
We now interpret them on the non-Euclidean hypothesis: this is always possible ; only the non-Euclidean distances of our different bodies in this new interpretation will not generally be the same as the Euclidean distances in the primitive interpretation.
Will our experiments, interpreted in this new manner, still be in accord with our 'law of relativity'! And if there were not this accord, should we not have also the right to say experi- ence bad proven the falsity of the non-Euclidean geometry?
It is easy to see that this is an idle fear; in fact, to apply the law of relativity in all rigor, it must be applied to the entire universe. For if only a part of this universe were considered, and if the absolute position of this part happened to vary, the distances to the other bodies of the universe would likewise vary, their influence on the part of the universe considered would con- sequently augment or diminish, which might modify the laws of the phenomena happening there.
But if our system is the entire universe, experience is power- less to give information about its absolute position and orienta- tion in space. All that our instruments, however perfected they may be, can tell us will be the state of the various parts of the tmiverse and their mutual distances.
So our law of relativity may be thus enunciated :
The readings we shall be able to make on our instruments at any instant will depend only on the readings we could have made on these same instruments at the initial instant.
Now such an enunciation is independent of every interpreta- tion of experimental facts. If the law is true in the Euclidean interpretation, it will also be true in the non-Euclidean interpre- tation.
Allow me here a short digression. I have sx)oken above of the data which define the position of the various bodies of the system ; I should likewise have spoken of those which define their velocities; I should then have had to distinguish the velocities with which the mutual distances of the different bodies vary;
EXPERIENCE AND GEOMETRY 86
and, on the other hand, the velocities of translation and rotation of the system, that is to say, the velocities (with which its absolute position land orientation vary.
To fully satisfy the mind, the law of relativity should be expressible thus :
The state of bodies and their mutual distances at any instant, as well as the velocities with which these distances vary at this same instant, will depend only on the state of those bodies and their mutual distances at the initial instant, and the velocities with which these distances vary at this initial instant, but they will not depend either upon the absolute initial position of the system, or upon its absolute orientation, or upon the velocities with which this absolute position and orientation varied at the initial instant.
Unhappily the law thus enunciated is not in accord with ex- periments, at least as they are ordinarily interpreted.
Suppose a man be transported to a planet whose heavens were always covered with a thick curtain of clouds, so that he could never see the other stars ; on that planet he would live as if it were isolated in space. Yet this man could become aware that it turned, either by measuring its oblateness (done ordinarily by the aid of astronomic observations, but capable of being done by purely geodetic means) , or by repeating the experiment of Fou- cault's pendulum. The absolute rotation of this plcmet could therefore be made evident.
That is a fact which shocks the philosopher, but which the physicist is compelled to accept.
We know that from this fact Newton inferred the existence of absolute space ; I myself am quite unable to adopt this view. I shall explain why in Part III. For the moment it is not my intention to enter upon this diflSculty.
Therefore I must resign myself, in the enunciation of the law of relativity, to including velocities of every kind among the data which define the state of the bodies.
However that may be, this difficulty is the same for Euclid's geometry as for Lobachevski's; I therefore need not trouble my- self with it, and have only mentioned it incidentally.
86 SCIENCE AND HYPOTHESIS
What is important is the conclusion: experiment can not de- cide between Euclid and Lobachevski.
To sum up, whichever way we look at it, it is impossible to discover in geometric empiricism a rational meaning.
6. Experiments only teach us the relations of bodies to one another; none of them bears or can bear on the relations of bodies with space, or on the mutual relations of different parts of space.
"Yes," you reply, "a single experiment is insuflScient, be- cause it gives me only a single equation with several unknowns ; but when I shall have made enough experiments I shall have equations enough to calculate all my unknowns."
To know the height of the mainmast does not sufSce for calcu- lating the age of the captain. When you have measured every bit of wood in the ship you will have many equations, but you will know his age no better. All your measurements bear- ing only on your bits of wood can reveal to you nothing except concerning these bits of wood. Just so your experiments, how- ever numerous they may be, bearing only on the relations of bodies to one another, will reveal to us nothing about the mutual relations of the various parts of space.
7. Will you say that if the experiments bear on the bodies, they bear at least upon the geometric properties of the bodies? But, first, what do you understand by geometric properties of the bodies? I assume that it is a question of the relations of the bodies with space ; these properties are therefore inaccessible to experiments which bear only on the relations of the bodies to one another. This alone would suffice to show that there can be no question of these properties.
StiU let us begin by coming to an understanding about the sense of the phrase: geometric properties of bodies. When I say a body is composed of several parts, I assume that I do not enunciate therein a geometric property, and this would remain true even if I agreed to give the improper name of points to the smallest parts I consider.
When I say that such a part of such a body is in contact with such a part of such another body, I enunciate a proposition which concerns the mutual relations of these two bodies and not their relations with space.
EXPEBIENCE AND GEOMETRY 87
I suppose you will grant me these are not geometric properties; at least I am sure you will grant me these properties are inde- pendent of all knowledge of metric geometry.
This presupposed, I imagine that we have a solid body formed of eight slender iron rods, OA, OB, OC, OD, OE, OF, 00, OH, united at one of their extremities 0. Let us besides have a second «olid body, for example a bit of wood, to be marked with three little flecks of ink which I shall call a, p, y. I further suppose it ascertained that apy may be brought into contact with AOO (I mean a with A, and at the same time fi with O and y with 0), then that we may bring successively into contact aPy with BOO, COO, DOO, EOO, FOO, then with AHO, BEO, CEO, DEO, EEO, FEO, then ay successively with AB, BC, CD, DE, EF, FA.
These are determinations we may make without having in advance any notion about form or about the metric properties of space. They in no wise bear on the 'geometric properties of bodies.' And these determinations will not be possible if the bodies experimented upon move in accordance with a group having the same structure as the Lobachevskian group (I mean according to the same laws as solid bodies in LobachevsM's geom- etry). They suffice therefore to prove that these bodies move in accordance with the Euclidean group, or at least that they do not move according to the Lobachevskian group.
That they are compatible with the Euclidean group is easy to see. For they could be made if the body apy was a rigid solid of our ordinary geometry presenting the form of a right- angled triangle, and if the points ABCDEFOE were the summits of a polyhedron formed of two regular hexagonal pyramids of our ordinary geometry, having for common base ABCDEF and for apices the one 0 and the other E.
Suppose now that in place of the preceding determination it is observed that as above aPy can be successively applied to AOO, BOO, COO, DOO, EOO, AEO, BEO, CEO, DEO, EEO, FEO, then that ap (and no longer ay) can be successively applied to AB, BC, CD, DE, EF and FA.
These are determinations which could be made if non-Euclid- ean geometry were true, if the bodies aPy and OABCDEFOE were rigid solids, and if the first were a right-angled triangle
88 SCIENCE AND HYPOTHESIS
and the second a double regular hexagonal pyramid of snitaible dimensions.
Therefore these new determinations are not possible if the bodies move according to the Euclidean group ; but they become so if it be supposed that the bodies move according to the Loba- chevskian group. They would suffice, therefore (if one made them), to prove that the bodies in question do not move accord- ing to the Euclidean group.
Thus, without making any hypothesis about form, about the nature of space, about the relations of bodies to space, and with- out attributing to bodies any geometric property, I have made observations which have enabled me to show in one case that the bodies experimented upon move according to a group whose structure is Euclidean, in the other case that they move according to a group whose structure is Lobachevskian.
And one may not say that the first aggregate of determinations would constitute an experiment proving that space is Euclidean, and the second an experiment proving that space is non-Euclidean.
In fact one could imagine (I say imagine) bodies moving so as to render possible the second series of determinations. And the proof is that the first mechanician met could construct such bodies if he cared to take the pains and make the outlay. You will not conclude from that, however, that space is non-Euclidean.
Nay, since the ordinary solid bodies would continue to exist when the mechanician had constructed the strange bodies of which I have just spoken, it would be necessary to conclude that space is at the same time Euclidean and non-Euclidean.
Suppose, for example, that we have a great sphere of radius B and that the temperature decreases from the center to the surface of this sphere according to the law of which I have spoken in describing the non-Euclidean world.
"We might have bodies whose expansion would be negligible and whioh would act like ordinary rigid solids ; and, on the other hand, bodies very dilatable and which would act like non-Euclidean solids. We might have two double pyramids OABCDEFOH and O'A'B'C'D'E'F'G'W and two triangles afiy and a'p^y'. The first double pyramid might be rectilinear and the second curvilinear;.
EXPERIENCE AND GEOMETRY 89
the triangle aPy might be made of inexpansible matter and the other of a very dilatable matter.
It would then be possible to make the first observations with the double pyramid OAH and the triangle aPy, and the second with the double pyramid O'A'H' and the triangle a'fify. And then experiment would seem to prove first that the Euclidean geometry is true and then that it is false.
Experiments therefore have a bearing, not on space, hut on bodies.
Supplement
8. To complete the matter, I ought to speak of a very delicate question, which would require long development; I shall confine myself to summarizing here what I have expounded in the Revue de MStaphysique et de Morale and in The Monist. When we say space has three dimensions, what do we mean t
We have seen the importance of those 'internal changes' revealed to us by our muscular sensations. They may serve to characterize the various attitudes of our body. Take arbitrarily as origin one of these attitudes A. When we pass from this initial attitude to any other attitude B, we feel a series of mus- cular sensations, and this series 8 will define B. Observe, how- ever, that we shall often regard two series 8 and 8' as defining the same attitude B (since the initial and final attitudes A and B remaining the same, the intermediary attitudes and the corre- sponding sensations may differ). How then shall we recognize the equivalence of these two series ! Because they may serve to compensate the same external change, or more generally because, when it is a question of compensating an external change, one of the series can be replaced by the other. Among these series, we have distinguished those which of themselves alone can com- pensate an external change, and which we have called 'displace- ments.' As we can not discriminate between two displacements which are too close together, the totality of these displacements presents the characteristics of a physical continuum ; experience teaches us that they are those of a physical continuum of six dimensions; but we do not yet know how many dimensions space itself has, we must first solve another question.
What is a point of space! Everybody thinks he knows, but
90 SCIENCE AND HYPOTHESIS
that is an illusion. What we see when we try to represent to our- selves a point of space is a black speck on white paper, a speck of chalk on a blackboard, always an object. The question should therefore be understood as follows :
What do I mean when I say the object B is at the same point that the object A occupied just now t Or further, what criterion will enable me to apprehend thist
I mean that, although I have not budged (which my muscular sense tells me) , my first finger which just now touched the object A touches at present the object B. I could have used other criteria ,* for instance another finger or the sense of sight. But the first criterion is sufficient; I know that if it answers yes, all the other criteria will give the same response. I know it by experience, I can not know it a priori. For the same reason I say that touch can not be exercised at a distance ; this is another way of enunci- ating the same experimental fact. And if, on the contrary, I say that sight acts at a distance, it means that the criterion furnished by sight may respond yes while the others reply no.
And in fact, the object, although moved away, may form its image at the same point of the retina. Sight responds yes, the object has remained at the same point and touch answers no, because my finger which just now touched the object, at present touches it no longer. If experience had shown us that one finger may respond no when the other says yes, we should likewise say that touch acts at a distance.
In short, for each attitude of my body, my first finger deter- mines a point, and this it is, and this alone, which defines a point of space.
To each attitude corresponds thus a point ; but it often happens that the same point corresponds to several different attitudes (in this case we say our finger has not budged, but the rest of the body has moved). We distinguish, therefore, among the changes of attitude those where the finger does not budge. How are we led thereto f It is because often we notice that in these changes the object which is in contact with the finger remains in contact with it.
Range, therefore, in the same class all the attitudes obtainable from each other by one of the changes we have thus distinguished.
EXPERIENCE AND GEOMETRY 91
To all the attitudes of the class will correspond the same point of space. Therefore to each class will correspond a point and to each point a class. But one may say that what experience arrives at is not the point, it is this class of changes or, better, the cor- responding class of muscular sensations.
And when we say space has three dimensions, we simply mean that the totality of these classes appears to us with the character- istics of a physical continuum of three dimensions.
One might be tempted to conclude that it is experience which has taught us how many dimensions space has. But in reality here also our experiences have bearing, not on space, but on our body and its relations with the neighboring objects. Moreover they are excessively crude.
In our mind pre-existed the latent idea of a certain number of groups — ^those whose theory Lie has developed. Which group shall we choose, to make of it a sort of standard with which to com- pare natural phenomena? And, this group chosen, which of its sub-groups shall we take to characterize a point of space t Ex- perience has guided us by showing us which choice best adapts itself to the properties of our body. But its role is limited to that.
Ancestral Experience
It has often been said that if individual experience could not create geometry the same is not true of ancestral experience. But what does that meant Is it meant that we could not experi- mentally demonstrate Euclid's postulate, but that our ancestors have been able to do it f Not in the least. It is meant that by natural selection our mind has adapted itself to the conditions of the external world, that it has adopted the geometry most advan- tageous to the species: or in other words the most convenient. This is entirely in conformity with our conclusions ; geometry is not true, it is advantageous.
PART III
FORCE
CHAPTER VI
The Classic Mechanics
The English teach mechanics as an experimental science; on the continent it is always expounded as more or less a deductive and a priori science. The English are right, that goes without saying; but how could the other method have been persisted in so longf Why have the continental savants who have sought to get out of the ruts of their predecessors been usually unable to free themselves completely !
On the other hand, if the principles of mechanics are only of experimental origin, are they not therefore only approximate and provisional! Might not new experiments some day lead us to modify or even to abandon them ?
Such are the questions which naturally obtrude themselves, and the diflSculty of solution comes principally from the fact that the treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is con- vention, what is hypothesis.
That is not all :
1** There is no absolute space and we can conceive only of relative motions ; yet usually the mechanical facts are enunciated as if there were an absolute space to which to refer them.
2° There is no absolute time; to say two durations are equal is an assertion which has by itself no meaning and which can acquire one only by convention.
3° Not only have we no direct intuition of the equality of two durations, but we have not even direct intuition of the
92
\
THE CLASSIC MECHANICS 98
fiimnltaneity of two events occurring in different places: this I liave explained in an article entitled La mesure du iemps.^ '
4** Finally, our Euclidean geometry is itself only a sort of convention of language; mechanical facts might be enunciated with reference to a non-Euclidean space which would be a guide less convenient than, but just as legitimate as, our ordinary space ; the enunciation would thus become much more complicated, but it would remain possible.
Thus absolute space, absolute time, geometry itself, are not conditions which impose themselves on mechanics ; all these things are no more antecedent to mechanics than the French language is logically antecedent to the verities one expresses in French.
We might try to enunciate the fundamental laws of mechanics in a language independent of all these conventions; we should thus without doubt get a better idea of what these laws are in themselves ; this is what M. Andrade has attempted to do, at least in part, in his Leqons de mecanique physique.
The enunciation of these laws would become of course much more complicated, because all these conventions have been devised expressly to abridge and simplify this enunciation.
As for me, save in what concerns absolute space, I shall ignore all these difficulties ; not that I fail to appreciate them, far from that; but we have sufficiently examined them in the first two parts of the book.
I shall therefore admit, provisionally, absolute time and Eu- clidean geometry.
The Principle op Inertia. — ^A body acted on by no force can only move uniformly in a straight line.
Is this a truth imposed a priori upon the mindf If it were 80, how should the Greeks have failed to recognize it? How could they have believed that motion stops when the cause which gave birth to it ceases ! Or again that every body if nothing prevents, will move in a circle, the noblest of motions?
If it is said that the velocity of a body can not change if there is no reason for it to change, could it not be maintained just as well that the position of this body can not change, or that the
^Eevue de M4taphysique et de Morale, t. YI., pp. 1-13 (January, 1898).
94 SCIENCE AND HYPOTHESIS
curvature of its trajectory can not change, if no external canse intervenes to modify themt
Is the principle of inertia, which is not an a priori truth, therefore an experimental factt But has any one ever experi- mented on bodies withdrawn from the action of every force t and, if so, how was it known that these bodies were subjected to no force t The example ordinarily cited is that of a baU rolling a very long time on a marble table ; but why do we say it is sub- jected to no force t Is this because it is too remote from all other bodies to experience any appreciable action from themt Yet it is not farther from the earth than if it were thrown freely into the air ; and every one knows that in this case it would experience the influence of gravity due to the attraction of the earth.
Teachers of mechanics usually pass rapidly over the example of the ball ; but they add that the principle of inertia is verified indirectly by its consequences. They express themselves badly; they evidently mean it is possible to verify various consequences of a more general principle, of which that of inertia is only a particular case.
I shall propose for this general principle the following enun- ciation :
The acceleration of a body depends only upon the position of this body and of the neighboring bodies and upon their velocities.
Mathematicians would say the movements of all the material molecules of the universe depend on differential equations of the second order.
To make it clear that this is really the natural generalization of the law of inertia, I shall beg you to permit me a bit of fiction. The law of inertia, as I have said above, is not imposed upon us a priori; other laws would be quite as compatible with the prin- ciple of suflScient reason. If a body is subjected to no force, in lieu of supposing its velocity not to change, it might be supposed that it is its position or else its acceleration which is not to change.
Well, imagine for an instant that one of these two hypothetical laws is a law of nature and replaces our law of inertia. What would be its natural generalization? A moment's thought will show us.
THE CLASSIC MECHANICS 95
In the first case, we must suppose that the velocity of a body depends only upon its position and upon that of the neighboring bodies; in the second case that the change of acceleration of a body depends only upon the position of this body and of the neighboring 'bodies, upon their velocities and upon their acceler- ations.
Or to speak the language of mathematics, the differential equations of motion would be of the first order in the first case, and of the third order in the second case.
Let us slightly modify our fiction. Suppose a world analogous to our solar system, but where, by a strange chance, the orbits of all the planets are without eccentricity and without inclination. Suppose further that the masses of these planets are too slight for their mutual perturbations to be sensible. Astronomers in- habiting one of these planets could not fail to conclude that the orbit of a star can only be circular and parallel to a certain plane ; the position of a star at a given instant would then suffice to de- termine its velocity and its whole path. The law of inertia which they would adopt would be the first of the two hypothetical laws I have mentioned.
Imagine now that this system is some day traversed with great velocity by a body of vast mass, coming from distant constella- tions. All the orbits would be profoundly disturbed. Still our astronomers would not be too greatly astonished ; they would very well divine that this new star was alone to blame for all the mischief. *'But," they would say, *'when it is gone, order will of itself be reestablished ; no doubt the distances of the planets from the sun will not revert to what they were before the cata- clysm, but when the perturbing star is gone, the orbits will again become circular."
It would only be when the disturbing body was gone and when nevertheless the orbits, in lieu of again becoming circular, became elliptic, that these astronomers would become conscious of their error and the necessity of remaking all their mechanics.
I have dwelt somewhat upon these hypotheses because it seems to me one can clearly comprehend what our generalized law of inertia really is only in contrasting it with a contrary hypothesis.
Well, now, has this generalized law of inertia been verified by
96 SCIENCE AND HYPOTHESIS
experiment, or can it bef When Newton wrote the Prindpia he quite regarded this truth as experimentally acquired and dem- onstrated. It was so in his eyes, not only through the anthropo- morphism of which we shall speak further on, but through the work of Galileo. It was so even from Kepler's laws themselves; in accordance with these laws, in fact, the path of a planet is completely determined by its initial position and initial velocity; this is just what our generalized law of inertia requires.
For this principle to be only in appearance true, for one to have cause to dread having some day to replace it by one of the analogous principles I have just now contrasted with it, would be necessary our having been misled by some amazing chance, like that which, in the fiction above developed, led into error our imaginary astronomers.
Such a hypothesis is too unlikely to delay over. No one will believe that such coincidences can happen; no doubt the prob- ability of two eccentricities being both precisely null, to within errors of observation, is not less than the probability of one being precisely equal to 0.1, for instance, and the other to 0.2, to within errors of observation. The probability of a simple event is not less than that of a complicated event ; and yet, if the first happens, we shall not consent to attribute it to chance ; we should not believe that nature had acted expressly to deceive us. The hypothesis of an error of this sort being discarded, it may therefore be admitted that in so far as astronomy is concerned, our law has been veri- fied by experiment.
But astronomy is not the whole of physics.
May we not fear lest some day a new experiment should come to falsify the law in some domain of physics t An experimental law is always subject to revision; one should always expect to see it replaced by a more precise law.
Yet no one seriously thinks that the law we are speaking of will ever be abandoned or amended. Whyt Precisely because it can never be subjected to a decisive test.
First of all, in order that this trial should be complete, it would be necessary that after a certain time all the bodies in the universe should revert to their initial positions with their initial
THU CLASSIC MECHANICS 97
velocities. It might then be seen whether, starting from this moment, they would resume their original paths.
But this test is impossible, it can be only partially applied, and, however well it is made, there will always be some bodies which will not revert to their initial positions ; thus every deroga- tion of the law will easily find its explanation.
This is not all ; in astronomy we see the bodies whose motions we study and we usually assume that they are not subjected to the action of other invisible bodies. Under these conditions our law must indeed be either verified or not verified.
But it is not the same in physics ; if the physical phenomena are due to motions, it is to the motions of molecules which we do not see. If then the acceleration of one of the bodies we see appears to us to depend on something else besides the positions or velocities of other visible bodies or of invisible molecules whose existence we have been previously led to admit, nothing prevents our supposing that this something else is the position or the velocity of other molecules whose presence we have not before suspected. The law will find itself safeguarded.
Permit me to employ mathematical language a moment to express the same thought under another form. Suppose we ob- serve n molecules and ascertain that their 3n coordinates satisfy a system of 3n differential equations of the fourth order (and not of the second order as the law of inertia would require) . We know that by introducing 3n auxiliary variables, a system of 3n equations of the fourth order can be reduced to a system of 6n equations of the second order. If then we suppose these 3n auxiliary variables represent the coordinates of n invisible mole- cules, the result is again in conformity with the law of inertia.
To sum up, this law, verified experimentally in some particular cases, may unhesitatingly be extended to the most general cases, since we know that in these general cases experiment no longer is able either to confirm or to contradict it.
The Law op Acceleration. — The acceleration of a body is equal to the force acting on it divided by its mass. Can this law be verified by experiment! For that it would be necessary to
8
98 SCIENCE AND HYPOTHESIS
measure the three magnitudes which figure in the enunciation: acceleration, force and mass.
I assume that acceleration can be measured, for I pass over the difSculty arising from the measurement of time. But how measure force, or mass t We do not even know what they are.
What is mass? According to Newton, it is the product of the volume