# The Foundations of Science

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SCIENCE AND EDUCATION
A SERIES OF VOLUMES FOR THE PROMOTION OF SCIENTIFIC RESEARCH AND EDUCATIONAL PROGRESS
Edited by J. McKEEN CATTELL
VOLUME I—THE FOUNDATIONS OF SCIENCE
UNDER THE SAME EDITORSHIP
SCIENCE AND EDUCATION. A series of volumes for the promotion of scientific research and educational progress.
Volume I. The Foundations of Science. By H. Poincaré. Containing the authorized English translation by George Bruce Halsted of “Science and Hypothesis,” “The Value of Science,” and “Science and Method.”
Volume II. Medical Research and Education. By Richard Mills Pearce, William H. Welch, W. H. Howell, Franklin P. Mall, Lewellys F. Barker, Charles S. Minot, W. B. Cannon, W. T. Councilman, Theobald Smith, G. N. Stewart, C. M. Jackson, E. P. Lyon, James B. Herrick, John M. Dodson, C. R. Bardeen, W. Ophils, S. J. Meltzer, James Ewing, W. W. Keen, Henry H. Donaldson, Christian A. Herter, and Henry P. Bowditch.
Volume III. University Control. J. McKeen Cattell and other authors.
AMERICAN MEN OF SCIENCE. A Biographical Directory.
SCIENCE. A weekly journal devoted to the advancement of science. The official organ of the American Association for the Advancement of Science.
THE POPULAR SCIENCE MONTHLY. A monthly magazine devoted to the diffusion of science.
THE AMERICAN NATURALIST. A monthly journal devoted to the biological sciences, with special reference to the factors of evolution.
THE SCIENCE PRESS
NEW YORKGARRISON, N. Y.
THE FOUNDATIONS OF SCIENCE
SCIENCE AND HYPOTHESIS
THE VALUE OF SCIENCE
SCIENCE AND METHOD
BY
H. POINCARÉ
AUTHORIZED TRANSLATION BY
GEORGE BRUCE HALSTED
WITH A SPECIAL PREFACE BY POINCARÉ, AND AN INTRODUCTION BY JOSIAH ROYCE, HARVARD UNIVERSITY
THE SCIENCE PRESS
NEW YORK AND GARRISON, N. Y.
1913
Copyright, 1913
By The Science Press
PRESS OF
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.
CONTENTS
PAGE :
| Henri Poincaré | xi |
| Author’s Preface to the Translation | 3 |
| SCIENCE AND HYPOTHESIS ||
| Introduction by Royce | 9 |
| Introduction | 27 |
| Part I. Number and Magnitude ||
| CHAPTER I.—On the Nature of Mathematical Reasoning | 31 |
| Syllogistie Deduction | 31 |
| Verification and Proof | 32 |
| Elements of Arithmetic | 33 |
| Reasoning by Recurrence | 37 |
| Induction | 40 |
| Mathematical Construction | 41 |
| CHAPTER II.—Mathematical Magnitude and Experience | 43 |
| Definition of Incommensurables | 44 |
| The Physical Continuum | 46 |
| Creation of the Mathematical Continuum | 46 |
| Measurable Magnitude | 49 |
| Various Remarks (Curves without Tangents) | 50 |
| The Physical Continuum of Several Dimensions | 52 |
| The Mathematical Continuum of Several Dimensions | 58 |
| Part II. Space ||
| CHAPTER III.—The Non-Euclidean Geometries | 55 |
| The Bolyai-Lobachevski Geometry | 56 |
| Riemann’s Geometry | 57 |
| The Surfaces of Constant Curvature | 58 |
| Interpretation of Non-Euclidean Geometries | 59 |
| The Implicit Axioms | 60 |
| The Fourth Geometry | 62 |
| Lie’s Theorem | 62 |
| Riemann’s Geometries | 63 |
| On the Nature-of Axioms | 63 |
| CHAPTER IV.—Space and Geometry | 66 |
| Geometric Space and Perceptual Space | 66 |
| Visual Space | 67 |
| Tactile Space and Motor Space | 68 |
| Characteristics of Perceptual Space | 69 |
| Change of State and Change of Position | 70 |
| Conditions of Compensation | 72 ​ |
| Solid Bodies and Geometry | 73 |
| Law of Homogeneity | 74 |
| The Non-Euclidean World | 75 |
| The World of Four Dimensions | 78 |
| Conclusions | 79 |
| Chapter V.—Experience and Geometry | 81 |
| Geometry and Astronomy | 81 |
| The Law of Relativity | 83 |
| Bearing of Experiments | 86 |
| Supplement (What is a Point?) | 89 |
| Ancestral Experience | 91 |
| Part III. Force ||
| Chapter VI.—The Classic Mechanics | 92 |
| The Principle of Inertia | 93 |
| The Law of Acceleration | 97 |
| Anthropomorphic Mechanics | 103 |
| The School of the Thread | 104 |
| Chapter VII.—Relative Motion and Absolute Motion | 107 |
| The Principle of Relative Motion | 107 |
| Newton’s Argument | 108 |
| Chapter VIII.—Emergy and Thermodynamics | 115 |
| Energetics | 115 |
| Thermodynamics | 119 |
| General Conclusions on Part III | 123 |
| Part IV. Nature ||
| Chapter IX.—Hypotheses in Physics | 127 |
| The Rôle of Experiment and Generalization | 127 |
| The Unity of Nature | 130 |
| The Rôle of Hypothesis | 133 |
| Origin of Mathematical Physics | 136 |
| Chapter X.—The Theories of Modern Physics | 140 |
| Meaning of Physical Theories | 140 |
| Physics and Mechanism | 144 |
| Present State of the Science | 148 |
| Chapter XI.—The Caleulus of Probabilities | 155 |
| Classification of the Problems of Probability | 158 |
| Probability in Mathematics | 161 |
| Probability in the Physical Sciences | 164 |
| Rouge et noir | 167 |
| The Probability of Causes | 169 |
| The Theory of Errors | 170 |
| Conclusions | 172 |
| Chapter XII.—Optics and Electricity | 174 |
| Fresnel’s Theory | 174 |
| Maxwell’s Theory | 175 |
| The Mechanical Explanation of Physical Phenomena | 177 ​ |
| Chapter XIII.—Electrodynamics | 184 |
| Ampére’s Theory | 184 |
| Closed Currents | 185 |
| Action of a Closed Current on a Portion of Current | 186 |
| Continuous Rotations | 187 |
| Mutual Action of Two Open Currents | 189 |
| Induction | 190 |
| Theory of Helmholtz | 191 |
| Difficulties Raised by these Theories | 193 |
| Maxwell’s Theory | 193 |
| Rowland’s Experiment | 194 |
| The Theory of Lorentz | 196 |
| THE VALUE OF SCIENCE ||
| Translator’s Introduction | 201 |
| Does the Scientist Create Science? | 201 |
| The Mind Dispelling Optical Illusions | 202 |
| Euclid not Necessary | 202 |
| Without Hypotheses, no Science | 203 |
| What Outcome? | 203 |
| Introduction | 205 |
| Part I. The Mathematical Sciences ||
| Chapter I.—Intuition and Logie in Mathematics | 210 |
| Chapter II.—The Measure of Time | 223 |
| Chapter III.—The Notion of Space | 235 |
| Qualitative, Geometry | 238 |
| The Physical Continuum of Several Dimensions | 240 |
| The Notion of Point | 244 |
| The Notion of Displacement | 247 |
| Visual Space | 252 |
| Chapter IV.—Space and its Three Dimensions | 256 |
| The Group of Displacements | 256 |
| Identity of Two Points | 259 |
| Tactile Space | 264 |
| Identity of the Different Spaces | 268 |
| Space and Empiricism | 271 |
| Rôle of the Semicircular Canals | 276 |
| Part II. The Physical Sciences ||
| Chapter V.—Analysis and Physics | 279 |
| Chapter VI.—Astronomy | 289 |
| Chapter VII.—The History of Mathematical Physics | 297 |
| The Physics of Central Forces | 297 |
| The Physics of the Principles | 299 |
| Chapter VIII.—The Present Crisis in Physics | 303 |
| The New Crisis | 303 |
| Carnot’s Principle | 303 ​ |
| The Principle of Relativity | 305 |
| Newton’s Principle | 308 |
| Lavoisier’s Principle | 310 |
| Mayer’s Principle | 312 |
| Chapter IX.—The Future of Mathematical Physies | 314 |
| The Principles and Experiment | 314 |
| The Role of the Analyst | 314 |
| Aberration and Astronomy | 315 |
| Electrons and Spectra | 316 |
| Conventions preceding Experiment | 317 |
| Future Mathematical Physics | 319 |
| Part III. The Objective Value of Science ||
| Chapter X.—Is Science Artificial? | 321 |
| The Philosophy of LeRoy | 321 |
| Science, Rule of Action | 323 |
| The Crude Fact and the Scientific Fact | 325 |
| Nominalism and the Universal Invariant | 333 |
| Chapter XI.—Science and Reality | 340 |
| Contingence and Determinism | 340 |
| Objectivity of Science | 347 |
| The Rotation of the Earth | 353 |
| Science for Its Own Sake | 354 |
| SCIENCE AND METHOD ||
| Introduction | 359 |
| Book I. Science and the Scientist ||
| Chapter I.—The Choice of Facts | 362 |
| Chapter II.—The Future of Mathematics | 369 |
| Chapter III.—Mathematical Creation | 383 |
| Chapter IV.—Chance | 395 |
| Book II. Mathematical Reasoning ||
| Chapter I.—The Relativity of Space | 413 |
| Chapter II.—Mathematical Definitions and Teaching | 430 |
| Chapter III.—Mathematics and Logic | 448 |
| Chapter IV.—The New Logics | 460 |
| Chapter V.—The Latest Efforts of the Logisticians | 472 |
| Book III. The New Mechanics ||
| Chapter I.—Mechanics and Radium | 486 |
| Chapter II.—Mechanics and Optics | 496 |
| Chapter III.—The New Mechanics and Astronomy | 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 POINCARÉ
Sir George Darwin, worthy son of an immortal father, said, referring to what Poincaré was to him and to his work: ‘‘He must be regarded as the presiding genius—or, shall I say, my patron saint?”
Henri Poincaré was born April 29, 1854, at Nancy, where his father was a physician highly respected. His schooling was broken into by the war of 1870–71, to get news of which he learned to read the German newspapers. He outclassed the other boys of his age in all subjects and in 1873 passed highest into the École Polytechnique, where, like John Bolyai at Maros Vásárhely, he followed the courses in mathematics without taking a note and without the syllabus. He proceeded in 1875 to the School of Mines, and was Nommé, March 26, 1879. But he won his doctorate in the University of Paris, August 1, 1879, and was appointed to teach in the Faculté des Sciences de Caen, December 1, 1879, whence he was quickly called to the University of Paris, teaching there from October 21, 1881, until his death, July 17, 1912. So it is an error to say he started as an engineer. At the early age of thirty-two he became a member of 1’Académie des Sciences, and, March 5, 1908, was chosen Membre de 1’Académie Française. July 1, 1909, the number of his writings was 436.
His earliest publication was in 1878, and was not important. Afterward came an essay submitted in competition for the Grand Prix offered in 1880, but it did not win. Suddenly there came a change, a striking fire, a bursting forth, in February, 1881, and Poincaré tells us the very minute it happened. Mounting an omnibus, ‘‘at the moment when I put my foot upon the step, the idea came to me, without anything in my previous thoughts seeming to foreshadow it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.’’ Thereby was opened a perspective new and immense. Moreover, the magic wand of his whole ​life-work had been grasped, the Aladdin’s lamp had been rubbed, non-Euclidean geometry, whose necromancy was to open up a new theory of our universe, whose brilliant exposition was commenced in his book Science and Hypothesis, which has been translated into six languages and has already had a circulation of over 20,000. The non-Euclidean notion is that of the possibility of alternative laws of nature, which in the Introduction to the Electricité et Optique, 1901, is thus put: ‘‘If therefore a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of others which will account equally well for all the peculiarities disclosed by experiment.’’
The scheme of laws of nature so largely due to Newton is merely one of an infinite number of conceivable rational schemes for helping us master and make experience; it is commode, convenient; but perhaps another may be vastly more advantageous. The old conception of true has been revised. The first expression of the new idea occurs on the title page of John Bolyai’s marvelous Science Absolute of Space, in the phrase ‘‘haud unquam a priori decidenda.’’
With bearing on the history of the earth and moon system and the origin of double stars, in formulating the geometric criterion of stability, Poincaré proved the existence of a previously unknown pear-shaped figure, with the possibility that the progressive deformation of this figure with increasing angular velocity might result in the breaking up of the rotating body into two detached masses. Of his treatise Les Méthodes nouvelles de la Méchanique céleste, Sir George Darwin says: ‘‘It is probable that for half a century to come it will be the mine from which humbler investigators will excavate their materials.’’ Brilliant was his appreciation of Poincaré in presenting the gold medal of the Royal Astronomical Society. The three others most akin in genius are linked with him by the Sylvester medal of the Royal Society, the Lobachevski medal of the Physico-Mathematical Society of Kazan, and the Bolyai prize of the Hungarian Academy of Sciences. His work must be reckoned with the greatest mathematical achievements of mankind.
The kernel of Poincaré’s power lies in an oracle Sylvester often quoted to me as from Hesiod: The whole is less than its part.
​He penetrates at once the divine simplicity of the perfectly general case, and thence descends, as from Olympus, to the special concrete earthly particulars.
A combination of seemingly extremely simple analytic and geometric concepts gave necessary general conclusions of immense scope from which sprang a disconcerting wilderness of possible deductions. And so he leaves a noble, fruitful heritage.
Says Love: ‘‘His right is recognized now, and it is not likely that future generations will revise the judgment, to rank among the greatest mathematicians of all time.’’
George Bruce Halsted
AUTHOR’S PREFACE TO THE TRANSLATION
I am exceedingly grateful to Dr. Halsted, who has been so good as to present my book to American readers in a translation, clear and faithful.
Every one knows that this savant has already taken the trouble to translate many European treatises and thus has powerfully contributed to make the new continent understand the thought of the old.
Some people love to repeat that Anglo-Saxons have not the same way of thinking as the Latins or as the Germans; that they have quite another way of understanding mathematics or of understanding physics; that this way seems to them superior to all others; that they feel no need of changing it, nor even of knowing the ways of other peoples.
In that they would beyond question be wrong, but I do not believe that is true, or, at least, that is true no longer. For some time the English and Americans have been devoting themselves much more than formerly to the better understanding of what is thought and said on the continent of Europe.
To be sure, each people will preserve its characteristic genius, and it would be a pity if it were otherwise, supposing such a thing possible. If the Anglo-Saxons wished to become Latins, they would never be more than bad Latins; just as the French, in seeking to imitate them, could turn out only pretty poor Anglo-Saxons.
And then the English and Americans have made scientific conquests they alone could have made; they will make still more of which others would be incapable. It would therefore be deplorable if there were no longer Anglo-Saxons.
But continentals have on their part done things an Englishman could not have done, so that there is no need either for wishing all the world Anglo-Saxon.
Each has his characteristic aptitudes, and these aptitudes ​should be diverse, else would the scientific concert resemble a quartet where every one wanted to play the violin.
And yet it is not bad for the violin to know what the violoncello is playing, and vice versa.
This it is that the English and Americans are comprehending more and more; and from this point of view the translations undertaken by Dr. Halsted are most opportune and timely.
Consider first what concerns the mathematical sciences. It is frequently said the English cultivate them only in view of their applications and even that they despise those who have other aims; that speculations too abstract repel them as savoring of metaphysic.
The English, even in mathematics, are to proceed always from the particular to the general, so that they would never have an idea of entering mathematics, as do many Germans, by the gate of the theory of aggregates. They are always to hold, so to speak, one foot in the world of the senses, and never burn the bridges keeping them in communication with reality. They thus are to be incapable of comprehending or at least of appreciating certain theories more interesting than utilitarian, such as the non-Euclidean geometries. According to that, the first two parts of this book, on number and space, should seem to them void of all substance and would only baffle them.
But that is not true. And first of all, are they such uncompromising realists as has been said? Are they absolutely refractory, I do not say to metaphysic, but at least to everything metaphysical?
Recall the name of Berkeley, born in Ireland doubtless, but immediately adopted by the English, who marked a natural and necessary stage in the development of English philosophy.
Is this not enough to show they are capable of making ascensions otherwise than in a captive balloon?
And to return to America, is not the Monist published at Chicago, that review which even to us seems bold and yet which finds readers?
And in mathematics? Do you think American geometers are concerned only about applications? Far from it. The part of the science they cultivate most devotedly is the theory of ​groups of substitutions, and under its most abstract form, the farthest removed from the practical.
Moreover, Dr. Halsted gives regularly each year a review of all productions relative to the non-Euclidean geometry, and he has about him a public deeply interested in his work. He has initiated this public into the ideas of Hilbert, and he has even written an elementary treatise on ‘Rational Geometry,’ based on the principles of the renowned German savant.
To introduce this principle into teaching is surely this time to burn all bridges of reliance upon sensory intuition, and this is, I confess, a boldness which seems to me almost rashness.
The American public is therefore much better prepared than has been thought for investigating the origin of the notion of space.
Moreover, to analyze this concept is not to sacrifice reality to I know not what phantom. The geometric language is after all only a language. Space is only a word that we have believed a thing. What is the origin of this word and of other words also? What things do they hide? To ask this is permissible; to forbid it would be, on the contrary, to be a dupe of words; it would be to adore a metaphysical idol, like savage peoples who prostrate themselves before a statue of wood without daring to take a look at what is within.
In the study of nature, the contrast between the Anglo-Saxon spirit and the Latin spirit is still greater.
The Latins seek in general to put their thought in mathematical form; the English prefer to express it by a material representation.
Both doubtless rely only on experience for knowing the world; when they happen to go beyond this, they consider their foreknowledge as only provisional, and they hasten to ask its definitive confirmation from nature herself.
But experience is not all, and the savant is not passive; he does not wait for the truth to come and find him, or for a chance meeting to bring him face to face with it. He must go to meet it, and it is for his thinking to reveal to him the way leading thither. For that there is need of an instrument; well, just there begins the difference—the instrument the Latins ordinarily choose is not that preferred by the Anglo-Saxons. ​For a Latin, truth can be expressed only by equations; it must obey laws simple, logical, symmetric and fitted to satisfy minds in love with mathematical elegance.
The Anglo-Saxon to depict a phenomenon will first be engrossed in making a model, and he will make it with common materials, such as our crude, unaided senses show us them. He also makes a hypothesis, he assumes implicitly that nature, in her finest elements, is the same as in the complicated aggregates which alone are within the reach of our senses. He concludes from the body to the atom.
Both therefore make hypotheses, and this indeed is necessary, since no scientist has ever been able to get on without them. The essential thing is never to make them unconsciously.
From this point of view again, it would be well for these two sorts of physicists to know something of each other; in studying the work of minds so unlike their own, they will immediately recognize that in this work there has been an accumulation of hypotheses.
Doubtless this will not suffice to make them comprehend that they on their part have made just as many; each sees the mote without seeing the beam; but by their criticisms they will warn their rivals, and it may be supposed these will not fail to render them the same service.
The English procedure often seems to us crude, the analogies they think they discover to us seem at times superficial; they are not sufficiently interlocked, not precise enough; they sometimes permit incoherences, contradictions in terms, which shock a geometric spirit and which the employment of the mathematical method would immediately have put in evidence. But most often it is, on the other hand, very fortunate that they have not perceived these contradictions; else would they have rejected their model and could not have deduced from it the brilliant results they have often made to come out of it.
And then these very contradictions, when they end by perceiving them, have the advantage of showing them the hypothetical character of their conceptions, whereas the mathematical method, by its apparent rigor and inflexible course, often inspires in us a confidence nothing warrants, and prevents our looking about us. ​From another point of view, however, the two conceptions are very unlike, and if all must be said, they are very unlike because of a common fault.
The English wish to make the world out of what we see. I mean what we see with the unaided eye, not the microscope, nor that still more subtile microscope, the human head guided by scientific induction.
The Latin wants to make it out of formulas, but these formulas are still the quintessenced expression of what we see. In a word, both would make the unknown out of the known, and their excuse is that there is no way of doing otherwise.
And yet is this legitimate, if the unknown be the simple and the known the complex?
Shall we not get of the simple a false idea, if we think it like the complex, or worse yet if we strive to make it out of elements which are themselves compounds?
Is not each great advance accomplished precisely the day some one has discovered under the complex aggregate shown by our senses something far more simple, not even resembling it—as when Newton replaced Kepler’s three laws by the single law of gravitation, which was something simpler, equivalent, yet unlike?
One is justified in asking if we are not on the eve of just such a revolution or one even more important. Matter seems on the point of losing its mass, its solidest attribute, and resolving itself into electrons. Mechanics must then give place to a broader conception which will explain it, but which it will not explain.
So it was in vain the attempt was made in England to construct the ether by material models, or in France to apply to it the laws of dynamic.
The ether it is, the unknown, which explains matter, the known; matter is incapable of explaining the ether.
Poincaré
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