The Foundations of Science: Science and Hypothesis, the Value of Science, Science and Method

Volume Ii. Medical Research and Education. by Richard Mills

Pearce, William H. Welch, W. H. Howell, Franklin P. Mall, Lewellys F. Barker, Charles S. Minot, W. B. Cannon, W. T. Councilman Theobald Smith, G. N. Stewart, C. M. Jackson, E. P. Lyon, James B. Herrick, John M. Dodson, C. R. Bardeen, W. Ophuls, S. J. Meltzer, James Ewing, W. W. Keen, Henry H. Donaldson, Christian A. Herter, and Henry P. Bowditch.

Volume Iii. University Control. by J. Mckeen Cattell and Other

authors.

AMERICAN MEN OF SCIENCE. A Biographical Directory.

SCIENCE. A weekly journal devoted to the advancement of science. The official organ of the American Association for the Advancement of Science.

THE POPULAR SCIENCE MONTHLY. A monthly magazine devoted to the diffusion of science.

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THE SCIENCE PRESS

NEW YORK GARRISON, N. Y.

THE FOUNDATIONS OF SCIENCE

SCIENCE AND HYPOTHESIS THE VALUE OF SCIENCE SCIENCE AND METHOD

BY H. POINCARÉ

AUTHORIZED TRANSLATION BY GEORGE BRUCE HALSTED

WITH A SPECIAL PREFACE BY POINCARÉ, AND AN INTRODUCTION BY JOSIAH ROYCE, HARVARD UNIVERSITY

THE SCIENCE PRESS NEW YORK AND GARRISON, N. Y. 1913

Copyright, 1913 BY THE SCIENCE PRESS

PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA.

CONTENTS

PAGE Henri Poincaré ix Author's Preface to the Translation 3

SCIENCE AND HYPOTHESIS

Introduction by Royce 9 Introduction 27

Part I. _number and Magnitude_

CHAPTER I.--On the Nature of Mathematical Reasoning 31 Syllogistic Deduction 31 Verification and Proof 32 Elements of Arithmetic 33 Reasoning by Recurrence 37 Induction 40 Mathematical Construction 41

Chapter Ii.--Mathematical Magnitude and Experience 43

Definition of Incommensurables 44 The Physical Continuum 46 Creation of the Mathematical Continuum 46 Measurable Magnitude 49 Various Remarks (Curves without Tangents) 50 The Physical Continuum of Several Dimensions 52 The Mathematical Continuum of Several Dimensions 53

Part Ii. _space_

CHAPTER III.--The Non-Euclidean Geometries 55 The Bolyai-Lobachevski Geometry 56 Riemann's Geometry 57 The Surfaces of Constant Curvature 58 Interpretation of Non-Euclidean Geometries 59 The Implicit Axioms 60 The Fourth Geometry 62 Lie's Theorem 62 Riemann's Geometries 63 On the Nature of Axioms 63

Chapter Iv.--Space and Geometry 66

Geometric Space and Perceptual Space 66 Visual Space 67 Tactile Space and Motor Space 68 Characteristics of Perceptual Space 69 Change of State and Change of Position 70 Conditions of Compensation 72 Solid Bodies and Geometry 72 Law of Homogeneity 74 The Non-Euclidean World 75 The World of Four Dimensions 78 Conclusions 79

Chapter V.--Experience and Geometry 81

Geometry and Astronomy 81 The Law of Relativity 83 Bearing of Experiments 86 Supplement (What is a Point?) 89 Ancestral Experience 91

Part Iii. _force_

CHAPTER VI.--The Classic Mechanics 92 The Principle of Inertia 93 The Law of Acceleration 97 Anthropomorphic Mechanics 103 The School of the Thread 104

Chapter Vii.--Relative Motion and Absolute Motion 107

The Principle of Relative Motion 107 Newton's Argument 108

CHAPTER VIII.--Energy and Thermodynamics 115 Energetics 115 Thermodynamics 119 General Conclusions on Part III 123

PART IV. Nature

CHAPTER IX.--Hypotheses in Physics 127 The Rôle of Experiment and Generalization 127 The Unity of Nature 130 The Rôle of Hypothesis 133 Origin of Mathematical Physics 136

Chapter X.--the Theories of Modern Physics 140

Meaning of Physical Theories 140 Physics and Mechanism 144 Present State of the Science 148

CHAPTER XI.--The Calculus of Probabilities 155 Classification of the Problems of Probability 158 Probability in Mathematics 161 Probability in the Physical Sciences 164 Rouge et noir 167 The Probability of Causes 169 The Theory of Errors 170 Conclusions 172

Chapter Xii.--Optics and Electricity 174

Fresnel's Theory 174 Maxwell's Theory 175 The Mechanical Explanation of Physical Phenomena 177

CHAPTER XIII.--Electrodynamics 184 Ampère's Theory 184 Closed Currents 185 Action of a Closed Current on a Portion of Current 186 Continuous Rotations 187 Mutual Action of Two Open Currents 189 Induction 190 Theory of Helmholtz 191 Difficulties Raised by these Theories 193 Maxwell's Theory 193 Rowland's Experiment 194 The Theory of Lorentz 196

THE VALUE OF SCIENCE

Translator's Introduction 201 Does the Scientist Create Science? 201 The Mind Dispelling Optical Illusions 202 Euclid not Necessary 202 Without Hypotheses, no Science 203 What Outcome? 203 Introduction 205

Part I. _the Mathematical Sciences_

CHAPTER I.--Intuition and Logic in Mathematics 210

CHAPTER II.--The Measure of Time 223

CHAPTER III.--The Notion of Space 235 Qualitative Geometry 238 The Physical Continuum of Several Dimensions 240 The Notion of Point 244 The Notion of Displacement 247 Visual Space 252

Chapter Iv.--Space and Its Three Dimensions 256

The Group of Displacements 256 Identity of Two Points 259 Tactile Space 264 Identity of the Different Spaces 268 Space and Empiricism 271 Rôle of the Semicircular Canals 276

Part Ii. _the Physical Sciences_

CHAPTER V.--Analysis and Physics 279

CHAPTER VI.--Astronomy 289

CHAPTER VII.--The History of Mathematical Physics 297 The Physics of Central Forces 297 The Physics of the Principles 299

CHAPTER VIII.--The Present Crisis in Physics 303 The New Crisis 303 Carnot's Principle 303 The Principle of Relativity 305 Newton's Principle 308 Lavoisier's Principle 310 Mayer's Principle 312

Chapter Ix.--the Future of Mathematical Physics 314

The Principles and Experiment 314 The Rôle of the Analyst 314 Aberration and Astronomy 315 Electrons and Spectra 316 Conventions preceding Experiment 317 Future Mathematical Physics 319

Part Iii. _the Objective Value of Science_

CHAPTER X.--Is Science Artificial? 321 The Philosophy of LeRoy 321 Science, Rule of Action 323 The Crude Fact and the Scientific Fact 325 Nominalism and the Universal Invariant 333

Chapter Xi.--Science and Reality 340

Contingence and Determinism 340 Objectivity of Science 347 The Rotation of the Earth 353 Science for Its Own Sake 354

SCIENCE AND METHOD

Introduction 359

Book I. _science and the Scientist_

CHAPTER I.--The Choice of Facts 362

CHAPTER II.--The Future of Mathematics 369

CHAPTER III.--Mathematical Creation 383

CHAPTER IV.--Chance 395

BOOK II. Mathematical Reasoning

CHAPTER I.--The Relativity of Space 413

CHAPTER II.--Mathematical Definitions and Teaching 430

CHAPTER III.--Mathematics and Logic 448

CHAPTER IV.--The New Logics 460

CHAPTER V.--The Latest Efforts of the Logisticians 472

BOOK III. The New Mechanics

CHAPTER I.--Mechanics and Radium 486

CHAPTER II.--Mechanics and Optics 496

CHAPTER III.--The New Mechanics and Astronomy 512

BOOK IV. Astronomic Science

CHAPTER I.--The Milky Way and the Theory of Gases 523

CHAPTER II.--French Geodesy 535

General Conclusions 544

Index 547

HENRI POINCARÉ

SIR GEORGE DARWIN, worthy son of an immortal father, said, referring to what Poincaré was to him and to his work: "He must be regarded as the presiding genius--or, shall I say, my patron saint?"

Henri Poincaré was born April 29, 1854, at Nancy, where his father was a physician highly respected. His schooling was broken into by the war of 1870-71, to get news of which he learned to read the German newspapers. He outclassed the other boys of his age in all subjects and in 1873 passed highest into the École Polytechnique, where, like John Bolyai at Maros Vásárhely, he followed the courses in mathematics without taking a note and without the syllabus. He proceeded in 1875 to the School of Mines, and was Nommé, March 26, 1879. But he won his doctorate in the University of Paris, August 1, 1879, and was appointed to teach in the Faculté des Sciences de Caen, December 1, 1879, whence he was quickly called to the University of Paris, teaching there from October 21, 1881, until his death, July 17, 1912. So it is an error to say he started as an engineer. At the early age of thirty-two he became a member of l'Académie des Sciences, and, March 5, 1908, was chosen Membre de l'Académie Française. July 1, 1909, the number of his writings was 436.

His earliest publication was in 1878, and was not important. Afterward came an essay submitted in competition for the Grand Prix offered in 1880, but it did not win. Suddenly there came a change, a striking fire, a bursting forth, in February, 1881, and Poincaré tells us the very minute it happened. Mounting an omnibus, "at the moment when I put my foot upon the step, the idea came to me, without anything in my previous thoughts seeming to foreshadow it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry." Thereby was opened a perspective new and immense. Moreover, the magic wand of his whole life-work had been grasped, the Aladdin's lamp had been rubbed, non-Euclidean geometry, whose necromancy was to open up a new theory of our universe, whose brilliant exposition was commenced in his book Science and Hypothesis, which has been translated into six languages and has already had a circulation of over 20,000. The non-Euclidean notion is that of the possibility of alternative laws of nature, which in the Introduction to the Électricité et Optique, 1901, is thus put: "If therefore a phenomenon admits of a complete mechanical explanation, it will admit of an infinity of Others which will account equally well for all the peculiarities disclosed by experiment."

The scheme of laws of nature so largely due to Newton is merely one of an infinite number of conceivable rational schemes for helping us master and make experience; it is commode, convenient; but perhaps another may be vastly more advantageous. The old conception of true has been revised. The first expression of the new idea occurs on the title page of John Bolyai's marvelous Science Absolute of Space, in the phrase "haud unquam a priori decidenda."

With bearing on the history of the earth and moon system and the origin of double stars, in formulating the geometric criterion of stability, Poincaré proved the existence of a previously unknown pear-shaped figure, with the possibility that the progressive deformation of this figure with increasing angular velocity might result in the breaking up of the rotating body into two detached masses. Of his treatise Les Méthodes nouvelles de la Méchanique céleste, Sir George Darwin says: "It is probable that for half a century to come it will be the mine from which humbler investigators will excavate their materials." Brilliant was his appreciation of Poincaré in presenting the gold medal of the Royal Astronomical Society. The three others most akin in genius are linked with him by the Sylvester medal of the Royal Society, the Lobachevski medal of the Physico-Mathematical Society of Kazan, and the Bolyai prize of the Hungarian Academy of Sciences. His work must be reckoned with the greatest mathematical achievements of mankind.

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