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Mathematics and Science: Last Essays HENRI POINCARE Mathematics and Science: Last Essays (Dernieres Pensees) Translated from the French by JOHN W. BOLDUC State Teachers College at Westfield, Massachusetts Dover Publications, Inc., New York Copyright 1963 by Dover Publications, Inc Origmally published in French under the title of Demises Pensdes, copyright (P) 1913 by Ernest Flammanon All rights reserved under Pan American and Inter national Copyright Conventions. Published simultaneously in Canada by McClelland and Stewart, Ltd. Published in the United Kingdom by Constable and Company Limited, 10 Change Stieet, London, W.C.2. This Dover edition, firsL published in 1963, is a new English translation of the first edition of Demises Fences, published by Ernest Flammanon in 1913. This English translation is published by special arrangement with Ernest Flammarion. Library of Congress Catalog Card Number: 63-31678 Manufactured in the United Stales of America Dover Publications, Inc, i So Varick Street New York 14, N.Y. TRANSLATOR'S NOTE Just as the poet must seek the appropriate word to convey an idea with sufficient vigor and yet obtain the cadence and rhyme necessary for the finished product, so also the translator must achieve the proper expression in one language to convey accurately and with equal vigor the author's ideas as expressed in his original language. In this process the languages in the translator's mind tend to lose their identity and one language easily assumes the idiosyncrasies of the other. I am therefore particularly grateful to Dr. Wallace L. Goldstein for his assistance in indicating flaws in grammatical constructions which would have resulted from the merging of the two languages. His help was equally important in proof-reading the manuscript and in the preparation of the index. Any errors in the final product, however, are my own. J. W. B. FOREWORD Under the title of Last Essays are gathered the various articles and lectures which Mr. Henri Poincar6 himself had intended should form the fourth volume of his writings on the philosophy of science. All previous essays and articles had already been included in that series. It would be superfluous to point to the amazing success of the first three volumes. In these Poincare", as the most illustrious modern mathematician, proved to be an eminent philosopher and an author whose writings profoundly influence human thought. It is very likely that if Henri Poincare" had published this volume himself, he would have modified certain details, and eliminated to the some repetitions. But it seemed to us that the respect due memory of this great man should forbid any editing of his text. It seemed equally superfluous to preface this volume with com mentaries on the works of Henri Poincare". These have been evaluated by scholars and any commentary could not possibly increase the glory of this great genius. 65410l)'l G. L. B. TABLE OF CONTENTS I. THE EVOLUTION OF LAWS 1 II. SPACE AND TIME . 15 III. WHY SPACE HAS THREE DIMENSIONS ... 25 IV. THE LOGIC OF INFINITY 45 V. MATHEMATICS AND LOGIC 65 VI. THE QUANTUM THEORY 75 VII. THE RELATIONS BETWEEN MATTER AND ETHER . 89 VIII. ETHICS AND SCIENCE 102 IX. THE MORAL ALLIANCE 114 INDEX 119 Chapter I THE EVOLUTION OF LAWS Mr. Boutroux, in his writings on the contingency of the laws of Nature, queried, whether natural laws are not susceptible to change and if the world evolves continuously, whether the laws themselves which govern this evolution are alone exempt from all variation. Such a concept has no chance of ever being accepted by the scientist; in the sense in which he would understand it, the scientist could not accept it without denying the justness and the very possibility of science. But the philosopher reserves the right to ask such a question, to consider the various solutions which it entails, to examine their consequences, and to try to reconcile them to the reasonable demands of the scientist. I should like to consider a few of the aspects which the problem can assume. I shall thus arrive not at so-called conclusions, but at various reflections which may perhaps not be devoid of interest. If, in the process, I take the liberty of considering somewhat at length certain related questions, I hope that the reader will bear with me. First, let us assume the point of view of the mathematician. Let us grant for a moment that the physical laws have undergone variations during the course of the ages, and let us ask ourselves whether we would have the means of noticing these variations. Let us not forget, first of all, that the few centuries during which man has lived and thought were preceded by periods of incomparably longer duration when man did not exist as yet; they will no doubt be followed by eras when the human species will have ceased to exist. If we wish to believe in an evolution of laws, unquestionably it can only have been very slow, so that, during the few years that man has been able to reason, the laws of nature could have under gone only insignificant changes. If the laws did evolve in the past, we must understand by that the geological past. Were the laws of former eras those of today ? Will the laws of tomorrow still be the 1 2 Last Essays same? When we ask such a question, what meaning must we attach to the terms "formerly," "today," and "tomorrow?" By "today" we mean those periods which are recorded in history; by "formerly" we mean the millions of years which preceded recorded history and during which the ichthyosauri lived quietly without philosophizing; "tomorrow" means the millions of years which will follow, during which the earth will cool and man will have neither eyes to see nor brain with which to think. With this in mind, what is a law? It is a constant link between the antecedent and the consequent, between the current state of the world and the immediately succeeding state. Knowing the current state of each part of the universe, the ideal scientist who would know all the laws of nature would possess fixed rules by which to deduce the state in which these same parts shall be tomorrow. It is conceivable that this process could be carried on indefinitely. Knowing the state of the world on Monday, we can foretell its state on that of we can deduce its state on Tuesday ; knowing Tuesday, Wednesday by the same process; and so on. But this is not all; if there is a constant link between the state of the world on Monday and that of Tuesday, it is possible to infer the second from the first. But it will also be possible to perform the reverse; that is, if the state of the world that exists on Tuesday is known, it will be possible to infer that of Monday; from the state on Monday we shall infer that of Sunday; and so on. It is thus possible to trace the course of time backward as well as forward. Knowing both the present and the laws, we can foretell the future, but we can understand the past as well. The process is essentially reversible. Since we are assuming at this juncture the point of view of the mathematician, we must give to this concept all the precision that it requires, even if it becomes necessary to use mathematical language. We should then say that the body of laws is equivalent to a system of differential equations which link the speed ofvariations ofthe different elements of the universe to the present values of these elements. Such a system involves, as we know, an infinite number of solu tions, But if we take the initial values of all the elements, that is, their values at the instant t = (which would correspond in ordinary language to the "present"), the solution is completely determined, so that we can calculate the values of all the elements at any period whatever, whether we suppose />0, which corresponds to the "future," or whether we suppose t<0, which corresponds to the "past." What is important to remember is that the manner of inferring the past from the present does not differ from that of inferring the future from the present. The Evolution of Laws 3 What means do we therefore have of knowing the geological past; that the is, history of those former eras during which the laws could have varied ? This past could not have been observed directly and we know it only by the traces which it has left in the present. We know it only through the present, and we can draw inferences about the past only by the process which we have just described, and which will permit us in the same manner to draw inferences about the future. But is this process capable of revealing changes in the laws? Obviously not; for we can apply this principle precisely that the laws have not only by supposing changed ; we know directly merely the state of the world on Monday, for example, and the rules which link the state on Sunday with that on Monday. The application of these rules will therefore enable us to know the state on Sunday; but if we wish to pursue this further and to deduce the state of the world on Saturday, it is absolutely necessary that we admit that the same rules which permitted us to proceed from Monday to Sunday were still valid between Sunday and Saturday, Without this, the only conclusion which we would be permitted to draw is that it is impossible to know what occurred on Saturday.
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