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Applicability of Mathematics As a Philosophical Problem The Applicability of Mathematics as a Philosophical Problem The Appl icabiIity of Mathematics as a Philosophical Problem Mark Steiner Harvard University Press Cambridge, Massachusetts London, England 1998 To the memory ofmyfather and my mother Copyright © 1998 by the President and Fellows of Harvard College All rights reserved Printed in the United States of America Library of Congress Cataloging-in-Publication Data Steiner, Mark. The applicability of mathematics as a philosophical problem / Mark Steiner. p. cm. Includes bibliographical references and index. ISBN 0-674-04097-X (cloth) ISBN 0-674-00970-3 (paper) 1. Mathematical physics. 2. Mathelnatics-Philosophy. QC20.S743 1998 530. I5-dc2l 98-3468 This book has been digitally reprinted. The content remains identical to that ofprevious printings. Contents Acknowledgments vii Introduction 1. The Semantic Applicability of Mathematics: Frege's Achievements 13 2. The Descriptive Applicability of Mathematics 24 3. Mathematics, Analogies, and Discovery in Physics 48 4. Pythagorean Analogies in Physics 76 5. Formalisms and Formalist Reasoning in Quantum Mechanics 116 6. Formalist Reasoning: The Mystery of Quantization 136 Appendix A. A "Nonphysical" Derivation of Quantum Mechanics 177 Appendix B. Nucleon-Pion Scattering 197 Appendix C. Nonrelativistic Schroedinger Equation with Spin 200 References 203 Index 211 Acknowledgments I am grateful to Daniel Amit, Paul Benacerraf, Bernard Berofsky, Jed Buchwald, Sylvain Cappell, Alan Chalmers, Percy Deift, Burton Dreben, Shmuel Elitzur, Juliet Floyd, Harry Furstenberg, Joel Gersten, Shelly Goldstein, Shaughan Lavine, Jesper Liitzen, David Malament, Nathan Malkin, Benoit Mandelbrot, Sidney Morgenbesser, Yuval Ne'eman, Robert Nozick, Itamar Pitowsky, Carl Posy, Hilary Putnam, Sam Schweber, Stewart Shapiro, Barry Simon, Shlomo Sternberg, and Larry Zalcman for their much appre­ ciated advice and criticism. Tragically, I can no longer thank Yael Cohen for all the help she gave me in working on this book. Nessa Olshansky-Ashtar made several editorial suggestions; thanks, Nessa. Angela Blackburn's professional copyediting improved the book markedly; it was a treat to work with her. This book has had a very long gestation period. In some sense, it is my final attempt to persuade my advisor and mentor, Professor Paul Benacerraf, that there really is a philosophical problem about the applicability of mathematics in the natural sciences. His influence on this book consists as much in what I left out as in what I put in. To many of the people on this list, particularly the physicists and mathematicians, I am indebted for much more than advice and criti­ cism. I could never have written a book on modern physics without them-in some cases, they simply taught me the material. In other cases, they corrected dangerous misconceptions and shared their insights with me. I do not know whether to thank them or to apolo­ gize for taking up so much of their valuable time; in some cases, their viii Acknowledgments e-mail correspondence with me was more voluminous than this book. (Which reminds me to thank whoever invented e-mail, with­ out which this book certainly could not have been written.) The reader will note that some of the most beautiful examples of mathematical reasoning in physics in this book are borrowed from the writings of Professor Shlomo Sternberg. I am grateful to him for the patient way he explained these writings to me, even when it took him four or five attempts till I "got it." My research was supported by grants from the Israel Academy of Sciences and the United States National Science Foundation; for this support I express my appreciation. An invitation to the Dibner Institute at the Massachusetts Institute of Technology, for which I am grateful, afforded me the opportunity to consult with eminent scholars there and in the Boston area generally. Finally, my thanks go to the editors of the Journal of Philosophy and Philosophia Mathematica for permission to reproduce material from Steiner 1995 and Steiner 1989 in Chapters 1 and 3, respectively. And to my wife, Rachel, and the rest of the family, I offer thanks for the love and patience which waiting for the book required. Introduction This book has two separate objectives. The first is to examine in what ways mathematics can be said to be applicable in the natural sciences or, if you prefer, to the empirical world. Mathematics is applicable in many senses, and this ambiguity has bred confusion and error-even among "analytic" philosophers: because there are many senses of "application" and "applicability," there are many questions about the application of mathematics that ought to be, but have not been, distinguished by philosophers. As a result, we do not always know what problem they are dealing with. For example, we often use pure mathematics in reasoning about the empirical world. This raises two questions: what is the logical form of such reasoning, and what are its metaphysical presupposi­ tions? The problem of logical form arises because number words func­ tion in mathematics as proper nouns (names of numbers, numerals), while in empirical descriptions ("three green leaves") they often func­ tion as adjectives.! This is an equivocation, which appears to make it impossible to reason using mathematics in an empirical situation. Thus, some writers have felt themselves forced to distinguish between "pure mathematics" and "empirical" or "applied" mathematics. I will point out that Frege solved the problem, making this distinction entirely superfluous, by showing how number words can function as names, not adjectives, everywhere. Names of what? Names, apparently, of numbers, thought of as abstract or nonphysical objects. And this brings us to the metaphys- 1 This is true in English and other European languages, but not in the Semitic lan­ guages. Thanks to my brother, Richard Steiner, for clarifying the situation for me. 2 Introduction ical question concerning application: how can facts about numbers be relevant to the empirical world? Frege had a keen answer to this, too: they aren't. Numbers are related, not to empirical objects, but to empirical concepts! It is the empirical concepts that are used to describe the world; numbers are used to characterize those very descriptions. (Of course, there are other objections to Platonism than its alleged inability to account for applications, but this is not a book about Platonism.) Thus, two frequently asked questions about the application of mathematics were answered definitively over a hundred years ago. If philosophers sometimes write as if this were not so, the reason may be because, as I suggested above, the different concepts of applica­ bility have not been made clear. To do this is precisely my first objective. *** To the extent they have discussed mathematical applicability at all, contemporary philosophers have usually explored its implications for mathematics itself. Following the great physicists, however, I would like to explore its implications for our view of the universe and the place in it of the human mind (or minds like the human mind, if there are any), and this is the second objective of the book. Galileo, of course, remarked in the seventeenth century that the Universe is a book written in the language of mathematics (actually: geometry). But the best known contemporary essay on the applica­ tion of mathematics and its meaning for humans is by the celebrated physicist, Eugene Wigner, who spoke of the "unreasonable effective­ ness of mathematics in natural science," which is a gift we "neither understand nor deserve." In fact, my main argument concerning the applicability of mathematics, though different from Wigner's, could easily be confused with his. Let me, therefore, outline the main argu­ ment of my book and show how it differs from, and escapes common objections to, Wigner's.2 2 I do this at the suggestion of Burton Dreben, Juliet Floyd, and Robert Nozick, whom I had the good fortune to consult with while it was still possible to rewrite the Introduction. Further valuable advice came from Hilary Putnam, Sam Schweber, Alan Chalmers, Jed Buchwald, and Jesper Liitzen. The opportunity to meet all these people was made possible by an invitation to spend the year at the Dibner Institute of Introduction 3 At the end of the nineteenth century, physics was at a crossroads.. Scientists were attempting to describe the unseen world of the very small, and they already knew that the atomic world obeys different laws than those governing the macroscopic world. Of course, as Charles Peirce pointed out, once the laws were conjectured (abduc­ tion), one could verify them indirectly by checking their conse­ quences for the macroscopic world (induction). But first they had to be guessed, and Peirce himself (like John Locke before him) was pes­ simistic about the ability of the human species even to conjecture the laws of the atom. Evolution, he argued, could not have equipped the human species with the ability to come up with the laws of objects which play no role in our daily life. (This was exactly Locke's argu­ ment, except that he said "God" instead of "Evolution.") How, then, did scientists arrive at the atomic and subatomic laws of nature? My answer: by mathematical analogy. Of course, not only by mathematical analogy and not only the atomic problem. The usual procedure of experimental inquiry went on just as in the past. What was new in the conduct of research was an increased reliance by scientists also on mathematical analogies. To be sure, classic inquirers like Newton and Maxwell had already used mathematical analogies, as I will argue. Now, however, scientists used mathemati­ cal analogies because they had no real alternative. Scientists looked for laws bearing a similar (not necessarily identical) mathematical form to the laws they were trying to augment, refine, or even replace. Often these analogies were Pythagorean, meaning that the analogies were then inexpressible in any other language but that of pure math­ ematics.
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