Modern Birkhäuser Classics

Many of the original research and survey monographs, as well as textbooks, in pure and applied mathematics published by Birkhäuser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new genera- tions of students, scholars, and researchers.

The Mathematical Experience, Study Edition

Philip J. Davis Reuben Hersh Elena Anne Marchisotto

Reprint of the 1995 Edition Updated with Epilogues by the Authors Philip J. Davis Reuben Hersh Division of Applied Mathematics Department of Mathematics Brown University and Statistics Providence, RI 02912 University of New Mexico USA Albuquerque, NM 87131 [email protected] USA [email protected]

Elena Anne Marchisotto Department of Mathematics California State University, Northridge Northridge, CA 91330 [email protected]

ISBN 978-0-8176-8294-1 e-ISBN 978-0-8176-8295-8 DOI 10.1007/978-0-8176-8295-8 Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011939204

© Springer Science+B usiness Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper www.birkhauser-science.com Philip J. Davis Reuben Hersh Elena Anne Marchisotto With an Introduction by Gian-Carlo Rota

The Mathematical Experience Study Edition

Birkhauser Boston • Basel • Berlin Philip J. Davis Reuben Hersh Division of Applied Mathematics Department of Mathematics Brown University and Statistics Providence, RI 02912 University of New Mexico Albuquerque, NM 87131 Elena Anne Marchisotto Department of Mathematics California State University, Northridge Northridge, CA 913308313

Library of Congress Cataloging-in-Publication Data

Davis, Philip J., 1923- The mathematical experience I Philip J. Davis, Reuben Hersh, Elena Anne Marchisotto : with an introduction by Gian-Carlo Rota. -- Study ed. p. em. Includes bibliographical references and index. ISBN 0-8176-3739-7 (h : acid-free).-- ISBN 3-7643-3739-7 (h acid-free) I. Mathematics--Philosophy. 2. Mathematics--History. 3. Mathematics--Study and teaching. I. Hersh, Reuben, 1927- 11. Marchisotto, Elena. III. Title. QA8.4.D37 1995 95-20875 51 O--dc20 CIP

Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhiiuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of$6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhiiuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-3739-7 ISBN 3-7643-3739-7 First edition was designed by Mike Fender, Cambridge, MA and set in Baskerville and I.T.C. Caslon Light 223 by Progressive Typographers, York, P A Text for Study Edition was typeset by Martin Stock, Cambridge, MA Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the U.S.A. 9 8 7 6 5 4 3 2 1 For my parents, Mildred and Philip Hersh

* * * *

For my brother, Hyman R. Davis

* * * *

For m y paren t s, Helen Cullura and Charles Joseph Corie

Contents

Preface Xlll Preface to the Study Edition XV Acknowledgements xvii Introduction XXI Overture 1 I. The Mathematical Landscape What is Mathematics? 6 Where is Mathematics 8 The Mathematical Community 9 The Tools of the Trade 13 How Much Mathematics is Now Known? 17 Ulam's Dilemma 20 How Much Mathematics Can There Be? 24 Appendix A-Brief Chronological Table to 1910 26 Appendix B-The Classification of Mathematics 1868 and 1979 Compared 29 Assignments and Problem Sets 31 2. Varieties of Mathematical Experience The Current Individual and Collective Consciousness 36 The Ideal Mathematician 38 A Physicist Looks at Mathematics 48 I. R. Shafarevitch and the New Neoplatonism 56 Unorthodoxies 59 The Individual and the Culture 64 Assignments and Problem Sets 70 3. Outer Issues Why Mathematics Works: A Conventionalist Answer 76 Mathematical Models 85

lX Contents

Utility 87 1. Varieties of Mathematical Uses 87 2. On the Utility of Mathematics to Mathematics 88 3. On the Utility of Mathematics to Other Scientific or Technological Fields 91 4. Pure vs. Applied Mathematics 93 5. From Haryism to Mathematical Maoism 95 Underneath the Fig Leaf 97 I. Mathematics in the Marketplace 97 2. Mathematics and War 101 3. Number Mysticism 104 4. Hermetic Geometry 108 5. Astrology 109 6. Religion 116 Abstraction and Scholastic Theology 121 Assignments and Problem Sets 128 4. Inner Issues Symbols 138 Abstraction 142 Generalization 150 Formalization 152 Mathematical Objects and Structure; Existence 156 Proof 163 Infinity, or the Miraculous Jar of Mathematics 168 The Stretched String 174 The Coin of Tyche 179 The Aesthetic Component 184 Pattern, Order, and Chaos 188 Algorithmic vs. Dialectic Mathematics 196 The Drive to Generality and Abstraction The Chinese Remainder Theorem: A Case Study 203 Mathematics as Enigma 212 Unity within Diversity 214 Assignments and Problem Sets 217 5. Selected Topics in Mathematics Group Theory and the Classification of Finite Simple Groups 227 The Prime Number Theorem 233 Non-Euclidean Geometry 241

X Contents

Non-Cantorian Set Theory 247 Appendix A 261 Nonstandard Analysis 261 Fourier Analysis 279 Assignments and Problem Sets 295 6. Teaching and Learning Confessions of a Prep School Math Teacher 304 The Classic Classroom Crisis of Understanding and Pedagogy 306 P6lya's Craft of Discovery 317 The Creation of New Mathematics: An Application of the Lakatos Heuristic 323 Comparative Aesthetics 330 Nonanalytic Aspects of Mathematics 333 Assignments and Problem Sets 349 7. From Certainty to Fallibility Platonism, Formalism, Constructivism 356 The Philosophical Plight of the Working Mathematician 359 The Euclid Myth 360 Foundations, Found and Lost 368 The Formalist Philosophy of Mathematics 377 Lakatos and the Philosophy of Dubitability 383 Assignments and Problems Sets 398 8. Mathematical Reality The Riemann Hypothesis 405 1r and -IT 411 Mathematical Models, Computers, and Platonism 417 Why Should I Believe a Computer? 422 Classification of Finite Simple Groups 429 Intuition 433 Four-Dimensional Intuition 442 True Facts About Imaginary Objects 448 Assignments and Problem Sets 454 Glossary 458 Bibliography 463 Index 481 Epilogues 489

Preface

HE OLDEST MATHEMATICAL tablets we have date from 2400 B.c., but there is no reason T to suppose that the urge to create and use mathe matics is not coextensive with the whole of civili zation. In tour or five millennia a vast body of practices and concepts known as mathematics has emerged and has been linked in a variety of ways with our day-to-day life. What is the nature of mathematics? What is its meaning? What are its concerns? What is its methodology? How is it created? How is it used? How does it fit in with the varieties of human experience? What benefits flow from it? What harm? What importance can be ascribed to it? These difficult questions are not made easier by the fact that the amount of material is so large and the amount of interlinking is so extensive that it is simply not possible for any one person to comprehend it all, let alone sum it up and compress the summary between the covers of an aver age-sized book. Lest we be cowed by this vast amount of material, let us think of mathematics in another way. Math ematics has been a human activity for thousands of years. To some small extent, everybody is a mathematician and does mathematics consciously. To buy at the market, to measure a strip of wall paper or to decorate a ceramic pot with a regular pattern is doing mathematics. Further, everybody is to some small extent a philosopher of mathe matics. Let him only exclaim on occasion: "But figures can't lie!" and he joins the ranks of Plato and of Lakatos. In addition to the vast population that uses mathematics on a modest scale, there are a small number of people who are professional mathematicians. They practice mathemat-

Xlll Preface

ics, foster it, teach it, create it, and use it in a wide variety of situations. It should be possible to explain to nonprofes sionals just what these people ar~ doing, what they say they are doing, and why the rest of the world should support them at it. This, in brief, is the task we have set for our selves. The book is not intended to present a systematic, self-contained discussion of a specific corpus of mathemati cal material, either recent or classical. It is intended rather to capture the inexhaustible variety presented by the math ematical experience. The major strands of our exposition will be the substance of mathematics, its history, its philoso phy, and how mathematical knowledge is elicited. The book should be regarded not as a compression but rather as an impression. It is not a mathematics book; it is a book about mathematics. Inevitably it must contain some mathe matics. Similarly, it is not a history or a philosophy book, but it will discuss mathematical history and philosophy. It follows that the reader must bring to it some slight p1ior knowledge of these things and a seed of interest to plant and water. The general reader with this background should have no difficulty in getting through the major por tion of the book. But there are a number of places where we have brought in specialized material and directed our exposition to the professional who uses or produces math ematics. Here the reader may feel like a guest who has been invited to a family dinner. After polite general con versation. the family turns to narrow family concerns, its delights and its worries, and the guest is left up in the air, but fascinated. At such places the reader should judiciously and lightheartedly push on. For the most part, the essays in this book can be read in dependently of each other. Some comment is necessary about the use of the word "I" in a book written by two people. In some instances it will be obvious which of the authors wrote the "I." In any case, mistaken identity can lead to no great damage, for each author agrees, in a general way, with the opinions of his colleague.

XIV Preface to the Study Edition

The first Mathematical Experience appeared in 1981. At that time, only a few years ago, it was commonly believed that it was impossible to make contemporary mathematics meaningful to the intelligent non-mathematician. Since then, dozens of popular books on contemporary mathematics have been pub lished. James Gleick's Chaos was a long-run best seller. John Casti is producing a continuing series of such books. In technology and invention, it's a commonplace that know ing what's possible is the most important ingredient of suc cessful innovation. Perhaps the first Mathematical Experience changed people's idea about what's possible in exposition of advanced contemporary mathematics. Alert readers recognized the book as a work of philosophy -a humanist philosophy of mathematics. It was far out, ''mav erick" (see Philip Kitcher), virtually out of contact with offi cial academic philosophy of mathematics. In the past 15 years, humanist philosophy of mathematics has bloomed. There are anthologies, symposia, a journal. The far-out maverick of 15 years ago might be the mainstream in a few years. The first Mathematical Experience was a trade book, not a text book. It was sold in book stores, not in professor's offices. But we heard over and over of college teachers using it, in the United States, Europe, Australia, Hong Kong, Israel. It's used in two different ways: "Math for liberal arts students" in col leges of art and science, and courses for future teachers, es pecially secondary math teachers, in colleges of education. In mathematics teaching, it's a commonplace that "Mathe matics isn't a spectator sport." You learn by doing, especially doing problems. Like all truisms, this is half true. Mathemat ics education as doing, doing, doing-no thinking, no con versation, no contemplation-can seem dreary. An artist isn't prohibited from occasional art appreciation-quite the con trary. You can'tlearn practical skill as a spectator, but you can learn good taste, among other things.

XV Preface

The first edition invited the reader to appreciate mathe matics, contemplate it, participate in a conversation about it. It contained no problems. If a teacher selected it, he/she had to supply what the book lacked. The study edition will be more convenient for both teacher and student. It aims for bal ance between doing and thinking. There are plenty of prob lems, mostly created by Professor Elena Anne Marchisotto, who also supplies generous discussion guides, essay topics, and bibliographies. We've also introduced "projects": con nected sequences of problems, rising in difficulty from very easy to a little less easy. They provide extra problem-solving enjoyment, and they make points about the nature of math ematics. We've written a section on differential and integral calculus-a complete course in 15 pages-and a section on the fascinating topic of complex numbers-fascinating from both mathematical and philosophic viewpoints. The Standards of the National Council ofTeachers of Math ematics appeared after the first Mathematical Experience. To a large extent, they validated our enterprise. We were following the Standards before they were written. The study edition does so even more than the first. No longer are "critical thinking" and "problem solving" just features of mathematics. They've become catchwords in American classrooms. The study edition of The Mathematical Experience is a part of the dominant trend in American educa tion.

xvi Acknowledgements

OME OF THE MATERIAL of this book was ex cerpted from published articles. Several of these Shave joint authorship: "Non-Cantorian Set The ory" by Paul Cohen and Reuben Hersh and "Non Standard Analysis" by Martin Davis and Reuben Hersh both appeared in the Scientific American. "Nonanalytic As pects of Mathematics" by Philip J. Davis and James A. An derson appeared in the SIAM Review. To Professors An derson, Cohen, and M. Davis and to these publishers, we extend our grateful acknowledgement for permission to include their work here. Individual articles by the authors excerpted here include "Number," "Numerical Analysis," and "Mathematics by Fiat?" by Philip J. Davis which appeared in the Scientific American, "The Mathematical Sciences," M.I.T. Press, and the Two Year College Mathematical journal respectively; "Some Proposals for Reviving the Philosophy of Mathe matics" and "Introducing Imre Lakatos" by Reuben Hersh, which appeared in Advances in Mathematics and the Mathematical Intelligencer, respectively. We appreciate the courtesy of the following organiza tions and individuals who allowed us to reproduce material in this book: The Academy of Sciences at Gottingen, Ambix, Dover Publishers, Mathematics of Computation, M.I.T. Press, New Yorker Magazine, Professor A. H. Schoen feld, and John Wiley and Sons. The section on Fourier analysis was written by Reuben Hersh and Phyllis Hersh. In critical discussions of philo sophical questions, in patient and careful editing of rough drafts, and in her unfailing moral support of this

XVII Acknowledgements project, Phyllis Hersh made essential contributions which it is a pleasure to acknowledge. The following individuals and institutions generously al lowed us to reproduce graphic and artistic material: Pro fessors Thomas Banchoff and Charles Strauss, th~ Brown University Library, the Museum of Modern Art, The Lummus Company, Professor Ron Resch, Routledge and Kegan Paul, Professor A. J. Sachs, the University of Chi cago Press, the Whitworth Art Gallery, the University of Manchester, the University of Utah, Department of Com puter Science, the Yale University Press. We wish to thank Professors Peter Lax and Gian-Carlo Rota for encouragement and suggestions. Professor Ga briel Stoltzenberg engaged us in a lively and productive correspondence on some of the issues discussed here. Pro fessor Lawrence D. Kugler read the manuscript and made many valuable criticisms. Professor Jose Luis Abreu's par ticipation in a Seminar on the Philosophy of Mathematics at the University of New Mexico is greatly appreciated. The participants in the Seminar on Philosophical Issues in Mathematics, held at Brown University, as well as the students in courses given at the University of New Mexico and at Brown, helped us crystallize our views and this help is gratefully acknowledged. The assistance of Professor Igor Najfeld was particularly welcome. We should like to express our appreciation to our col leagues in the History of Mathematics Department at Brown University. In the course of many years of shared lunches, Professors David Pingree, Otto Neugebauer, A. J. Sachs, and Gerald Toomer supplied us with the "three I's": information, insight, and inspiration. Thanks go to Profes sor Din-Yu Hsieh for information about the history of Chi nese mathematics. Special thanks to Eleanor Addison for many line draw ings. We are grateful to Edith Lazear for her careful and critical reading of Chapters 7 and 8 and her editorial com ments. We wish to thank Katrina Avery, Frances Beagan, Jo seph M. Davis, Ezoura Fonseca, and Frances Gajdowski for xviii their efficient help in the preparation and handling of the manuscript. Ms. Avery also helped us with a number of classical references. We would like to thank Professor Julian Gevirtz for a careful reading of the first printing which helped us find a number of misprints and errors.

P. J. DAVIS R. HERSH

xix

Introduction

DEDICATED TO MARK KAC "oh philosophie alimentaire!" -Sartre

HE TURN OF THE CENTURY, the Swiss his torian Jakob Burckhardt, who, unlike most A historians, was fond of guessing the future, once confided to his friend Friedrich Nietzsche the prediction that the Twentieth Century would be "the age of oversimplification". Burckhardt's prediction has proved frighteningly accu rate. Dictators and demagogues of all colors have captured the trust of the masses by promising a life of bread and bliss, to come right after the war to end all wars. Philoso phers have proposed daring reductions of the complexity of existence to the mechanics of elastic billiard balls; others, more sophisticated, have held that life is language, and that language is in turn nothing but strings of marble like units held together by the catchy connectives of Fre gean logic. Artists who dished out in all seriousness check erboard patterns in red, white, and blue are now fetching the highest bids at Sotheby's. The use of such words as "mechanically" "automatically" and "immediately" is now accepted by the wizards of Madison Avenue as the first law of advertising. Not even the best minds of Science have been immune to the lure of oversimplification. Physics has been driven by the search for one, only one law which one day, just around the corner, will unify all forces: gravitation and

XXI Introduction electricity and strong and weak interactions and what not. Biologists are now mesmerized by the prospect that the se cret of life may be gleaned from a double helix dotted with large molecules. Psychologists have prescribed in turn sex ual release, wonder drugs and primal screams as the cure for common depression, while preachers would counter with the less expensive offer to join the hosannahing cho rus of the born-again. It goes to the credit of mathematicians to have been the slowest to join this movement. Mathematics, like theology and all free creations of the Mind, obeys the inexorable laws of the imaginary, and the Pollyannas of the day are of little help in establishing the truth of a conjecture. One may pay lip service to Descartes and Grothendieck when they wish that geometry be reduced to algebra, or toRus sell and Gentzen when they command that mathematics become logic, but we know that some mathematicians are more endowed with the talent of drawing pictures, others with that ofjuggling symbols and yet others with the ability of picking the flaw in an argument. Nonetheless, some mathematicians have given in to the simplistics of our day when it comes to the understanding of the nature of their activity and of the standing of mathe matics in the world at large. With good reason, nobody likes to be told what he is really doing or to have his inti mate working habits analyzed and written up. What might Senator Proxmire say if he were to set his eyes upon such an account? It might be more rewarding to slip into the Senator's hands the textbook for Philosophy of Science 301, where the author, an ambitious young member of the Philosophy Department, depicts with impeccable clarity the ideal mathematician ideally working in an ideal world. We often hear that mathematics consists mainly in "proving theorems". Is a writer's job mainly that of "writing sentences"? A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking and frustra tion, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks. Few people, if any, had dared write this out loud before Davis and Hersh. Theorems are not to

xxii Introduction mathematics what successful courses are to a meal. The nu tritional analogy is misleading. To master mathematics is to master an intangible view, it is to acquire the skill of the vir tuoso who cannot pin his performance on criteria. The theorems of geometry are not related to the field of Geom etry as elements are to a set. The relationship is more sub tle, and Davis and Hersh give a rare honest description of this relationship. After Davis and Hersh, it will be hard to uphold the Glas perlenspiel view of mathematics. The mystery of mathemat ics, in the authors' amply documented account, is that con clusions originating in the play of the mind do find striking practical 2pplications. Davis and Hersh have chosen to de scribe the mystery rather than explain it away. Making mathematics accessible to the educated layman, while keeping high scientific standards, has always been considered a treacherous navigation between the ~cylla of professional contempt and the Charybdis of public misun derstanding. Davis and Hersh have sailed across the Strait under full sail. They have opened a discussion of the math ematical experience that is inevitable for survival. Watch ing from the stern of their ship, we breathe a sigh of relief as the vortex of oversimplification recedes into the dis tance.

GIAN-CARLO ROTA August 9, 1980

xxm

"The knowledge at which geometry aims is the knowledge of the eternal." PLATO, REPUBLIC, VII, 527

"That sometimes clear . . . and sometimes vague stuff ... which is ... mathematics." IMRE LAKATOS, 1922-1974

"What is laid down, ordered, factual, is never enough to embrace the whole truth: life always spills over the rim of every cup." BORIS PASTERNAK, 1890-1960