Modern Birkhäuser Classics
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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] Originally published as a hardcover edition under the same title by Birkhäuser Boston, ISBN 978-0-8176-3739-2, ©1995 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 Printed on acid-free paper © 1981 , First Edition, Birkhiiuser Birkhiiuser }J ® Published 1995. Study Edition Birkhiiuser 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 Xl 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.