The Mathematical Experience The Mathematical Experience Philip J Davis Reuben Hersh With an Introduction by Gian-Carlo Rota HOVGHTON MIFFLIN COMPANY BOSTON CopyrigJII © 1981 by Birkhliuser Boston All rights rcser\'ed Nil I'"rl of Ihis work ma)' bc reproduced or I ransmilled in lUI), form or b)' an)' mcans, cieci runic 01' mechanical, including photocopying lllld rccnrding, or by allY informmion siorage or rClric\'l.1 S)·slelll excepl as may he expressly permillcd h)' Ihc 1!l76 Cop)Tiglli ACI or in , -riling from Ihc publishcr Requests for permission should he lHldl'cssed in "Tiling 10 I-)oughlon ~[irnin COJ1lpany "ark Sireet, Boston, Massachusells 1J21OH Ubrary rif C/JIIKmt.l Cntalogillg ill I'/llilicalioll Data navis, Philip J date The malhematical experience Reprint Originally puhlishcd: BnSlOn: Uirkhauser 19111 niblingraphy: p Includes index I ~Ialhemal ics -Philosoph)' ~I alhemlllics- H iswl'y :t ~lathematirs-Slll(ly :lIId leaching I Hcrsh Rcuben date J I Titlc, QAHA.1>37 1982 5)() RI·203(H ISBi\ O·:l9:'·!~2131·X (pbk.) AACR2 "rillled in the Uniled Sillies of ,\mericli ALIO ~ I Reprinled by arrangemcnl wilh nirkhliuser BnSlO1l Houghton Mimi" COJ1lP:IIlY 1':.pcl'hilck [982 For my parents, Mildred and Philip Hersh **** For my brother, Hyman R Davis Contents Preface Acknowledgements Introduction Overture The Mathematical Landscape What is Mathematics? Where is Mathematics? The Mathematical Community The Tools of the Trade How Much Mathematics is Now Known? Ulam's Dilemma How Much Mathematics Can There Be? Appendix A-Brief Chronological Table to 1910 Appendix B-The Classification of Mathematics 1868 and 1979 Compared Varieties of Mathematical Experience The Current Individual and Collective ConsCIousness The Ideal Mathematician A Physicist Looks at Mathematics I R Shafarevitch and the New Neo~~n~m Unorthodoxies The Individual and the Culture Outer Issues Why Mathematics Works: A Conventionalist Answer XI XIII XVII 13 17 20 24 26 29 32 34 44 ~ 55 60 68 Contents Mathematical Models Utility Varieties of Mathematical Uses On the Utility of Mathematics to Mathematics On the Utilil)' of Mathematics to Other Scientific or Technological Fields Pure vs Applied Mathematics From Hardyism to Mathematical Maoism Underneath the Fig Leaf Mathematics in the Marketplace Mathematics and War Number Mysticism He17fletic Geometry Astrology Religion Abstraction and Scholastic Theology ii i9 i9 80 83 85 8i 89 89 93 911 100 101 IOH II :~ Inner Issues Symbols Abstraction Generalization Formalization Mathematical Objects and Structures; Existence Proof Infinity, or the Miraculous Jar of Mathematics The Stretched String The Coin of Tyche The Aesthetic Component Pattern, Order, and Chaos Algorithmic vs Dialectic Mathematics The Drive to Generality and Abstraction The Chinese Remainder Theorem: A Case Study Mathematics as Enigma Unity within Diversity Selected Topics in Mathematics Group Theory and the Classification of Finite Simple Groups 19 '> 1211 134 131) ~ 140 14i 152 158 163 168 I-