The Mathematical Experience

This is the classic introduction for the educated lay reader to the richly diverse world of mathematics: its history, philosophy, principles, and personalities.

Philip J. Davis is a professor of applied mathematics at Brown University. Reuben Hersh is a professor of mathematics at the University of New Mexico in Albuquerque.

Preface

xi

(2)

Acknowledgements

xiii

(4)

Introduction

xvii

Overture

1

(5)

1. The Mathematical Landscape

6

(26)

What is Mathematics?

6

(2)

Where is Mathematics?

8

(1)

The Mathematical Community

9

(4)

The Tools of the Trade

13

(4)

How Much Mathematics is Now Known?

17

(3)

Ulam's Dilemma

20

(4)

How Much Mathematics Can There Be?

24

(2)

Appendix A -- Brief Chronological Table to 1910

26

(3)

Appendix B -- The Classification of Mathematics, 1868 and 1979 Compared

29

(3)

2. Varieties of Mathematical Experience

32

(36)

The Current Individual and Collective Consciousness

32

(2)

The Ideal Mathematician

34

(10)

A Physicist Looks at Mathematics

44

(8)

I. R. Shafarevitch and the New Neoplatonism

52

(3)

Unorthodoxies

55

(5)

The Individual and the Culture

60

(8)

3. Outer Issues

68

(54)

Why Mathematics Works: A Conventionalist Answer

68

(9)

Mathematical Models

77

(2)

Utility

79

(10)

1. Varieties of Mathematical Uses

79

(1)

2. On the Utility of Mathematics to Mathematics

80

(3)

3. On the Utility of Mathematics to Other Scientific or Technological Fields

83

(2)

4. Pure vs. Applied Mathematics

85

(2)

5. From Hardyism to Mathematical Maoism

87

(2)

Underneath the Fig Leaf

89

(24)

1. Mathematics in the Marketplace

89

(4)

2. Mathematics and War

93

(3)

3. Number Mysticism

96

(4)

4. Hermetic Geometry

100

(1)

5. Astrology

101

(7)

6. Religion

108

(5)

Abstraction and Scholastic Theology

113

(9)

4. Inner Issues

122

(81)

Symbols

122

(4)

Abstraction

126

(8)

Generalization

134

(2)

Formalization

136

(4)

Mathematical Objects and Structures; Existence

140

(7)

Proof

147

(5)

Infinity, or the Miraculous Jar of Mathematics

152

(6)

The Stretched String

158

(5)

The Coin of Tyche

163

(5)

The Aesthetic Component

168

(4)

Pattern, Order, and Chaos

172

(8)

Algorithmic vs. Dialectic Mathematics

180

(7)

The Drive to Generality and Abstraction The Chinese Remainder Theorem: A Case Study

187

(9)

Mathematics as Enigma

196

(2)

Unity within Diversity

198

(5)

5. Selected Topics in Mathematics

203

(69)

Group Theory and the Classification of Finite Simple Groups

203

(6)

The Prime Number Theorem

209

(8)

Non-Euclidean Geometry

217

(6)

Non-Cantorian Set Theory

223

(14)

Appendix A

237

(1)

Nonstandard Analysis

237

(18)

Fourier Analysis

255

(17)

6. Teaching and Learning

272

(46)

Confessions of a Prep School Math Teacher

272

(2)

The Classic Classroom Crisis of Understanding and Pedagogy

274

(11)

Polya's Craft of Discovery

285

(6)

The Creation of New Mathematics: An Application of the Lakatos Heuristic

291

(7)

Comparative Aesthetics

298

(3)

Nonanalytic Aspects of Mathematics

301

(17)

7. From Certainty to Fallibility

318

(45)

Platonism, Formalism, Constructivism

318

(3)

The Philosophical Plight of the Working Mathematician

321

(1)

The Euclid Myth

322

(8)

Foundations, Found and Lost

330

(9)

The Formalist Philosophy of Mathematics

339

(6)

Lakatos and the Philosophy of Dubitability

345

(18)

8. Mathematical Reality

363

(49)

The Riemann Hypothesis

363

(6)

Pie and Pie

369

(6)

Mathematical Models, Computers, and Platonism

375

(5)

Why Should I Believe a Computer?

380

(7)

Classification of Finite Simple Groups

387

(4)

Intuition

391

(9)

Four-Dimensional Intuition

400

(6)

True Facts About Imaginary Objects

406

(6)

Glossary

412

(5)

Bibliography

417

(18)

Index

435

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The Mathematical Experience

Summary

Author Biography

Table of Contents

Supplemental Materials