The mathematical experience

Title ......Page 3
Copyright ......Page 4
Dedication ......Page 5
Contents ......Page 7
Preface ......Page 11
Acknowledgements ......Page 13
Introduction ......Page 17
Epigraph ......Page 21
Overture ......Page 23
1. The Mathematical Landscape ......Page 27
What is Mathematics? ......Page 28
Where is Mathematics? ......Page 30
The Mathematical Community ......Page 31
The Tools of the Trade ......Page 35
How Much Mathematics is Now Known? ......Page 39
Ulam\'s Dilemma ......Page 42
How Much Mathematics Can There Be? ......Page 46
Appendix A—Brief Chronological Table to 1910 ......Page 48
Appendix B—The Classification of Mathematics. 1868 and 1979 Compared ......Page 51
2. Varieties of Mathematical Experience ......Page 53
The Current Individual and Collective Consciousness ......Page 54
The Ideal Mathematician ......Page 56
A Physicist Looks at Mathematics ......Page 66
I. R. Shafarevitch and the New Neoplatonism ......Page 74
Unorthodoxies ......Page 77
The Individual and the Culture ......Page 82
3. Outer Issues ......Page 89
Why Mathematics Works: A Conventionalist Answer ......Page 90
Mathematical Models ......Page 99
1. Varieties of Mathematical Uses ......Page 101
2. On the Utility of Mathematics to Mathematics ......Page 102
3. On the Utility of Mathematics to Other Scientific or Technological Fields ......Page 105
4. Pure vs. Applied Mathematics ......Page 107
5. From Hardyism to Mathematical Maoism ......Page 109
1. Mathematics in the Marketplace ......Page 111
2. Mathematics and War ......Page 115
3. Number Mysticism ......Page 118
4. Hermetic Geometry ......Page 122
5. Astrology ......Page 123
6. Religion ......Page 130
Abstraction and Scholastic Theology ......Page 135
4. Inner Issues ......Page 143
Symbols ......Page 144
Abstraction ......Page 148
Generalization ......Page 156
Formalization ......Page 158
Mathematical Objects and Structures; Existence ......Page 162
Proof ......Page 169
Infinity, or the Miraculous Jar of Mathematics ......Page 174
The Stretched String ......Page 180
The Coin of Tyche ......Page 185
The Aesthetic Component ......Page 190
Pattern, Order, and Chaos ......Page 194
Algorithmic vs. Dialectic Mathematics ......Page 202
The Drive to Generality and Abstraction. The Chinese Remainder Theorem: A Case Study ......Page 209
Mathematics as Enigma ......Page 218
Unity within Diversity ......Page 220
5. Selected Topics in Mathematics ......Page 223
Group Theory and the Classification of Finite Simple Groups ......Page 225
The Prime Number Theorem ......Page 231
Non-Euclidean Geometry ......Page 239
Non-Cantorian Set Theory ......Page 245
Nonstandard Analysis ......Page 259
Fourier Analysis ......Page 277
6. Teaching and Learning ......Page 293
Confessions of a Prep School Math Teacher ......Page 294
The Classic Classroom Crisis of Understanding and Pedagogy ......Page 296
Polya\'s Craft of Discovery ......Page 307
The Creation of New Mathematics: An Application of the Lakatos Heuristic ......Page 313
Comparative Aesthetics ......Page 320
Nonanalytic Aspects of Mathematics ......Page 323
7. From Certainty to Fallibility ......Page 339
Platonism, Formalism, Constructivism ......Page 340
The Philosophical Plight of the Working Mathematician ......Page 343
The Euclid Myth ......Page 344
Foundations, Found and Lost ......Page 352
The Formalist Philosophy of Mathematics ......Page 361
Lakatos and the Philosophy of Dubitability ......Page 367
8. Mathematical Reality ......Page 383
The Ricmann Hypothesis ......Page 385
Pi and Pi^ ......Page 391
Mathematical Models, Computers, and Platonism ......Page 397
Why Should I Believe a Computer? ......Page 402
Classification of Finite Simple Groups ......Page 409
Intuition ......Page 413
Four-Dimensional Intuition ......Page 422
True Facts About Imaginary Objects ......Page 428
Glossary ......Page 434
Bibliography ......Page 439
Index ......Page 457