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| Preface | |
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| Acknowledgements | |
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| Introduction | |
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| Overture | |
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| The Mathematical Landscape | |
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| What is Mathematics? | |
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| Where is Mathematics? | |
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| The Mathematical Community | |
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| The Tools of the Trade | |
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| How Much Mathematics is Now Known? | |
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| Ulam's Dilemma | |
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| How Much Mathematics Can There Be? | |
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| Brief Chronological Table to 1910 | |
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| The Classification of Mathematics. 1868 and 1979 Compared | |
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| Varieties of Mathematical Experience | |
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| The Current Individual and Collective Consciousness | |
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| The Ideal Mathematician | |
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| A Physicist Looks at Mathematics | |
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| I. R. Shafarevitch and the New Neoplatonism | |
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| Unorthodoxies | |
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| The Individual and the Culture | |
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| Outer Issues | |
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| Why Mathematics Works: A Conventionalist Answer | |
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| Mathematical Models | |
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| Utility | |
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| Varieties of Mathematical Uses | |
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| On the Utility of Mathematics to Mathematics | |
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| On the Utility of Mathematics to Other Scientific or Technological Fields | |
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| Pure vs. Applied Mathematics | |
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| From Hardyism to Mathematical Maoism | |
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| Underneath the Fig Leaf | |
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| Mathematics in the Marketplace | |
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| Mathematics and War | |
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| Number Mysticism | |
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| Hermetic Geometry | |
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| Astrology | |
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| Religion | |
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| Abstraction and Scholastic Theology | |
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| Inner Issues | |
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| Symbols | |
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| Abstraction | |
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| Generalization | |
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| Formalization | |
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| Mathematical Objects and Structures; Existence | |
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| Proof | |
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| Infinity, or the Miraculous Jar of Mathematics | |
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| The Stretched String | |
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| The Coin of Tyche | |
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| The Aesthetic Component | |
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| Pattern, Order, and Chaos | |
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| Algorithmic vs. Dialectic Mathematics | |
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| The Drive to Generality and Abstraction The Chinese Remainder Theorem: A Case Study | |
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| Mathematics as Enigma | |
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| Unity within Diversity | |
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| Selected Topics in Mathematics | |
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| Group Theory and the Classification of Finite Simple Groups | |
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| The Prime Number Theorem | |
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| Non-Euclidean Geometry | |
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| Non-Cantorian Set Theory | |
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| Appendix A | |
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| Nonstandard Analysis | |
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| Fourier Analysis | |
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| Teaching and Learning | |
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| Confessions of a Prep School Math Teacher | |
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| The Classic Classroom Crisis of Understanding and Pedagogy | |
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| Polya's Craft of Discovery | |
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| The Creation of New Mathematics: An Application of the Lakatos Heuristic | |
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| Comparative Aesthetics | |
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| Nonanalytic Aspects of Mathematics | |
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| From Certainty to Fallibility | |
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| Platonism, Formalism, Constructivism | |
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| The Philosophical Plight of the Working Mathematician | |
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| The Euclid Myth | |
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| Foundations, Found and Lost | |
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| The Formalist Philosophy of Mathematics | |
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| Lakatos and the Philosophy of Dubitability | |
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| Mathematical Reality | |
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| The Riemann Hypothesis | |
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| [pi] and [characters not reproducible] | |
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| Mathematical Models, Computers, and Platonism | |
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| Why Should I Believe a Computer? | |
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| Classification of Finite Simple Groups | |
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| Intuition | |
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| Four-Dimensional Intuition | |
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| True Facts About Imaginary Objects | |
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| Glossary | |
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| Bibliography | |
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| Index | |