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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 | |