TABLE OF CONTENTS: Foreword xi
Acknowledgments xvii
Introduction 1 CHAPTER 1: THE FOUNDATIONS 8
1.1. The 1920s: Sets 10
1.2. The 1940s: Structures 14
1.3. The 1960s: Categories 17
1.4. The 1980s: Functions 21 CHAPTER TWO: PURE MATHEMATICS 25
2.1. Mathematical Analysis: Lebesgue Measure (1902) 29
2.2. Algebra: Steinitz Classification of Fields (1910) 33
2.3. Topology: Brouwer's Fixed-Point Theorem (1910) 37
2.4. Number Theory: Gelfand Transcendental Numbers (1929) 39
2.5. Logic: Gödel's Incompleteness Theorem (1931) 43
2.6. The Calculus of Variations: Douglas's Minimal Surfaces (1931) 47
2.7. Mathematical Analysis: Schwartz's Theory of Distributions (1945) 52
2.8. Differential Topology: Milnor's Exotic Structures (1956) 56
2.9. Model Theory: Robinson's Hyperreal Numbers (1961) 59
2.10. Set Theory: Cohen's Independence Theorem (1963) 63
2.11. Singularity Theory: Thom's Classification of Catastrophes (1964) 66
2.12. Algebra: Gorenstein's Classification of Finite Groups (1972) 71
2.13. Topology: Thurston's Classification of 3-Dimensional Surfaces (1982) 78
2.14. Number Theory: Wiles's Proof of Fermat's Last Theorem (1995) 82
2.15. Discrete Geometry: Hales's Solution of Kepler's Problem (1998) 87 CHAPTER THREE: APPLIED MATHEMATICS 92
3.1. Crystallography: Bieberbach's Symmetry Groups (1910) 98
3.2. Tensor Calculus: Einstein's General Theory of Relativity (1915) 104
3.3. Game Theory: Von Neumann's Minimax Theorem (1928) 108
3.4. Functional Analysis: Von Neumann's Axiomatization of Quantum Mechanics (1932) 112
3.5. Probability Theory: Kolmogorov's Axiomatization (1933) 116
3.6. Optimization Theory: Dantzig's Simplex Method (1947) 120
3.7. General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954) 122
3.8. The Theory of Formal Languages: Chomsky's Classification (1957) 125
3.9. Dynamical Systems Theory: The KAM Theorem (1962) 128
3.10. Knot Theory: Jones Invariants (1984) 132 CHAPTER FOUR: MATHEMATICS AND THE COMPUTER 139
4.1. The Theory of Algorithms: Turing's Characterization (1936) 145
4.2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950) 148
4.3. Chaos Theory: Lorenz's Strange Attractor (1963) 151
4.4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976) 154
4.5. Fractals: The Mandelbrot Set (1980) 159 CHAPTER FIVE: OPEN PROBLEMS 165
5.1. Arithmetic: The Perfect Numbers Problem (300 BC) 166
5.2. Complex Analysis: The Riemann Hypothesis (1859) 168
5.3. Algebraic Topology: The Poincare Conjecture (1904) 172
5.4. Complexity Theory: The P=NP Problem (1972) 176 Conclusion 181
References and Further Reading 187
Index 189
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