Odifreddi, Piergiorgio 　著

内容

目次

Foreword xi Acknowledgments xvii Introduction 1 CHAPTER 1: THEFOUNDATIONS 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: PUREMATHEMATICS 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: GelfandTranscendental Numbers (1929) 39 2.5. Logic: Godel'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) 522.8. Differential Topology: Milnor's Exotic Structures (1956) 56 2.9. ModelTheory: Robinson's Hyperreal Numbers (1961) 59 2.10. Set Theory: Cohen'sIndependence Theorem (1963) 63 2.11. Singularity Theory: Thom'sClassification of Catastrophes (1964) 66 2.12. Algebra: Gorenstein'sClassification of Finite Groups (1972) 71 2.13. Topology: Thurston'sClassification 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: APPLIEDMATHEMATICS 92 3.1. Crystallography: Bieberbach's Symmetry Groups (1910) 983.2. Tensor Calculus: Einstein's General Theory of Relativity (1915) 1043.3. Game Theory: Von Neumann's Minimax Theorem (1928) 108 3.4. FunctionalAnalysis: Von Neumann's Axiomatization of Quantum Mechanics (1932) 112 3.5.Probability Theory: Kolmogorov's Axiomatization (1933) 116 3.6. OptimizationTheory: Dantzig's Simplex Method (1947) 120 3.7. General Equilibrium Theory:The Arrow-Debreu Existence Theorem (1954) 122 3.8. The Theory of FormalLanguages: Chomsky's Classification (1957) 125 3.9. Dynamical SystemsTheory: The KAM Theorem (1962) 128 3.10. Knot Theory: Jones Invariants(1984) 132 CHAPTER FOUR: MATHEMATICS AND THE COMPUTER 139 4.1. The Theoryof Algorithms: Turing's Characterization (1936) 145 4.2. ArtificialIntelligence: Shannon's Analysis of the Game of Chess (1950) 148 4.3. ChaosTheory: Lorenz's Strange Attractor (1963) 151 4.4. Computer-Assisted Proofs:The Four-Color Theorem of Appel and Haken (1976) 154 4.5. Fractals: TheMandelbrot Set (1980) 159 CHAPTER FIVE: OPEN PROBLEMS 165 5.1. Arithmetic:The Perfect Numbers Problem (300 BC) 166 5.2. Complex Analysis: The RiemannHypothesis (1859) 168 5.3. Algebraic Topology: The Poincare Conjecture(1904) 172 5.4. Complexity Theory: The P=NP Problem (1972) 176 Conclusion181 References and Further Reading 187 Index 189