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About this book
About this book
The twentieth century was a time of unprecedented development in mathematics, as well as in all sciences: more theorems were proved and results found in a hundred years than in all of previous history. In this book, Piergiorgio Odifreddi distills this unwieldy mass of knowledge into a fascinating and authoritative overview of the subject.
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: 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) 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
Piergiorgio Odifreddi is Professor of Mathematical Logic at the University of Turin and has been a visiting professor at Cornell University for many years. He is the author of the textbook "Classical Recursion Theory". He is also a regular contributor to the Italian daily "La Repubblica". Freeman Dyson, Professor Emeritus of Physics at the Institute for Advanced Study, is the author of several books, including "Disturbing the Universe".
Odifreddi's overview is of course a personal one, but it is hard to argue with either his choices or his organization. This is a perfect handle on an otherwise bewildering proliferation of ideas. -- Ben Longstaff New Scientist Odifreddi clearly and concisely describes important 20th-century developments in pure and applied mathematics... Unlike similar volumes, this book keeps descriptions general and contains a short section on the philosophical foundations of mathematics to help non-mathematicians easily navigate the material. Library Journal This is an astonishingly readable, succinct, and wonderful account of twentieth-century mathematics! It is a great book for mathematics majors, students in liberal-arts courses in mathematics, and the general public. I am amazed at how easily the author has set out the achievements in a broad array of mathematical fields. The writing appears effortless. -- Paul Campbell Mathematics Magazine Piergiogio Odifreddi's book successfully portrays the major developments in 20th century mathematics by an examination of the mathematical problems that have gained prominence during the past 100 years... [T]he literary style is such that the contents are made accessible to a very wide readership, but with no hint of oversimplification. -- P.N. Ruane MathDL