The Mathematical Experience




  1. The Mathematical Landscape
    • What is Mathematics?
    • Where is Mathematics?
    • The Mathematical Community
    • The Tools of the Trade
    • How Much Mathematics is Now Known? Ulam's Dilemma
    • How Much Mathematics Can There Be? Appendix A-Brief Chronological Table to 1910
    • Appendix B-The Classification of Mathematics. 1868 and 1979 Compared
  2. Varieties of Mathematical Experience
    • The Current Individual and Collective Consciousness
    • The Ideal Mathematician
    • A Physicist Looks at Mathematics
    • I.R. Shafarevitch and the New Neoplatonism
    • Unorthodoxies
    • The Individual and the Culture
  3. Outer Issues
    • Why Mathematics Works: A Conventionalist Answer
    • Mathematical Models
    • Utility
      1. Varieties of Mathematical Uses
      2. On the Utility of Mathematics to Mathematics
      3. On the Utility of Mathematics to Other Scientific or Technological Fields
      4. Pure vs. Applied Mathematics
      5. From Hardyism to Mathematical Maoism
    • Underneath the Fig Leaf
      1. Mathematics in the Marketplace
      2. Mathematics and War
      3. Number Mysticism
      4. Hermetic Geometry
      5. Astrology
      6. Religion
    • Abstraction and Scholastic Theology
  4. Inner Issues
    • Symbols
    • Abstraction
    • Generalization
    • Formalization
    • Mathematical Objects and Structures; Existence
    • Proof
    • Infinity, or the Miraculous Jar of Mathematics
    • The Stretched String
    • The Coin of Tyche
    • The Aesthetic Component
    • Pattern, Order, and Chaos
    • Algorithmic vs. Dialectic Mathematics
    • The Drive to Generality and Abstraction
    • The Chinese Remainder Theorem: A Case Study
    • Mathematics as Enigma
    • Unity within Diversity
  5. Selected Topics in Mathematics
    • Group Theory and the Classification of Finite Simple Groups
    • The Prime Number Theorem
    • Non-Euclidean Geometry
    • Non-Cantorian Set Theory
    • Appendix A
    • Nonstandard Analysis
    • Fourier Analysis
  6. Teaching and Learning
    • Confessions of a Prep School Math Teacher
    • The Classic Classroom Crisis of Understanding and Pedagogy
    • Polya's Craft of Discovery
    • The Creation of New Mathematics: An Application of the Lakatos Heuristic
    • Comparative Aesthetics
    • Nonanalytic Aspects of Mathematics
  7. From Certainty to Fallibility
    • Platonism, Formalism, Constructivism The Philosophical Plight of the Working Mathematician
    • The Euclid Myth
    • Foundations, Found and Lost
    • The Formalist Philosophy of Mathematics
    • Lakatos and the Philosophy of Dubitability
  8. Mathematical Reality
    • The Riemann Hypothesis
    • Pi and Pi Hat
    • Mathematical Models, Computers, and Platonism
    • Why Should I Believe a Computer?
    • Classification of Finite Simple Groups
    • Intuition
    • Four-Dimensional Intuition
    • True Facts About Imaginary Objects



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