The Mathematical Experience
PHILIP J. DAVIS & REUBEN HERSH
Contents
Preface
Acknowledgments
Introduction
Overture
- 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
- 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
- Outer Issues
- Why Mathematics Works: A Conventionalist Answer
- Mathematical Models
- Utility
- Varieties of Mathematical Uses
- On the Utility of Mathematics to Mathematics
- On the Utility of Mathematics to Other Scientific or Technological Fields
- Pure vs. Applied Mathematics
- From Hardyism to Mathematical Maoism
- Underneath the Fig Leaf
- Mathematics in the Marketplace
- Mathematics and War
- Number Mysticism
- Hermetic Geometry
- Astrology
- Religion
- Abstraction and Scholastic Theology
- 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
- 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
- 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
- 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
- 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
Glossary
Bibliography
Index
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