| Contents Notes |
Traces the history of mathematics, offers profiles of major mathematicians and their discoveries, and looks at the philosophy of mathematics. 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? -- 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 -- Abstraction and scholastic theory -- 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 -- 6. Teaching and learning -- Confessions of a prep school math teacher -- The classic classroom crises of understanding and pedagogy -- Pólya'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 mathematics -- Lakatos and the philosophy of dubitability -- 8. Mathematical reality -- The Riemann hypothesis -- [pi] and [ãpi] -- 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. |
