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Index
Cover
Title Page
Copyright Page
Dedication Page
Contents
Preface
Chapter I. Evolution, Mathematics, and the Evolution of Mathematics
1. Evolution
2. Mathematical Ability in the Animal World
3. Mathematical Ability in Humans
4. Mathematics that Yields Evolutionary Advantage
5. Mathematics with no Evolutionary Advantage
6. Mathematics in Early Civilizations
7. And then Came the Greeks
8. What Motivated the Greeks?
Chapter II. Mathematics and the Greeks’ View of the World
9. The Origin of Basic Science: Asking Questions
10. The First Mathematical Models
11. Platonism versus Formalism
12. Models of the Heavenly Bodies
13. On the Greek Perception of Science
14. Models of the Heavenly Bodies (Cont.)
Chapter III. Mathematics and the View of the World in Early Modern Times
15. The Sun Reverts to the Center
16. Giants’ Shoulders
17. Ellipses versus Circles
18. And then Came Newton
19. Everything you Wanted to know about Infinitesimal Calculus and Differential Equations
20. Newton's Laws
21. Purpose: The Principle of Least Action
22. The Wave Equation
23. On the Perception of Science in Modern Times
Chapter IV. Mathematics and the Modern View of the World
24. Electricity and Magnetism
25. And then Came Maxwell
26. Discrepancy between Maxwell's Theory and Newton's Theory
27. The Geometry of the World
28. And then Came Einstein
29. The Discovery of the Quantum State of Nature
30. The Wonder Equation
31. Groups of Particles
32. The Strings Return
33. Another Look at Platonism
34. The Scientific Method: Is there an Alternative?
Chapter V. The Mathematics of Randomness
35. Evolution and Randomness in the Animal World
36. Probability and Gambling in Ancient Times
37. Pascal and Fermat
38. Rapid Development
39. The Mathematics of Predictions and Errors
40. The Mathematics of Learning from Experience
41. The Formalism of Probability
42. Intuition versus the Mathematics of Randomness
43. Intuition versus the Statistics of Randomness
Chapter VI. The Mathematics of Human Behavior
44. Macro-considerations
45. Stable Marriages
46. The Aggregation of Preferences and Voting Systems
47. The Mathematics of Confrontation
48. Expected Utility
49. Decisions in a State of Uncertainty
50. Evolutionary Rationality
Chapter VII. Computations and Computers
51. Mathematics for Computations
52. From Tables to Computers
53. The Mathematics of Computations
54. Proofs with High Probability
55. Encoding
56. What Next?
Chapter VIII. Is there Really no Doubt?
57. Mathematics without Axioms
58. Rigorous Development without Geometry
59. Numbers as Sets, Logic as Sets
60. A Major Crisis
61. Another Major Crisis
Chapter IX. The Nature of Research in Mathematics
62. How Does a Mathematician Think?
63. On Research in Mathematics
64. Pure Mathematics vis-à-vis Applied Mathematics
65. The Beauty, Efficiency, and Universality of Mathematics
Chapter X. Why is Teaching and Learning Mathematics so Hard?
66. Why Learn Mathematics?
67. Mathematical Thinking: There is no Such Thing
68. A Teacher-Parent Meeting
69. A Logical Structure vis-à-vis a Structure for Teaching
70. What is Hard in Teaching Mathematics?
71. The Many Facets of Mathematics
Afterword
Sources
Index of Names
Index of Subjects
About the Author
Back Cover
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