Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

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 AN 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