泰勒斯的遗产

Preface

0 Introduction

PART Ⅰ： History and Philosophy of Mathematics

1 Egyptian Mathematics

2 Scales of Notation

3 Prime Numbers

4 Sumerian-Babylonian Mathematics

5 More about Mesopotamian Mathematics

6 The Dawn of Greek Mathematics

7 Pythagoras and His School

8 Perfect Numbers

9 Regular Polyhedra

10 The Crisis of Incommensurables

11 From Heraclitus to Democritus

12 Mathematics in Athens

13 Plato and Aristotle on Mathematics

14 Constructions with Ruler and Compass

15 The Impossibility of Solving the Classical Problems

16 Euclid

17 Non-Euclidean Geometry and Hilberts Axioms

18 Alexandria from 300 BC to 200 BC

19 Archimedes

20 Alexandria from 200 BC to 500 AD

21 Mathematics in China and India

22 Mathematics in Islamic Countries

23 New Beginnings in Europe

24 Mathematics in the Renaissance

25 The Cubic and Quartic Equations

26 Renaissance Mathematics Continued

27 The Seventeenth Century in France

28 The Seventeenth Century Continued

29 Leibniz

30 The Eighteenth Century

31 The Law of Quadratic Reciprocity

PART Ⅱ： Foundations of Mathematics

1 The Number System

2 Natural Numbers (Peanos Approach)

3 The Integers

4 The Rationals

5 The Real Numbers

6 Complex Numbers

7 The Fundamental Theorem of Algebra

8 Quaternions

9 Quaternions Applied to Number Theory

10 Quaternions Applied to Physics

11 Quaternions in Quantum Mechanics

12 Cardinal Numbers

13 Cardinal Arithmetic

14 Continued Fractions

15 The Fundamental Theorem of Arithmetic

16 Linear Diophantine Equations

17 Quadratic Surds

18 Pythagorean Triangles and Fermats Last Theorem

19 What Is a Calculation?

20 Recursive and Recursively Enumerable Sets

21 Hilberts Tenth Problem

22 Lambda Calculus

23 Logic from Aristotle to Russell

24 Intuitionistic Propositional Calculus

25 How to Interpret Intuitionistic Logic

26 Intuitionistic Predicate Calculus

27 Intuitionistic Type Theory

28 Godels Theorems

29 Proof of GSdels Incompleteness Theorem

30 More about Godels Theorems

31 Concrete Categories

32 Graphs and Categories

33 Functors

34 Natural Transformations

35 A Natural Transformation between Vector Spaces

References

Index

本书以专题方式讲述数学的历史和数学的哲学（非史论型著作），每个专题相对独立。全书以数学历史为线索，以数学为内容主体，以数学哲学为引申，易读、易懂，是本科生学习数学过程中非常好的课外读物。