泰勒斯的遗产 英文PDF电子书下载

0 Introduction 1

PART Ⅰ：History and Philosophy of Mathematics 5

1 Egyptian Mathematics 7

2 Scales of Notation 11

3 Prime Numbers 15

4 Sumerian-Babylonian Mathematics 21

5 More about Mesopotamian Mathematics 25

6 The Dawn of Greek Mathematics 29

7 Pythagoras and His School 33

8 Perfect Numbers 37

9 Regular Polyhedra 41

10 The Crisis of Incommensurables 47

11 From Heraclitus to Democritus 53

12 Mathematics in Athens 59

13 Plato and Aristotle on Mathematics 67

14 Constructions with Ruler and Compass 71

15 The Impossibility of Solving the Classical Problems 79

16 Euclid 83

17 Non-Euclidean Geometry and Hilbert's Axioms 89

18 Alexandria from 300 BC to 200 BC 93

19 Archimedes 97

20 Alexandria from 200 BC to 500 AD 103

21 Mathematics in China and India 111

22 Mathematics in Islamic Countries 117

23 New Beginnings in Europe 121

24 Mathematics in the Renaissance 125

25 The Cubic and Quartic Equations 133

26 Renaissance Mathematics Continued 139

27 The Seventeenth Century in France 145

28 The Seventeenth Century Continued 153

29 Leibniz 159

30 The Eighteenth Century 163

31 The Law of Quadratic Reciprocity 169

PART Ⅱ：Foundations of Mathematics 173

1 The Number System 175

2 Natural Numbers(Peano's Approach) 179

3 The Integers 183

4 The Rationals 187

5 The Real Numbers 191

6 Complex Numbers 195

7 The Fundamental Theorem of Algebra 199

8 Quaternions 203

9 Quaternions Applied to Number Theory 207

10 Quaternions Applied to Physics 211

11 Quaternions in Quantum Mechanics 215

12 Cardinal Numbers 219

13 Cardinal Arithmetic 223

14 Continued Fractions 227

15 The Fundamental Theorem of Arithmetic 231

16 Linear Diophantine Equations 233

17 Quadratic Surds 237

18 Pythagorean Triangles and Fermat's Last Theorem 241

19 What Is a Calculation? 245

20 Recursive and Recursively Enumerable Sets 251

21 Hilbert's Tenth Problem 255

22 Lambda Calculus 259

23 Logic from Aristotle to Russell 265

24 Intuitionistic Propositional Calculus 271

25 How to Interpret Intuitionistic Logic 277

26 Intuitionistic Predicate Calculus 281

27 Intuitionistic Type Theory 285

28 G？del's Theorems 289

29 Proof of G？del's Incompleteness Theorem 291

30 More about G？del's Theorems 293

31 Concrete Categories 295

32 Graphs and Categories 297

33 Functors 299

34 Natural Transformations 303

35 A Natural Transformation between Vector Spaces 307

References 311

Index 321