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