Where Do Numbers Come From?

Introduction 1
PART I THE RATIONALS 5
1 Counting Sheep 7
1.1 A Foundation Myth 7
1.2 What Were Numbers Used For? 12
1.3 A Greek Myth 15
2 The Strictly Positive Rationals 23
2.1 An Indian Legend 23
2.2 Equivalence Classes 27
2.3 Properties of the Strictly Positive Rationals 33
2.4 What Have We Actually Done? 37
3 The Rational Numbers 39
3.1 Negative Numbers 39
3.2 Defining the Rational Numbers 44
3.3 What Does Nature Say? 51
3.4 When Are Two Things the Same? 52
PART II THE NATURAL NUMBERS 59
4 The Golden Key 61
4.1 The Least Member 61
4.2 Inductive Definition 65
4.3 Applications 69
4.4 Prime Numbers 77
5 Modular Arithmetic 83
5.1 Finite Fields 83
5.2 Some Pretty Theorems 87
5.3 A New Use for Old Numbers 91
5.4 More Modular Arithmetic 98
5.5 Problems of Equal Difficulty 101
6 Axioms for the Natural Numbers 109
6.1 The Peano Axioms 109
6.2 Order 113
6.3 Conclusion of the Argument 117
6.4 Order Numbers Can Be Used as Counting Numbers 121
6.5 Objections 127
PART III THE REAL NUMBERS (AND THE
COMPLEX NUMBERS) 135
7 What Is the Problem? 137
7.1 Mathematics Becomes a Profession 137
7.2 Rogue Numbers 138
7.3 How Can We Justify Calculus? 147
7.4 The Fundamental Axiom of Analysis 151
7.5 Dependent Choice 156
7.6 Equivalent Forms of the Fundamental Axiom 159
8 And What Is Its Solution? 167
8.1 A Construction of the Real Numbers 167
8.2 Some Consequences 177
8.3 Are the Real Numbers Real? 182
9 The Complex Numbers 187
9.1 Constructing the Complex Numbers 187
9.2 Analysis for C 191
9.3 Continuous Functions from C 195
10 A Plethora of Polynomials 199
10.1 Preliminaries 199
10.2 The Fundamental Theorem of Algebra 205
10.3 Liouville Numbers 209
10.4 A Non-Archimedean Ordered Field 213
11 Can We Go Further? 221
11.1 The Quaternions 221
11.2 What Happened Next 226
11.3 Valedictory 230
APPENDICES 231
Appendix A Products of Many Elements 233
Appendix B nth Complex Roots 239
Appendix C How Do Quaternions Represent Rotations? 243
Appendix D Why Are the Quaternions So Special? 247
References 255
Index 257