Table of Contents
Introduction 8
Chapter 1 The Foundations of Mathematics 13
Arithmetic or Geometry 13
Being Versus Becoming 15
Universals 18
The Axiomatic Method 19
Number Systems 20
Calculus Reopens Foundational Questions 21
Non-Euclidean Geometries 23
Elliptic and Hyperbolic Geometries 24
Riemannian Geometry 25
Cantor 27
The Quest for Rigour 28
Set Theoretic Beginnings 28
Foundational Logic 30
Impredicative Constructions 32
Nonconstructive Arguments 33
Intuitionistic Logic 34
Other Logics 36
Formalism 37
Gödel 38
Recursive Definitions 41
Computers and Proof 43
Category Theory 44
Abstraction in Mathematics 44
Isomorphic Structures 46
Topos Theory 49
Intuitionistic Type Theories 50
Internal Language 51
Gödel and Category Theory 52
The Search for a Distinguished Model 53
Boolean Local Topoi 55
One Distinguished Model or Many Models 58
Chapter 2 Ancient Western Mathematics 60
Mathematics in Ancient Mesopotamia 60
The Numeral System and Arithmetic Operations 61
Geometric and Algebraic Problems 64
Pythagorean Theorem 66
Mathematical Astronomy 70
Mathematics in Ancient Egypt 72
The Numeral System and Arithmetic Operations 73
Geometry 77
Assessment of Egyptian Mathematics 79
Greek Mathematics 81
The Pre-Euclidean Period 81
The Elements 89
The Three Classical Problems 92
Geometry in the 3Rd Century BCE 96
Archimedes 96
Apollonius 97
Applied Geometry 104
Trisecting the Angle: The Quadratrix of Hippias 106
Later Trends in Geometry and Arithmetic 112
Greek Trigonometry and Mensuration 112
Number Theory 114
Survival and Influence of Greek Mathematics 117
Mathematics in the Islamic World (8th-15th Century) 119
Origins 119
Mathematics in the 9th Century 122
Mathematics in the 10th Century 124
Omar Khayyam 128
Islamic Mathematics to the 15th Century 130
Chapter 3 European Mathematics Since the Middle Ages 134
European Mathematics During the Middle Ages and Renaissance 134
The Transmission of Greek and Arabic Learning 135
The Universities 136
The Renaissance 139
Mathematics in the 17th and 18th Centuries 141
Institutional Background 141
Numerical Calculation 142
Analytic Geometry 146
The Calculus 153
Institutional Background 166
Analysis and Mechanics 168
History of Analysts 170
Other Developments 175
Theory of Equations 176
Foundations of Geometry 179
Mathematics in the 19th Century 182
Projective Geometry 183
Making the Calculus Rigorous 186
Fourier Series 189
Elliptic Functions 190
The Theory of Numbers 193
The Theory of Equations 198
Gauss 202
Non-Euclidean Geometry 203
Riemann 206
Riemann's Influence 209
Differential Equations 212
Linear Algebra 214
The Foundations of Geometry 218
The Foundations of Mathematics 220
Mathematics in the 20th and 21st Centuries 222
Cantor 222
Mathematical Physics 226
Algebraic Topology 229
Developments in Pure Mathematics 234
Mathematical Physics and the Theory of Groups 239
Probabilistic Mathematics 244
Chapter 4 Indian and East Asian Mathematics 254
Vedic Number Words and Geometry 254
The Post-Vedic Context 256
Indian Numerals and the Decimal Place-Value System 257
The Role of Astronomy and Astrology 259
Classical Mathematical Literature 261
The Changing Structure of Mathematical Knowledge 264
Mahavira and Bhaskara II 265
Teachers and Learners 267
The School of Madhava in Kerala 268
Mathematics in China 270
The Nine Chapters 273
The Commentary of Liu Hui 280
The "Ten Classics" 282
Theory of Root Extraction and Equations 284
The Method of the Celestial Unknown 287
Chinese Remainder Theorem 289
Mathematics in Japan 291
The Introduction of Chinese Books 291
The Elaboration of Chinese Methods 292
Abacus 293
Chapter 5 The Philosophy of Mathematics 296
Mathematical Platonism 299
Formal Definition 299
Nontraditional Versions 301
Mathematical Anti-Platonism 304
Realistic Anti-Platonism 305
Nominalism 307
Logicism, Intuitionism, and Formalism 312
Mathematical Platonism: For and Against 315
The Fregean Argument for Platonism 315
Infinity 320
The Epistemological Argument Against Platonism 323
Ongoing Impasse 326
Glossary 328
Bibliography 331
Index 342
