The Britannica Guide to the History of Mathematics

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ISBN: 9781615301270 | 1615301275
Cover: Hardcover
Copyright: 8/15/2010

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The field of mathematics today represents an ongoing global effort, spanning both countries and centuries. While some developments emerged in multiple cultures, independent of each other, others involved an extensive exchange of ideas among individuals around the world. Through this in-depth narrative, students will learn how major mathematical concepts were first derived, as well as how they evolved with the advent of later thinkers shedding new light on various applications. Everything from Euclidean geometry to the philosophy of mathematics is illuminated as readers are transported to the ancient civilizations of Mesopotamia, Egypt, and beyond to discover the history of mathematical thought.

Introduction	p. 12
Ancient Western Mathematics	p. 21
Ancient Mathematical Sources	p. 22
Mathematics in Ancient Mesopotamia	p. 23
The Numeral System and Arithmetic Operations	p. 24
Geometric and Algebraic Problems	p. 27
Mathematical Astronomy	p. 29
Mathematics in Ancient Egypt	p. 31
The Numeral System and Arithmetic Operations	p. 32
Geometry	p. 36
Assessment of Egyptian Mathematics	p. 38
Greek Mathematics	p. 39
The Development of Pure Mathematics	p. 39
The Pre-Euclidean Period	p. 40
The Elements	p. 46
The Three Classical Problems	p. 49
Geometry in the 3rd Century BCE	p. 51
Archimedes	p. 52
Apollonius	p. 54
Applied Geometry	p. 60
Later Trends in Geometry and Arithmetic	p. 66
Greek Trigonometry and Mensuration	p. 67
Number Theory	p. 67
Survival and Influence of Greek Mathematics	p. 70
Mathematics in the Islamic World (8th-15th Century)	p. 72
Origins	p. 72
Mathematics in the 9th Century	p. 74
Mathematics in the 10th Century	p. 76
Omar Khayyam	p. 79
Islamic Mathematics to the 15th Century	p. 81
European Mathematics Since the Middle Ages	p. 84
European Mathematics During the Middle Ages and Renaissance	p. 84
The Transmission of Greek and Arabic Learning	p. 86
The Universities	p. 87
The Renaissance	p. 89
Mathematics in the 17th and 18th Centuries	p. 91
The 17th Century	p. 91
Institutional Background	p. 92
Numerical Calculation	p. 92
Analytic Geometry	p. 95
The Calculus	p. 101
The 18th Century	p. 112
Institutional Background	p. 112
Analysis and Mechanics	p. 114
History of Analysis	p. 116
Other Developments	p. 120
Theory of Equations	p. 120
Foundations of Geometry	p. 122
Mathematics in the 19th and 20th Centuries	p. 125
Projective Geometry	p. 126
Making the Calculus Rigorous	p. 129
Fourier Series	p. 133
Elliptic Functions	p. 134
The Theory of Numbers	p. 136
The Theory of Equations	p. 140
Gauss	p. 143
Non-Euclidean Geometry	p. 144
Riemann	p. 147
Riemann's Influence	p. 151
Differential Equations	p. 154
Linear Algebra	p. 156
The Foundations of Geometry	p. 158
The Foundations of Mathematics	p. 161
Cantor	p. 162
Mathematical Physics	p. 165
Algebraic Topology	p. 169
Developments in Pure Mathematics	p. 173
Mathematical Physics and the Theory of Groups	p. 177
South and East Asian Mathematics	p. 182
Ancient Traces	p. 182
Vedic Number Words and Geometry	p. 183
The Post-Vedic Context	p. 184
Indian Numerals and the Decimal Place-Value System	p. 185
The "Classical" Period	p. 186
The Role of Astronomy and Astrology	p. 187
Classical Mathematical Literature	p. 189
The Changing Structure of Mathematical Knowledge	p. 191
Mahavira and Bhaskara II	p. 192
Teachers and Learners	p. 193
The School of Madhava in Kerala	p. 194
Exchanges with Islamic and Western Mathematics	p. 195
Mathematics in China	p. 195
The Textual Sources	p. 196
The Great Early Period, 1st-7th Centuries	p. 198
The Nine Chapters	p. 198
The Commentary of Liu Hui	p. 204
The "Ten Classics"	p. 206
Scholarly Revival, 11th-13th Centuries	p. 207
Theory of Root Extraction and Equations	p. 208
The Method of the Celestial Unknown	p. 209
Chinese Remainder Theorem	p. 211
Fall into Oblivion, 14th-16th Centuries	p. 211
Mathematics in Japan	p. 213
The Introduction of Chinese Books	p. 213
The Elaboration of Chinese Methods	p. 214
The Foundations of Mathematics	p. 217
Ancient Greece to the Enlightenment	p. 217
Arithmetic or Geometry	p. 217
Being Versus Becoming	p. 218
Universals	p. 221
The Axiomatic Method	p. 222
Number Systems	p. 223
The Reexamination of Infinity	p. 224
Calculus Reopens Foundational Questions	p. 225
Non-Euclidean Geometries	p. 226
Elliptic and Hyperbolic Geometries	p. 227
Riemannian Geometry	p. 228
Cantor	p. 229
The Quest for Rigour	p. 230
Formal Foundations	p. 230
Set Theoretic Beginnings	p. 230
Foundational Logic	p. 232
Impredicative Constructions	p. 233
Nonconstructive Arguments	p. 234
Intuitionistic Logic	p. 235
Other Logics	p. 237
Formalism	p. 237
Gödel	p. 238
Recursive Definitions	p. 241
Computers and Proof	p. 243
Category Theory	p. 244
Abstraction in Mathematics	p. 244
Isomorphic Structures	p. 246
Topos Theory	p. 247
Intuitionistic Type Theories	p. 248
Internal Language	p. 249
Gödel and Category Theory	p. 250
The Search for a Distinguished Model	p. 251
Boolean Local Topoi	p. 252
One Distinguished Model or Many Models	p. 254
The Philosophy of Mathematics	p. 256
Mathematical Platonism	p. 258
Traditional Platonism	p. 258
Nontraditional Versions	p. 260
Mathematical Anti-Platonism	p. 263
Realistic Anti-Platonism	p. 263
Nominalism	p. 266
Logicism, Intuitionism, and Formalism	p. 270
Mathematical Platonism: For and Against	p. 272
The Fregean Argument for Platonism	p. 273
The Epistemological Argument Against Platonism	p. 277
Ongoing Impasse	p. 280
Glossary	p. 282
Bibliography	p. 285
Index	p. 294
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