Mathematical thought from ancient to modern times. V.3

Contents......Page 12
2. The Theory of Congruences......Page 18
3. Algebraic Numbers......Page 23
4. The Ideals of Dedekind......Page 27
5. The Theory of Forms......Page 31
6. Analytic Number Theory......Page 34
1. The Renewal of Interest in Geometry......Page 39
2. Synthetic Euclidean Geometry......Page 42
3. The Revival of Synthetic Projective Geometry......Page 45
4. Algebraic Projective Geometry......Page 57
5. Higher Plane Curves and Surfaces......Page 60
2. The Status of Euclidean Geometry About 1800......Page 66
3. The Research on the Parallel Axiom......Page 68
4. Foreshadowings of Non-Euclidean Geometry......Page 72
5. The Creation of Non-Euclidean Geometry......Page 74
6. The Technical Content of Non-Euclidian Geometry......Page 79
7. The Claims of Lobatchevsky and Bolyai to Priority......Page 82
8. The Implications of Non-Euclidean Geometry......Page 84
2. Gauss's Differential Geometry......Page 87
3. Riemann's Approach to Geometry......Page 94
4. The Successors of Riemann......Page 101
5. Invariants of Differential Forms......Page 104
2. Surfaces as Models of Non-Euclidean Geometry......Page 109
3. Projective and Metric Geometry......Page 111
4. Models and the Consistency Problem......Page 118
5. Geometry from the Transformation Viewpoint......Page 122
6. The Reality of Non-Euclidean Geometry......Page 126
1. Background......Page 129
2. The Theory of Algebraic Invariants......Page 130
3. The Concept of Birational Transformations......Page 137
4. The Function-Theoretic Approach to Algebraic Geometry......Page 139
5. The Uniformization Problem......Page 142
6. The Algebraic-Geometric Approach......Page 144
7. The Arithmetic Approach......Page 147
8. The Algebraic Geometry of Surfaces......Page 148
1. Introduction......Page 152
2. Functions and Their Properties......Page 154
3. The Derivative......Page 159
4. The Integral......Page 161
5. Infinite Series......Page 166
6. Fourier Series......Page 171
7. The Status of Analysis......Page 177
1. Introduction......Page 184
2. Algebraic and Transcendental Numbers......Page 185
3. The Theory of Irrational Numbers......Page 187
4. The Theory of Rational Numbers......Page 192
5. Other Approaches to the Real Number System......Page 195
6. The Concept of an Infinite Set......Page 197
7. The Foundation of the Theory of Sets......Page 199
8. Transfinite Cardinals and Ordinals......Page 203
9. The Status of Set Theory by 1900......Page 207
1. The Defects in Euclid......Page 210
2. Contributions to the Foundations of Projective Geometry......Page 212
3. The Foundations of Euclidean Geometry......Page 215
4. Some Related Foundational Work......Page 220
5. Some Open Questions......Page 222
1. The Chief Features of the Nineteenth-Century Developments......Page 228
2. The Axiomatic Movement......Page 231
3. Mathematics as Man's Creation......Page 233
4. The Loss of Truth......Page 237
5. Mathematics as the Study of Arbitrary Structures......Page 241
6. The Problem of Consistency......Page 243
7. A Glance Ahead......Page 244
1. The Origins......Page 245
3. Early Work on Content and Measure......Page 246
4. The Lebesgue Integral......Page 249
5. Generalizations......Page 255
1. Introduction......Page 257
2. The Beginning of a General Theory......Page 261
3. The Work of Hilbert......Page 265
4. The Immediate Successors of Hilbert......Page 275
5. Extensions of the Theory......Page 278
1. The Nature of Functional Analysis......Page 281
2. The Theory of Functionals......Page 282
3. Linear Functional Analysis......Page 286
4. The Axiomatization of Hilbert Space......Page 296
1. Introduction......Page 301
2. The Informal Uses of Divergent Series......Page 303
3. The Formal Theory of Asymptotic Series......Page 308
4. Summability......Page 314
1. The Origins of Tensor Analysis......Page 327
2. The Notion of a Tensor......Page 328
3. Covariant Differentiation......Page 332
4. Parallel Displacement......Page 335
5. Generalizations of Riemannian Geometry......Page 338
1. The Nineteenth-Century Background......Page 341
2. Abstract Group Theory......Page 342
3. The Abstract Theory of Fields......Page 351
4. Rings......Page 355
5. Non-Associative Algebras......Page 358
6. The Range of Abstract Algebra......Page 361
1. The Nature of Topology......Page 363
2. Point Set Topology......Page 364
3. The Beginnings of Combinational Topology......Page 368
4. The Combinational Work of Poincaré......Page 375
5. Combinatorial Invariants......Page 381
6. Fixed Point Theorems......Page 382
7. Generalizations and Extensions......Page 384
1. Introduction......Page 387
2. The Paradoxes of Set Theory......Page 388
3. The Axiomatization of Set Theory......Page 390
4. The Rise of Mathematical Logic......Page 392
5. The Logistic School......Page 397
6. The Intuitionist School......Page 402
7. The Formalist School......Page 408
8. Some Recent Developments......Page 413
List of Abbreviations......Page 418
B......Page 420
C......Page 421
F......Page 422
H......Page 423
K......Page 424
M......Page 425
P......Page 426
S......Page 427
X......Page 428
Z......Page 429