History and Philosophy of Modern Mathematics

Measure and Integration: A Concise Introduction to Real Analysis......Page 5
CONTENTS......Page 9
Preface......Page 13
Acknowledgments......Page 15
Introduction......Page 17
1.1 History of the Idea......Page 19
1.2 Deficiencies of the Riemann Integral......Page 21
1.3 Motivation for the Lebesgue Integral......Page 24
2.1 Fields, Monotone Classes, and Borel Fields......Page 29
2.2 Additive Measures......Page 36
2.3 Carathיodory Outer Measure......Page 38
2.4 E. Hopf\'s Extension Theorem......Page 42
2.4.1 Fields, σ-Fields, and Measures Inherited by a Subset......Page 47
3.1 The Finite Interval […N, N )......Page 49
3.2 Measurable Sets, Borel Sets, and the Real Line......Page 52
3.2.1 Lebesgue Measure on R......Page 54
3.3 Measure Spaces and Completions......Page 56
3.3.2 A Nonmeasurable Set......Page 59
3.4 Semimetric Space of Measurable Sets......Page 61
I: Logic and the Foundations of Mathematics......Page 68
Poincarי against the Logicists......Page 70
4.1 Measurable Functions......Page 73
4.1.1 Baire Functions of Measurable Functions......Page 74
4.2 Limits of Measurable Functions......Page 76
4.3 Simple Functions and Egoroffs Theorem......Page 79
4.3.1 Double Sequences......Page 81
4.3.2 Convergence in Measure......Page 83
4.4 Lusins Theorem......Page 84
5.1 Special Simple Functions......Page 87
5.2 Extending the Domain of the Integral......Page 90
5.2.1 The Class L+ of Nonnegative Measurable Functions......Page 92
5.2.2 The Class L of Lebesgue Integrable Functions......Page 96
5.2.3 Convex Functions and Jensen\'s Inequality......Page 99
5.3 Lebesgue Dominated Convergence Theorem......Page 101
5.4 Monotone Convergence and Fatous Theorem......Page 107
5.5 Completeness of L1(X, Afr, μ ) and the Pointwise Convergence Lemma......Page 110
5.6 Complex-Valued Functions......Page 118
6.1 Product Measures......Page 121
6.2 Fubinis Theorem......Page 126
6.3 Comparison of Lebesgue and Riemann Integrals......Page 135
7.1 Functions of Bounded Variation......Page 141
II: Reinterpretations in the History of Mathematics......Page 146
7.3 Lebesgues Theorem and Vitalis Covering Theorem......Page 149
7.4 Absolutely Continuous and Singular Functions......Page 157
8 General Countably Additive Set Functions......Page 169
8.1 Hahn Decomposition Theorem......Page 170
8.2 Radon-Nikodym Theorem......Page 174
8.3 Lebesgue Decomposition Theorem......Page 179
9.1 The Banach Space Lp(X, Afr, μ )......Page 183
9.2 The Dual of a Banach Space......Page 188
9.3 The Dual Space of Lp(X, Afr, μ )......Page 192
9.4 Hilbert Space, Its Dual, and L2( X, Afr, μ )......Page 196
9.5 Riesz-Markov-Saks-Kakutani Theorem......Page 203
10 Translation Invariance in Real Analysis......Page 213
10.1 An Orthonormal Basis for L2(T)......Page 214
10.2 Closed, Invariant Subspaces of L2(T)......Page 221
10.2.1 Integration of Hilbert Space Valued Functions......Page 222
10.2.2 Spectrum of a Subset of L2(T)......Page 224
10.3 Schwartz Functions: Fourier Transform and Inversion......Page 226
10.4.1 The Fourier Transform in L2(R)......Page 231
10.4.2 Translation-Invariant Subspaces of L2(R)......Page 234
10.4.3 The Fourier Transform and Direct Integrals......Page 236
10.5 Irreducibility of L2(R) Under Translations and Rotations......Page 237
10.5.1 Position and Momentum Operators......Page 239
10.5.2 The Heisenberg Group......Page 240
A.1 The Limits to Countable Additivity......Page 243
Logos, Logic, and Logistikי: Some Philosophical Remarks on Nineteenth-Century Transformation of Mathematics......Page 247
Index......Page 249
Contents......Page 6
Preface......Page 8
An Opinionated Introduction......Page 12
Logical Truth and Analyticity in Carnap\'s \"Logical Syntax of Language\"......Page 91
The Emergence of First-Order Logic......Page 104
Kronecker\'s Place in History......Page 148
Felix Klein and His \"Erlanger Programm\"......Page 154
Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics......Page 186
How Can Mathematicians and Mathematical Historians Help Each Other?......Page 210
III: Case Studies in the History and Philosophy of Mathematics......Page 228
Fitting Numbers to the World: The Case of Probability Theory......Page 230
Ten Misconceptions about Mathematics and Its History......Page 269
Mathematics and the Sciences......Page 287
Mathematical Naturalism......Page 302
IV: The Social Context of Modern Mathematics......Page 336
Partisans and Critics of a New Science: The Case of Artificial Intelligence and Some Historical Parallels......Page 338
The Emergence of Princeton as a World Center for Mathematical Research, 1896-1939......Page 355
Contributors......Page 376
A......Page 380
B......Page 381
C......Page 382
D......Page 383
F......Page 384
G......Page 385
H......Page 386
J......Page 387
L......Page 388
M......Page 389
N......Page 390
P......Page 391
R......Page 392
T......Page 393
V......Page 394
Z......Page 395