Preface ........................................................ xi
A remark on notation ........................................... xi
Acknowledgments ............................................... xii
Chapter 1 Logic and foundations ................................ 1
§1.1 Material implication ...................................... 1
§1.2 Errors in mathematical proofs ............................. 2
§1.3 Mathematical strength ..................................... 4
§1.4 Stable implications ....................................... 6
§1.5 Notational conventions .................................... 8
§1.6 Abstraction ............................................... 9
§1.7 Circular arguments ....................................... 11
§1.8 The classical number systems ............................. 12
§1.9 Round numbers ............................................ 15
§1.10 The "no-self-defeating object" argument, revisited ....... 16
§1.11 The "no-self-defeating object" argument, and the
vagueness paradox ........................................ 28
§1.12 A computational perspective on set theory ............... 35
Chapter 2 Group theory ........................................ 51
§2.1 Torsors .................................................. 51
§2.2 Active and passive transformations ....................... 54
§2.3 Cayley graphs and the geometry of groups ................. 56
§2.4 Group extensions ......................................... 62
§2.5 A proof of Gromov's theorem .............................. 69
Chapter 3 Analysis ............................................ 79
§3.1 Orders of magnitude, and tropical geometry ............... 79
§3.2 Descriptive set theory vs. Lebesgue set theory ........... 81
§3.3 Complex analysis vs. real analysis ....................... 82
§3.4 Sharp inequalities ....................................... 85
§3.5 Implied constants and asymptotic notation ................ 87
§3.6 Brownian snowflakes ...................................... 88
§3.7 The Euler-Maclaurin formula, Bernoulli numbers, the
zeta function, and real-variable analytic continuation ... 88
§3.8 Finitary consequences of the invariant subspace
problem ................................................. 104
§3.9 The Guth-Katz result on the Erdцs distance problem ...... 110
§3.10 The Bourgain-Guth method for proving restriction
theorems ................................................ 123
Chapter 4 Non-Standard analysis .............................. 133
§4.1 Real numbers, non-standard real numbers, and finite
precision arithmetic .................................... 133
§4.2 Non-Standard analysis as algebraic analysis ............. 136
§4.3 Compactness and contradiction: the correspondence
principle in ergodic theory ............................. 137
§4.4 Non-Standard analysis as a completion of standard
analysis ................................................ 150
§4.5 Concentration compactness via non-standard analysis ..... 168
Chapter 5 Partial differential equations ..................... 181
§5.1 Quasilinear well-posedness .............................. 181
§5.2 A type diagram for function spaces ...................... 189
§5.3 Amplitude-frequency dynamics for semilinear dispersive
equations ............................................... 194
§5.4 The Euler-Arnold equation ............................... 203
Chapter 6 Miscellaneous ...................................... 217
§6.1 Multiplicity of perspective ............................. 217
§6.2 Memorisation vs. derivation ............................. 220
§6.3 Coordinates ............................................. 222
§6.4 Spatial scales .......................................... 227
§6.5 Averaging ............................................... 228
§6.6 What colour is the sun? ................................. 231
§6.7 Zeno's paradoxes and induction .......................... 232
§6.8 Jevons' paradox ......................................... 233
§6.9 Bayesian probability .................................... 236
§6.10 Best, worst, and average-case analysis .................. 242
§6.11 Duality ................................................. 244
§6.12 Open and closed conditions .............................. 246
Bibliography .................................................. 249
Index ......................................................... 255
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