Doing Mathematics: Convention, Subject, Calculation, Analogy

Contents......Page 8
List of Figures and Tables......Page 10
Preface......Page 14
ONE Introduction......Page 20
CONVENTION......Page 24
THE FIELDS OF TOPOLOGY......Page 26
STRATEGY AND STRUCTURE......Page 30
ANALOGY AND SYZYGY......Page 33
CONVENTION AND CULTURE IN MATHEMATICS......Page 37
THE CENTRAL LIMIT THEOREM......Page 39
THE ESTABLISHMENT OF MEANS AND VARIANCES......Page 40
JUSTIFYING MEANS AND VARIANCES......Page 41
EXCEPTIONS AND ALTERNATIVES TO MEANS-VARIANCE ANALYSIS......Page 44
THEOREMS FROM PROBABILITY AS JUSTIFICATIONS FOR MEANS-VARIANCE ANALYSIS......Page 48
ITO LEVY AND KHINCHIN: A TECHNICAL INTERLUDE......Page 49
DEFENDING RANDOMNESS FROM IMPOSSIBLE LONGSHOTS AND COINCIDENCES......Page 53
MAKING VARIANCES REAL BY IDENTIFYING THEM WITH ENDURING OBJECTS AND OTHER FEATURES OF THE WORLD......Page 54
VARIANCES AS THE CANONICAL ACCOUNT OF FLUCTUATION......Page 57
NOISE......Page 58
EINSTEIN AND FISHER......Page 60
V......Page 61
ANALYSIS AND ALGEBRA......Page 63
THE SUBJECTS OF TOPOLOGY......Page 65
PREVIEW......Page 69
BEFORE THE BIRTH: STORIES OF NINETEENTH-CENTURY PRECURSORS......Page 70
GENERAL TOPOLOGY: EXAMPLES AND COUNTEREXAMPLES DIAGRAMS AND CONSTRUCTIONS......Page 74
PARTS IN ANALYSIS......Page 77
ALGEBRAIC TOPOLOGY AS PLAYING WITH BLOCKS......Page 79
IV......Page 81
OPEN SETS AS THE CONTINUITY OF NEIGHBORHOODS......Page 82
DIMENSION AND HOMOLOGY AS BRIDGING CONCEPTS......Page 85
THE TENSIONS IN PRACTICE: THE LOCAL AND THE GLOBAL......Page 87
THE TENSION IN ANALYSIS: THE HARD AND THE SOFT......Page 88
HARD ANALYSIS: THE ISING MODEL OF FERROMAGNETISM AND THE STABILITY OF MATTER......Page 92
KEEPING TRACK OF UNSMOOTHNESS AND DISCONTINUITY......Page 98
KEEPING TRACK OF THE SHAPE OF OBJECTS......Page 101
DIAGRAMMATICS AND MACHINERY......Page 103
PICTURES AND DIAGRAMS AS WAYS OF KEEPING TRACK OF FLOWS AND SYSTEMIC RELATIONSHIPS......Page 104
THE RIGHT PARTS......Page 107
NATURALITY......Page 108
THE TENSIONS AND THE LESSONS......Page 113
Appendix: The Two-Dimensional Ising Model of a Ferromagnet......Page 114
SOME OF THE SOLUTIONS TO THE TWO-DIMENSIONAL ISING MODEL......Page 117
FOUR Calculation: Strategy Structure and Tactics in Applying Classical Analysis......Page 125
DECIPHERING STRATEGY TACTICS AND STRUCTURE IN MATHEMATICS PAPERS......Page 126
RIGOROUS AND EXACT MATHEMATICAL PHYSICS......Page 129
C.N. YANG ON THE SPONTANEOUS MAGNETIZATION OF THE ISING LATTICE......Page 134
FIGURING OUT SOME OF YANG\'S PAPER......Page 136
STRATEGY AND STRUCTURE IN PROOFS OF THE STABILITY OF MATTER......Page 142
SOME OF THE MATHEMATICS AND THE PHYSICS......Page 146
MATHEMATICAL TECHNIQUES......Page 149
STAGING SEQUENCING DEMONSTRATING......Page 151
NEW IDEAS AND NEW SEQUENCES: THE ELLIOTT LIEB-WALTER THIRRING PROOF......Page 160
CHARLES FEFFERMAN\'S CLASSICAL ANALYSIS......Page 167
1. THE STABILITY OF MATTER......Page 169
2. MATTER AS A GAS......Page 173
3. STABILITY OF MATTER WITH A GOOD CONSTANT......Page 175
MORE ON FEFFERMAN-SECO......Page 179
THE FEFFERMAN-SECO ARGUMENT: A TECHNICAL INTERLUDE......Page 181
\"WHY ARE THESE MATHEMATICAL OBJECTS USEFUL IN DESCRIBING THIS SYSTEM?\"......Page 185
TACTICS ASCENDANT: UNAVOIDABLE ALGEBRAIC MANIPULATION AND LENGTHY COMPUTATIONAL PROOFS......Page 188
THE WU-MCCOY-TRACY-BAROUCH PAPER......Page 191
WHY WIENER-HOPF AND TOEPLITZ?......Page 198
FIGURING OUT MORE OF WHAT IS REALLY GOING ON......Page 202
CLUES ALONG THE WAY: IDENTIFYING THE FORMAL AND THE ABSTRACT WITH SUBSTANTIVE MEANING......Page 205
ANALOGY IN MATHEMATICAL WORK......Page 208
THE MATHEMATICAL ANALOGY HISTORICALLY......Page 211
THE ANALOGY IN PRACTICE......Page 213
A SYZYGY BETWEEN FIELDS OF MATHEMATICS EXPRESSED IN VARIOUS WAYS......Page 217
THE ONSAGER PROGRAM IN LANGLANDS-PROGRAM TERMS......Page 220
THE ONSAGER PROGRAM AND THE LANGLANDS PROGRAM......Page 222
THE IMPORT OF THE ANALOGIES AND THE SYZYGY......Page 229
PRIME FACTORIZATIONS AND ELEMENTARY EXCITATIONS......Page 231
THE ANALOGY MADE CONCRETE......Page 233
How THESE IDEAS CAME TO BE SEEN AS AN INTERRELATED COMPLEX......Page 236
IV......Page 237
ANDRE WEIL\'S ROSETTA STONE LEARNING RIEMANNIAN AND TRANSLATING......Page 238
SYZYGY AND FUNCTORIALITY......Page 244
THE PROGRAMS IN PRACTICE......Page 246
RIEMANN AND MAXWELL AS AN EPITOME......Page 250
\"WHAT IS REALLY GOING ON\" IN THE MATHEMATICS......Page 252
MATHEMATICS AND THE CITY IN THE NINETEENTH CENTURY......Page 256
III......Page 259
ORDER AND STRUCTURE......Page 262
CENTER AND PERIPHERY......Page 265
PROCESS PATTERN AND PALIMPSEST......Page 266
EMPLOYING YOUR BODY TO PROVE A THEOREM: THE SNAKE LEMMA OF ALGEBRAIC TOPOLOGY......Page 268
TRAIN TRACKS AND PAIRS OF PANTS......Page 270
GOD\'S TRANSCENDENCE AS ANALOGIZED TO MATHEMATICAL NOTIONS......Page 272
ASCENT TO THE ABSOLUTE......Page 275
SETS AS MODELS OF GOD\'S INFINITUDE......Page 278
THE CITY OF MATHEMATICS......Page 279
Epilog......Page 283
A: The Spontaneous Magnetization of a Two-Dimensional Ising Model\"......Page 285
B: On the Dirac and Schwinger Corrections to the Ground-State Energy of an Atom......Page 294
C: Sur la forme des espaces topologiques et sur les points fixes des representations......Page 309
D: Une lettre a Simone Weil......Page 312
Notes......Page 326
Bibliography......Page 420
Index......Page 454