Doing Mathematics: Convention, Subject, Calculation, Analogy

Description

When one attempts to write an expository presentation, one should try to write clearly, directly, and in the simplest language possible. This book does not succeed in that endeavor. Rather, the author is to me a model of obfuscation, arcane language, wandering threads of reference, obscure relationships, and opaque explanation. Perhaps a philospher who likes to couch his writings in such wrapping would relate to this style of presentation, but certainly not a reader interested in how mathematicians actually do mathematics. I would assume that rules out professional mathematicians by default. In some ways this could have been a good book, if the author had been less interested in showing us how much obscura he knows and more in explaining in a truly understandable fashion how people do mathematics . A reader who is looking for insights into mathematicians and their methods would not go far wrong reading Hardy's A Mathematician's Apology, or Littlewoood's Miscellany. 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