Why Is There Philosophy of Mathematics At All?

Ian Hacking

Cambridge University Press, Jan 30, 2014 - Science - 212 pages

This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.

Contents

• A cartesian introduction: 1
• On jargon: 2
• Descartes: 3
• A Application: 4
• Descartes Geometry: 5
• An astonishing identity: 6
• The application of geometry to arithmetic: 8
• The application of mathematics to mathematics: 9
• 26: 10
• The same stuff?: 11
• Overdetermined?: 12
• Unity behind diversity: 13
• On mentioning honours the Fields Medals: 15
• Analogy and André Weil 1940: 16
• The Langlands programme: 18
• Application analogy correspondence: 20
• B Proof: 21
• Eternal truths: 22
• Mere eternity as against necessity: 23
• Voevodskys extreme: 25
• Cartesian proof: 26
• caveat emptor: 28
• making it all obvious: 29
• Proofs and refutations: 30
• On squaring squares and not cubing cubes: 32
• From dissecting squares to electrical networks: 34
• Intuition: 35
• Descartes against foundations?: 37
• The two ideals of proof 38 37 35 34: 38
• who checks whom? 40 2 What makes mathematics mathematics?: 40
• We take it for granted: 41
• Arsenic: 42
• Some dictionaries: 43
• What the dictionaries suggest: 45
• A Japanese conversation: 47
• A sullen antimathematical protest 48 A miscellany: 48
• An institutional answer: 51
• A neurohistorical answer: 52
• The Peirces father and son: 53
• logicism: 54
• Bourbaki: 55
• Only Wittgenstein seems to have been troubled: 57
• Aside on method on using Wittgenstein: 59
• A semantic answer: 60
• More miscellany 61 Proof: 61
• On advance 66: 67
• Hilbert and the Millennium: 68
• Symmetry: 71
• The Butterfly Model: 72
• Could mathematics be a fluke of history?: 73
• The Latin Model: 74
• Inevitable or contingent?: 75
• Play: 76
• Mathematical games ludic proof: 77
• Why is there philosophy of mathematics?: 78
• A perennial topic: 79
• What is the philosophy of mathematics anyway?: 80
• in or out?: 81
• Ancient and Enlightenment: 83
• Food for thought Matière à penser: 86
• The Monster: 87
• Exhaustive classification: 88
• Moonshine: 89
• The experience of outthereness: 90
• Parables: 91
• The neurobiological retort: 92
• My own attitude: 93
• Naturalism: 94
• Plato: 96
• application: 97
• The jargon: 98
• Necessity: 99
• Russell trashes necessity: 100
• Necessity no longer in the portfolio: 102
• Aside on Wittgenstein: 103
• Kants question: 104
• Russells version: 105
• Russell dissolves the mystery: 106
• number a secondorder concept: 107
• a Vienna: 108
• b Quine: 109
• Ayer Quine and Kant: 110
• Logicizing philosophy of mathematics: 111
• A nifty onesentence summary Putnam redux: 112
• John Stuart Mill on the need for a sound philosophy of mathematics: 113
• Proofs: 115
• A Little contingencies: 116
• infinity: 117
• complex numbers: 119
• Changing the setting: 121
• The discovery of proof: 122
• Kants tale: 123
• Pythagoras: 126
• Unlocking the secrets of the universe: 127
• Plato theoretical physicist: 129
• Harmonics works: 130
• Why there was uptake of demonstrative proof: 131
• Plato kidnapper: 132
• Another suspect? Eleatic philosophy: 133
• Logic and rhetoric: 135
• esoteric and exoteric: 136
• Civilization without proof: 137
• Class bias: 138
• the ideal of proof impede the growth of knowledge?: 139
• What gold standard?: 140
• Proof demoted: 141
• A style of scientific reasoning: 142
• Applications 1 Past and present A The emergence of a distinction 2 3 4 5 6 7 8 9 Plato on the difference between philosophical and practical mathem...: 144
• Pure and mixed: 146
• Newton: 148
• Probability swinging from branch to branch: 149
• Rein and angewandt: 150
• Pure Kant 151 10 11: 151
• Pure Gauss: 152
• The German nineteenth century told in aphorisms Applied polytechniciens: 153
• Military history: 156
• William Rowan Hamilton: 158
• Cambridge pure mathematics: 160
• Hardy Russell and Whitehead: 161
• Wittgenstein and von Mises: 162
• SIAM: 163
• B A very wobbly distinction 17 18 Kinds of application: 164
• Robust but not sharp: 168
• Symmetry: 171
• The representationaldeductive picture: 172
• Articulation Moving from domain to domain: 174
• Rigidity: 175
• Maxwell and Buckminster Fuller: 176
• The maths of rigidity: 179
• Aerodynamics: 181
• Rivalry: 182
• The British institutional setting: 184
• The German institutional setting Mechanics: 186
• Geometry pure and applied A general moral: 188
• Another style of scientific reasoning: 189
• Platos name 1 2 3 4 5 6 7 Hauntology Platonism: 191
• Websters Born that: 193
• Sources: 194
• Semantic ascent: 195
• Organization: 196
• Alain Connes Platonist 8 9 10 11 12 13 Offduty and offthecuff: 197
• Connes archaic mathematical reality: 198
• Aside on incompleteness and platonism: 201
• Two attitudes structuralist and Platonist: 202
• What numbers could not: 203
• Pythagorean Connes: 205
• Timothy Gowers antiPlatonist 14 15 16 17 18 19 20 21 A very public mathematician: 206
• Does mathematics need a philosophy?: 207
• On becoming an antiPlatonist: 208
• Does mathematics need a philosophy?: 209
• Ontological commitment: 211
• Truth: 212
• Observable and abstract numbers: 213
• Gowers versus Connes: 215
• The standard semantical account: 216
• The famous maxim: 218
• Chomskys doubts On referring: 220
• Counterplatonisms 1 Two more platonisms and their opponents: 223
• A Totalizing platonism as opposed to intuitionism Disclosures References Index 2 3 4 5 6 7 8 9 10 11 Paul Bernays 18881977: 224
• The setting: 225
• Totalities: 227
• Other totalities: 228
• Arithmetical and geometrical totalities: 230
• different philosophical concerns: 231
• Two more mathematicians Kronecker and Dedekind: 232
• Some things Dedekind said: 233
• What was Kronecker protesting?: 235
• The structuralisms of mathematicians and philosophers distinguished: 236
• Todays platonismnominalism 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Disclaimer A brief history of nominalism: 238
• The nominalist programme: 239
• Why deny?: 241
• Russellian roots: 242
• Ontological commitment Commitment: 244
• The indispensability argument Presupposition: 246
• Contemporary platonism in mathematics: 250
• Intuition: 251
• Whats the point of platonism?: 253
• The only kind of thinking that has ever advanced human culture: 254
• Where do I stand on todays platonismnominalism? The last word: 256
• 122: 266
• 223: 274
• 137: 281
• 139: 282
• 150: 283
• 161: 284
• 174: 285
• 164: 286
• 168: 287

About the author

Ian Hacking is a retired professor of Collège de France, Chair of Philosophy and History of Scientific Concepts, and retired University Professor of Philosophy at the University of Toronto. His most recent books include The Social Construction of What? (1999), An Introduction to Probability and Inductive Logic (Cambridge University Press, 2001), The Emergence of Probability (Cambridge University Press, 2006), Scientific Reason (2009) and Exercises in Analysis (Cambridge University Press, 2009).

Bibliographic information

• Title: Why Is There Philosophy of Mathematics At All?
• Author: Ian Hacking
• Publisher: Cambridge University Press, 2014
• ISBN: 1107729823, 9781107729827
• Length: 212 pages
• Subjects: Mathematics / History & Philosophy; Mathematics / Logic; Philosophy / General; Science / Philosophy & Social Aspects