Why Is There Philosophy of Mathematics At All

Cover
About  the Book
WHY IS THERE PHILOSOPHY OF MATHEMATICS AT ALL?
Copyright
     © Ian Hacking, 2014
     ISBN 978-1-107-05017-4 Hardback
     ISBN 978-1-107-65815-8 Paperback
Dedication
Contents
Foreword
Chapter 1. A cartesian introduction
     1 Proofs, applications, and other mathematical activities
     2 On jargon
     3 Descartes
     A Application
          4 Arithmetic applied to geometry
          5 Descartes' Geometry
          6 An astonishing identity
          7 Unreasonable effectiveness
          8 The application of geometry to arithmetic
          9 The application of mathematics to mathematics
          10 The same stuff?
          11 Over-determined?
          12 Unity behind diversity
          13 On mentioning honours - the Fields Medals
          14 Analogy - and Andre Weil 1940
          15 The Langlands programme
          16 Application, analogy, correspondence
     B Proof
          17 Two visions of proof
          18 A convention
          19 Eternal truths
          20 Mere eternity as against necessity
          21 Leibnizian proof
          22 Voevodsky' s extreme
          23 Cartesian proof
          24 Descartes and Wittgenstein on proof
          25 The experience of cartesian proof: caveat emptor
          26 Grothendieck's cartesian vision: making it all obvious
          27 Proofs and refutations
          28 On squaring squares and not cubing cubes
          29 From dissecting squares to electrical networks
          30 Intuition
          31 Descartes against foundations?
          32 The two ideals of proof
          33 Computer programmes: who checks whom?
Chapter 2. What makes mathematics mathematics?
     1 We take it for granted
     2 Arsenic
     3 Some dictionaries
     4 What the dictionaries suggest
     5 A Japanese conversation
     6 A sullen anti-mathematical protest
     7 A miscellany
     8 An institutional answer
     9 A neuro-historical answer
     10 The Peirces, father and son
     11 A programmatic answer: logicism
     12 A second programmatic answer: Bourbaki
     13 Only Wittgenstein seems to have been troubled
     14 Aside on method - on using Wittgenstein
     15 A semantic answer
     16 More miscellany
     17 Proof
     18 Experimental mathematics
     19 Thurston's answer to the question 'what makes?'
     20 On advance
     21 Hilbert and the Millennium
     22 Symmetry
     23 The Butterfly Model
     24 Could 'mathematics' be a 'fluke of history'?
     25 The Latin Model
     26 Inevitable or contingent?
     27 Play
     28 Mathematical games, ludic proof
Chapter 3. Why is there philosophy of mathematics?
     1 A perennial topic
     2 What is the philosophy of mathematics anyway?
     3 Kant: in or out?
     4 Ancient and Enlightenment
     A An answer from the ancients: proof and exploration
          5 The perennial philosophical obsession ...
          6 The perennial philosophical obsession ... is totally anomalous
          7 Food for thought (Matiere a penser)
          8 The Monster
          9 Exhaustive classification
          10 Moonshine
          11 The longest proof by hand
          12 The experience of out-thereness
          13 Parables
          14 Glitter
          15 The neurobiological retort
          16 My own attitude
          17 Naturalism
          18 Plato!
     B An answer from the Enlightenment: application
          19 Kant shouts
          20 The jargon
          21 Necessity
          22 Russell trashes necessity
          23 Necessity no longer in the portfolio
          24 Aside on Wittgenstein
          25 Kant's question
          26 Russell's version
          2 7 Russell dissolves the mystery
          28 Frege: number a second-order concept
          29 Kant's conundrum becomes a twentieth-century dilemma: (a) Vienna
          30 Kant's conundrum becomes a twentieth-century dilemma: (b) Quine
          31 Ayer, Quine, and Kant
          32 Logicizing philosophy of mathematics
          33 A nifty one-sentence summary (Putnam redux)
          34 John Stuart Mill on the need for a sound philosophy of mathematics
Chapter 4. Proofs
     1 The contingency of the philosophy of mathematics
     A Little contingencies
          2 On inevitability and 'success'
          3 Latin Model: infinity
          4 Butterfly Model: complex numbers
          5 Changing the setting
     B Proof
          6 The discovery of proof
          7 Kant's tale
          8 The other legend: Pythagoras
          9 Unlocking the secrets of the universe
          10 Plato, theoretical physicist
          11 Harmonics works
          12 Why there was uptake of demonstrative proof
          13 Plato, kidnapper
          14 Another suspect? Eleatic philosophy
          15 Logic (and rhetoric)
          16 Geometry and logic: esoteric and exoteric
          17 Civilization without proof
          18 Class bias
          19 Did the ideal of proof impede the growth of knowledge?
          20 What gold standard?
          21 Proof demoted
          22 A style of scientific reasoning
Chapter 5. Applications
     1 Past and present
     A THE EMERGENCE OF A DISTINCTION
          2 Plato on the difference between philosophical and practical mathematics
          3 Pure and mixed
          4 Newton
          5 Probability - swinging from branch to branch
          6 Rein and angewandt
          7 Pure Kant
          8 Pure Gauss
          9 The German nineteenth century, told in aphorisms
          10 Applied polytechniciens
          11 Military history
          12 William Rowan Hamilton
          13 Cambridge pure mathematics
          14 Hardy, Russell, and Whitehead
          15 Wittgenstein and von Mises
          16 SIAM
     B A VERY WOBBLY DISTINCTION
          17 Kinds of application
          18 Robust but not sharp
          19 Philosophy and the Apps
          20 Symmetry
          21 The representational-deductive picture
          22 Articulation
          23 Moving from domain to domain
          24 Rigidity
          25 Maxwell and Buckminster Fuller
          26 The maths of rigidity
          2 7 Aerodynamics
          28 Rivalry
          29 The British institutional setting
          30 The German institutional setting
          31 Mechanics
          32 Geometry, 'pure' and 'applied'
          33 A general moral
          34 Another style of scientific reasoning
Chapter 6. In Plato's name
     1 llauntology
     2 Platonism
     3 Webster's
     4 Born that way
     5 Sources
     6 Semantic ascent
     7 Organization
     A ALAIN CONNES, PLATONIST
          8 Off-duty and off-the-cuff
          9 Connes' archaic mathematical reality
          10 Aside on incompleteness and platonism
          11 Two attitudes, structuralist and Platonist
          12 What numbers could not be
          13 Pythagorean Connes
     B TIMOTHY GOWERS, ANTI-PLATONIST
          14 A very public mathematician
          15 Does mathematics need a philosophy? No
          16 On becoming an anti-Platonist
          17 Does mathematics need a philosophy? Yes
          18 Ontological commitment
          19 Truth
          20 Observable and abstract numbers
          21 Gowers versus Connes
          22 The 'standard' semantical account
          23 The famous maxim
          24 Chomsky's doubts
          25 On referring
Chapter 7 Counter-platonisms
     1 Two more platonisms - and their opponents
     A TOTALIZING PLATONISM AS OPPOSEDTO INTUITIONISM
          2 Paul Bernays (1888-1977)
          3 The setting
          4 Totalities
          5 Other totalities
          6 Arithmetical and geometrical totalities
          7 Then and now: different philosophical concerns
          8 Two more mathematicians, Kronecker and Dedekind
          9 Some things Dedekind said
          10 What was Kronecker protesting?
          11 The structuralisms of mathematicians and philosophers distinguished
     B TODAY'S PLATONISM/NOMINALISM
          12 Disclaimer
          13 A brief history of nominalism now
          14 The nominalist programme
          15 Whydeny?
          16 Russellian roots
          17 Ontological commitment
          19 The indispensability argument
          20 Presupposition
          21 Contemporary platonism in mathematics
          22 Intuition
          23 What's the point of platonism?
          24 Peirce: The only kind of thinking that has everadvanced human culture
          25 Where do I stand on today's platonism/ nominalism?
          26 The last word
Disclosures
References
Index