Why is there philosophy of mathematics at all? / Ian Hacking.

Hacking explores how mathematics became possible for the human race, and how it ensured our status as the dominant species.

Saved in:
Bibliographic Details
Main Author: Hacking, Ian
Language:English
Published: New York : Cambridge University Press, 2014.
Subjects:
Mathematics > Philosophy.
Proof theory.
Physical Description:xv, 290 pages ; 23 cm
Format: Book
Contents:
  • 1. A Cartesian introduction; 2. What makes mathematics mathematics?; 3. Why is there philosophy of mathematics?; 4. Proofs; 5. Applications; 6. In Plato's name; 7. Counter-Platonisms; Disclosures.
  • 1. A cartesian introduction. Proofs, applications, and other mathematical activities
  • On jargon
  • Descartes. A. Application. Arithmetic applied to geometry
  • Descartes’ Geometry
  • An astonishing identity
  • Unreasonable effectiveness
  • The application of geometry to arithmetic
  • The application of mathematics to mathematics
  • The same stuff?
  • Over-determined?
  • Unity behind diversity
  • On mentioning honours−the Fields Medals
  • Analogy–and André Weil 1940
  • The Langlands programme
  • Application, analogy, correspondence. B. Proof. Two visions of proof
  • A convention
  • Eternal truths
  • Mere eternity as against necessity
  • Leibnizian proof
  • Voevodsky’s extreme
  • Cartesian proof
  • Descartes and Wittgenstein on proof
  • The experience of cartesian proof: caveat emptor
  • Grothendieck’s cartesian vision: making it all obvious
  • Proofs and refutations
  • On squaring squares and not cubing cubes
  • From dissecting squares to electrical networks
  • Intuition
  • Descartes against foundations?
  • The two ideals of proof
  • Computer programmes: who checks whom?
  • 2. What makes mathematics mathematics? We take it for granted
  • Arsenic
  • Some dictionaries
  • What the dictionaries suggest
  • A Japanese conversation
  • A sullen anti-mathematical protest
  • A miscellany
  • An institutional answer
  • A neuro-historical answer
  • The Peirces, father and son
  • A programmatic answer: logicism
  • A second programmatic answer: Bourbaki
  • Only Wittgenstein seems to have been troubled
  • Aside on method – on using Wittgenstein
  • A semantic answer
  • More miscellany
  • Proof
  • Experimental mathematics
  • Thurston’s answer to the question ‘what makes?’
  • On advance
  • Hilbert and the Millennium
  • Symmetry
  • The Butterfly Model
  • Could ‘mathematics’ be a ‘fluke of history’?
  • The Latin Model
  • Inevitable or contingent?
  • Play
  • Mathematical games, ludic proof.
  • 3. Why is there philosophy of mathematics? A perennial topic
  • What is the philosophy of mathematics anyway?
  • Kant: in or out?
  • Ancient and Enlightenment
  • A. An answer from the ancients: proof and exploration. The perennial philosophical obsession . . .
  • The perennial philosophical obsession . . . is totally anomalous
  • Food for thought (Matière à penser)
  • The Monster
  • Exhaustive classification
  • Moonshine
  • The longest proof by hand
  • The experience of out-thereness
  • Parables
  • Glitter
  • The neurobiological retort
  • My own attitude
  • Naturalism
  • Plato! B. An answer from the Enlightenment: application. Kant shouts
  • The jargon
  • Necessity
  • Russell trashes necessity
  • Necessity no longer in the portfolio
  • Aside on Wittgenstein
  • Kant’s question
  • Russell’s version
  • Russell dissolves the mystery
  • Frege: number a second-order concept
  • Kant’s conundrum becomes a twentieth-century dilemma: (a) Vienna
  • Kant’s conundrum becomes a twentieth-century dilemma: (b) Quine
  • Ayer, Quine, and Kant
  • Logicizing philosophy of mathematics
  • A nifty one-sentence summary (Putnam redux)
  • John Stuart Mill on the need for a sound philosophy of mathematics.
  • 4. Proofs. The contingency of the philosophy of mathematics. A. Little contingencies. On inevitability and ‘success’
  • Latin Model: infinity
  • Butterfly Model: complex numbers
  • Changing the setting. B. Proof. The discovery of proof
  • Kant’s tale
  • The other legend: Pythagoras
  • Unlocking the secrets of the universe
  • Plato, theoretical physicist
  • Harmonics works
  • Why there was uptake of demonstrative proof
  • Plato, kidnapper
  • Another suspect? Eleatic philosophy
  • Logic (and rhetoric)
  • Geometry and logic: esoteric and exoteric
  • Civilization without proof
  • Class bias
  • Did the ideal of proof impede the growth of knowledge?
  • What gold standard?
  • Proof demoted
  • A style of scientific reasoning.
  • 5. Applications. Past and present. A. The emergence of a distinction
  • Plato on the difference between philosophical and practical mathematics
  • Pure and mixed
  • Newton
  • Probability – swinging from branch to branch
  • Rein and angewandt
  • Pure Kant
  • Pure Gauss
  • The German nineteenth century, told in aphorisms
  • Applied polytechniciens
  • Military history
  • William Rowan Hamilton
  • Cambridge pure mathematics
  • Hardy, Russell, and Whitehead
  • Wittgenstein and von Mises
  • SIAM. B. A very wobbly distinction. Kinds of application
  • Robust but not sharp
  • Philosophy and the Apps
  • Symmetry
  • The representational−deductive picture
  • Articulation
  • Moving from domain to domain
  • Rigidity
  • Maxwell and Buckminster Fuller
  • The maths of rigidity
  • Aerodynamics
  • Rivalry
  • The British institutional setting
  • The German institutional setting
  • Mechanics
  • Geometry, ‘pure’ and ‘applied’
  • A general moral
  • Another style of scientific reasoning.
  • 6. In Plato’s name. Hauntology
  • Platonism
  • Webster’s
  • Born that way
  • Sources
  • Semantic ascent
  • Organization. A. Alain Connes, Platonist
  • Off-duty and off-the-cuff
  • Connes’ archaic mathematical reality
  • Aside on incompleteness and platonism
  • Two attitudes, structuralist and Platonist
  • What numbers could not be
  • Pythagorean Connes. B. Timothy Gowers, anti-Platonist
  • A very public mathematician
  • Does mathematics need a philosophy? No
  • On becoming an anti-Platonist
  • Does mathematics need a philosophy? Yes
  • Ontological commitment
  • Truth
  • Observable and abstract numbers
  • Gowers versus Connes
  • The ‘standard’ semantical account
  • The famous maxim
  • Chomsky’s doubts
  • On referring.
  • 7. Counter-platonisms. Two more platonisms – and their opponents. A. Totalizing platonism as opposed to intuitionism
  • Paul Bernays (1888–1977)
  • The setting
  • Totalities
  • Other totalities
  • Arithmetical and geometrical totalities
  • Then and now: different philosophical concerns
  • Two more mathematicians, Kronecker and Dedekind
  • Some things Dedekind said
  • What was Kronecker protesting?
  • The structuralisms of mathematicians and philosophers distinguished. B. Today’s platonism/nominalism
  • Disclaimer
  • A brief history of nominalism now
  • The nominalist programme
  • Why deny?
  • Russellian roots
  • Ontological commitment
  • Commitment
  • The indispensability argument
  • Presupposition
  • Contemporary platonism in mathematics
  • Intuition
  • What’s the point of platonism?
  • Peirce: The only kind of thinking that has ever advanced human culture
  • Where do I stand on today’s platonism/nominalism?
  • The last word.