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:
| Main Author: | |
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| Language: | English |
| Published: |
New York :
Cambridge University Press,
2014.
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| Subjects: | |
| Physical Description: | xv, 290 pages ; 23 cm |
| Format: | Book |
MARC
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| 100 | 1 | |a Hacking, Ian. |0 http://id.loc.gov/authorities/names/n50020633 | |
| 245 | 1 | 0 | |a Why is there philosophy of mathematics at all? / |c Ian Hacking. |
| 264 | 1 | |a New York : |b Cambridge University Press, |c 2014. | |
| 300 | |a xv, 290 pages ; |c 23 cm | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 505 | 0 | |a 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. | |
| 505 | 0 | |a 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? | |
| 505 | 0 | |a 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. | |
| 505 | 0 | |a 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. | |
| 505 | 0 | |a 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. | |
| 505 | 0 | |a 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. | |
| 505 | 0 | |a 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. | |
| 505 | 0 | |a 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. | |
| 520 | |a Hacking explores how mathematics became possible for the human race, and how it ensured our status as the dominant species. | ||
| 650 | 0 | |a Mathematics |x Philosophy. |0 http://id.loc.gov/authorities/subjects/sh85082153 | |
| 650 | 0 | |a Proof theory. |0 http://id.loc.gov/authorities/subjects/sh85107437 | |
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