A critique of a formalist-mechanist version of the justification of arguments in mathematicians' proof practices

Philosophia Mathematica

Volumn 15, Issue 3, 2007, Pages 291-320

  Rav, Yehuda ^a  

^a UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34848821710     PISSN: 00318019     EISSN: 17446406     Source Type: Journal    
DOI: 10.1093/philmat/nkm023     Document Type: Article

References (61)

1. ABERDEIN, A. [2006]: 'The informal logic of mathematical proof', in [Hersh, 2006], pp. 56-70.
2. ATIYAH, M., et al. [1994]: 'Responses to: A. Jaffe and F. Quinn, "Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics"', Bulletin of the American Mathematical Society (N.S.) 30, 178-207.
3. AZZOUNI, J. [2004]: 'The derivation-indicator view of mathematical practice', Philosophia Mathematica (3) 12, 81-105.
4. BAILEY, D.H., and J.M. BORWEIN [2005]: 'Experimental mathematics: Examples, methods and implications', Notices of the American Mathematical Society 52, 502-514.
5. BARWISE, J. [1977]: 'An introduction to first-order logic', in J. Barwise, ed., Handbook of Mathematical Logic. Amsterdam: North-Holland, pp. 5-46.
6. BARWISE, J. [1989]: 'Mathematical proofs of computer system correctness', Notices of the American Mathematical Society 36 844-851.
7. BRIDGES, D., and R. MINES [1984]: 'What is constructive mathematics?', The Mathematical Intelligencer 6 No. 4, 32-38.
8. DE BRUIJN, N.G. [1970]: 'The mathematical language AUTOMATH, its usage and some of its extensions', Lecture Notes in Mathematics; 125. Berlin: Springer, pp. 29-61.
9. DE BRUIJN, N.G. [1991]: 'Checking mathematics with computer assistance', Notices of the American Mathematical Society 38 8-15.
10. CELLUCCI, C. [2002]: Filosofia e matematica. Rome: Laterza.
11. CELLUCCI, C., and D. GILLIES, eds. [2005]: Mathematical Reasoning and Heuristics. London: King's College Publications.
12. CORCORAN, J. [1973]: 'Gaps between logical theory and mathematical practice', in M. Bunge, ed., The Methodological Unity of Science. Dordrecht: Reidel, pp. 23-50.
13. CORFIELD, D. [2003]: Towards a Philosophy of Real Mathematics Cambridge: Cambridge University Press.
14. CUTLAND, N. [1980]: Computability: An Introduction to Recursive Function Theory. Cambridge: Cambridge University Press.
15. DAVIES, E.B. [2005]: 'A defence of mathematical pluralism', Philosophia Mathematica (3) 13, 252-276.
16. DAVIS, P.J., and R. HERSH [1981]: The Mathematical Experience. Boston: Birkhäuser.
17. DAWSON, J.W., Jr [2006]: 'Why do mathematicians re-prove theorems?', Philosophia Mathematica (3) 14, 269-286.
18. DUMMETT, M. [1993]: The Seas of Language. Oxford: Oxford University Press.
19. FALLIS, D. [2003]: 'Intentional gaps in mathematical proofs', Synthese 134, 45-69.
20. FEFERMAN, S. [1979]: 'What does logic have to tell us about mathematical proofs?' The Mathematical Intelligencer 2 (1997-80), No. 1, 20-24.
21. GLAS, E. [2006]: 'Mathematics as objective knowledge and as a human practice', in [Hersh, 2006], pp. 289-303.
22. GRABINER, J.V. [1974]: 'Is mathematical truth time-dependent?', American Mathematical Monthly 81, 354-365.
23. HAZEWINKEL, M. [1985]: 'Experimental mathematics', Mathematical Modelling 6, 175-211.
24. HAZEWINKEL, M., ed. [1988]: Encyclopedia of Mathematics. Dordrecht: Kluwer.
25. HERSH, R. [1979]: 'Some proposals for reviving the philosophy of mathematics', Advances in Mathematics 31, 31-50.
26. HERSH, R., ed. [2006]: 18 Unconventional Essays on the Nature of Mathematics. New York: Springer.
27. JAFFE, A. [1997]: 'Proof and the evolution of mathematics', Synthese 111, 133-146.
28. JAFFE, A., and F. QUINN [1993]: '"Theoretical mathematics": Toward a cultural synthesis of mathematics and theoretical mathematics', Bulletin of the American Mathematical Society (N.S.) 29, 1-13.
29. JAFFE, A., and F. QUINN [1994]: 'Response to: "Responses to: A. Jaffe and F. Quinn, 'Theoretical matTematics: toward a cultural synthesis of mathematics and theoretical physics'"', Bulletin of the American Mathematical Society (N.S.) 30, 208-211.
30. JANS, J.P. [1964]: Rings and Homology. New York: Holt, Rinehart and Winston.
31. JOSEPH, G.G. [1994]: 'Different ways of knowing: Contrasting styles of argument in Indian and Greek mathematical traditions', in P. Ernest, ed., Mathematics, Education and Philosophy: An International Perspective. London: Falmer Press, pp. 194-204.
32. JUTTING, L.S. VAN B. [1979]: 'Checking Landau's "Grundlagen" in the AUTOMATH system'. Ph.D. thesis, Eindhoven University of Technology (1977). Mathematical Centre Tracts 83. Amsterdam: Mathematisch Centrum.
33. KLEENE, S.C. [1952]: Introduction to Metamathematics. Princeton, N.J.: Van Nostrand.
34. KLEINER, I., and N. MOVSHOVITZ-HADAR [1997] 'Proof: A many-splendored thing', The Mathematical Intelligencer 19, No. 3, 16-26.
35. KLINE, M. [1972]: Mathematical Thought from Ancient to Modern Times. New York; Oxford: Oxford University Press.
36. KNORR, W.R. [1975]: The Evolution of the Euclidean Elements. Dordrecht: Reidel.
37. KREISEL, G. [1969]: 'The formalist-positivist doctrine of mathematical precision in the light of experience', L'Âge de la Science 3, 17-46.
38. KREISEL, G. [1971]: 'Review of The Collected Papers of Gerhard Gentzen (M.E. Szabo, ed., Amsterdam: North-Holland, 1969)', Journal of Philosophy 68, 238-265.
39. LAKATOS, I. [1976a]: Proofs and Refutations. Cambridge: Cambridge University Press.
40. LAKATOS, I. [1976b]: 'A renaissance of empiricism in the recent philosophy of mathematics', British Journal for the Philosophy of Science 27, 201-223.
41. LANDAU, E. [1965]: Die Grundlagen der Analysis. New York: Chelsea (Reprint).
42. LARVOR, B. [2001]: 'What is a dialectical philosophy of mathematics?' Philosophia Mathematica (3) 9, 212-229.
43. LENG, M. [2002]: 'Phenomenology and mathematical practice', Philosophia Mathematica (3) 10, 3-25.
44. LEIBNIZ, G. W. [1875-90]: Die philosophischen Schriften von Gottfried Wilhelm Leibniz. C.I. Gerhardt, ed. Berlin: Weidmann. Reprinted Hildesheim: Olms, 1979.
45. LIVINGSTON, E. [1999]: 'Cultures of proving', Social Studies of Science 29, 867-888.
46. LONGO, G. [2003]: 'Proofs and programs', Synthese 134, 85-117.
47. MARQUIS, J.-P. [1999]: 'Mathematical engineering and mathematical change', International Studies in the Philosophy of Science 13 245-259.
48. PRAWITZ, D. [ 1993]: 'Gentzen's analysis of first-order proofs', in R.I.G. Hughes, ed., A Philosophical Companion to First-Order Logic Indianapolis, Indiana: Hackett, pp. 202-211.
49. PRAWITZ, D. [2005]: 'Logical consequence from a constructivist point of view', in [Shapiro, 2005b], pp. 671-695.
50. RAV, Y. [1999]: 'Why do we prove theorems? Philosophia Mathematica (3) 7, 5-41.
51. RAV, Y. [2005]: Reflections on the proliferous growth of mathematical concepts and tools: Some case histories from mathematicians' workshops', in [Cellucci and Gillies, 2005], pp. 49-69.
52. ROTA, G.-C. [ 1997]: 'The phenomenology of mathematical proof', Synthese 111, 183-196.
53. SARTON, G. [1952]: A History of Science: Ancient Science through the Golden Age of Greece. Cambridge, Mass.: Harvard University Press.
54. SHAPIRO, S. [2005a]: 'Logical consequence, proof theory, and model theory', in [Shapiro, 2005b], pp. 651-670.
55. SHAPIRO, S. [2005b]: The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press.
56. TAKEUTI, G. [1975]: Proof Theory. Amsterdam: North-Holland.
57. TARSKI, A. [1956 (1935)]: 'On the concept of logical consequence', in Logic, Semantics, Metamathematics: Papers from 1923 to 1938. Oxford: Oxford University Press, pp. 409-420.
58. THOMAS, R.S.D. [1991]: 'Meaning in ordinary language and in mathematics', Philosophia Mathematica (2) 6, 3-38.
59. THURSTON, W.P. [1994]: 'On proof and progress in mathematics', Bulletin of the American Mathematical Society (N.S.) 30, 161-177.
60. VAN DER WAERDEN, B.L. [1954]: Science Awakening. Leyden: Nordhoff.
61. ZEILBERGER, D. [1993]: 'Theorems for a price: Tomorrow's semi-rigorous mathematical culture', Notices of the American Mathematical Society 40, 978-981.