Advertisement

Abstract

This paper is concerned with real proofs as opposed to formal proofs, and specifically with the ultimate reason of real proofs (‘Why Proof?’) and with the notion of real proof (‘What is a Proof?’).

Keywords

Biological Evolution Formal Proof Analytic Proof Peano Arithmetic Incompleteness Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Avigad: Mathematical method and proof. Synthese, 153 (2006) pp 105–159CrossRefGoogle Scholar
  2. 2.
    J. Azzouni: The derivation-indicator view of mathematical practice. Philosophia Mathematica, vol. 12 (2004) pp 81–105Google Scholar
  3. 3.
    R. Carnap: The logical syntax of language (Humanities Press, New York 1951)Google Scholar
  4. 4.
    C. Cellucci: Le ragioni della logica (Laterza, Rome 1998; fifth ed 2008)Google Scholar
  5. 5.
    C. Cellucci: The growth of mathematical knowledge: an open world view. In: The growth of mathematical knowledge, ed by E. Grosholz and H. Breger (Kluwer, Dordrecht 2000) pp 153–176Google Scholar
  6. 6. C. Cellucci: Filosofia e matematica (Laterza, Rome 2002) English translation: Introduction to 18 unconventional essays on the nature of mathematics, ed by R. Hersh (Springer, Berlin 2006) pp 17–36Google Scholar
  7. 7.
    C. Cellucci: Mathematical discourse vs mathematical intuition. In: Mathematical reasoning and heuristics, ed by C. Cellucci and D. Gillies (College Publications, London 2005) pp 137–165Google Scholar
  8. 8.
    C. Cellucci: The question Hume didn’t ask: why should we accept deductive inferences? In: Demonstrative and non-demonstrative reasoning in mathematics and natural science, ed by C. Cellucci and P. Pecere (Edizioni dell’Università, Cassino 2006) pp 207–235Google Scholar
  9. 9.
    C. Cellucci: Gödel aveva qualcosa da dire sulla natura del ragionamento? In: La complessità di Gödel, ed by G. Lolli and U. Pagallo (Giappichelli, Turin 2008) pp 31–64Google Scholar
  10. 10.
    C. Cellucci: The nature of mathematical explanation. Studies in History and Philosophy of Science (2008) To appearGoogle Scholar
  11. 11.
    C. Cellucci: Perché ancora la filosofia (Laterza, Rome 2008) To appearGoogle Scholar
  12. 12.
    W. S. Cooper: The evolution of reason. Logic as a branch of biology (Cambridge University Press, Cambridge 2001)Google Scholar
  13. 13.
    J. W. Dawson: Why do mathematicians re-prove theorems? Philosophia Mathematica 14 (2006) pp 269–286CrossRefGoogle Scholar
  14. 14.
    R. Descartes: Oeuvres, ed by C. Adam and P. Tannery (Vrin, Paris 1996)Google Scholar
  15. 15.
    K. Devlin: The math instinct. Why you’re a mathematical genius (along with lobsters, birds, cats, and dogs) (Thunder’s Mouth Press, New York 2005)Google Scholar
  16. 16.
    H. Diels: Die Fragmente der Vorsokratiker ed by W. Krantz (Weidmann, Berlin 1934)Google Scholar
  17. 17.
    G. Frege: The foundations of arithmetic. A logico-mathematical enquiry into the concept of number (Blackwell, Oxford 1953)Google Scholar
  18. 18.
    G. Frege: The basic laws of arithmetic, ed by M. Furth (University of California Press, Berkeley and Los Angeles 1964)Google Scholar
  19. 19.
    G. Frege: Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In: From Frege to Gödel. A source book in mathematical logic, 1879-1931, ed by J.van Heijenoort (Harvard University Press, Cambridge MA 1967) pp 5–82Google Scholar
  20. 20.
    G. Frege: On the foundations of geometry: second series. In: Collected papers on mathematics, logic, and philosophy ed by B. McGuinness (Blackwell, Oxford 1984) pp 293–340Google Scholar
  21. 21.
    G. Galilei: Opere, ed by A. Favaro (Barbera, Florence 1968)Google Scholar
  22. 22.
    G. Gentzen: Investigations into logical deduction. In: The collected papers of Gerhard Gentzen, ed by M. E. Szabo (North-Holland, Amsterdam 1969) pp 68–131Google Scholar
  23. 23.
    K. Gödel: What is Cantor’s continuum problem?-(1964). In: Collected works, vol. I, ed by S. Feferman et al. (Oxford University Press, Oxford 1990) pp 254–270Google Scholar
  24. 24.
    K. Gödel: Is mathematics syntax of language?-Version III. In: Collected works, vol. III, ed by S. Feferman et al. (Oxford University Press, Oxford 1995) pp 334–356Google Scholar
  25. 25.
    E. Grosholz: Representation and productive ambiguity in mathematics and the sciences (Oxford University Press, Oxford 2007)Google Scholar
  26. 26.
    R. W. Hamming: The unreasonable effectiveness of mathematics. The American Mathematical Monthly 87 (1980) pp 81–90CrossRefGoogle Scholar
  27. 27.
    R. W. Hamming: Mathematics on a distant planet. The American Mathematical Monthly 105 (1998) pp 640–650CrossRefGoogle Scholar
  28. 28.
    G. H. Hardy: Mathematical proof. Mind 38 (1929) pp 1–25CrossRefGoogle Scholar
  29. 29.
    W. D. Hart: Introduction to The philosophy of mathematics, ed by W. D. Hart(Oxford University Press, Oxford 1996) pp 1–13Google Scholar
  30. 30.
    M. Heidegger: Brief über den Humanismus. In: Gesamtausgabe, vol. 9: Wegmarken (Klostermann, Frankfurt am Mein 1976) pp 313–364Google Scholar
  31. 31.
    M. Heil and J. C. Silva Bueno: Within-plant signaling by volatiles leads to induction and priming of an indirect plant defense in nature. Proceedings of the National Academy of Sciences USA 104 (2007) pp 5467–5472CrossRefGoogle Scholar
  32. 32.
    R. Hersh: What is mathematics, really? (Oxford University Press, Oxford 1997)Google Scholar
  33. 33.
    D. Hilbert: Beweis des tertium non datur. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Math.-Phys. Klasse (1931) pp 120–125Google Scholar
  34. 34.
    D. Hilbert: The foundations of mathematics. In: From Frege to Gödel. A source book in mathematical logic, 1879-1931, ed by J.van Heijenoort (Harvard University Press, Cambridge MA 1967) pp 464–479Google Scholar
  35. 35.
    D. Hilbert: Letter to Frege 29.12.1899. In: Philosophical and mathematical correspondence (The University of Chicago Press, Chicago 1980) pp 38–41Google Scholar
  36. 36.
    D. Hilbert: Die Grundlagen der Geometrie. In: David Hilbert’s lectures on the foundations of geometry (1891-1902), ed by M. Hallett and U. Majer (Springer, Berlin 2004) pp 72–81Google Scholar
  37. 37.
    J. Hintikka and U. Remes: The method of analysis: its geometrical origin and its general significance (Reidel, Dordrecht 1974)Google Scholar
  38. 38.
    G. R. Hunt and R. D. Gray: Tool manufacture by New Caledonian crows: chipping away at human uniqueness. Acta Zoologica Sinica 52 (2006) Supplement, pp 622–625Google Scholar
  39. 39.
    I. Kant: Lectures on logic, ed. J. M. Young (Cambridge University Press, Cambridge 1992)Google Scholar
  40. 40.
    W. R. Knorr: The ancient tradition of geometric problems (Dover, Mineola NY 1993)Google Scholar
  41. 41.
    S. Kripke: Naming and necessity (Blackwell, Oxford 1980)Google Scholar
  42. 42.
    G. Lolli: QED. Fenomenologia della dimostrazione (Bollati Boringhieri, Turin 2005)Google Scholar
  43. 43.
    E. Mach: Knowledge and error. Sketches on the psychology of enquiry (Reidel, Dordrecht 1976)Google Scholar
  44. 44.
    A. Macintyre: The mathematical significance of proof theory. Philosophical Transactions of the Royal Society 363 (2005) pp 2419–2435CrossRefGoogle Scholar
  45. 45.
    P. Mäenpää: The art of analysis. Logic and history of problem solving. Dissertation (University of Helsinki, Helsinki 1993)Google Scholar
  46. 46.
    L. I. Meikle and J. D. Fleuriot: Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In: Theorem proving in higher order logics, LNCS 2758, ed by D. Basin and B. Wolff (Springer, Berlin 2003) pp 319–334Google Scholar
  47. 47.
    I. Newton: An account of the book entitled Commercium Epistolicum Collinii et aliorum, de analysi promota. Philosophical Transactions, vol. 29 (1715) pp 173–224 The quoted passage is also reprinted in I. B. Cohen, Introduction to Newton’s ‘Principia’ (Cambridge University Press, Cambridge 1971) p. 295Google Scholar
  48. 48.
    I. Newton: The mathematical papers, ed by D. T. Whiteside (Cambridge University Press, Cambridge 1967-81)Google Scholar
  49. 49.
    I. Newton: MS Add. 3968, f. 101. In: I. B. Cohen, Introduction to Newton’s ‘Principia’ (Cambridge University Press, Cambridge 1971) pp 293–294Google Scholar
  50. 50.
    Pappus of Alexandria: Book 7 of the Collection, ed by A. Jones (Springer, Berlin 1986)Google Scholar
  51. 51.
    T. J. Pennings: Do dogs know calculus? College Mathematics Journal 34 (2003) pp 178–182CrossRefGoogle Scholar
  52. 52.
    G. Pólya: Mathematics and plausible reasoning (Princeton University Press, Princeton 1954)Google Scholar
  53. 53.
    Proclus: In primum Euclidis Elementorum librum commentarii, ed by G. Friedlein (Olms, Hildesheim 1992)Google Scholar
  54. 54.
    Y. Rav: A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica 15 (2007) pp 291–320CrossRefGoogle Scholar
  55. 55.
    K. Ribet: From the Taniyama-Shimura conjecture to Fermat’s Last Theorem. Annales de la Faculté des Sciences de Toulouse — Mathématiques, vol. 11 (1990) pp 116–139Google Scholar
  56. 56.
    G. C. Rota: Indiscrete thoughts, ed by F. Palombi (Birkhäuser, Boston 1997)Google Scholar
  57. 57.
    B. Russell: Human society in ethics and politics (Allen & Unwin, London 1954)Google Scholar
  58. 58.
    A. Tarski: The semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research, vol. 4 (1944) pp 341–376CrossRefGoogle Scholar
  59. 59.
    B. Timmermans: La résolution des problèmes de Descartes à Kant (Presses Universitaires de France, Paris 1995)Google Scholar
  60. 60.
    J. van Benthem: Interview. In: Philosophy of mathematics: 5 questions, ed by V. F. Hendricks and H. Leitgeb (Automatic Press / VIP Press, New York 2007)Google Scholar
  61. 61.
    H. Vaihinger: Die Philosophie des Als Ob (Felix Meiner, Leipzig 1927)Google Scholar

Copyright information

© Springer-Verlag Italia 2008

Authors and Affiliations

  • Carlo Celluci
    • 1
  1. 1.Università di Roma ‘La Sapienza’Dipartimento di Studi Filosofici ed EpistemologiciRomaItaly

Personalised recommendations