Advertisement

On Saying What You Really Want to Say: Wittgenstein, Gödel, and the Trisection of the Angle

  • Juliet Floyd
Chapter
Part of the Synthese Library book series (SYLI, volume 251)

Abstract

Wittgenstein’s remarks on the first incompleteness theorem1 have often been denounced, and mostly dismissed. Despite indirect historical evidence to the contrary,2 it is a commonplace that Wittgenstein rejected Gödel’s proof because he did not, or even could not, understand it.3 Kreisel twice used the word “wild” when he reviewed Wittgenstein on Göde1.4 Dummett, in many respects an admirer of Wittgenstein’s philosophy, wrote that the remarks on Gödel and on the notion of consistency are “of poor quality or contain definite errors”.5 Gödel’s own comments were damning (see Section III below).

Keywords

Philosophical Investigation Mathematical Truth Incompleteness Theorem Mathematical Proposition Logical Syntax 
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.

Bibliography

  1. Anderson, A. R.: 1958, ‘Mathematics and the “Language-Game—, Review of Metaphysics 11, 446–458.Google Scholar
  2. Boyer, C. B.: 1968, A History of Mathematics, Princeton University Press, Princeton. Benacerraf, P. and Putnam, H. (eds.): 1964, Philosophy of Mathematics, Prentice Hall, N.J., pp. 481–490.Google Scholar
  3. Carnap, R.: 1937, The Logical Syntax of Language, Amethe Smeaton (trans. ), Routledge & Kegan Paul Ltd., London.Google Scholar
  4. Cavell, S.: 1969, Must We Mean What We Say?, Cambridge University Press.Google Scholar
  5. Cavell, S.: 1979, The Claim of Reason: Wittgenstein, Skepticism, Morality, and Tragedy, Oxford University Press.Google Scholar
  6. Cavell, S.: 1989, This New Yet Unapproachable America: Lectures After Emerson After Wittgenstein, Living Batch Press, Albuquerque, N.M.Google Scholar
  7. Cavell, S.: 1990, ‘Conditions Handsome and Unhandsome, The Carus Lectures 1988, ’ University of Chicago.Google Scholar
  8. Courant, R. and H. Robbins: 1978, What Is Mathematics? Oxford University Press.Google Scholar
  9. Dawson, John W., Jr.: 1991, ‘The Reception of Gödel’s Incompleteness Theorems’, in Thomas Drucker (ed.), Perspectives on the History of Mathematical Logic, Birkhäuser Boston, Inc., Boston, Mass., pp. 84–100.Google Scholar
  10. Diamond, C.: 1991, The Realistic Spirit: Wittgenstein, Philosophy and the Mind, M. I. T. Press, Cambridge, Mass.Google Scholar
  11. Dreben, B.: 1990, ‘Quine’, in R. B. Barrett and R. F. Gibson (eds.), Perspectives on Quine, Basil Blackwell, Cambridge, Mass., pp. 81–95.Google Scholar
  12. Dreben, B. and J. Floyd: 1991, ‘Tautology: How Not to Use A Word’, Synthese 87, 23–50.CrossRefGoogle Scholar
  13. Dudley, U.: 1987, A Budget of Trisections, Springer Verlag, New York.CrossRefGoogle Scholar
  14. Dummett, M.: 1959, ‘Wittgenstein’s Philosophy of Mathematics’ in Dummett, M.: 1978, Truth and Other Enigmas, Harvard University Press, Cambridge Mass., pp. 166–185.Google Scholar
  15. Floyd, J.: 1990, ‘The Rule of the Mathematical: Wittgenstein’s Later Discussions’ ( Harvard Ph.D. Dissertation).Google Scholar
  16. Floyd, J.: 1991, ‘Wittgenstein on 2,2,2…: On the Opening of Remarks on the Foundations of Mathematics’, Synthese 87 1: 143–180.Google Scholar
  17. Frege, G.: 1884, The Foundations of Arithmetic, first published as Die Grundlagen der Arithmetik, Verlag von Wilhelm Koebner, Breslau; published in English as The Foundations of Arithmetic, J. L. Austin (trans.), revised edition, Northwestern University Press, Evanston, IL, 1980.Google Scholar
  18. Friedlander, E.: 1992, ‘Expressions of Judgment’, Harvard Ph.D. Thesis.Google Scholar
  19. Gauss, C. F.: 1801, Disquisitiones Arithmeticae, A. A. Clarke (trans.), Yale University Press, 1966.Google Scholar
  20. Gerrard, S.: 1991, ‘Wittgenstein’s Philosophies of Mathematics’, Synthese 87, 125–142.CrossRefGoogle Scholar
  21. Gödel, K.: 1931, ‘On formally undecidable propositions of Principia mathematica and related systems I’, reprinted in S. Feferman, J. W. Dawson, Jr., S. C. Kleene, G. H. Moore, R. M. Solovay and J. van Heijenoort (eds.): 1986, Kurt Gödel: Collected Works Volume I, Oxford University Press.Google Scholar
  22. Goldfarb, W.: 1983, ‘I Want You to Bring Me a Slab’, Synthese 56, 265–282.CrossRefGoogle Scholar
  23. Goldfarb, W.: 1992, ‘Wittgenstein on Understanding’, in P. A. French, T. E. Uehling, Jr. and H. K. Wettstein (eds.), Midwest Studies in Philosophy XVII: The Wittgenstein Legacy, pp. 109–122.Google Scholar
  24. Goldfarb, W. and T. Ricketts: 1992, ‘Carnap and the Philosophy of Mathematics’, forthcoming in D. Bell and W. Vossenkuhl (eds.), Science and Subjectivity. Akademie Verlag, Berlin.Google Scholar
  25. Goodstein, R. L.: 1957, ‘Critical Notice of Remarks on the Foundations of Mathematics’, Mind,549–553.Google Scholar
  26. Herstein, I. N.: 1975, Topics in Algebra, Xerox, 2nd edition, Lexington, Mass.Google Scholar
  27. Hilmy, S. S.: 1987, The Later Wittgenstein, Basil Blackwell, Oxford.Google Scholar
  28. Hintikka, J.: 1991, ‘An Impatient Man and His Papers’ Synthese 87 2, 183–202.Google Scholar
  29. Hintikka, J.: 1993, ‘The Original Sinn of Wittgenstein’s Philosophy of Mathematics’, in Klaus Puhl (ed.), Wittgenstein’s Philosophy of Mathematics (Proceedings of the 15th International Wittgenstein Symposium), Hölder-Pichler-Tempsky, Vienna, pp. 24–51.Google Scholar
  30. Hintikka, J. and M. B. Hintikka: 1986, Investigating Wittgenstein, Blackwell, Oxford.Google Scholar
  31. Hobson, E. W.: 1953, Squaring the Circle: A History of the Problem, Chelsea Publishing Co., Boston.Google Scholar
  32. Hodges, A.: 1983, Alan Turing: The Enigma, Simon & Schuster, New York.Google Scholar
  33. Hylton, P.: 1990, Russell, Idealism, and the Emergence of Analytic Philosophy,Oxford University Press.Google Scholar
  34. Klein, F.: 1895, Famous Problems of Elementary Geometry, W. W. Beman, D. E. Smith, R. C. Archibald (trans. ), 2nd ed.Google Scholar
  35. Kreisel, G.: 1958, ‘Wittgenstein’s Remarks on the Foundations of Mathematics’, British Journal for the Philosophy of Science 9, 135–158.CrossRefGoogle Scholar
  36. Kreisel, G.: 1983, ‘Einige Erläuterungen zu Wittgensteins Kummer mit Hilbert und Gödel’, in P. Weingartner and J. Czermak (eds.), Epistemology and Philosophy of Science. Proceedings of the 7th International Wittgenstein Symposium, Hölder-Pichler-Tempsky, Vienna, pp. 295–303.Google Scholar
  37. Kripke, S. (1982): Wittgenstein on Rules and Private Language, Harvard University Press, Cambridge, MA.Google Scholar
  38. Lakatos, I.: 1977, Proofs and Refutations, the Logic of Mathematical Discovery, J. Worrall and E. Zahar (eds. ), Cambridge University Press.Google Scholar
  39. Mancosu, P.: 1991, ‘On the Status of Proofs by contradiction in the Seventeenth Century’, Synthese 88, 15–41.CrossRefGoogle Scholar
  40. Mancosu, P.: 1992, ‘Aristotelian Logic and Euclidean Mathematics: Seventeenth-Century Developments of the Quaestio de Certitudine Mathematicarum’, Studies in the History and Philosophy of Science 23(2), 241–265.Google Scholar
  41. Marion, M.: 1992, ‘Wittgenstein and the Dark Cellar of Platonism’, Address to the XVth International Wittgenstein Symposium, Kirchberg am Wechsel, Austria.Google Scholar
  42. Marion, M.: ‘Wittgenstein and Finitism’, forthcoming, Synthese.Google Scholar
  43. Moore, G. E.: 1959, Philosophical Papers, Allen and Unwin, London.Google Scholar
  44. Nedo and Ranchetti (eds.): 1983, Wittgenstein: Sein Leben in Bildern und Texten: Suhrkamp Verlag, Frankfurt am Main.Google Scholar
  45. Putnam, H.: 1975, Mathematics, Matter and Method: Philosophical Papers Volume I,Cambridge University Press.Google Scholar
  46. Putnam, H.: 1983, Realism and Reason: Philosophical Papers Volume II,Cambridge University Press.Google Scholar
  47. Putnam, H.: 1991, ‘Wittgenstein on Religious Belief’, in On Community, Leroy S. Rouner (ed.), Notre Dame Press, pp. 56–75.Google Scholar
  48. Russell, B.: 1938, Principles of Mathematics, 2nd ed., W. W. Norton Co., New York.Google Scholar
  49. Shanker, S. G.: 1987, Wittgenstein and the Turning Point in the Philosophy of Mathematics, State University Press of New York, Albany.Google Scholar
  50. Shanker, S. G.: 1988, ‘Wittgenstein’s Remarks on the Significance of Gödel’s Theorem’, in S. G. Shanker (ed.), Gödel’s Theorem in Focus, Croom Helm, London, pp. 176–177.Google Scholar
  51. von Wright, G. H.: 1983, Wittgenstein, University of Minnesota Press, Minneapolis.Google Scholar
  52. Wang, H.: 1986, Beyond Analytic Philosophy: Doing Justice to What We Know, M. I. T. Press, Cambridge, Mass.Google Scholar
  53. Wang, H.: 1987, Reflections on Gödel, M. I. T. Press, Cambridge, Mass.Google Scholar
  54. Wang, H.: 1991a, ‘Imagined Discussions with Gödel and with Wittgenstein’, Address to the Kurt Gödel Society Meeting at Kirchberg am Wechsel, Austria.Google Scholar
  55. Wang, H.: 1991b, ‘To and From Philosophy — Discussions with Gödel and Wittgenstein’, Synthese 88 (2), 229–277.CrossRefGoogle Scholar
  56. Wanzel, P.: 1837, ‘Recherches sur les moyens de reconnaître si un Problème de géométrie peut se résoudre avec la règle et le compas,’ Journal de Mathématiques Pure et Appliquées 2, 366–372.Google Scholar
  57. Watson, A.: 1938, ‘Mathematics and its Foundations’, Mind 47, 440–451.CrossRefGoogle Scholar
  58. Wittgenstein, L.: 1921, Logische-Philosophische Abhandlung, final chapter, Ostwald’s Annalen der Naturphilosophie; first published in English under the title Tractatus Logico- Philosophicus, 1922, C. K. Ogden (trans.), Routledge & Kegan Paul, Ltd., London; English translation reprinted with corrections 1933.Google Scholar
  59. Wittgenstein, L.: 1958a, Philosophical Investigations, G. E. M. Anscombe and R. Rhees (eds.), G. E. M. Anscombe (trans.), second ed., Blackwell, Oxford.Google Scholar
  60. Wittgenstein, L.: 1958b, The Blue and Brown Books, Blackwell, Oxford.Google Scholar
  61. Wittgenstein, L.: 1961, Notebooks 1914–16, G. H. von Wright and G. E. M. Anscombe (eds.), G. E. M. Anscombe (trans.), Blackwell, Oxford.Google Scholar
  62. Wittgenstein, L.: 1979, Ludwig Wittgenstein and the Vienna Circle, shorthand notes recorded by F. Waismann, B. McGuinness (ed.), J. Schulte and B. McGuiness (trans. ), Blackwell, Oxford.Google Scholar
  63. Wittgenstein, L.: 1974, Philosophical Grammar, R. Rhees (ed.), A. J. P. Kenny (trans. ), Blackwell, Oxford.Google Scholar
  64. Wittgenstein, L.: 1975, Philosophical Remarks, R. Rhees (ed.), R. Hargreaves and R. White (trans. ), Blackwell, Oxford.Google Scholar
  65. Wittgenstein, L.: 1978, Remarks on the Foundations of Mathematics, G. H. von Wright, R. Rhees, G. E. M. Anscombe (ed.), G. E. M. Anscombe (trans.) revised ed., Blackwell, Oxford.Google Scholar
  66. Wittgenstein, L.: 1979, Cambridge Lectures: 1932–1935, from the Notes of Alice Ambrose and Margaret Macdonald, Alice Ambrose (ed.), Rowman and Littlefield, Totowa, N.J., reprinted by the University of Chicago Press.Google Scholar
  67. Wittgenstein, L.: 1980a, Remarks on the Philosophy of Psychology, Volumes 1 and 11, G. E. M. Anscombe and G. H. von Wright (eds.), G. E. M. Anscombe (trans. ), University of Chicago Press.Google Scholar
  68. Wittgenstein, L.: 1980b, Culture and Value, G. H. von Wright and H. Nyman (eds.), Peter Winch (trans. ), University of Chicago Press.Google Scholar
  69. Wittgenstein, L.: 1980c, Wittgenstein’s Lectures, Cambridge 1930–32, from the notes of John King and Desmond Lee, ed. Desmond Lee, Blackwell, Oxford.Google Scholar
  70. Wittgenstein, L.: 1984, Werkausgabe in 8 Bänden, Suhrkamp Verlag, Frankfurt am Main.Google Scholar
  71. Wittgenstein, L.: 1989, Wittgenstein’s 1939 Lectures on the Foundations of Mathematics, C. Diamond (ed.), The University of Chicago Press (Originally published by Cornell University Press, Ithaca, NY, 1976 ).Google Scholar
  72. Wittgenstein, L.: The ‘Frühversion’, or early version of Philosophical Investigations, compiled by G. H. von Wright and Heikki Nyman (unpublished manuscript).Google Scholar
  73. Wright C. (1980): Wittgenstein on the Foundations of Mathematics, Harvard University Press, Cambridge, MA.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Juliet Floyd
    • 1
  1. 1.City College, City University of New YorkBoston UniversityUSA

Personalised recommendations