Skip to main content
SpringerLink
Log in

Wittgenstein and finitism

  • Published:
Synthese Aims and scope Submit manuscript

Im Aufbau der Mathematik gibt es zwei offene Stellen, wo es möglicherweise ins Unergründliche geht: der Fortgang in der Reihe der natürlichen Zahlen und das Kontinuum.

Hermann Weyl

Abstract

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aberth, O.: 1970, ‘Computable Analysis and Differential Equations’, in J. Myhill, A. Kino and R. E. Vesley (eds.): 1970,Intuitionism and Proof Theory, North-Holland, Amsterdam, pp. 47–52.

    Google Scholar 

  • Aberth, O.: 1980,Computable Analysis, McGraw-Hill, New York.

    Google Scholar 

  • Ambrose, A.: 1935, ‘Finitism in Mathematics’,Mind 44, pp. 186–203 & 317–40.

    Google Scholar 

  • Anscombe, G. E. M.: 1959,An Introduction to Wittgenstein's Tractatus, third edition, Hutchison, London.

    Google Scholar 

  • Baker, G. P. and P. M. S. Hacker: 1985,Wittgenstein: Rules, Grammar and Necessity, Blackwell, Oxford.

    Google Scholar 

  • Beeson, M. J.: 1985,Foundations of Constructive Mathematics, Springer, Berlin.

    Google Scholar 

  • Bishop, E.: 1967,Foundations of Contructive Analysis, McGraw-Hill, New York.

    Google Scholar 

  • Black, M.: 1964,A Companion to Wittgenstein's ‘Tractatus’, Cambridge University Press, Cambridge.

    Google Scholar 

  • Braithwaite, R. B.: 1931, ‘Editor's Introduction’, in F. P. Ramsey: 1931,The Foundations of Mathematics and other Essays, R. Braithwaite (ed.), Kegan Paul, Trench Trubner & Co., London., pp. ix-xiv.

    Google Scholar 

  • Brouwer, L. E. J.: 1923, ‘On the Significance of the Principle of Excluded Middle in Mathematics, especially in Function Theory’, in van Heijenoort (1967), pp. 335–45.

  • Brouwer, L. E. J.: 1927, ‘Intuitionistic Reflections on Formalism’, in van Heijenoort (1967), pp. 490–2.

  • Brouwer, L. E. J.: 1928, ‘Mathematik Wissenschaft und Sprache’, in L. E. J. Brouwer: 1976,Collected Works, A. Heyting (ed.), North-Holland, Amsterdam, vol. 1, pp. 417–28.

    Google Scholar 

  • Carnap, R.: 1934,The Logical Syntax of Language, Kegan Paul, Trench, Trubner & Co, London.

    Google Scholar 

  • Curry, H. B.: 1941, ‘A Formalisation of Recursive Arithmetic’,American Journal of Mathematics 63, pp. 263–82.

    Google Scholar 

  • Dedekind, R.: 1890, ‘Letter to Keferstein’, in van Heijenoort (1967), pp. 99–103.

  • Dummett, M. A. E.: 1959, ‘Wittgenstein's Philosophy of Mathematics’, in M. A. E. Dummett: 1978,Truth and other Enigmas, Duckworth, London, pp. 166–85.

    Google Scholar 

  • Dummett, M. A. E.: 1990, ‘The Source of the Concept of Truth’, in G. Boolos (ed.),Meaning and Method: Essays in Honor of Hilary Putnam, Cambridge University Press, Cambridge, pp. 1–15.

    Google Scholar 

  • Egidi, R.: 1991, ‘Ramsey and Wittgenstein on Scientific Theories’,Theoria 57, pp. 196–210.

    Google Scholar 

  • Feferman, S.: 1984, ‘Between Constructive and Classical Mathematics’, in M. Richter, E. Börger, W. Oberschelp, B. Schinzel and W. Thomas (eds.): 1984,Computation and Proof Theory, Springer, Lecture Notes in Mathematics 1104, Berlin, pp. 143–62.

    Google Scholar 

  • Fogelin, R: 1968, ‘Wittgenstein and Intuitionism’,American Philosophical Quarterly 5, pp. 267–74.

    Google Scholar 

  • Fogelin, R.: 1976,Wittgenstein, Routledge & Kegan Paul, London.

    Google Scholar 

  • Frascolla, P.: 1980a, ‘Il concetto di infinito nella filosofia della mathematica di Wittgenstein’,Annali della Scuola Normale Superiore di Pisa. Classe de lettere e filosofia 10, pp. 639–74.

    Google Scholar 

  • Frascolla, P.: 1980b, ‘The Constructivist Model in Wittgenstein's Philosophy of Mathematics’, in S. G. Shanker (ed.): 1986,Ludwig Wittgenstein. Critical Assessments, Croom Helm, London, vol. 3, pp. 242–9.

    Google Scholar 

  • Frascolla, P.: 1984, ‘Wittgenstein e il Principio del Terzo Escluso’,Rivista di Filosofia 75, pp. 293–9.

    Google Scholar 

  • Gandy, R. O.: 1982, ‘Limitations to Mathematical Knowledge’, in D. van Dalen, D. Lascar and J. Smiley (eds.): 1982,Logic Colloquium '80, North-Holland, Amsterdam, pp. 129–46.

    Google Scholar 

  • Gauthier, Y.: 1985, ‘A Theory of Local Negation: The Model and some Applications’,Archiv für mathematische Logik und Grundlagenforschung 25, pp. 127–43.

    Google Scholar 

  • Gauthier, Y.: 1991,De la logique interne, Vrin, Paris.

    Google Scholar 

  • Gödel, K.: 1931, ‘On Formally Undecidable Propositions ofPrincipia Mathematica and Related Systems I’, in van Heijenoort (1967), pp. 596–616.

  • Goodstein, R. L.: 1945, ‘Function Theory in an Axiom-Free Equation Calculus’,Proceedings of the London Mathematical Society 48, pp. 401–34.

    Google Scholar 

  • Goodstein, R. L.: 1951,Constructive Formalism, Leicester University Press, Leicester.

    Google Scholar 

  • Goodstein, R. L.: 1957,Recursive Number Theory, North-Holland, Amsterdam.

    Google Scholar 

  • Goodstein, R. L.: 1961,Recursive Analysis, North-Holland, Amsterdam.

    Google Scholar 

  • Goodstein, R. L.: 1969, ‘A Constructive Form of the Second Gauss Proof of the Fundamental Theorem of Algebra’, in B. Dejon and P. Henrici (eds.): 1969,Constructive Aspects of the Fundamental Theorem of Algebra, Wiley, London, pp. 69–76.

    Google Scholar 

  • Goodstein. R. L.: 1971,Development of Mathematical Logic, Logos Press, London.

    Google Scholar 

  • Goodstein, R. L.: 1972, ‘Wittgenstein's Philosophy of Mathematics’, in A. Ambrose and M. Lazerowitz (eds.): 1972,Ludwig Wittgenstein. Philosophy and Language, Allen & Unwin, London, pp. 271–86.

    Google Scholar 

  • Hacker, P. M. S.: 1986,Insight and Illusion, second edition, Clarendon Press, Oxford.

    Google Scholar 

  • Heyting, A.: 1930, ‘Die Formalen Regeln der Intuitionistischen Logik’,Sitzungsberichte der Preussischen Akademie der Wissenschaften 30, I, pp. 42–56, II, pp. 57–71, III, pp. 158–169.

    Google Scholar 

  • Hilbert, D.: 1922, ‘Neubegründung der Mathematik. Erste Mitteilung’, in Hilbert (1935), pp. 157–77.

  • Hilbert, D.: 1923, ‘Die logischen Grundlagen der Mathematik’, in Hilbert (1935), pp. 178–91.

  • Hilbert, D.: 1925, ‘On the Infinite’, in van Heijenoort (1967), pp. 367–92.

  • Hilbert, D. and P. Bernays: 1934,Grundlagen der Mathematik, Bd. 1, Springer, Berlin.

    Google Scholar 

  • Hilbert, D.: 1935,Gesammelte Abhandlungen, vol. 3, Springer, Berlin.

    Google Scholar 

  • Johnson, W. E.: 1922,Logic. Part II, Cambridge, Cambridge University Press.

    Google Scholar 

  • Kaufmann, F.: 1930, ‘The Infinite in Mathematics and its Elimination’, in B. McGuinness (ed.): 1978,The Infinite in Mathematics, D. Reidel, Dordrecht, pp. 1–164.

    Google Scholar 

  • Lawvere, W.: 1964, ‘An Elementary Theory of the Category of Sets’,Proceedings of the National Academy of Science 52, pp. 1506–11.

    Google Scholar 

  • Majer, U.: 1989, ‘Ramsey's Conception of Theories: An Intuitionistic Approach’,History of Philosophy Quarterly 6, pp. 233–58.

    Google Scholar 

  • Majer, U.: 1991, ‘Ramsey's Theory of Truth and the Truth of Theories: A Synthesis of Pragmatism and Intuitionism in Ramsey's Last philosophy’,Theoria 57, pp. 162–95.

    Google Scholar 

  • Marion, M.: 1996,Wittgenstein, Finitism, and the Foundations of Mathematics, Oxford University Press, forthcoming.

  • Mayberry, J.: 1987, ‘What are Numbers?’,Philosophical Studies 54, pp. 317–54.

    Google Scholar 

  • Myhill, J.: 1974, ‘The Undefinability of the Set of Natural Numbers in the Ramified Principia’, in G. Nakhnikian (ed.): 1974,Bertrand Russell's Philosophy, Duckworth, London, pp. 19–27.

    Google Scholar 

  • Noë, A.: 1994, ‘Wittgenstein, Phenomenology and What it Makes Sense to Say’,Philosophy and Phenomenological Research 54, pp. 1–42.

    Google Scholar 

  • Ramsey, F. P.: 1925, ‘Universals’, in Ramsey (1978), pp. 17–39.

  • Ramsey, F. P.: 1926, ‘The Foundations of Mathematics’, in Ramsey (1978), pp. 152–212.

  • Ramsey, F. P.: 1927, ‘Facts and Propositions’, in Ramsey (1978), pp. 40–57.

  • Ramsey, F. P.: 1929a, ‘General Propositions and Causality’, in Ramsey (1978), pp. 133–51.

  • Ramsey, F. P.: 1929b, ‘Principles of Finitist Mathematics’, in Ramsey (1991), pp. 197–202.

  • Ramsey, F. P.: 1929c, ‘The Formal Structure of Intuitionist Mathematics’, in Ramsey (1991), pp. 203–220.

  • Ramsey, F. P.: 1929d, ‘Notes on Theories’, in Ramsey (1991), pp. 228–236.

  • Ramsey, F. P.: 1978,Foundations, D. H. Mellor (ed.), Routledge & Kegan Paul, London.

    Google Scholar 

  • Ramsey, F. P.: 1991,Notes on Philosophy, Probability and Mathematics, M. C. Galavotti (ed.), Bibliopolis, Naples.

    Google Scholar 

  • Robinson, R.: 1947, ‘Primitive Recursive Functions’,Bulletin of the American Mathematical Society 53, pp. 925–42.

    Google Scholar 

  • Russell, B.: 1919,Introduction to Mathematical Philosophy, Allen & Unwin, London.

    Google Scholar 

  • Russell, B.: 1931, ‘Critical Notice onThe Foundations of Mathematics and other Logical Essays. By Frank Plumpton Ramsey’,Mind 40, pp. 476–82.

    Google Scholar 

  • Russell, B.: 1936, ‘The Limits of Empiricism’,Aristotelian Society Proceedings, N. S. 36, pp. 131–150.

    Google Scholar 

  • Russell, B.: 1968,The Autobiography of Bertrand Russell. 1914–1944, Allen & Unwin, London.

    Google Scholar 

  • Skolem, T. 1922, ‘Some Remarks on Axiomatized set Theory’, in van Heijenoort (1967), pp. 291–301.

  • Skolem, T.: 1923, ‘The Foundations of Elementary Arithmetic established by means of the Recursive Mode of Thought, without Use of Apparent Variables ranging over Infinite Domains’, in van Heijenoort (1967), pp. 303–33.

  • Soames, S.: 1983, ‘Generality, Truth-Functions, and Expressive Capacity in theTractatus’,The Philosophical Review 92, pp. 573–89.

    Google Scholar 

  • Sundholm, G.: 1995, ‘The General Form of the Operation in Wittgenstein'sTractatus’, forthcoming inGrazer Philosophische Studien.

  • van Heijenoort, J.: 1967,From Frege to Gödel. A Sourcebook in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge Mass.

    Google Scholar 

  • Watson, A. G. D.: 1938, ‘Mathematics and its Foundations’,Mind 47, pp. 440–51.

    Google Scholar 

  • Weyl, H.: 1921, ‘Über die neue Grundlagenkrise der Mathematik’, in Weyl (1968), vol. 2, pp. 143–80.

    Google Scholar 

  • Weyl, H.: 1929, ‘Consistency in Mathematics’, in Weyl (1968), vol. 3, pp. 150–70.

    Google Scholar 

  • Weyl, H.: 1940, ‘The Ghost of Modality’, in Weyl (1968), vol. 3, pp. 684–709.

    Google Scholar 

  • Weyl, H.: 1968,Gesammelte Abhandlungen, K. Chandrasekharan (ed.), Springer, Berlin.

    Google Scholar 

  • Whitehead, A. N. and B. Russell: 1925,Principia Mathematica, Volume 1, Cambridge, Cambridge University Press.

    Google Scholar 

  • Wittgenstein, L.: 1922,Tractatus Logico-Philosophicus, Routledge & Kegan Paul, London.

    Google Scholar 

  • Wittgenstein, L.: 1959, ‘Wittgenstein's Lectures in 1930–33’, in G. E. Moore:Philosophical Papers, Allen & Unwin, London, pp. 252–324.

    Google Scholar 

  • Wittgenstein, L.: 1965,Philosophical Remarks, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L.: 1974,Philosophical Grammar, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L.: 1978,Remarks on the Foundations of Mathematics, revised third edition, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L.: 1979a,Notebooks 1914–1916, revised second edition, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L.: 1979b,Ludwig Wittgenstein and the Vienna Circle, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L.: 1979c,Wittgenstein's Lectures, Cambridge 1932–1935, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L.: 1980a,Wittgenstein's Lectures, Cambridge 1930–1932, Blackwell, Oxford.

    Google Scholar 

  • Wittgenstein, L.: 1980b,Culture and Value, Blackwell, Oxford.

    Google Scholar 

  • Wright, C. J. G.: 1980,Wittgenstein on the Foundations of Mathematics, Duckworth, London.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, University of Ottawa, 65 University, P.O. Box 450 STN A, K1N 6N5, Ottawa, ON, Canada

    Mathieu Marion

Authors
  1. Mathieu Marion
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marion, M. Wittgenstein and finitism. Synthese 105, 141–176 (1995). https://doi.org/10.1007/BF01064216

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01064216

Keywords

Search

Navigation