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Mathematics as Grammar

  • Ole Ravn
  • Ole Skovsmose
Chapter
Part of the History of Mathematics Education book series (HME)

Abstract

This chapter addresses Wittgenstein’s interpretation of mathematics, although not in the form expressed in the Tractatus, but as it later become elaborated. Wittgenstein emphasises the profound social component in mathematical constructions. He interprets mathematics as basically a rule-following system, and he sees mathematical rules as being similar to grammatical rules. Rule-following can be interpreted as a social practice, or as a social construction.

As an alternative to the theory of language and mathematics as presented in Tractatus, the later Wittgenstein does not propound another theory of the same universal scope. In his opinion, no universal theory about language can be formulated. His central term in this clarification is “language game.” There are many different kind of games: football, handball, tennis, badminton, draughts, chess, bingo … But there is no essence underlying the fact that we call them games. Wittgenstein’s point is that it is a hopeless to try to provide a universal definition of “game.” And further: what can be said about “game” also applies to “mathematics.”. It is hopeless to try to answer the question: “What is mathematics?”

Keywords

Grammar Language game Mathematics as calculations Mathematics as measure Mathematics as rule-following Mathematics as social practice 

References

  1. Gefwert, C. (1998). Wittgenstein on mathematics, minds and mental machines. Aldershot, UK: Ashgate.Google Scholar
  2. Shanker, S. (1987). Wittgenstein and the turning-point in the philosophy of mathematics. London, UK: Croom Helm Ltd.Google Scholar
  3. Wittgenstein, L. (1974). Tractatus logico-philosophicus. London, UK: Routledge & Kegan Paul. (First published 1922).Google Scholar
  4. Wittgenstein, L. (1978). Remarks on the foundations of mathematics (3rd ed.). Oxford, UK: Basil Blackwell.Google Scholar
  5. Wittgenstein, L. (1997). Philosophical investigations. Oxford, UK: Blackwell Publishers. (First published 1953).Google Scholar
  6. Wrigley, M. (1986). Wittgenstein’s philosophy of mathematics. In S. Shanker (Ed.), Ludwig Wittgenstein: Critical assessments (Vol. III). London, UK: Croom Helm Ltd.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ole Ravn
    • 1
  • Ole Skovsmose
    • 1
    • 2
  1. 1.Department of Learning and PhilosophyAalborg UniversityAalborgDenmark
  2. 2.Department of Mathematics EducationState University of São Paulo, (Universidade Estadual Paulista, Unesp)São PauloBrazil

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