Mathematics as Grammar

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


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?”


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


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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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