Mathematics as Grammar

Tésis de Doctorado en Filosofía, Universidad de Indiana, Bloomington, 2000

This thesis makes sense of ‘grammar’s role in Wittgenstein's philosophy of mathematics during the early thirties. It constructs a formal model of Wittgenstein’s notion of grammar as expressed in his writings of that period, justifies the appropriateness of that model and then employs it to test Wittgenstein's claim that mathematical propositions are ultimately grammatical. Chapter 1 frames the dissertation’s topic in its historical and conceptual context. Chapter 2 traces the origins of Wittgenstein’s grammatical approach to mathematics in Frege’s philosophy of arithmetics. Chapter 3 explains the central role of calculation in Wittgenstein’s philosophy of mathematics. Chapter 4 presents Wittgenstein’s account of mathematical application [Anwendung]. Chapter 5 provides a formalized theory of grammatical analysis, and Chapter 6 applies it to a portion of language containing numerical expressions. It proves that if the object language contains the appropriate numerical expressions, the resulting grammar includes at least some rules which may be naturally interpreted as mathematical. In particular, it shows that Wittgenstein’s grammatical analysis of the ordinary use of numerical expressions yields familiar theorems of arithmetic.Chapter 7 rounds up Wittgenstein’s account of arithmetics in light of the formal results of the previous chapters. Finally, Chapter 8 explains Wittgenstein’s account of mathematical necessity as a special case of grammatical necessity.