Mathematical Sense: Wittgenstein’s Syntactical Structuralism

Abstract

On Wittgenstein’s purely syntactical, radical constructivist account of mathematics, the sense (meaning) of a mathematical ‘proposition’ is not at all like the sense of a contingent (empirical) proposition. In the latter case, a contingent proposition has a fully determinate sense as a function of linguistic conventions, which make it possible to fully understand the sense of a contingent proposition without knowing its truth-value. In the mathematical case, however, a so-called “mathematical proposition” only has sense when we have proved it (with understanding) and thereby located it within the syntactical structure of a mathematical calculus. This paper aims to (1) show how Wittgenstein’s radical position on a mathematical proposition and its sense results from his life-long view that mathematics is exclusively syntactical and invented bit-by-little-bit by human beings, (2) propose a particular conception of “mathematical sense” – and an interpretation of Wittgenstein’s remarks – that best resolves the internal tension between two of Wittgenstein’s principal views on mathematics, and (3) consider some objections to Wittgenstein’s view and how he does or might respond to them. The paper shows that, on Wittgenstein’s account, the sense of a mathematical proposition is its syntactical location and its syntactical connections within a purely syntactical calculus.