This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett 1959, p. 329). On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or false. Mathematical truths are thus not consequences of a prior adoption of a convention or rules as orthodox conventionalism has it. The goal of the paper is not merely exegetical, however, and argues that radical conventionalism is able to withstand some of the most difficult objections that have been brought forward against it, including those of Dummett himself, and thus that radical conventionalism has been prematurely excluded from consideration by philosophers of mathematics.

This paper develops a game-theoretic solution to the rule-following paradox whereby the meaning of a given term F is defined by the practice of using F. The game-theoretic machinery makes it clear how such a practice can define correctness conditions for indefinitely many cases, and thus rule out deviant interpretations.

This paper is a close reading of Wittgenstein's remarks on inconsistency in the Lectures on the Foundations of Mathematics, especially Wittgenstein's discussion with Alan Turing. I argue that Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of it. Such a practice may well continue without trouble, and if there is trouble, the contradiction is not the problem, as any contradiction is parasitic on a falsehood, the real culprit.