Not a difference of opinion: Wittgenstein and Turing on contradictions in mathematics

Abstract

In his 1939 Cambridge Lectures on the Foundations of Mathematics, Wittgenstein proclaims that he is not out to persuade anyone to change their opinions. I seek to further our understanding of this point by investigating an exchange between Wittgenstein and Turing on contradictions. In defending the claim that contradictory calculi are mathematically defective, Turing suggests that applying such a calculus would lead to disasters such as bridges falling down. In the ensuing discussion, it can seem as if Wittgenstein challenges Turing's claim that such disasters would occur. I argue that this is not what Wittgenstein is doing. Rather, he is scrutinizing the meaning and philosophical import of Turing's claim—showing how Turing is wavering between making an empirical prediction and a logical observation, and that it is only through this wavering that Turing can believe that he has provided a proper explanation of why contradictory calculi are mathematically defective.