Typically, Wittgenstein is assumed to have been apathetic to the developments in computability theory through the 1930s. Wittgenstein’s disparaging remarks about Gödel’s incompleteness theorems, and mathematical logic in general, are well documented. It seems safe to assume the same would apply for Turing’s work. The chief aim of this thesis is to debunk this picture. I will show that: a) Wittgenstein read, understood and engaged with Turing’s proofs regarding the Entscheidungsproblem. b) Wittgenstein’s remarks on this topic are highly perceptive and have pedagogical value, shedding light on Turing’s work. c) Wittgenstein was highly supportive of Turing’s work as it manifested Wittgenstein’s prevailing approach to mathematics. d) Adopting a Wittgensteinian approach to Turing’s proofs enables us to answer several live problems in the modern literature on computability. Wittgenstein was notably resistant to Cantor’s diagonal proof regarding uncountability, being a finitist and extreme anti-platonist. He was interested, however, in the diagonal method. He made several remarks attempting to adapt the method to work in purely intensional, rule-governed terms. These are unclear and unsuccessful. Turing’s famous diagonal application realised this pursuit. Turing’s application draws conclusions from the diagonal procedure without having to posit infinite extensions. Wittgenstein saw this, and made a series of interesting remarks to that effect. He subsequently gave his own (successful) intensional diagonal proof, abstracting from Turing’s. He endorsed Turing’s proof and reframed it in terms of games to highlight certain features of rules and rule-following. I then turn to the Church-Turing thesis (CTT). I show how Wittgenstein endorsed the CTT, particularly Turing’s rendition of it. Finally, I show how adopting a family-resemblance approach to computability can answer several questions regarding the epistemological status of the CTT today
