ABSTRACT. I consider the sense in which language is “finite” for Wittgenstein, and also some of the implications of this question for Alan Turing’s definition of the basic architecture of a universal computing machine, as well as some of the vast technological, social, and political consequences that have followed from it. I shall argue that similar considerations about the relationship between finitude and infinity in symbolism play a decisive role in two of these thinkers’ most important results, the “rule-following considerations” for Wittgenstein and the proof of the insolubility of Hilbert’s decision problem for Turing. Fortuitously, there is a recorded historical en- counter between Wittgenstein and Turing, for Turing participated in Wittgenstein’s “lectures” on the foundations of mathematics in Cambridge in 1939; their interactions are documented in the text Wittgenstein’s Lectures on the Foundations of Mathematics edited by Cora Diamond. Although my aim here is not to adduce biographical details, I think their exchange nevertheless evinces a deep and interesting problem of concern to both. We may put this problem as that of the relationship of language’s finite symbolic corpus to (what may seem to be) the infinity of its meaning. (pp. 215–247)

Keywords: Wittgenstein, Turing, language, philosophy, mathematics, symbolism