Wittgenstein, Turing, and the ‘Finitude’ of Language

Abstract

I argue that Wittgenstein’s rule-following considerations and Turing’s 1936
demonstration of the insolubility of Hilbert’s decision problem have a common
root in thinking about the ability of finite signs to capture infinite
procedures or meanings. In the 1939 Lectures on the Foundations of Mathematics,
in direct response to Turing, Wittgenstein clearly rejects finitism and suggests
that the problem of the meaning of the infinite is to be addressed by
considering the finite symbolic “grammar” of infinite procedures and
capabilities. This bears similarities to Turing’s own application of the
“diagonal procedure,” which also captures, I argue, the problem of how infinite
procedures are represented by finite signs.