Wittgenstein, Turing, and "Surveyability"

Wittgenstein, Turing, and "Surveyability"

Juliet Floyd

Wittgenstein, Turing, and “Surveyability” Recent debates about the philosophical status of formalization and mechanization of proof may be illuminated by considering the mutual impact Hilbert, Wittgenstein and Turing had on one another around issues concerning the evolution of notations in symbolic logic. When Wittgenstein remarked in 1937 that ‘a proof must be surveyable’ he was reworking ideas of Frege, Hilbert and Turing. “Surveyability” for Wittgenstein was neither a verificationist requirement nor a refutation of the claim that all proofs must have corresponding formal proofs, much less a refutation of logicism. Instead, it places front and center what mathematicians do, i.e., it explores what logicism comes to in an everyday sense. The idea, consonant with certain trends in so-called “philosophy of mathematical practice”, is not to provide or ask for a “foundation” for mathematics in any ordinary sense, but rather to take a pragmatic approach to the very idea of “foundations”. For Wittgenstein, as for Frege and Hilbert, communicability is a central norm of proof, and the idea of “surveyability” furthers it. Turing’s everyday machine model of computation in his 1936 paper explicitly incorporates this ideal. I argue that this ideal is furthered in ordinary language philosophy, as exemplified by Austin. Wittgenstein’s path toward his greatest work, Philosophical Investigation,s shifted in response to Turing’s work on the Entscheidungsproblem: he concertedly developed the notions of mathematical “technique” and “form of life” beginning in 1937, going beyond the notions of “practice” and “language- game”. In 1939 he and Turing discussed these ideas in his Cambridge lectures on the foundations of mathematics, sparking some of Turing’s subsequent work on types. The relevant ideas here draw out new ways of looking at Turing’s 1936 paper, as well as his more speculative writings in the late 1940s about “intelligent machinery” and his 1950 “Turing Test”. These ideas are relevant for the place of language in our contemporary world, which is in many ways Turing’s world