Wittgenstein, Turing and Gödel

In 1946, recalling his discussions with Turing in Cambridge before the war, Wittgenstein stressed that Turing’s ‘machines’ are really “humans who calculate” (RPP I 1096), alluding to the Halting Problem and Turing's idea of circle-free machines. Was this intended to embrace or to reject Turing’s model of human calculative activity? What form of anthropomorphism was (and is) at stake? The question becomes intriguing when we reflect that while Gödel held that it was Turing’s “precise and unquestionably adequate” definition of the notion of a formal system that allowed his own incompleteness theorems to be proved rigorously (in full generality) for the first time. Gödel also held that Turing made a “philosophical error” in holding that human mental procedures cannot go beyond mechanical procedures. I shall contrast the viewpoints of Wittgenstein, Gödel and Turing, emphasizing the logical systems that each one of them provided, explaining how these systems themselves embody distinctive philosophical conceptions of rigor, generality, and logic.