Famously, after obtaining his Ph.D. from Princeton University in June of 1938, Alan Turing (1912-54) turned down an offer from John von Neumann (1902-55) to come work with him as a postdoctoral assistant at the Institute for Advanced Study (Macrae, 1999). Rather than stay in America (see essay below), Turing instead returned to Cambridge, where he had been a fellow at King’s College since 1936.

For about a year, starting in September 1938, Turing worked part-time at the Government Code and Cypher School (GC&CS), engaging in research on the cryptanalysis of the German Enigma cipher machine. In his spare time, he attended various lectures at Cambridge, including a graduate seminar held by philosopher Ludwig Wittgenstein (1889-1951) on the foundations of mathematics.

Wittgenstein’s seminar, as he himself described it, regarded investigations into *“the role that logic plays in mathematics, or the relation supposed to hold between logic and mathematics”.* In particular, Wittgenstein was interested in interrogating mathematicians’ attitudes towards contradictions, an attitude of *“avoiding them at all costs”*, or as he put it, of *“a germ which shows general illness”* (Vanrie, 2024; Berg, 2021). Finding a contradiction in a system, Wittgenstein argued, is to mathematicians akin to *“finding a germ in an otherwise healthy body”* and arguing that this *“shows that the whole system or body is diseased”*. This view ran counter to his own views, which—in his later work—maintained that mathematics is a human invention and that contradictions more-so indicate flaws in this invention than truths about the absurdity of reality. Wittgenstein in his seminar, thus, sought to discredit ‘contradictory calculus’, a contraption he regarded as inherently defective.

Turing disagreed. His most celebrated result, entitled ‘*On Computable Numbers, with an Application to the Entscheidungsproblem’* employed proof by contradiction in showing that Hilbert’s “decision problem” did not have a solution—that some problems are indeed undecidable. The result had concrete implications for the development of the computer. Turing—in other words—was building a career on a result obtained using the very contraption that Wittgenstein was looking to suppress.

This essay is about the colliding of two worlds in one Cambridge classroom in 1939—the anti-Platonist late Wittgenstein and the realist views of Turing and, in turn, his hero Kurt Gödel (1906-1978).

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