When Turing met Wittgenstein (1939)
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).