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  • Print publication year: 2016
  • Online publication date: March 2016

2 - The Forgotten Turing

from Part One - Inside Our Computable World, and the Mathematics of Universality


In fond memory of Robin Oliver Gandy: 1919–1995


Alan Turing is remembered for many things. He is widely known as code breaker and cryptographer, and as – at least in some sense – inventor of the computer. He is if anything even more famous as the father of artificial intelligence. Beyond that he was a mathematician, mathematical logician and pioneer in the study of morphogenesis.

Logicians remember Turing for his celebrated Entscheidungsproblem paper (Turing, 1937a), for work on the λ -calculus (Turing, 1937b) and, if they are cognoscenti, for his second great paper (Turing, 1939) in the Proceedings of the London Mathematical Society. All that work was completed by 1940. It is not widely appreciated that Turing's interest in logic continued to the end of his life. His later interest in the theory of types, a central area in the foundations of mathematics, is largely forgotten. That interest has had more influence than is evident. I am going to tell the story: it is an odd one.

The one and only student

My D.Phil. supervisor at Oxford was Robin Gandy: he was the only person to take a doctorate under Turing's supervision, and also one of his closest friends.

While Turing had only one student, Gandy had many. I was one of the the cohort from the early 1970s. Gandy rather liked the thought that his students were intellectual grandchildren of Turing. We mostly cared rather less about that, but from Gandy's reminiscences we all caught a glimpse of Turing through the eyes of someone who had known him very well. Turing met Gandy in 1940 at a party in King's College, Cambridge. Gandy was a third year student, while Turing, who had intermitted his Fellowship of the College at the start of the war, was already engaged in his celebrated work at Bletchley Park. During the war, Gandy became a friend and then much later Turing's student. By the end of Turing's life Gandy had begun his academic career as a Lecturer in Applied Mathematics at Leicester. You can read about their unique relationship in the Turing biography by Andrew Hodges (1983). Here I shall focus on Turing's influence on Gandy as his Ph.D. supervisor.

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A., Church (1940). A formulation of the simple theory of types.Journal of Symbolic Logic 5, 56–68.
R.O., Gandy (1953). On axiomatic systems in mathematics and theories of physics. Ph.D. dissertation, University of Cambridge.
R.O., Gandy (1956). On the axiom of extensionality – Part I.Journal of Symbolic Logic 21, 36–48.
R.O., Gandy (1959). On the axiom of extensionality – Part II.Journal of Symbolic Logic 24, 287–300.
R.O., Gandy (1980). Church's Thesis and principles for mechanisms. In The Kleene Symposium, J., Barwise, H.J., Keisler, and K., Kunen (editors), pp.123–148, North-Holland.
A.P., Hodges (1980). Alan Turing. The Enigma of Intelligence, Burnett Books. See also the website
S.C., Kleene (1959). Countable functionals. In Constructivity in Mathematics, A., Heyting (editor). North-Holland.
G., Kreisel (1959). Interpretation of analysis by means of functionals of finite type. In Constructivity in Mathematics, A., Heyting (editor). North-Holland.
Y., Moschovakis and M., Yates (1996). In memoriam: Robin Oliver Gandy, 1919–1995.Bulletin of Symbolic Logic 2, 367–370.
M.H.A., Newman and A.M., Turing (1942). A formal theorem in Church's theory of types.Journal of Symbolic Logic 7, 28–33.
D.S., Scott (1962). More on the axiom of extensionality. In Essays on the Foundations of Mathematics, Dedicated to A.A. Fraenkel on his Seventieth Anniversary, Y., Bar- Hillel, E.I.J., Poznanski, M.O., Rabin and A., Robinson (editors). North-Holland, 115–131.
A.M., Turing (1937). On computable numbers, with an application to the Entscheidungsproblem.Proceedings of the London Mathematical Society 42(2), 230–265; A Correction. ibid. 43, 544–546 (1938).
A.M., Turing (1937a). Computability and λ-definability.Journal of Symbolic Logic 2, 15-163.
A.M., Turing (1939). Systems of logic based on ordinals.Proceedings of the London Mathematical Society 45(2), 161–228.
A.M., Turing (1942). The use of dots as brackets in Church's system.Journal of Symbolic Logic 7, 146–156.
A.M., Turing (1948). Practical forms of type theory.Journal of Symbolic Logic 13, 80–94.