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Discovering America
Turing spent most of the years 1937-38 at Princeton University, based in the world-leading research group in mathematical logic headed by Alonzo Church.
Princeton University had become a great intellectual centre, absorbing exiles from a Nazifying Europe, and great future developments in mathematics, science and economics were started there.
In particular, Turing met John von Neumann there.
The oral history project of the mathematics department marks the importance of this period. The interview with Alonzo Church draws out rather little regarding Turing, whom Church calls 'a loner.' He was indeed, but Alan Turing had quite a good social life at the Graduate College, and also travelled between South Carolina and New England. He did not, however, find any way of satisfying his need for a sexual relationship. This was a period when he yearned for young men around him — but they were all unattainable.
The Cambridge mathematician G. H. Hardy, also gay, was a visitor while Turing was there. Hardy became famous for his view of 'pure' mathematics. In a piece of writing on Cambridge Scientific Minds, I have described more about the similarities and contrasts between Hardy and Turing in mathematics and life.
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This fine drawing by Jin Wicked
can be seen in full on her page. Read more in this article about how it subtly represents Alan Turing's life and ideas.
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Discovering the uncomputable
The work he did in 1937-8, his most difficult and most abstruse, charted new territory in trying to bring the uncomputable into some kind of order with 'systems of logics based on ordinals'.
His ordinal logics rested on the idea of the transfinite. Get a wonderful picture of the transfinite from Rudy Rucker's description.
This page by Barry Cooper describes modern research on 'Turing degrees' of uncomputable numbers.
Turing extended his machine formalism with abstract elements he called 'oracles.' Turing wrote:
We shall not go any further into the nature of this oracle apart from saying that it cannot be a machine.
The whole point of the 'oracle' is that it is a mathematical tool for the analysis of what cannot be done mechanically. Since the 1950s it has become a standard idea ('relative computability') of mathematical logic. But the philosopher B. J. Copeland has claimed that Turing would have supported a project to 'construct' machines equipped with oracles, and holds out the prospect of 'the biggest revolution in computing since 1948.' This is both historically and scientifically absurd. See a comment on Copeland's picture of an oracle and see this on-line lecture for more discussion.
Turing may have taken the word 'oracle' from Shaw's science-fiction play Back to Methuselah. Turing was a great fan of Bernard Shaw's plays, and saw a production of Back to Methuselah at Cambridge in 1933. The oracle is a fount of superhuman wisdom in the imaginary future of the year 31920.
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More mathematics, real and imaginary
Turing's work at Princeton also involved work on complex analysis and the Riemann zeta function. Turing hoped to locate a zero of the Riemann zeta-function off the critical line. A book, The Music of the Primes, by my Oxford colleague Marcus du Sautoy, discusses the history of the Riemann Hypothesis (Hilbert's Eighth Problem) and what Turing achieved.
Turing's setting in Princeton in this period of the growing Nazi threat has inspired a passage in the novel Cryptonomicon by the (science-)fiction writer Neal Stephenson. See this excerpt.
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Dr Turing
Turing's work on 'ordinal logics' was completed in 1938 and successfully submitted as a Ph.D. thesis. As a result, you can trace his 'genealogy' on the
Mathematics Genealogy Project. It runs back through doctoral supervision to Church, Veblen, E.H. Moore, H.A. Newton, Chasles, Poisson, and Lagrange. Stretching the definition a little, the line goes back to Euler, Johann Bernoulli, Jacob Bernoulli and Leibniz, who can be said to have started symbolic logic.
Alan Turing did not like using the title, but when he next returned to America, in very different circumstances, he found it essential. See this report where he made a joke of his title "Dr A.M. Turing, Ph.D."
Turing and von Neumann
Alan Turing could have stayed in America. He was offered a post-doctoral position by John von Neumann. This was nothing to do with computing or computers. It was because Turing had also done new research in the theory of continuous groups, which was von Neumann's leading research area.
It is unlikely that they ever discussed computing.
See source documents on What von Neumann knew of Turing in 1937-39.
See the Turing Digital Archive for a handwritten letter from von Neumann to Turing. (It is item 12 on this page. It is undated and an editor has written '1951?' on it, but the material concerns Turing's 1938 work.)
He could have been a pacifist exile in America. But he returned to Britain and to war.
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He also enjoyed the first Technicolor feature film in 1938. And he quoted: Dip the apple in the brew... |
Loss of Innocence, 1938-39
Turing's work in logic had already stimulated an interest in cryptography. He had also built actual physical machines.
At about the time of the Munich agreement he began helping the British government with the problem of deciphering German communications. From now on, the finite, practical, human logic of the Enigma cipher machine came to dominate his life. From now on he served the State.
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An Enigma rotor |
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No-one could have guessed where this would lead, not even
Ludwig Wittgenstein
with whom Turing argued about the philosophy of mathematics. See Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939 for a transcript.
Turing and Wittgenstein did not discuss the philosophy of Mind, then or later. Many people have wondered what they would have said to each other.
John Casti has written an imaginary conversation, The Cambridge Quintet, involving such a dialogue; see also a page of comment by Chris Mitchell.
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See his Tractatus on a single screen! |
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More exiles...
The real Alan Turing in late August 1939, sailing at Bosham, Sussex. Behind him is Fred Clayton, another young Fellow of King's College, Cambridge. Between them are two refugee boys, Robert and Karl from Austria. Alan Turing and Fred had helped them to find asylum in Britain.
While they were there the pact between Hitler and Stalin was signed and war became inevitable. |
War and Peace
The coming of war meant that Turing never again concentrated on mathematical logic, and he did not follow up the ideas he had in 1937-38. The war exiled him from the opportunity to develop his advanced mathematical ideas, when he was at the height of his powers. Something was lost. But something else was gained: a connection between the logical and the physical.
An article by me, The military use of Alan Turing, explores Alan Turing's personal and moral position, as well as his intellectual work, at the time when he decided to give himself to the struggle against Nazi Germany.
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