Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game

by Andrew Hodges

  I think you know the history of Turing’s paper on Computable numbers. Just as it was reaching its final state an offprint arrived, from Alonzo Church of Princeton, of a paper anticipating Turing’s results to a large extent.

  I hope it will nevertheless be possible to publish the paper. The methods are to a large extent different, and the result is so important that different treatments of it should be of interest. The main result of both Turing and Church is that the Entscheidungsproblem on which Hilbert’s disciples have been working for a good many years – i.e. the problem of finding a mechanical way of deciding whether a given row of symbols is the enunciation of a theorem provable from the Hilbert axioms – is insoluble in its general form….

  Alan reported to his mother on 29 May:

  I have just got my main paper ready and sent in. I imagine it will appear in October or November. The situation with regard to the note for Comptes Rendus was not so good. It appears that the man I wrote to, and whom I asked to communicate the paper for me had gone to China, and moreover the letter seems to have been lost in the post, for a second letter reached his daughter.

  Meanwhile a paper has appeared in America, written by Alonzo Church, doing the same things in a different way. Mr Newman and I have decided however that the method is sufficiently different to warrant the publication of my paper too. Alonzo Church lives at Princeton so I have decided quite definitely about going there.

  He had applied for a Procter Fellowship. Princeton offered three of these, one in the gift of Cambridge, one of Oxford, one of the Collège de France. He was not to be successful, for the Cambridge one went that year to R. A. Lyttleton, the mathematician and astronomer. But he must have found that his King’s fellowship would provide just enough funds.

  Meanwhile, it was now necessary for the publication of the paper that he should include a demonstration that its definition of ‘computable’ – that is, as anything that could be computed by a Turing machine – was exactly equivalent to what Church had called ‘effectively calculable’, meaning that it could be described by a formula in the lambda-calculus. So he studied Church’s work from the papers which he and S. C. Kleene had produced in 1933 and 1935, and sketched out the required demonstration in an appendix to the paper which was finished on 28 August. The correspondence of ideas was quite straightforward, since Church had used a definition (that of a formula being ‘in normal form’) which corresponded to the Turing definition of ‘satisfactory’ machines, and had then used a Cantor diagonal argument to produce an unsolvable problem.

  If he had been a more conventional worker, he would not have attacked the Hilbert problem without having read up all of the available literature, including Church’s work. He then might not have been pre-empted – but then, he might never have created the new idea of the logical machine, with its simulation of ‘states of mind’, which not only closed the Hilbert problem but opened up quite new questions. It was the advantage and the disadvantage of working as what Newman called ‘a confirmed solitary’. Both with the Central Limit Theorem and with the Entscheidungsproblem, he had been the Captain Scott of mathematics, coming in a splendid second place. And while he was not the person to think of mathematics and science as a sort of competitive game, it was obviously a disappointment. It meant months of delay, and obscured the originality of his own attack. It confused his moment of coming out into the world.

  As for the Central Limit Theorem, his fellowship dissertation was entered for the Cambridge mathematical essay competition, the Smith’s Prize, that summer. This caused a flurry down at Guildford, where Mrs Turing and John spent a frantic half-hour on hands and knees doing up the parcel, which Alan had left until the last minute before sending off. John had married in August 1934 and Alan had by now become an uncle. Neither his brother, nor his parents, had the faintest inkling of the philosophical problems which underlay his work, or which underlay his life. News of Alan’s successes came as glowing reports from a higher and higher Sixth Form. Mrs Turing, with her interest in the spiritual world, would have been the most sensitive to Alan’s concern with free will, but even she never saw this fundamental connection. For Alan never expatiated on his inner problems, and only occasionally did rather cryptic hints of them emerge.

  The university, like King’s, took a charitable view of Alan’s rediscovery of the theorem, and it won him the prize and hence £31. By now he had taken up sailing as a holiday pastime, and thought of putting the money towards buying a boat. But he decided against it, perhaps needing it for his year in America.

  Victor Beuttell came to stay with Alan at Cambridge in the early summer. Alan was returning the hospitality that the Beuttells had offered him but another reason for Victor’s visit was that he had now joined the family firm and had been set to work on developing the K-ray system. The geometry that he had discussed with Alan at school had helped him, but he was hoping to have Alan’s advice on the new problem which was to make a double-sided system so that both sides of a poster could be illuminated evenly by a single light source. (It was required by a brewery chain.) Alan, however, said he was too preoccupied with his own work, and instead they went off to watch the May Bumps boat races.

  Once they were talking about art and sculpture, and it was in this connection that Alan suddenly amazed Victor by saying that he found the male form beautiful, and the female unattractive. Victor now found himself a double crusader, and tried to convince Alan that Jesus had indicated the right course by befriending Mary Magdalene. Alan had no answer to this, but then this was not a problem of reason. All he could do was to express the sensation of being in a Looking-Glass world, in which from his point of view the conventional ideas were the wrong way about. This was probably the first time that he opened the subject outside the King’s ambience.

  It was difficult for Victor, who was a not particularly mature twenty-one, to know how to react. An element of trust now came into his staying in Alan’s room, though Alan remained ‘a perfect gentleman’. But Victor did not reject Alan’s friendship. Instead, they continued to agree to disagree on this subject as they did on religion. They talked of what hereditary or environmental influences might determine it one way or the other. But whatever these were, it was clear that here was part of Alan that was so; that part of his reality was shaped that way. For him, without a God, there was nothing to appeal to but some inner consistency. As in mathematics, that consistency could not be proved by a rule-book, and there was no deus ex machina to hand down decisions as to right and wrong. The axioms of his life were becoming clear by now, although how to live them out was quite another question. He had wanted the commonest in nature; he liked ordinary things. But he found himself to be an ordinary English homosexual atheist mathematician. It would not be easy.

  Alan also paid a visit to the Clock House before going out west, the first for three years. Mrs Morcom was now semi-invalid, but still mentally as vigorous as ever. She noted in her diary:

  September 9 (Wednesday) … Alan Turing came … He has come for a farewell visit before going out to America for 9 months (Princetown) to study under 2 great authorities on his subject: Godel (Warsaw) Alonso Church and Kleene. We had talk before dinner and again later to bring us up to date with our news…. He and Edwin played billiards.

  September 10: … Alan and Veronica to farms and Dingleside …. V and Alan tea up here with me. Had long talk with Alan about his work and whether in his subject (some abstruse branch of logic) one would come to ‘dead end’ etc.

  September 11: Alan went down to church alone to see Chris’ window and the little garden which he hadn’t seen before since it was finished – only the day he came to the dedication of the window … Alan taught me game called ‘Go’ – rather like Peggity.

  September 12: … Rupert and Alan had tea in my room and then I took them all by surprise by coming down to dinner. There were 10 of us – a jolly party. Gramophone concert … Men billiards.

  September 13 … Alan did problems with R[eginald] … Alan Rup[ert]
and 2 girls bathed at Cadbury’s pool … Rup[ert] and Alan tea with me … Alan tried to explain what he is working at … they went off to catch 7.45 New Street.

  Alan lost Rupert when it came to the satisfactory and the unsatisfactory description numbers. It would have been hard for Mrs Morcom to feel that this ‘abstruse branch of logic’ had anything to do with the scientific imagination of her lost son, so that Alan had done something that Christopher had been called away from.

  Mrs Turing saw Alan off at Southampton on 23 September, when he embarked on the Cunard liner, the Berengaria. He had picked up a sextant in the Farringdon Road market to amuse himself on the voyage. He also went equipped with all the standard upper-middle-class British prejudices about America and Americans, and the five days on the Atlantic did little to disabuse him. From ‘41°20' N, 62°W’, he complained:2

  It strikes me that Americans can be the most insufferable and insensitive creatures you could wish. One of them has just been talking to me and telling me of all the worst aspects of America with evident pride. However they may not all be like that.

  The towers of the Manhattan skyline swam into view next morning, on 29 September, and Alan entered the New World:

  We were practically in New York at 11.00 a.m. on Tuesday but what with going through quarantine and passing the immigration officers we were not off the boat until 5.30 p.m. Passing the immigration officers involved waiting in a queue for over two hours with screaming children round me. Then, after getting through the customs I had to go through the ceremony of initiation to the U.S.A., consisting of being swindled by a taxi-driver. I considered his charge perfectly preposterous, but as I had already been charged more than double English prices for sending my luggage, I thought it was possibly right.

  Alan inherited his father’s belief that to take a taxi was the height of extravagance. But America, with its infinite variety, was not all ‘like that’, and Princeton, where he arrived late that evening on the train had little in common with the ‘mass of canaille’ of the cheapest Tourist Class. For if Cambridge embodied class, then Princeton spoke wealth. Perhaps of all the elite American universities, Princeton was the most self-contained, insulated from the squalor of the depression. One could look out and never know that America had a problem. In fact, it hardly looked like America at all, for with its mock Gothic architecture, its restriction to male students, its rowing on the artificial Carnegie Lake, Princeton tried to outdo the detachment of Oxford and Cambridge. It was the Emerald City in the land of Oz. And as if the isolation from ordinary America were not enough, the Graduate College was separated off from the undergraduate life, to stand upon its gentle prominence, overlooking a clean sweep of fields and woods. The tower of the Graduate College imitated that of Magdalen College Oxford, and it was popularly called the Ivory Tower, because of that benefactor of Princeton, the Procter who manufactured Ivory Soap.

  Mathematics at Princeton had been greatly augmented by the endowment of five million dollars for the foundation, in 1932, of the Institute for Advanced Study. Until 1940 the Institute had no separate building of its own. Those whom it funded, almost all mathematicians and theoretical physicists, shared the space of Fine Hall, home of the regular Princeton mathematical faculty. Although for technical purposes the distinction had to be drawn, in practice no one knew nor cared who was Princeton University and who was IAS. The doubled department had attracted some of the greatest names in world mathematics, and especially the exiles from Germany. It was in some ways an all-American foundation, in others like some immigrant ship still traversing the Atlantic. The richly funded Princeton fellowships also attracted research students of a world class, although more from England than from any other country. There were none from King’s, but Alan’s friend Maurice Pryce from Trinity was in residence for a second year. Here, amidst the huddled élite of the exiled European intelligentsia, lay the opportunity for Alan Turing to follow up his major result. His first report home, on 6 October, betrayed no lack of self-confidence.

  The mathematics department here comes fully up to expectations. There is a great number of the most distinguished mathematicians here. J. v. Neumann, Weyl, Courant, Hardy, Einstein, Lefschetz, as well as hosts of smaller fry. Unfortunately there are not nearly so many logic people here as last year. Church is here of course, but Gödel, Kleene, Rosser and Bernays who were here last year have left. I don’t think I mind very much missing any of these except Gödel. Kleene and Rosser are, I imagine, just disciples of Church and have not much to offer that I could not get from Church. Bernays is I think getting rather ‘vieux jeu’ that is the impression I get from his writing, but if I were to meet him I might get a different impression.

  Of these, Hardy was only visiting from Cambridge for a term.

  At first he was very standoffish or possibly shy. I met him in Maurice Pryce’s rooms the day I arrived, and he didn’t say a word to me. But he is getting much more friendly now.

  Hardy was something of a Turing of an earlier generation; he was another ordinary English homosexual atheist, who just happened to be one of the best mathematicians in the world. He was more fortunate than Alan in that his chief interest, the theory of numbers, fell cleanly within the classical framework of pure mathematics. He did not have Alan’s problem, of having to create his own subject. And his work was much more regular, more professional, than ever Alan’s was. But both were refugees from the system, for whom Keynesian Cambridge was the only possible home, although neither belonged to the more glamorous circles. Both were passive resisters, though Hardy was slightly less passive; he had been president of the Association of Scientific Workers out of principle, and had Lenin’s picture in his rooms. As the older man, his views were that much more firmly cast. Bertrand Russell once wittily distinguished catholic from protestant sceptics, according to the tradition they had rejected, and on this model Alan was, at this stage, more of a Church of England atheist. Hardy, however, played upon the English refusal to take ideas seriously, by becoming an atheist evangelical. At the same time, he found the pleasures of ritual in his devotion to the game of cricket. There was no one who knew more about it, although when in America he transferred his allegiance to baseball. He would organise cricket matches at Trinity, with Disbelief playing against Belief and the Almighty challenged to rain out the unbelievers. Hardy delighted in making a game out of anything, especially atheism.

  Alan would have attended his advanced lectures and classes at Cambridge, and therefore felt aggrieved at being ignored. Although ‘friendly’, the relationship was not one that overcame a generation and multiple layers of reserve. And if this was true of his acquaintanceship with Hardy, who saw the world through such very similar eyes, it was all the more so of Alan’s other professional contacts with elders. Although he was emerging as a figure of the serious academic world, he found it hard to shed the outlook and manners of an undergraduate.

  The list of names in Alan’s letter in itself meant little except that he might attend their lectures and seminars. Einstein would be seen occasionally in the corridors, but was almost incommunicado. S. Lefschetz was a pioneer in topology, which was at the centre of Princeton mathematics, and indeed a principal growth point of modern mathematics, but Alan’s personal contact with him was probably characterised by an occasion when Lefschetz questioned whether he would understand L. P. Eisenhart’s lecture course on Riemannian geometry, a question Alan considered insulting. Courant and Weyl, with von Neumann, covered the whole mainstream of pure and applied mathematics, bringing something of Hilbert’s Göttingen to life again on the western shore. But of them it was probably only von Neumann who had contact with Alan, through shared interests in group theory.

  As for the logicians, Gödel had returned to Czechoslovakia. Kleene and Rosser had made more substantial contributions to logic than Alan’s letter suggested, but had taken up positions elsewhere, and he would never meet either of them. The Swiss logician P. Bernays, a close associate of Hilbert, and another exile from Göttingen, h
ad returned to Zürich. Thus the impression that Alan had given to Mrs Morcom, of working with two or three major authorities, was incorrect. It was a matter of working with Church alone, except inasmuch as there were graduates studying logic on a lower level. And Church was a retiring man himself, not given to a great deal of discussion. In short, Princeton did not cure Alan of being a ‘confirmed solitary’. He wrote:

  I have seen Church two or three times and I get on with him very well. He seems quite pleased with my paper and thinks it will help him to carry out a programme of work he has in mind. I don’t know how much I shall have to do with this programme of his, as I am developping [sic] the thing in a slightly different direction, and shall probably start writing a paper on it in a month or two. After that I may write a book.

  Whatever these plans were, they did not come to fruition; there was no paper which fell into this description, nor a book.

  He conscientiously attended Church’s lectures, which were rather on the ponderous and laborious side. In particular, he took notes of Church’s theory of types, reflecting his continued interest in that aspect of mathematical logic. There were something like ten students present, including a younger American, Venable Martin, whom Alan befriended and helped with understanding the course. Alan remarked:

  The graduate students include a very large number who are working in mathematics and none of them mind talking shop. It is very different from Cambridge in that way.

  At Cambridge it was thought in very bad taste at High Table, or anywhere, for a person to speak only of his speciality. But this was not a feature of the English university that Princeton had imported along with the architecture. The English students, all from Oxford or Cambridge, would be amused at such American greetings as ‘Hi, pleased to meet you, what courses are you taking?’ English work was hidden under a decent show of well-bred amateurishness. This pretended negligence astonished the earnest devotees of the work ethic. But for Alan, who was excluded from the smarter circles of Cambridge society for his lack of sophistication, the more straightforward approach was an attraction. In that way America suited him – but not in other respects. To his mother, he wrote on