Alan Turing: The Enigma The Centenary Edition

by Andrew Hodges

  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 élite 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 was an exact replica of 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, had 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 gree
tings 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 14 October:

  Church had me out to dinner the other night. Considering that the guests were all university people I found the conversation rather disappointing. They seem, from what I can remember of it, to have discussed nothing but the different states they came from. Description of travel and places bores me intensely.

  He enjoyed the play of ideas, and in the same letter he let slip a hint of ideas in which Bernard Shaw himself might have found a plot:

  You have often asked me about possible applications of various branches of mathematics. I have just discovered a possible application of the kind of thing I am working on at present. It answers the question ‘What is the most general kind of code or cipher possible’, and at the same time (rather naturally) enables me to construct a lot of particular and interesting codes. One of them is pretty well impossible to decode without the key, and very quick to encode. I expect I could sell them to H.M. Government for quite a substantial sum, but am rather doubtful about the morality of such things. What do you think?

  Ciphering would be a very good example of a ‘definite method’ applied to symbols, something that could be done by a Turing machine. It would be essential to the nature of a cipher that the encipherer behave like a machine, in accordance with whatever rules had been fixed in advance with the receiver of the message.

  As for a ‘most general code or cipher possible’, in a sense any Turing machine could be regarded as encoding what it read on its tape, into what it wrote on the tape. However, to be useful there would have to be an inverse machine, which could reconstruct the original tape. His result, whatever it was, must have started on these lines. But as for the ‘particular and interesting codes’ he offered no further clue.

  Nor did he touch again on the conflict indicated by the word ‘morality’: what was he to do? Mrs Turing, of course, was a Stoney; she assumed that science existed for the sake of useful applications, and she was not the person to doubt the moral authority of His Majesty’s Government. But the intellectual tradition to which Alan belonged was quite different. It was not only for the detachment of Cambridge, but for a very significant section of modern mathematical opinion that G.H. Hardy spoke when he wrote:3

  The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholely ‘useless’ (and this is true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work. … The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill.

  In making explicit his response to the growing separation of mathematics from applied science, Hardy attacked the shallowness of the current ‘left-wing’ Lancelot Hogben interpretation of mathematics in terms of social and economic utility, an interpretation based on the ‘dull and elementary’ aspects of the subject. Hardy spoke more for himself, however, in holding that ‘useful’ mathematics had in any case worked more for ill than for good, being preponderantly military in application. He held the total uselessness of his own work in the theory of numbers to be a positive virtue, rather than a matter for apology:

  No-one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.

  Hardy’s own near-pacifist convictions stemmed from before the Great War, but no one touched by the Anti-War movements of the 1930s could fail to be unaware of a view that military applications were to be shunned. If Alan had now discovered something like a ‘warlike purpose’ in the play of symbols, he was faced, at least in embryo, with a mathematician’s dilemma. Behind the off-hand, teasing words to his mother, there lay a serious question.

  Meanwhile the English students were brightening the Graduate College life with amusements of their own:

  One of the Commonwealth Fellows, Francis Price (not to be confused with Maurice Pryce …) arranged a hockey match the other day between the Graduate College and Vassar, a women’s college (amer.)/university (engl.) some 130 miles away. He got up a team of which only half had ever played before. We had a couple of practice games and went to Vassar in cars on Sunday. It was raining slightly when we arrived, and what was our horror when we were told the ground was not fit for play. However, we persuaded them to let us play a pseudo-hockey match in their gymn. at wh[ich] we defeated them 11–3. Francis is trying to arrange a return match, which will certainly take place on a field.

  The amateurism was deceptive, since Shaun Wylie, the topologist, and Francis Price, the physicist, both from New College, Oxford, were players of a national standard. Alan was hardly in the same class (even if he was not now ‘watching the daisies grow’), but enjoyed the games. Soon they were playing three times a week amongst themselves, and sometimes against local girls’ schools.

  The effete English playing a women’s game might well have amazed the native Princeton students, but within the establishment there was a somewhat embarrassing anglophilia, in that all the most stuffy and mannered aspects of the English system were admired. In the summer of 1936, the Princeton chapel had been packed for a memorial service for George V. There was a professor in the Graduate College who harped upon his admiration for the royalty in a way that to educated English ears seemed only vulgar. As for George V’s successor, the revelations of Edward VIII’s Mediterranean cruise and Mrs Simpson created a particular sensation at Princeton. Alan wrote to his mother on 22 November:

  I am sending you some cuttings about Mrs Simpson as representative sample of what we get over here on this subject. I don’t suppose you have even heard of her, but some days it has been ‘front page stuff’ here.

  Indeed, the British newspapers maintained their silence until 1 December, when the bishop of Bradford remarked that the King stood in need of God’s grace, and Baldwin showed his hand. On 3 December, Alan wrote:

  I am horrified at the way people are trying to interfere with the King’s marriage. It may be that the King should not marry Mrs Simpson, but it is his private concern. I should tolerate no interference by bishops myself, and I don’t see that the King need either.

  But the King’s marriage was not a private matter, but one that reflected upon the British state. It was a prophetic episode for Alan, ‘horrified’ at government interference with an individual life. For his class, the horror was rather that the King himself had betrayed King and Country, a logical paradox more upsetting than any that Russell or Gödel had found.

  On 11 December the Windsors went into their butterfly life of exile, and the reign of George VI began. Alan wrote that day:

  I suppose this business of the King’s abdication has come as rather a shock to you. I gather practically nothing was known of Mrs Simpson in England till about ten days ago. I am rather divided in my opinion of the whole matter. At first I was wholly in favour of the King retaining the throne and marrying Mrs Simpson, and if this were the only issue it would still be my opinion. However I have heard tales recently which seem to alter it rather. It appears that the King was extremely lax about state documents, leaving them about and letting Mrs Simpson and friends see them. There had been distressing leakages. Also one or two other things of same character, but this is the one I mind about most. However, I respect David Windsor for his attitude.

  Alan
’s respect extended to the acquisition of a gramophone record of the abdication speech. He further wrote on 1 January:

  I am sorry that Edward VIII has been bounced into abdicating. I believe the Government wanted to get rid of him and found Mrs Simpson a good opportunity. Whether they were wise to try to get rid of him is another matter. I respect Edward for his courage. As for the Archbishop of Canterbury I consider his behaviour disgraceful. He waited until Edward was safely out of the way arid then unloaded a whole lot of quite uncalled-for abuse. He didn’t dare do it whilst Edward was King. Further he had no objections to the King having Mrs Simpson as a mistress, but marry her, that wouldn’t do at all. I don’t see how you can say that Edward was guilty of wasting his ministers’ time and wits at a critical moment. It was Baldwin who opened the subject.

  The archbishop’s broadcast, of 13 December, had denounced the King for abandoning his duties for a mere ‘craving for private happiness’; the pursuit of happiness had never been accorded a high priority by the British rulers. Alan’s views on marriage and morals were those of a modernist; in a discussion at King’s with his theological contemporary Christopher Stead he had said that people should let their natural feelings take their course – and as for bishops, a class of person particularly dear to Mrs Turing, they epitomised for him the ancien régime. He talked to Venable Martin, his American friend from Church’s logic class, about the ‘very shabby way’ in which the King had been treated.

  As for work, on 22 November he wrote to Philip Hall:

  I have not made any very startling discoveries over here, but I shall probably be publishing two or three small papers: just bits and pieces. One of them will be a proof of Hilbert’s inequality if it really turns out to be new, and another on groups which I did about a year ago and Baer thinks is worth publishing. I shall write these things up and then have another go at the Math[ematical] logic.