Prof

by Dermot Turing

  I am told over here that the postage to America from England is only 11/2d although it is 5 back, so suggest you ask at P.O.

  Yours

  Alan

  Letters home seem to have been more frequent during Alan’s time at Princeton. Those that survive are dated more evenly throughout the week, rather than turned out as Sunday duty; certainly more were preserved (27 over two years, compared with 18 for the previous four), and they are longer. It gives a slightly clearer picture of what was happening, though Alan’s choice of subject, as always, tended towards the technical and neutral, but perhaps the frequency of letters implies willingness to keep the link to home alive. At Mother’s suggestion, Alan kept pace with political and other developments in England and Europe by reading the New Statesman and Nation (sent over each week by Mother), which was then fizzing over the abdication crisis. The paper was not completely devoted to politics, though, and had a puzzles section at the back, including a weekly logic challenge. On 19 December 1936, Problem No. 207 appeared, set by Sir Arthur Eddington, whose lectures had set Alan off on his fellowship dissertation on the Gaussian Error Function. Problem No. 207, ‘Looking Glass Zoo’, was set in the language of Lewis Carroll, and featured a trip by some boys and girls to the zoo, who had to identify certain animals, and say which were male and female, from an array of apparently nonsensical information. Various solutions were offered in subsequent editions, including one by Alan’s mentor M.H.A. Newman, but the prize-winning entry was by ‘Champ’, whom Alan readily identified as his friend David Champernowne. Conveniently the printed solution came with the Champ’s address, so Alan was able to re-establish contact. Champ’s solution reads, in part:

  Humpty Dumpty explains. ‘There couldn’t have been more than three girls,’ reflected Humpty Dumpty, ‘because a girl is always a square root of minus one, and there are only twelve of those, they taught us that at school.’ … ‘I suppose so,’ said I, ‘but what do you mean by saying “a girl is the square root of minus one”?’ ‘When a girl thinks twice, she thinks contrariwise, but when a boy thinks twice he thinks Truth.’

  Square root of minus one? Because the problem could be solved using the calculus of complex numbers, which have a ‘real’ part (a normal number like 5 or π) plus an ‘imaginary’ part which is a multiple of √(-1). It seems likely that the ironic and topsy-turvy logic of Alice would have been lost on Alonzo Church. Nor was there much prospect of sharing this sort of fun with the great brains at the Institute for Advanced Study. The Institute was a very new foundation within Princeton, a sort of university within a university, with a Carrollian logic of its own. In 1936 it was still housed in Fine Hall alongside the university’s maths faculty; it didn’t get its own site until later. The IAS wanted to attract mathematical, and in particular Jewish mathematical, talent from an increasingly hostile Europe. There were large salaries on offer ($16,000 for senior professors), causing some cynics to say the acronym stood for Institute for Advanced Salaries. The idea was that, freed from the burdens of teaching, the world’s finest intellects would stand around Institute blackboards and argue together, producing the world’s finest ideas. In fact, many stood around and intrigued, and looked for permanent postings at other American universities. New arrivals could find the place isolated and lonely. Excepting John von Neumann, the mathematicians did not, unlike in central Europe, talk to each other much, and when they did it was not about maths. The intrigues along the corridors were not a problem for non-Institute university people like Alonzo Church and Alan Turing, who could comfortably get on with their work alone, but the bubbling pot of excitement that Alan had hoped for had in fact gone rather tepid.

  Perhaps therefore it is not surprising that Alan’s presentation of his paper on Computable Numbers, on 2 December 1936, attracted only ‘rather bad attendance’, which Alan put down to his lack of reputation: a professor visiting from elsewhere had a good turnout, despite a humdrum presentation, maybe because the audience were trying to catch his eye. Alan was still grumbling about the reception given to his paper, or rather the lack of it, in February. At that stage, there was nobody pushing Alan forward at Princeton. Nevertheless, Alan learned to speak and write in lambda-calculus, turning out a number of papers. He also started working on ‘group theory’, which was John von Neumann’s idea. Catching the notice of von Neumann was important, not just because von Neumann was by many accounts the most brilliant mathematician of the twentieth century, but also because he was urbane and charming and could influence funding for a second year in Princeton, if Alan wanted it.

  GRADUATE COLLEGE

  PRINCETON UNIVERSITY

  22 Feb

  My dear Mother,

  I went to the Eisenharts regular Sunday tea yesterday and there they took me in relays to try and persuade me to stay another year. The Dean weighed in with hints that the Procter Fellowship was mine for the asking (this is worth $2,000 p.a.). I said I thought King’s would probably prefer that I return, but gave vague promise that I would sound them on the matter. Whether I want to stay is another matter. The people I know here will all be leaving, and I don’t much care about the idea of spending a long summer in this country. I should like to know if you have any opinions on the subject. I think it is most likely I shall come back to England. […]

  Yours

  Alan

  There were three Procter fellowships each year: one each for Oxford, Cambridge, and the Collège de France. Alan had missed out on the Cambridge one for 1936–37 and had subsisted on his King’s fellowship stipend, and abstinence from taxis. Alan applied for a lectureship at Cambridge, but by the spring it was clear it wasn’t going to be given; a further year at Princeton, with the agreement of King’s, seemed a sensible plan. Von Neumann wrote supporting the application, and so it was settled.

  Alan took a short American holiday before going back to England for the summer. Terrifyingly, this required mastery of a motor car. Maurice Pryce, a friend of Alan’s from Cambridge (who had got a lectureship, and so wasn’t coming back), sold Alan the car and took on the job of driving instructor. The holiday included a 400-mile round-trip along the coast to visit a cousin of Ethel Turing on her mother’s side: a retired clergyman called Jack Crawford. Cousin Jack rated alongside the much-loved Aunt Sibyl for what Alan called the Relations Merit Diploma: he had studied science in Dublin, had a telescope in a little observatory, and talked about the grinding of mirrors. Fortunately for all concerned, Maurice Pryce came too. Alan reported to his mother rather unconvincingly, ‘I am getting rather more competent with the car so should get up there without serious difficulty’, but this was a full five months later, when he was planning a second visit to the Crawfords for Thanksgiving.

  Alan arrived back in Cambridge for the summer at the end of June 1937. Here there were some new and exciting things. The moral scientists were keen to introduce Alan to Ludwig Wittgenstein, who had been in and out of Cambridge for the previous 20 years or so. Wittgenstein did not attend meetings of the Moral Sciences Club, allegedly because of complaints that he talked so much that nobody else could get a word in edgeways. Wittgenstein had an engineering background and was a logician, and a meeting of minds between Turing and Wittgenstein seemed a good idea.

  Alan was also getting interested in a pure mathematical problem linked to the spacing-out of prime numbers, called the Riemann Hypothesis. Alan took delivery of a bundle of papers on the subject, which he began swotting up. Proving the Riemann Hypothesis was in the list of 23 vital unsolved problems in mathematics put forward by David Hilbert in 1900, and it is still regarded today as the most famous unsolved problem in mathematics. Alan Turing had already had a go at another of Hilbert’s designs, and the Riemann problem had been around since 1859. Alan had been thinking about it since, as Alan put it in a letter to Stanley Skewes (another mathematician from King’s): ‘you made the mistake of talking to me about it from time to time when you were rowing two and I at bow until eventually I thought I had better find out what it was all a
bout’.

  What it was all about was the square root of minus one. The Riemann zeta-function is a sum of an infinite series. For any complex number s:

  And the interesting question is, for what values of s the sum of the series adds up to exactly zero? Riemann’s hypothesis is that whenever the sum is zero, the ‘real’ part of the complex number s is 1/2. The Riemann Hypothesis was particularly interesting in 1937, because a Cambridge mathematician called E.C. Titchmarsh had used a mechanical desk calculator to crank through no fewer than 1,041 points and confirmed that they did satisfy the Riemann Hypothesis. To do maths this way was heretical: rather than prove or disprove Riemann by classical analysis, Titchmarsh had been searching for a counter-example by brute-force computing. He had failed, which made the puritans smirk; but Alan Turing wasn’t a puritan, and machine methodology was something which always appealed to him.

  Soon, though, it was time to go back to Princeton, where there was still more to do. Princeton was suiting Alan rather better in his second year there – ‘there is only one feature of American life which I find really tiresome, the impossibility of getting a bath in the ordinary sense’. Alan’s mentor M.H.A. Newman was in Princeton as well, having joined the Institute for a year. Alan also made friends with a Canadian physicist called Malcolm MacPhail, who noted that:

  Turing actually designed an electric multiplier and built the first three or four stages to see if it could be made to work. For the purpose he needed relay-operated switches which, not being commercially available at that time, he built himself. The Physics Department at Princeton had a small but well equipped machine shop for its graduate students to use, and my small contribution to the project was to lend Turing my key to the shop, which was probably against all the regulations, and show him how to use the lathe, drill, press etc. without chopping off his fingers. And so, he machined and wound the relays; and to our surprise and delight the calculator worked.

  For some reason Alan had also been prompted to think again about codes and ciphers. Malcolm MacPhail again:

  It was probably in the fall of 1937 that Turing first became alarmed about a possible war with Germany. He was at that time supposedly working hard on his famous thesis but nevertheless found time to take up the subject of cryptanalysis with characteristic vigour. On this topic we had many discussions. He assumed that words would be replaced by numbers taken from an official code book and messages would be transmitted as numbers in the binary scale. But, to prevent the enemy from deciphering captured messages even if they had the code book, he would multiply the number corresponding to a specific message by a horrendously long but secret number and transmit the product. The length of the secret number was to be determined by the requirement that it should take 100 Germans working eight hours a day on desk calculators 100 years to discover the secret factor by routine search!

  Alan relaxing with Malcolm MacPhail, his accomplice in building an electric multiplier, and Venable Martin, another friend, in Dean Eisenhart’s garden in 1938.

  Church-Turing Thesis

  The main task, however, was to write a thesis for a Ph.D degree. This was going to be on the subject of ‘ordinal logics’, a subject in which John von Neumann had dabbled as a high-school student (and, being von Neumann, on which he had published a precocious paper). Ordinals are a bit like Russian dolls, each one a bit bigger, and including all the preceding ones. In an extension of his work on the Entscheidungsproblem, Alan was applying the concept to systems of logic, each one wider and more comprehensive than the last, to explore the boundaries of what might be formally provable, and what might not.

  Initially Alan thought the thesis would be done by Christmas 1937, but it had an unhappy gestation, not made easier when Professor Church reviewed the draft. Many years later, Alonzo Church was interviewed by William Aspray about his graduate students at Princeton. Initially, Church didn’t include Turing in his list of students whom he remembered – this may be because Turing was a rather unusual ‘student’, given that he was really more in the nature of a visiting professor, albeit one working towards a Ph.D.

  ASPRAY: Did you have much contact with him [Alan Turing] while he was writing his paper?

  CHURCH: I had a lot of contact with him. I discussed his dissertation with him rather carefully.

  ASPRAY: Can you tell me something about his personality?

  CHURCH: I did not have enough contact with him to know. He had the reputation of being a loner and rather odd.

  On 30 March 1938 Alan wrote to a fellow King’s mathematics don, Philip Hall: ‘I am writing a thesis for a Ph.D, which is proving rather intractable, and I am always rewriting parts of it.’ Seven weeks later, things were not much better, telling Mother: ‘my Ph.D. thesis has been delayed a good deal more than I had expected. Church made a number of suggestions which resulted in the thesis being expanded to an appalling length. I hope the length of it won’t make it difficult to get it published.’ Alan also had problems with the typist, who evidently struggled with Alan’s curious Gothic German notations and the horrors of the lambda-calculus. All the rewrites and retypes didn’t make the thesis into a straightforward, accessible piece like Computable Numbers. Some years later, Dr Robin Gandy wrote to M.H.A. Newman about it. Gandy said: ‘Alan considered that his paper on ordinal logics had never received the attention it deserved (He wouldn’t admit that it was a stinker to read).’ Occasionally Alan conceded that symbolic logic could be heavy-going:

  It has long been recognised that mathematics and logic are virtually the same and that they may be expected to merge imperceptibly into one another. Actually this merging process has not gone at all far, and mathematics has profited very little from researches in symbolic logic. Symbolic logic is a very alarming mouthful for most mathematicians, and the logicians are not very much interested in making it more palatable.

  If the mathematicians themselves found this line of work hard to digest, heaven help the rest of us. Yet, holding its head clear of the obscurities, the thesis did contain ‘a new idea that was to change the face of the general theory of computation’. This was the concept of the ‘oracle’, a source of wisdom to which you could go for a solution to the mathematically unsolvable. Human beings depend on their instincts and intuition when they know something is right but cannot prove it logically – Alan, always rooted in the real world, reintroduced this piece of common sense into the arid world of symbolic logic. Without going into it very far in his thesis, Alan’s invention of the oracle showed that there were different degrees of unsolvability, a field of enquiry which proved fruitful for a later generation of mathematical logicians.

  Meanwhile Alan was fretting about whether his fellowship at King’s would be renewed. The situation was complicated. Usually King’s would diligently inform the applicants, but Alan was not in residence and was without access to the high-table grapevine. Gingerly he went to see the Dean at Princeton to ask about a possible job in America, and within a few days John von Neumann had invited Alan to join the Institute for Advanced Study (with a salary of $1,500). So there was a job for Alan either way, but Alan’s preference was to return to Britain, ‘unless you are actually at war before July’. To find out for certain, Alan indulged in another unthinkable extravagance, and sent a cable to King’s. King’s had in fact renewed the fellowship – they just hadn’t told Alan – and so he declined von Neumann’s offer. Eventually, Alan’s thesis was accepted on 17 May 1938 and at the end of the month he was examined on it, receiving the doctorate in June. Alan sent a copy in the form he intended it to be published to Philip Hall, saying, ‘I also expect to find the back lawn criss-crossed with 8 ft trenches’.

  There were no trenches yet across the back lawn when Alan returned to King’s in the summer of 1938, but Hitler’s ‘year of no surprises’ had come to an end in December 1937. In 1938 he launched his major programme of expanding the Reich. Anschluss with Austria occurred on 11–12 March. The Munich crisis developed in late September, and in November there occurred the no
torious Kristallnacht, a night of state-sponsored vandalism of all things Jewish. Despite Chamberlain’s ‘peace with honour’ window-dressing there were tasks to be done. Alan’s friend Fred Clayton, who occupied rooms just below him, had helped a Viennese Jewish boy called Karl come to Britain, one of the small number grudgingly allowed in by the British government after the Kristallnacht. Karl was languishing in a refugee camp in Harwich. Alan went with Fred to visit the camp and this led to Alan sponsoring another refugee boy, Robert Augenfeld, soon anglicised as ‘Bob’, helping him to get settled at a school and with a foster-family. And there was preparation for the war which the dons at King’s knew was unavoidable. Shortly after his return to England in July 1938, Alan had been visited by one of those senior dons.

  Discomfort before the storm. August 1939: Fred Clayton; boatman; Fred’s protégé Karl; and Alan squeezed into the stern with Bob Augenfeld.

  One day in the summer of 1938, after the Nazis had taken over Austria, I was sitting in my rooms at King’s when there was a knock on the door. In came F.E. Adcock, accompanied by a small, birdlike man with bright blue eyes whom he introduced as Commander Denniston. He asked whether, in the event of war, I would be willing to do confidential work for the Foreign Office. It sounded interesting, and I said I would. I was thereupon asked to sign the Official Secrets Act form. By now I had guessed what it was all about. It was well known to us that Adcock had been a member of Admiral ‘Blinker’ Hall’s Room 40 at the Admiralty in the First World War, which had done pioneering work on the decoding of enemy messages.

  Alas, this isn’t Alan’s account of his recruitment, but that of L.P. Wilkinson, who was a King’s don at the same time. It is tempting to infer, from Alan’s conversations about codebreaking with Malcolm McPhail in Princeton the previous year, that he had already been approached informally by Adcock in the summer of 1937. There is nothing in the files to verify this; what is certain is that in 1938 Denniston was doing his rounds, and Alan signed up formally then. Together with Wilkinson, and another fellow of King’s, D.W. Lucas, he attended a cryptology course at the Government Code & Cypher School, then housed in London near St James’s Park underground station. This included an introduction to the fiendish German encipherment machine called Enigma. At some point he was also introduced to A.D. (Dilly) Knox, the veteran codebreaker from Great War days, who was trying to break it. During the coming months, Alan was allowed to take secret work back to King’s, to pit his wits against Enigma, subject always to taking appropriate security precautions. At Cambridge, the older college rooms have two doors, an inner and an outer one: when the outer one (the ‘oak’) is shut, the occupant is out, or not to be disturbed; if the outer one is open, you can knock on the inner, painted door for admittance. Security for Alan seems to have consisted of ‘sporting his oak’; not that any casual spies would have found it easy to decrypt the chaos and clutter in his rooms. For us, Alan’s codebreaking activities can wait for another chapter. He had plenty else going on.