On hearing this story, Alan’s reaction was wholehearted. One wet Sunday in February 1939 he cycled with Fred to the camp at Harwich. He had conceived the idea of sponsoring a boy who wanted to go to school and university. Most of the boys were only too glad to be free of school for good. Of the very few exceptions, one was Robert Augenfeld – ‘Bob’ from the moment of his arrival in England – who had decided when he was ten that he wanted to be a chemist. He came from a Viennese family of considerable distinction and his father, who had been an aide-de-camp in the First World War, had instructed him to insist he should continue with his education. He had no means of support in England, and Alan agreed to sponsor him. It was impractical, for Alan’s fellowship stipend would run to nothing of the kind, although he had probably saved some of Procter’s money. His father wrote asking ‘Is it wise, people will misunderstand?’ which annoyed Alan, although David Champernowne thought his father had a good point.
But the immediate practical problems were soon solved. Rossall, a public school on the Lancashire coast, had offered to take in a number of refugee boys without fee. Fred’s protégé Karl was going to take a place there, and this was arranged for Bob as well. Bob had to travel up north to be interviewed, where Rossall accepted him with the proviso that he should first improve his English at a preparatory school. On the way he had been looked after by the Friends in Manchester, and they in turn approached a rich, Methodist, mill-owning family to take him in. (Karl was fostered in just the same way.) This settled his future, and although Alan was ultimately responsible for him, and Bob always felt a great debt, he did not have to pay for more than some presents and school kit to help the boy get started. His recklessness had been justified, although it certainly helped that Bob was mentally as tough as Alan, having survived the loss of everything he knew, and being determined to fight for his own future education.
Meanwhile Alan was becoming more closely involved with the problems of GC and CS. At Christmas there was another training session at the headquarters in Broadway. Alan went down and stayed at an hotel in St James’s Square with Patrick Wilkinson, the slightly senior classics don at King’s, who had also been drawn in. Thereafter, every two or three weeks, he would make visits to help with the work. He found himself attached to Dillwyn Knox, the Senior Assistant, and to young Peter Twinn, a physics postgraduate from Oxford, who had joined as a new permanent Junior when a vacancy was advertised in February. Alan would be allowed to take back to King’s some of the work they were doing on the Enigma. He said he ‘sported his oak’ when he studied it, as well he might. It was wise of Denniston not to wait until hostilities opened before letting his reserve force see the problems. But they were getting nowhere. A general knowledge of the Enigma machine was not enough upon which to base an attack.
It would have amazed Mrs Turing, if she had known that her younger son was being entrusted with state secrets. Alan had by this time developed a skilful technique for dealing with his family, and his mother in particular. They all thought of him as devoid of common sense, and he in turn would rise to the role of absent-minded professor. ‘Brilliant but unsound’, that was Alan to his mother, who undertook to keep him in touch with all those important matters of appearance and manners, such as buying a new suit every year (which he never wore), Christmas presents, aunts’ birthdays, and getting his hair cut. She was particularly quick to note and comment on anything that smacked of lower-middle-class manners. Alan tolerated this at home, using his persona as the boy genius to advantage. He avoided confrontation – in the case of religious observance by singing Christmas hymns while he worked over Easter and vice versa, or by referring in conversation to ‘Our Lord’ with a perfectly straight face. He was not exactly telling lies, but successfully avoiding hurt by deception. This was not something he would do for anyone else, but for him, as for most people, the family was the last bastion of deceit.
There was, however, another side to the relationship: Mrs Turing did sense that he had done something incomprehensibly important, and was most impressed by the interest aroused in his work abroad. Once a letter came from Japan! For some reason she was particularly struck by the fact that Scholz was going to mention Alan’s work in the 1939 revision of the German Encyklopädie der mathematischen Wissenschaften.38 It needed such official-sounding reverberations for her to feel that anything had happened. Alan in turn was not above using his mother as a secretary; she sent out some of the reprints of Computable Numbers while he was in America. He also made an effort to explain mathematical logic and complex numbers to her – but with a complete lack of success.
It was in the spring of 1939 that he gave his first Cambridge lecture course. He started with fourteen Part III students, but ‘no doubt the attendance will drop off as the term advances,’ he wrote home. He must have kept at least one, for he had to set questions on his course for the examination in June. One of these asked for a proof of the result of Computable Numbers. It must have been very pleasing to be able to set as an examination problem in 1939, the question that Newman had posed as unanswered only four years before.
But at the same time, Alan joined Wittgenstein’s class on Foundations of Mathematics. Although this had the same title as Alan’s course, it was altogether different. The Turing course was one on the chess game of mathematical logic; extracting the neatest and tightest set of axioms from which to begin, making them flower according to the exact system of rules into the structures of mathematics, and discovering the technical limitations of that procedure. But Wittgenstein’s course was on the philosophy of mathematics; what mathematics really was.
Wittgenstein’s classes were unlike any others; for one thing, the members had to pledge themselves to attend every session. Alan broke the rules and received a verbal rap on the knuckles as a result: he missed the seventh class, very possibly because of his journey to the Clock House where, on 13 February, an entire side chapel of the parish church was dedicated to Christopher, on the ninth anniversary of his death. This particular course extended over thirty-one hours, twice a week for two terms. There were about fifteen in the class, Alister Watson among them, and each had to go first for a private interview with Wittgenstein in his austere Trinity room. These interviews were renowned for their long and impressive silences, for Wittgenstein despised the making of polite conversation to a far more thorough-going degree than did Alan. At Princeton, Alan had told Venable Martin of how Wittgenstein was ‘a very peculiar man’, for after they had talked about some logic, Wittgenstein had said that he would have to go into a nearby room to think over what had been said.
Sharing a brusque, outdoor, spartan, tie-less appearance (though Alan remained faithful to his sports jacket, in contrast to the leather jacket worn by the philosopher), they were rather alike in this intensity and seriousness. Neither one could be defined by official positions (Wittgenstein, then fifty, had just been appointed Professor of Philosophy in succession to G.E. Moore), for they were unique individuals, creating their own mental worlds. They were both interested only in fundamental questions, although they went in different directions. But Wittgenstein was much the more dramatic figure. Born into the Austrian equivalent of the Carnegies, he had given away a family fortune, spent years in village school-teaching, and lived alone for a year in a Norwegian hut. And even if Alan was a son of Empire, the Turing household had precious little in common with the Palais Wittgenstein.
Wittgenstein39 wanted to ask about the relationship of mathematics to ‘words of ordinary everyday language’. What, for instance, did the chess-like ‘proofs’ of pure mathematics, have to do with ‘proof’ as in ‘The proof of Lewy’s guilt is that he was at the scene of his crime with a pistol in his hand’? As Wittgenstein kept saying, the connection was never clear. Principia Mathematica only pushed the problem to another place: it still required people to agree on what it meant to have ‘a proof; it required people to agree what counting and recognising and symbols meant. When Hardy said that 317 was a prime because it was so, what did
this mean? Did it only mean that people would always agree if they did their sums right? How did they know what were the ‘right’ rules? Wittgenstein’s technique was to ask questions which brought words like proof, infinite, number, rule, into sentences about real life, and to show that they might make nonsense. As the only working mathematician in the class, Alan tended to be treated as responsible for everything that mathematicians ever said or did, and he rather nobly did his best to defend the abstract constructions of pure mathematics against Wittgenstein’s attack.
In particular, there was an extended argument between them about the whole structure of mathematical logic. Wittgenstein wanted to argue that the business of creating a watertight, automatic logical system had nothing to do with what was ordinarily meant by truth. He fastened upon the feature of any completely logical system, that a single contradiction, and a self-contradiction in particular, would allow the proof of any proposition:
WITTGENSTEIN: … Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone – much more extraordinary than you might think. … Because the thing works like this: if a man says ‘I am lying’ we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn’t matter. … it is just a useless language-game, and why should anybody be excited?
TURING: What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.
WITTGENSTEIN: Yes – and more: nothing has been done wrong. … where will the harm come?
TURING: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort.
WITTGENSTEIN: … The question is: Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc., outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics? Turing says, ‘Because something may go wrong with the application.’ But nothing need go wrong. And if something does go wrong – if the bridge breaks down – then your mistake was of the kind of using a wrong natural law. …
TURING: You cannot be confident about applying your calculus until you know that there is no hidden contradiction in it.
WITTGENSTEIN: There seems to me to be an enormous mistake there. … Suppose I convince Rhees of the paradox of the Liar, and he says, ‘I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 × 2 = 369.’ Well, we should not call this ‘multiplication’, that is all. …
TURING: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere.
WITTGENSTEIN: But nothing has ever gone wrong that way yet. …
But Alan would not be convinced. For any pure mathematician, it would remain the beauty of the subject, that argue as one might about its meaning, the system stood serene, self-consistent, self-contained. Dear love of mathematics! Safe, secure world in which nothing could gó wrong, no trouble arise, no bridges collapse! So different from the world of 1939.
He did not complete his research into the Skewes problem, which was left as an error-strewn manuscript40 and never taken up by him again. But he continued to pursue the more central problem, that of examining the zeroes of the Riemann zeta-function. The theoretical part, that of finding and justifying a new method of calculating the zeta-function, was finished at the beginning of March, and was submitted for publication.41 This left the computational part to be done. In this respect there had been a development. Malcolm MacPhail had written42 in connection with the electric multiplier:
How is your University fixed with storage batteries and lathes and so on which you can use for your machine? It’s such a pity that you will have to alter it. Hope you do not find it too much bunched together to be hard to work with. By the way if you are going to have time to work on it this fall and want some help don’t hesitate to ask my brother. I told him about the machine and how it worked. He’s very enthusiastic about your method of drawing wiring diagrams which rather surprised me. You know how conservative and old-fashioned engineers tend to be.
It so happened that his brother, Donald MacPhail, was a research student attached to King’s, studying mechanical engineering. The multiplier made no progress, but Donald MacPhail did now join in the zeta-function machine project.
Alan was not the only person to be thinking about mechanical computation in 1939. There were a number of ideas and initiatives, reflecting the growth of new electrical industries. Several projects were on hand in the United States. One of these was the ‘differential analyser’ that the American engineer Vannevar Bush had designed at the Massachusetts Institute of Technology in 1930. This could set up physical analogues of certain differential equations – the class of problem of most interest in physics and engineering. A similar machine had then been built by the British physicist D.R. Hartree out of Meccano components at Manchester University. This in turn had been followed by the commissioning of another differential analyser at Cambridge, where in 1937 the mathematical faculty had sanctioned a new Mathematical Laboratory to house it. One of Alan’s fellow ‘B-stars’ of 1934, the applied mathematician M.V. Wilkes, had been appointed as its junior member of staff.
Such a machine would have been useless for the zeta-function problem. Differential analysers could simulate only one special kind of mathematical system, and that only to a limited and very approximate extent. Similarly the Turing zeta-function machine would be entirely specific to the even more special problem on hand. It had no connection whatever with the Universal Turing Machine. It could hardly have been less universal. On 24 March he applied43 to the Royal Society for a grant to cover the cost of constructing it, and on their questionnaire wrote,
Apparatus would be of little permanent value. It could be added to for the purpose of carrying out similar calculations for a wider range of t* and might be used for some other investigations connected with the zeta-function. I cannot think of any applications that would not be connected with the zeta-function.
Hardy and Titchmarsh were quoted as referees for the application, which won the requested £40. The idea was that although the machine could not perform the required calculation exactly, it could locate the places where the zeta-function took a value near zero, which could then be tackled by a more exact hand computation. Alan reckoned it would reduce the amount of work by a factor of fifty. Perhaps as important, it would be a good deal more fun.
The Liverpool tide-predicting machine made use of a system of strings and pulleys to create an analogue of the mathematical problem of adding a series of waves. The length of the string, as it wrapped itself round the pulleys, would be measured to obtain the total sum required. They started with the same idea for the zeta-function summation, but then changed to a different design. In this, a system of meshing gear wheels would rotate to simulate the circular functions required. The addition was to be done not by measuring length, but weight. There would in fact be thirty wave-like terms to be added, each simulated by the rotation of one gear wheel. Thirty weights were to be attached to the corresponding wheels, at a distance from their centres, and then the moment of the weights would vary in a wave-like way as the wheels rotated. The summation would be performed by balancing the combined effect of the weights by a single counterweight.
The frequencies of the thirty waves required ran through the logarithms of the integers up to 30. To represent these irrational quantities by gear wheels they had to be approximated by fractions. Thus for instance the frequency determined by the logarithm of 3 was represented in the machine by gears giving a ratio of 34 × 31/57 × 35. This required four gear wheels, with 34, 31, 57 and 35 teeth respectively, to move each other so that one of them could act as the generator of t
he ‘wave’. Some of the wheels could be used two or three times over, so that about 80, rather than 120 gear wheels were required in all. These wheels were ingeniously arranged in meshing groups, and mounted on a central axis in such a way that the turning of a large handle would set them in simultaneous motion. The construction of the machine demanded a great deal of highly accurate gear-cutting to make this possible.
Donald MacPhail drew up a blueprint of the design,44 dated 17 July 1939. But Alan did not leave the engineering work to him. In fact his room, in the summer of 1939, was liable to be found with a sort of jigsaw puzzle of gear wheels across the floor. Kenneth Harrison, now a Fellow, was invited in for a drink and found it in this state. Alan tried and lamentably failed to explain what it was all for. It was certainly far from obvious that the motion of these wheels would say anything about the regularity with which the prime numbers thinned out, in their billions of billions out to infinity. Alan made a start on doing the actual gear-cutting, humping the blanks along to the engineering department in a rucksack, and spurning an offer of help from a research student. Champ lent a hand in grinding some of the wheels, which were kept in a suitcase in Alan’s room, much impressing Bob when he came down from his school at Hale in August.
Kenneth Harrison had been much amazed, for he well knew from conversations with Alan that a pure mathematician worked in a symbolic world and not with things. The machine seemed to be a contradiction. It was particularly remarkable in England, where there existed no tradition of high status academic engineering, as there was in France and Germany and (as with Vannevar Bush) in the United States. Such a foray into the practical world was liable to be met with patronising jokes within the academic world. For Alan Turing personally, the machine was a symptom of something that could not be answered by mathematics alone. He was working within the central problems of classical number theory, and making a contribution to it, but this was not enough. The Turing machine, and the ordinal logics, formalising the workings of the mind; Wittgenstein’s enquiries; the electric multiplier and now this concatenation of gear wheels – they all spoke of making some connection between the abstract and the physical. It was not science, not ‘applied mathematics’, but a sort of applied logic, something that had no name.
Alan Turing: The Enigma The Centenary Edition Page 26