Alan Turing: The Enigma (35 page)

Presumably
Alan had again applied for a lectureship, but if so he had again been disappointed. However, he had offered to the faculty a course for the spring term on Foundations of Mathematics. (Newman was not giving one this year.) This they accepted,
³⁷
awarding the rather nominal £10 fee, as was the custom for mathematically respectable, but not officially commissioned Part III lectures. He was also asked to assess the claims of Friedrich Waismann, the philosopher from the Vienna Circle, exiled in Britain and expelled for misbehaviour from Wittgenstein’s retinue, who wanted to lecture on Foundations of Arithmetic. So Alan had carved out a small niche for himself.

On 13 November 1938, Neville Chamberlain attended the Armistice Day service in the University Church, and a bishop gratifyingly referred to the ‘courage, insight and perseverance of the Prime Minister in his interviews with Herr Hitler that saved the peace of Europe six weeks ago.’ But some Cambridge opinion was more in touch with reality. In King’s, Professor Clapham chaired a committee for the reception of Jewish refugees allowed in by the government after the November wave of violence in Germany. These were events with a particular meaning for Alan’s friend Fred Clayton, who between 1935 and 1937 had spent time studying first in Vienna and then in Dresden, with experiences very different from the jolly hockey-sticks of Princeton.

They meant two very difficult and hurtful things. On the one hand, he was highly conscious of the implications of the Nazi regime. On the other, there were two boys, one the younger son of a Jewish widow living in the same house in Vienna, one at the school where he had taught in Dresden. The November 1938 events had put the Vienna family in great peril, and he received appeals for help from Frau S—. He tried to help her get her sons to England, and this was achieved just before Christmas by the Quakers’ Relief Action. They found themselves in a refugee camp on the coast at Harwich, and wrote to Fred, who soon made a visit. In the dank, freezing, slave-market atmosphere some other young refugees rendered some German and English songs, and the passage from Schiller’s
Don Karlos
about Elizabeth receiving those fleeing from the Netherlands. Fred was already very fond of Karl, an affection which fatherless Karl returned, and went away to help find someone to foster him.

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.

Wittgenstein
³⁹
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.