To English minds, the wonder was that any state or party could interest itself in abstract ideas.
Meanwhile to the New Statesman, Hitler’s rancour at the Treaty of Versailles only vindicated what Keynes and Lowes Dickinson had always said. The difficulty was that being fair to Germany now meant making concessions to a barbarous regime. Conservative opinion, however, perceived the new Germany in terms of a balance of nation states, in which it was a renewed potential threat to Britain, but also a strong ‘bulwark’ against the Soviet Union. It was in this context that the Cambridge Anti-War movement revived in November 1933. Alan wrote:
12/11/33
There has been a lot happening this week. The Tivoli Cinema had arranged to shew a film called ‘Our Fighting Navy’ which was blatant militarist propaganda. The Anti-War movement organized a protest. The organization wasn’t very good and we only got 400 signatures of wh[ich] 60 or more were from King’s. The film was eventually withdrawn, but this was on account of the shindy that the militarists made outside the cinema when they had heard of our protest and had got it into their heads that we were going to break up the Cinema.
A further comment, that ‘There was a very successful A[nti]-W[ar] demonstration yesterday’, referred to the Armistice Day wreath-laying ceremony, which this year had more the flavour of a political statement. This was not wholly pacifist in spirit. Alan’s friend James Atkins had decided that he was a pacifist, and Alan himself that he was not. But very influential was the suggestion that the First World War had been whipped up by the self-interest of the armament manufacturers. There was great feeling, in which probably Alan shared, that glorification of weapons should not be allowed to make a second great war more likely.
It was Eddington, who as a Quaker was a pacifist and internationalist, who stimulated the next outward and visible step in Alan’s career. This time it was not in connection with the ‘Jabberwocky’ of quantum mechanics, but through his course of lectures on the methodology of science26 which Alan attended in the autumn of 1933. Eddington touched upon the tendency of scientific measurements to be distributed, when plotted on a graph, on what was technically called a ‘normal’ curve. Whether it was the wingspans of Drosophilae, or Alfred Beuttell’s winnings at Monte Carlo, the readings would tend to bunch around a central value, and die away on either side, in a specific way. To explain why this should be so was a problem of fundamental importance in the theory of probability and statistics. Eddington offered an outline of why it was to be expected, but this did not satisfy Alan who, sceptical as ever, wanted to prove an exact result by rigorous pure-mathematical standards.
By the end of February 1934 he had succeeded. It did not require a conceptual advance, but still this was the first substantial result of his own. Typically, for him, it was one that connected pure mathematics with the physical world. But when he showed his work to someone else, he was told that the Central Limit Theorem, as the result was called, had already been proved in 1922 by a certain Lindeberg.27 Working in his self-contained way, he had not thought to find out first whether his objective had already been attained. But he was also advised that, provided due explanation was given, it might still be acceptable as original work for a King’s fellowship dissertation.
From 16 March to 3 April 1934, Alan joined a Cambridge party to go skiing in the Austrian Alps. It had a vaguely Quaker, internationalist link with Frankfurt University, whose ski-hut near Lech on the Austro-German border they used. The flavour of cooperation was soured by the fact that the German ski coach was an ardent Nazi. On his return, Alan wrote:
29/4/34
… We had a very amusing letter from Micha, the German leader of the skiing party … He said ‘… but in thoughts I am in your middle’ …
I am sending some research I did last year to Czüber* in Vienna, not having found anyone in Cambridge who is interested in it. I am afraid however that he may be dead, as he was writing books in 1891.
But first the final Tripos examination had to be got out of the way; Part II from 28 to 30 May and then the Schedule B papers28 from 4 to 6 June. In between the examinations he had to rush down to Guildford to see his father. Mr Turing, who was now sixty, underwent a prostate operation after which he was never again in the good health he had so far enjoyed.
He passed with distinction, making him what was called a ‘B-star Wrangler’ along with eight others. It was only an examination, and Alan deprecated the fuss that his mother made over sending telegrams, and tried to persuade her not to come to the Degree Day formalities on 19 June. But it did mean the award by King’s of a research studentship at £200 per annum, and this enabled him to stay on to try for a fellowship – a serious ambition of which he could now feel more confident than he had in 1932. Several others of his year stayed, including Fred Clayton and Kenneth Harrison. David Champernowne had switched to economics and had not yet taken his degree. James had found himself disoriented by the abstract nature of Part II, and gained a Second. He was not sure how to begin his career, and for the next few months, during which he came to visit Alan several times, did some private tuition work.
By the end of Alan’s undergraduate period, his depression was lifting and new industry was arising, just as in the world outside. He had begun to put down firm Cambridge roots, and to cut a figure as one less subdued and more ready with wit and good humour. It was still true that he belonged neither to an ‘aesthete’ nor to an ‘athlete’ compartment. He had continued to row in the boat club, and got on amiably with the other members, once downing a pint of beer in one go. He played bridge with others of his year, though with the usual defect of serious mathematicians he could not be trusted to add up the scores. The visitor to his room would find a disarray of books and notes and unanswered letters about socks and underpants from Mrs Turing. Round the walls were stuck various mementoes – Christopher’s picture, for one – but also, for those with eyes to see, magazine pictures with male sex-appeal. He also liked to root around in sales and street markets, and picked up a violin in London, on Farringdon Road, for which he took some lessons. This did not produce very aesthetic results, but there was a little of the ‘aesthete’ side in him, inasmuch as it debunked the pompous and stiff-upper-lip models of behaviour. It was all somewhat mystifying to Mrs Turing, when at Christmas 1934 Alan asked for a teddy bear, saying he had never had one as a little boy. The Turings usually dutifully exchanged more useful and improving presents. But he had his way, and Porgy the bear was installed.
Graduation meant little change in his general way of life, except that he gave up rowing and resumed running. After the degree day he took a cycling trip to Germany, asking an acquaintance, Denis Williams, to come with him. A first-year student of the Moral Sciences Tripos, Denis knew Alan from the Moral Science Club, the King’s boat club and the skiing trip. They took their bicycles on the train as far as Cologne, and then did thirty miles or so a day. One purpose of the trip was to visit Göttingen, where Alan consulted some authority, presumably in connection with the Central Limit Theorem.
A peculiar gangster regime there might be in Berlin, but Germany was still best for student travel, with cheap fares and youth hostels. They could hardly avoid seeing the swastika flags draped everywhere, but to English eyes they seemed less sinister than ridiculous. Once they stopped in a mining village, where they heard the miners singing on their way to work – a welcome contrast to the contrived Nazi displays. In the youth hostel Denis chatted with a German traveller, bidding goodbye amiably with a ‘Heil Hitler’, as foreign students generally did simply as a matter of polite conformity to local custom. (There had also been cases of them being assaulted when they failed to do so.) Alan came in and happened to see this. He said to Denis, ‘You shouldn’t have said that, he was a Socialist.’ He must have spoken with the German earlier, and Denis was struck by the fact that someone had identified himself to Alan as an opponent of the regime. But it was not that Alan reacted as a signed-up anti-fascist, it was that he could not go through with a ritua
l with which he did not agree. To Denis it was more like another incident on their trip, when two working-class boys from England happened to catch up with them and Denis said that it would be polite to invite them over to have a drink. ‘Noblesse oblige’, said Alan, which made Denis feel very small and insincere.
They happened to be in Hanover a day or two after 30 June 1934, when the SA was overthrown. Alan’s knowledge of German, although it was culled from mathematical textbooks, was better than Denis’s, and he translated from the newspaper an account of how Roehm first had been given the chance to commit suicide and had then been shot. They were rather surprised by the attention given to his demise by the English press. But then, this was a symbolic event with resonances going beyond the plain fact that Hitler thereby gained supreme power. It removed a major contradiction within the Nazi party, trumpeting its intention to turn Germany into a giant stud farm. To grateful conservatives it was the end of ‘decadent’ Germany. Later, when Hitler was thoroughly unpopular, the opposite connection could be drawn, and Nazidom painted as itself ‘decadent’ and ‘perverse’. Behind it lay the powerful leitmotiv that Hitler so skilfully orchestrated: that of the homosexual traitor.
For some Cambridge students a sight of the new Germany, and a brush with its crudities, might engender a powerful anti-fascist commitment. That step was not for Alan Turing. He was always friendly to the anti-fascist cause, but nothing would make him a ‘political’ person. His was the other road to freedom, that of dedication to his craft. Let others do what they could; he would achieve something right, something true. He would continue the civilisation that the anti-fascists defended.
In the summer and autumn of 1934, he continued to work on his dissertation.29 The deadline for its submission was 6 December, but Alan handed it in a month early, and was ready for a next step. Eddington, who had played so important a part in his early development, had suggested his dissertation problem to him. The next suggestion came from Hilbert, although not so directly. In the spring of 1935, while his dissertation went the rounds of the King’s Fellows, Alan went to a Part III course on Foundations of Mathematics. It was given by M. H. A. Newman.
Newman, then nearly forty, was with J. H. C. Whitehead the foremost British exponent of topology. This branch of mathematics could be described as the result of abstracting from geometry such concepts as ‘connected’, ‘edge’ and ‘neighbouring’ which did not depend upon measurement.* In the 1930s it was unifying and generalising much of pure mathematics. Newman was a progressive figure in a Cambridge where classical geometry was more strongly represented.
The basis of topology was the theory of sets, and so Newman had been drawn into the foundations of set theory. He had also attended the 1928 international congress at which Hilbert represented the Germany excluded in 1924. Hilbert had revived his call for an investigation into the foundations of mathematics. And it was in Hilbert’s spirit, rather than as a continuation of Russell’s ‘logistic’ programme, that Newman lectured. Indeed, the Russell tradition had petered out, for Russell himself had left Cambridge in 1916 when first convicted and deprived of his Trinity College lectureship; and of his contemporaries, Wittgenstein had turned in a different direction, Harry Norton had gone mad, and Frank Ramsey had died in 1930. This left Newman as the only person in Cambridge with a deep knowledge of modern mathematical logic, although there were others, Braithwaite and Hardy amongst them, who were interested in the various approaches and programmes.
The Hilbert programme was essentially an extension of the work on which he had started in the 1890s. It did not attempt to answer the question which Frege and Russell had tackled, that of what mathematics really was. In that respect it was less philosophical, less ambitious. On the other hand, it was more far-reaching in that it asked profound and difficult questions about the systems such as Russell produced. In fact Hilbert posed the question as to what were, in principle, the limitations of a scheme such as that of Principia Mathematica. Was there a way of finding out what could, and what could not, be proved within such a theory? Hilbert’s approach was called the formalist approach, because it treated mathematics as if a game, a matter of form. The allowable steps of proof were to be considered like the allowable moves in a game of chess, with the axioms as the starting position of the game. In this analogy, ‘playing chess’ corresponded to ‘doing mathematics’, but statements about chess (such as ‘two knights cannot force checkmate’) would correspond to statements about the scope of mathematics. And it was with such statements that the Hilbert programme was concerned.
At that 1928 congress, Hilbert made his questions quite precise. First, was mathematics complete, in the technical sense that every statement (such as ‘every integer is the sum of four squares’) could either be proved, or disproved. Second, was mathematics consistent, in the sense that the statement ‘2 + 2 = 5’ could never be arrived at by a sequence of valid steps of proof. And thirdly, was mathematics decidable? By this he meant, did there exist a definite method which could, in principle, be applied to any assertion, and which was guaranteed to produce a correct decision as to whether that assertion was true.
In 1928, none of these questions was answered. But it was Hilbert’s opinion that the answer would be ‘yes’ in each case. In 1900 Hilbert had declared ‘that every definite mathematical problem must necessarily be susceptible of an exact settlement … in mathematics there is no ignorabimus’; and when he retired in 1930 he went further:30
In an effort to give an example of an unsolvable problem, the philosopher Comte once said that science would never succeed in ascertaining the secret of the chemical composition of the bodies of the universe. A few years later this problem was solved…. The true reason, according to my thinking, why Comte could not find an unsolvable problem lies in the fact that there is no such thing as an unsolvable problem.
It was a view more positive than the Positivists. But at the very same meeting, a young Czech mathematician, Kurt Gödel, announced results which dealt it a serious blow.
Gödel was able to show31 that arithmetic must be incomplete: that there existed assertions which could neither be proved nor disproved. He started with Peano’s axioms for the integers, but enlarged through a simple theory of types, so that the system was able to represent sets of integers, sets of sets of integers, and so on. However, his argument would apply to any formal mathematical system rich enough to include the theory of numbers, and the details of the axioms were not crucial.
He then showed that all the operations of ‘proof’, these ‘chess-like’ rules of logical deduction, were themselves arithmetical in nature. That is, they would only employ such operations as counting and comparing, in order to test whether one expression had been correctly substituted for another – just as to see whether a chess move was legal or not would only be a matter of counting and comparing. In fact, Gödel showed that the formulae of his system could be encoded as integers, so that he had integers representing statements about integers. This was the key idea.
Gödel continued to show how to encode proofs as integers, so that he had a whole theory of arithmetic, encoded within arithmetic. It was an exploitation of the fact that if mathematics were regarded purely as a game with symbols, then it might as well employ numerical symbols as any other. He was able to show that the property of ‘being a proof’ or of ‘being provable’ was no more and no less arithmetical than the property of ‘being square’ or ‘being prime’.
The effect of this encoding process was that it became possible to write down arithmetical statements which referred to themselves, like the person saying ‘I am lying.’ Indeed Gödel constructed one particular assertion which had just such a property, for in effect it said ‘This statement is unprovable.’ It followed that this assertion could not be proved true, for that would lead to a contradiction. Nor could it be proved false, for the same reason. It was an assertion which could not be proved or disproved by logical deduction from the axioms, and so Gödel had proved that arithmetic was incom
plete, in Hilbert’s technical sense.
There was more to it than this, for one remarkable thing about Gödel’s special assertion was that since it was not provable, it was, in a sense, true. But to say it was ‘true’ required an observer who could, as it were, look at the system from outside. It could not be shown by working within the axiomatic system.
Another point was that the argument assumed that arithmetic was consistent. If, in fact, arithmetic were inconsistent, then every assertion would be provable. So, more precisely, Gödel had shown that formalised arithmetic must either be inconsistent, or incomplete. He was also able to show that arithmetic could not be proved consistent within its own axiomatic system. To do so, all that would be required would be a proof that there was a single proposition (say, 2 + 2 = 5) which could not be proved true. But Gödel was able to show that such a statement of existence had the same character as the sentence that asserted its own unprovability. And in this way, he had polished off the first two of Hilbert’s questions. Arithmetic could not be proved consistent, and it was certainly not consistent and complete. This was an amazing new turn in the enquiry, for Hilbert had thought of his programme as one of tidying up loose ends. It was upsetting for those who wanted to find in mathematics something that was absolutely perfect and unassailable; and it meant that new questions came into view.
Newman’s lectures finished with the proof of Gödel’s theorem, and thus brought Alan up to the frontiers of knowledge. The third of Hilbert’s questions still remained open, although it now had to be posed in terms of ‘provability’ rather than ‘truth’. Gödel’s results did not rule out the possibility that there was some way of distinguishing the provable from the non-provable statements. Perhaps the rather peculiar Gödelian assertions could somehow be separated off. Was there a definite method, or as Newman put it, a mechanical process which could be applied to a mathematical statement, and which would come up with the answer as to whether it was provable?
Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game Page 17
