In Alan’s case, there was a suggestion in the nickname of his role at school, as the tolerated ‘Maths Brain’ with his star globe and pendulum, who had performed the feat of cycling from Southampton. As at school, trivial examples of ‘eccentricity’ circulated in Bletchley circles. Near the beginning of June he would suffer from hay fever, which blinded him as he cycled to work, so he would use a gas mask to keep the pollen out, regardless of how he looked. The bicycle itself was unique, since it required the counting of revolutions until a certain bent spoke touched a certain link (rather like a cipher machine), when action would have to be taken to prevent the chain coming off. Alan had been delighted at having, as it were, deciphered the fault in the mechanism, which meant that he saved himself weeks of waiting for repairs, at a time when the bicycle had again become what it was when invented – the means of freedom. It also meant that no one else could ride it. He made a more explicit defence of his tea-mug (again irreplaceable, in war-time conditions) by attaching it with a combination lock to a Hut 8 radiator pipe. But it was picked, to tease him.
Trousers held up by string, pyjama jacket under his sports coat – the stories, whether true or not, went the rounds. And now that he was in a position of authority, the nervousness of his manner was more open to comment. There was his voice, liable to stall in mid-sentence with a tense, high-pitched ‘Ah-ah-ah-ah-ah’ while he fished, his brain almost visibly labouring away, for the right expression, meanwhile preventing interruption. The word, when it came, might be an unexpected one, a homely analogy, slang expression, pun or wild scheme or rude suggestion accompanied with his machine-like laugh; bold but not with the coarseness of one who had seen it all and been disillusioned, but with the sharpness of one seeing it through strangely fresh eyes. ‘Schoolboyish’ was the only word they had for it. Once a personnel form came round the Huts, and some joker filled in for him, ‘Turing A. M. Age 21’, but others, including Joan, said it should be ‘Age 16’.
He cared little for appearances, least of all for his own, generally looking as though he had just got up. He disliked shaving with a razor and used an old electric shaver instead – probably because cuts could make him pass out with the sight of blood. He had a permanent five o’clock shadow, which emphasised a dark and rough complexion which needed more than the cursory attention it received. His teeth were noticeably yellow, although he did not smoke. But what people noticed most were his hands, which were strange anyway, with odd ridges on his fingernails. These were never clean or cut, and, well before the war, he had made them much worse by a nervous habit of picking at the side, raising an unpleasant peeling scar.
To some extent, his lack of concern for appearances, like his low-budget mode of life, was an intensification of what people meant by ‘donnish’, and as such was far more striking to those outside university circles, than to those long familiar with bicycling dons eking out their stipends. It departed from the ‘don’ typology in his peculiar youthfulness of manner, but Alan Turing still presented the world outside Oxford and Cambridge with a crash course in King’s College values, and the reaction to his oddness was mostly a concentrated form of the mixture of baffled respect and head-shaking suspicion with which English intellectuals were traditionally regarded. This was particularly true at Guildford, where the engagement was perceived in terms of types, he as the don shy of women, and she as the ‘country vicar’s daughter’* and bluestocking ‘female mathematician’. It was demeaning, but the repetition of superficial anecdotes about his usually quite sensible solutions to life’s small challenges served the useful purpose of deflecting attention away from the more dangerous and difficult questions about what an Alan Turing might think about the world in which he lived. English ‘eccentricity’ served as a safety valve for those who doubted the general rules of society. More sensitive people at Bletchley were aware of layers of introspection and subtlety of manner that lay beneath the occasional funny stories. But perhaps he himself welcomed the chortling over his habits, which created a line of defence for himself, without a loss of integrity. He, this unsophisticated outsider at the centre, could be left alone at the point where it mattered.
In the summer of 1941 that much more worldly observer Malcolm Muggeridge had cause to visit Bletchley and notice that31
Every day after luncheon when the weather was propitious the cipher-crackers played rounders on the manor-house lawn, assuming the quasi-serious manner dons affect when engaged in activities likely to be regarded as frivolous or insignificant in comparison with their weightier studies. Thus they would dispute some point about the game with the same fervour as they might the question of free-will or determinism…. Shaking their heads ponderously, sucking air noisily into their noses between words – ‘I thought mine was the surer stroke’, or: ‘I can assert without contradiction that my right foot was already …’
Alan did indeed have that way of sucking in his breath before speaking, while in Hut 8 they were, when off-duty, talking about games, free will and determinism.
He was currently reading a new book by Dorothy Sayers, The Mind of the Maker.32 It was not his usual taste in reading, this being Sayers’ attempt to interpret the Christian doctrine of divine creation through her own experience as a novelist, but he would have enjoyed the challenge of her sophisticated attitude to free will, which she saw from God’s point of view, in the light of her knowledge that fictional characters had to find their own integrity and unpredictability, and were not determined by a master plan at the outset. One image which caught Alan’s fancy was that of Laplacian determinism suggesting that ‘God, having created his Universe, has now screwed the cap on His pen, put His feet on the mantelpiece and left the work to get on with itself.’
This was not so new, but it must have made striking reading while the Bombes ticked away, getting on with the work by themselves – and while the Wrens did their appointed tasks, without knowing what any of it was for. He was fascinated by the fact that people could be taking part in something clever, in a quite mindless way.
Machines, and people acting like machines, had replaced a good deal of human thought, judgment, and recognition. Few knew how the system worked, and for anyone else, it was a mystic oracle, producing an unpredictable judgment. Mechanical, determinate processes were producing clever, astonishing decisions. There was a connection here with the framework of ideas that had gone into Computable Numbers. This, of course, was far from forgotten. Alan explained the Turing machine idea to Joan, and gave her an off-print of one of Church’s papers, though she perhaps disappointed him in her response. He also gave a talk on the subject of his discovery. Meanwhile Turing machines, reading and writing, had sprung into an exceedingly practical form of life, and were producing a kind of intelligence.
A subject closely analogous to cryptanalysis, and which could be spoken of when off-duty, was chess. Alan’s interest was not limited to chess as recreation; he was concerned to abstract a point of principle from his effort to play the game. He became very interested in the question of whether there was a ‘definite method’ for playing chess – a machine method, in fact, although this would not necessarily mean the construction of a physical machine, but only a book of rules that could be followed by a mindless player – like the ‘instruction note’ formulation of the concept of computability. In such discussions Alan would often jokingly refer to a ‘slave’ player.
The analogy between chess and mathematics had already been employed and in each case the same problem arose, that of how to choose the right move to reach a given goal – in the case of chess, to achieve checkmate. Gödel had shown that in mathematics there was no way at all to reach some goals, and Alan had shown that there was no mechanical way to decide whether, for a given goal, there was a route or not. But the question could still be asked as to how mathematicians, chess-players or code-breakers did in practice make those ‘intelligent’ steps, and to what extent they could be simulated by machines.
Although his solution of the Entscheidungsproblem a
nd his work on ordinal logics had focussed attention upon the limitations of mechanical processes, it was now that the underlying materialist stream of thought began to make itself more clear, less interested in what could not be done by machines, than in discovering what could. He had demolished the Hilbert programme, but he still exuded the Hilbert spirit of attack upon unsolved problems, and enjoyed a confidence that nothing was beyond rational investigation – including rational thought itself.
Jack Good, like Alan, had the Bletchley mind, not being simply ‘a mathematician’, but a person who enjoyed exploring the connections between logical skills and the physical world. Chess interested him too, and unlike Alan he was a Cambridgeshire county player. He had already in 1938 published a light-hearted article on mechanised chess-playing in the first issue of Eureka, the house magazine of the Cambridge mathematics students. Besides playing chess, Alan taught Jack Good the game of Go, and before long found himself being beaten at that as well.
Over meal-times on night shift they would talk about the problem of mechanising chess. They latched on to a basic idea, which they agreed to be obvious. It was that a chess player might often see wonderful moves that could be made if only the opponent would do such-and-such, but in serious play, White would assume that Black would always exploit the situation to maximum advantage. White’s strategy therefore would be to make the move least advantageous to Black – the move making Black’s best move the least successful of all the possible best moves – the minimum maximum, in fact.
This was not a new idea. The theory of games had been studied mathematically since the 1920s, and this principle, second nature to chess-players,* had been abstracted and formalised in the manner of modern mathematics. The word ‘minimax’ had been coined for the idea of the least bad course of action. It applied not only to games like chess, but also to those which involved guessing and bluffing. Much of the mathematical work had been done by von Neumann, taking up ideas first published by the French mathematician E. Borel in 1921. Borel had defined ‘pure’ and ‘mixed’ strategies in game-playing. Pure strategies were definite rules, setting out the proper action in any contingency; a mixed strategy would consist of two or more different pure strategies, to be chosen at random, but with specified probabilities for each strategy according to the contingency.
Von Neumann had been able to show that for any game with two players, with fixed rules, there would exist optimal strategies, usually mixed, for each player. Alan would very likely have attended his talk on the game of Poker, given at Princeton in 1937, which illustrated the result.33 It was von Neumann’s beautiful, if depressing theorem, that in any two-person game* both players would be locked into their ‘minimax’ strategies, both finding that all they could do was to make the best of a bad job, and to give the opponent the worst of a good job, and that these two objectives would always coincide.
Poker, with its bluffing and guessing, was a better illustration of the von Neumann theory than chess.† A game without concealment, such as chess, von Neumann called a game of ‘perfect information’, and he proved that any such game would always possess an optimal ‘pure strategy’. In the case of chess, this would be a complete set of rules for what to do in every contingency. There being far more possible chess positions than plugboard positions for the Enigma, however, the general von Neumann theory had nothing of practical value to say about the game. It was an example of where a high-powered, abstract approach failed to be of use. Alan and Jack Good’s approach was quite different in nature, being for one thing concerned not so much with a theory of the game, but with a discussion of human thought processes. It was an ad hoc discussion, ‘dull and elementary’ by pure-mathematical standards, and pursued quite independently of the existing theory of games. They could have done it at school.
In their analysis it first had to be assumed that there was some sensible scoring system, awarding a numerical value to the various possible future positions on the basis of pieces held, pieces threatened, squares controlled and so forth. With this agreed, the most crude ‘definite method’ would be simply to make the move that maximised the score. The next level of refinement would take the opponent’s reply into account, using the ‘minimax’ idea to choose the ‘least bad’ move. In chess there would normally be about thirty possible moves for each player, so that even this crude system would require about a thousand separate assessments. A further step of looking ahead would require thirty thousand.
Reducing the figure of thirty to a mere two for the sake of a diagram, the player (White) making a three-move look-ahead is confronted by a ‘tree’ such as:
A human playing White might reason that it would be good to reach position E, but Black will not be so obliging, and will respond to move B by reply F. A second best, for White, would be position D, but again it must be assumed that Black will play C to prevent this. Of the two evils, positions C and F, C is the lesser one, since at least it guarantees for White a position of value 27. So White plays move A.
A ‘machine’ could simulate this train of thought by a method of ‘backing up’ the tree. Having worked out all the scores for three moves ahead, it would then label the intermediate positions on a minimax basis. It would assign 27 to C, 45 to D, 81 to E, 16 to F (the best in each case), then 27 to A, 16 to B (the worst in each case), and finally select move A for White.
This basic idea created a ‘machine’ that could effect a decision procedure bearing some relation to human intelligence. It was small beer compared with the Hilbert problem, which had required thinking about decision procedures for the whole of mathematics. But on the other hand, it was something that could actually work. As a practical model for mechanical ‘thinking’ it fascinated Alan to the point of obsession.
Such a three-move analysis would be hopelessly ineffective in real chess, in which players would think not in terms of moves but of chains of moves, as for example when a sequence of obligatory captures was set in motion. Alan and Jack Good saw this and decided that the ‘look-ahead depth’ would have to be variable, continuing as far as any capture was possible at all, so that evaluations would only be made on ‘quiescent’ positions. Even so, such a scheme would fail to cope with more subtle play, involving traps leading to pins or forks, a fact which they discussed. It was a crude, brute force attack on chess-playing, but it was a first step in mechanising a fairly sophisticated thought process – at least, a first non-secret step.
They thought these ideas too obvious to be worth publishing. Alan did, however, continue to pursue his own mathematical work and to submit it for publication in America. A true intellectual, he would have been ashamed to let human crime and folly defeat him. ‘Before the war my work was in logic and my hobby was cryptanalysis,’ he once said, ‘and now it is the other way round.’ He had to thank Newman for stimulating his thoughts on this ‘hobby’ of mathematical logic, for they corresponded34 in 1940 and 1941, in which latter year Newman again gave Cambridge lectures on Foundations of Mathematics.
Most of Alan’s efforts were directed towards a new formulation of the theory of types. Russell had regarded types as rather a nuisance, adopted faute de mieux in order to save Frege’s set theory. Other logicians had felt that a hierarchy of logical categories was really quite a natural idea, and that it was the attempt to lump together every conceivable entity into ‘sets’ that was strange. Alan inclined to the latter view. He would prefer a theory which agreed with the way in which mathematicians actually thought, and which worked in a practical way. He also wanted to see mathematical logic used to make the work of mathematicians more rigorous. In a less technical essay written in this period,35 ‘The Reform of Mathematical Notation’, he explained that despite all the efforts of Frege and Russell and Hilbert
… mathematics has profited very little from researches in symbolic logic. The chief reason for this seems to be a lack of liaison between the logician and the mathematician-in-the-street. Symbolic logic is a very alarming mouthful for most mathematicians, and the logicians
are not very much interested in making it more palatable.
His own effort to bridge the gap began with an attempt
… to put the theory of types into a form in which it can be used by the mathematician-in-the-street without having to study symbolic logic, much less use it. The statement of the type principle given below was suggested by lectures of Wittgenstein, but its shortcomings should not be laid at his door.
The type principle is effectively taken care of in ordinary language by the fact that there are nouns as well as adjectives. We can make the statement ‘All horses are four-legged’, which can be verified by examination of every horse, at any rate if there are only a finite number of them. If however we try to use words like ‘thing’ or ‘thing whatever’ trouble begins. Suppose we understand ‘thing’ to include everything whatever, books, cats, men, women, thoughts, functions of men with cats as values, numbers, matrices, classes of classes, procedures, propositions … Under these circumstances what can we make of the statement ‘All things are not prime multiples of 6’…. What do we mean by it? Under no circumstances is the number of things to be examined finite. It may be that some meaning can be given to statements of this kind, but for the present we do not know of any. In effect then the theory of types requires us to refrain from the use of such nouns as ‘thing’, ‘object’ etc., which are intended to convey the idea ‘anything whatever’.
The technical work of separating mathematical ‘nouns’ from ‘adjectives’ was based upon that of Church, whose lectures he had followed at Princeton, and who published a description of his type theory in 1940. Part of Alan’s work was done in collaboration with Newman through correspondence; their joint paper36 being received at Princeton on 9 May 1941. It must have crossed the Atlantic just as the München was captured. Alan produced a further paper37 of a highly technical nature, ‘The Use of Dots as Brackets in Church’s system’, and submitted it just a year later. This promised as forthcoming two more papers, but these never emerged.
Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game Page 35