Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game

by Andrew Hodges

  The suggestions he made were simply the activities that had been pursued on and off duty in Hut 8 and Hut 4, rather surprisingly brought into the open:

  (i) Various games e.g. chess, noughts and crosses, bridge, poker

  (ii) The learning of languages

  (iii) Translation of languages

  (iv) Cryptography*

  (v) Mathematics.

  Of these (i), (iv) and to a lesser extent (iii) and (v) are good in that they require little contact with the outside world. For instance in order that the machine should be able to play chess its only organs need be ‘eyes’ capable of distinguishing the various positions on a specially made board, and means for announcing its own moves. Mathematics should preferably be restricted to branches where diagrams are not much used. Of the above possible fields the learning of languages would be the most impressive, since it is the most human of these activities. This field seems however to depend rather too much on sense organs and locomotion to be feasible.

  The field of cryptography will perhaps be the most rewarding. There is a remarkably close parallel between the problems of the physicist and those of the cryptographer. The system on which a message is enciphered corresponds to the laws of the universe, the intercepted messages to the evidence available, the keys for a day or a message to important constants which have to be determined. The correspondence is very close, but the subject matter of cryptography is very easily dealt with by discrete machinery, physics not so easily.

  There was more to Intelligent Machinery than this. One feature was that he laid down definitions of what was meant by ‘machine’, in such a way that it connected the 1936 Turing machine with the real world. He distinguished first:

  ‘Discrete’ and ‘Continuous’ machinery. We may call a machine ‘discrete’ when it is natural to describe its possible states as a discrete set…. The states of ‘continuous’ machinery on the other hand form a continuous manifold …. All machinery can be regarded as continuous, but when it is possible to regard it as discrete it is usually best to do so.

  and then:

  ‘Controlling’ and ‘Active’ machinery. Machinery may be described as ‘controlling’ if it only deals with information. In practice this condition is much the same as saying that the magnitude of the machine’s effects may be as small as we please…. ‘Active’ machinery is intended to produce some definite physical effect.

  He then gave examples:

  A Bulldozer

  Continuous Active

  A Telephone

  Continuous Controlling

  A Brunsviga

  Discrete Controlling

  A Brain is probably

  Continuous Controlling, but is very similar to much discrete machinery

  The ENIAC, ACE, etc.

  Discrete Controlling

  A Differential Analyser

  Continuous Controlling

  A ‘Brunsviga’ was a standard make of desk calculator, and the point was that such a machine, like an Enigma, a Bombe, a Colossus, the ENIAC or the planned ACE, was best regarded as a ‘controlling’ device. In practice it would have a physical embodiment, but the nature of the embodiment, and the magnitude of its physical effects, were essentially irrelevant. The Turing machine was the abstract version of such a ‘discrete controlling’ machine, and the cipher machines and decipherment machines were physical versions of them. They had taken up much of his working life. And the fundamental thesis of Intelligent Machinery was that the brain could also be ‘best regarded as’ a machine of this kind.

  The paper also included a short calculation which bridged the two descriptions of a machine such as a computer, the logical description and the physical description. He showed that in a job taking more than 101017 steps, a physical storage mechanism would be virtually certain to jump into the ‘wrong’ discrete state, because of the ever-present effects of random thermal noise. This was hardly a practical constraint. He might have made a similar calculation regarding the effect of quantum indeterminacy, and the upshot would have been the same. The determinism of the logical machine, although it could never be rendered with absolute perfection, was still effectively independent of all the ‘Jabberwocky’ of physics. This part of the paper integrated his several interests in logic and physics, mapped out where his own work stood within a wider framework, and summed up a long chapter of unfulfilled ambitions.

  A final section suggested approaches to ‘intelligent machinery’ which were not based on this crude ‘teaching’, but upon his real experience as a pure mathematician. He considered the process of transforming problems from one formulation into another, solving a problem by proving a theorem in some other logical system, and translating the result back into the original form. This corresponded closely with the real work of mathematics, that of detecting analogies, and searching for openings towards a proof within some framework of ideas. ‘Further research into intelligence of machinery will probably be very greatly concerned with “searches” of this kind,’ he wrote. ‘We may perhaps call such searches “intellectual searches”. They might very briefly be defined as “searches carried out by brains for combinations with particular properties”.’ Of course, this was not exactly unrelated to the task of cryptanalysis, that of finding patterns in the apparently patternless.

  He drew a Darwinian parallel:

  It may be of interest to mention two other kinds of search in this connection. There is the genetical or evolutionary search by which a combination of genes is looked for, the criterion being survival value. The remarkable success of this search confirms to some extent the idea that intellectual activity consists mainly of various kinds of search.

  The remaining form of search is what I should like to call the ‘cultural search’. As I have mentioned, the isolated man does not develop any intellectual power. It is necessary for him to be immersed in an environment of other men, whose techniques he absorbs during the first twenty years of his life. He may then perhaps do a little research of his own and make a very few discoveries which are passed on to other men. From this point of view the search for new techniques must be regarded as carried out by the human community as a whole, rather than by individuals.

  This was a rare revelation of his self-perception. It was a dignified and generous response to the lessons of 1937 and 1945, when others had come forward with ideas equivalent to his own – so much more realistic than the usual worrying about ‘priority’, with its implicit fear of cheating and copying, and so free of the male competitiveness which was by 1948 becoming more and more evident in science.54 He never claimed more than that ‘some years ago I made an investigation into what could be done by a rule-of-thumb process,’ when referring to his own part. And of course this had been yet another of the lessons of 1941, that it was the search of the whole Bletchley community that mattered so much. But that very fact might perhaps have made him wonder more as to whether the operation of the brain ‘without interference’ was really the right way in which to focus attention. The very existence of these social or cultural levels of description was an indication that individual ‘intelligence’ was not the whole story. This question was not developed in this essay. Meanwhile, his ability to stand back from the individual struggle was certainly required in adjusting himself to work with the rival computer that had been developed at Manchester.

  He wrote off to F. C. Williams for information, and received a reply probably on 8 July. By this time, the fact was that they had already, on 21 June 1948, successfully run the first program on the first working stored program electronic digital computer in the world. Darwin had talked about ‘formidable mathematical difficulties’, but at Manchester they had just got on with it, and produced a computer behind Darwin’s back. It used, for storage, the cathode ray tube that Williams had developed, and at this point the total store consisted of just 1024 binary digits stored on one tube. Alan drew attention to this figure in a table of ‘memory capacities’ in this report:

  Brunsviga

  9
0

  ENIAC without cards and with fixed programme

  600

  ENIAC with cards

  ∞*

  ACE as proposed

  60,000

  Manchester machine as actually working (817/48)

  1,100

  It was a pointed contrast between one machine still merely ‘proposed’, and another that actually worked. But the figures also pointed to the fact that F. C. Williams had pursued his project in a more modest way. The Manchester computer was small, and might even be called small-minded. But it was the first embodiment of a Universal Turing Machine, albeit with a very short ‘tape’. Alan wrote out a little routine55 to perform long division, and posted it north immediately.

  Jack Good and Donald Michie looked in on Alan at King’s, and rather annoyed him by peeping at the uncompleted version of Intelligent Machinery while he was out. Afterwards, walking along King’s Parade, Alan dropped a very deliberate remark to Jack Good about a boy in Paris.* His drift got across to Jack, who had not known before. They were also in correspondence during this summer period. Jack wrote:

  25 July 48

  Dear Prof,

  When I was last in Oxford I met a lecturer in physiology who said that he thought the number of neurons in the brain was only about two million. This seems amazingly little to me even allowing [for] the fact that the number of processes from each neuron is something like 40. I wonder if you could tell me the right answer, with or without a reference.

  I understand that by next October we’ll have swapped towns, Judging by the international situation I think you’ve had the better of the bargain….

  How near were you to getting into the Olympics?

  Jack was leaving his Manchester lectureship to join the branch of the Civil Service now known as the Government Communications Headquarters, and located at Eastcote, in north-west London. As for the international situation, the new lines were rapidly hardening. Yugoslavia had been expelled from the Cominform – a break which led Robin, like many other sympathisers with the pre-war USSR, to move much further away from the Communist party. The airlift to West Berlin was under way, and for the first time there was serious talk of war with Russia.

  The US Air Force had begun its temporary stay on British soil, and Americans were overtaking plucky British losers in the Empire Stadium, where a scraggy, rationed Britain was hosting the Olympic Games.† Alan went with Anderson, an acquaintance from the Hare and Hounds Club, and they saw the Czech athlete Zátopek win the 10,000 metre race on 30 July. The Marathon was won for Argentina but in a time still only seventeen minutes better than Alan’s. He replied to Jack:

  Dear Jack,

  I have repeatedly looked in books on neurology for the very important number N you asked about, and never found any figure offered. My own estimate is 3.108
  I’ve had something wrong with my leg for some months, so wasn’t able to run in any Marathons this season.

  Yours Prof

  An injury to his hip had put paid to his chances for the Olympic marathon team, for which otherwise he might have qualified, and to his regret, prevented any further development in serious long-distance running.

  Alan sent off another routine, for factorising numbers, to Manchester on 2 August. Then he went on holiday with Neville in Switzerland. It was the first escape from austerity England, and they could hardly believe the fresh country foods. The trip was made on the travel allowance of £25, in the form of five crisp five-pound notes. They did it by cycling and staying in youth hostels: they trod glaciers and scaled mountains and had the usual smouldering arguments of people on holiday together, as when Alan let his bicycle break down through inattention, or was keen on another young man in the hostel. It was not quite E. M. Forster’s greenwood, but as near as he had ever approached it.

  The summer continued with a week in the Lake District at Pigou’s cottage, with Peter Matthews. Pigou was very keen on mountaineering, and even more, with a pre-Carpenter innocence, on Wilfrid Noyce the young mountaineer.* Alan took care to practise rock-climbing on the King’s College gate with Peter Matthews before they went. It was like some 1890s Cambridge reading party, with old Pigou clocking up the times and the victors at chess. He had a collection of First World War medals which, though a pacifist, he had been awarded for ambulance service, and used to award them after a farewell dinner to whoever had done best on the slopes. Alan did some easy climbing, but mostly trotted round Buttermere in short shorts. From Jack Good:

  16 Sep 48

  Dear Prof,

  Pardon the use of the typewriter: I have come to prefer discrete machines to continuous ones.

  When I was in Cambridge recently I hunted unsuccessfully … for an estimate … of N, the number of neurons in the human brain. Soon after this Donald succeeded in finding a reference. He tells me that … N=10000000000 roughly.

  I visited Oxford last week-end. Donald showed me a ‘chess machine’ invented by Shaun [Wylie] and himself. It suffers from the very serious disadvantage that it does not analyse more than one move ahead. I am convinced that such a machine would play a very poor game, however accurately it scored the position with respect to matter and space. In fact it could easily be beaten by playing ‘psychologically’ i.e. by taking into account the main weakness of the machine….

  When in Oxford I succeeded in hypnotising Donald …. Would you agree that a very typical property of the brain is the ability to think in analogies? This means taking only a part of the evidence into account…. Do you know of any reference to Russian electronic computers? …

  Donald Michie was now studying physiology at Oxford. He had followed up their Bletchley speculations by teaming up with Shaun Wylie to devise a chess program they called the Machiavelli. Meanwhile Alan and David Champernowne had worked out one they called the Turochamp.56 It followed the minimax system, and the important idea of pursuing chains of captures until no more could be made. It had a scoring system in which pawn mobility, castling, and getting a rook on to the seventh rank were included as well as captures. None of this went much beyond what Alan had discussed back in 1941 with Jack Good, or indeed with Champ in 1944. Going for a walk, probably at Christmas that year, they had made a bet on whether a machine could beat Champ himself at chess by 1957. The odds were put at 13 to 10 in favour of the machine succeeding. The Turochamp certainly did not reach this standard, although it beat his wife, a beginner at chess. It was not taken at all seriously, or written out in detail. But it would have been a system of this kind which gave Alan ‘a sense of pitting one’s wits against something’, as he wrote in Intelligent Machinery. Champ also took on the system for poker that Alan had more carefully worked out, and had the pleasure of beating it by sheer good luck. Alan replied to Jack:

  Sept 18 48

  Dear Morcom,

  I am glad to hear my estimate of no. of neurons is not too essentially wrong.

  The chess machine designed by Champ and myself is rather on your lines. Unfortunately we made no definite record of what it was, but I am going to write one down definitely in the next few days with a view to playing the Shaun-Michie machine.

  To a large extent I agree with you about ‘thinking in analogies’, but I do not think of the brain as ‘searching for analogies’ so much as having analogies forced upon it by its own limitations …

  The report was handed in. Mike Woodger was very excited by the prospects it opened up, and gladly drew the diagrams neatly for the typed version. Darwin was less impressed, probably highly embarrassed by the references to Dorothy Sayers, God, and robots taking country walks. At the Executive Committee meeting on 28 September he explained sniffily that ‘Dr Turing had now produced a report which, although not suitable for publication, demonstrated that during his stay there he had been engaged in rather fundamental studies.’ The un
suitable paper disappeared into the NPL files. Ironically, it was on 20 September 1948 that von Neumann gave a first published lecture57 on the ‘theory of automata’ – in effect, the theory of discrete controlling machines – in which he drew attention, after eleven years, to the fundamental importance of the Universal Turing Machine.

  Robin had rented Blackett’s holiday home in Wales on occasion, and did so again this year. By inviting Alan, he made possible a third holiday before the summer was out. Another of the party was Nicholas Furbank, the friend of E. M. Forster, who had lately been writing a book on Samuel Butler. This too was rather like an old-style Cambridge reading party – they were amused by the resemblance – with organised walks, and funny nicknames, and reading Thomas Love Peacock’s gothic Melincourt aloud.

  Alan seemed very happy. They played rationalist Twenty Questions as they filed along the hill paths and old railway tunnels. Alan developed a theory of how to choose the next question so as to maximise the expected weight of evidence of the answer. He also recounted quaint tales of the Pigou regime, whose antiquated brand of hearty male misogyny left him bemused. ‘The standard at the Pigou cottage is very high,’ he said. ‘I ran all round Buttermere faster than Noel-Baker in ’28, and I only got a bronze.’ One day they took a taxi and a bus at dawn for an expedition to the real mountain slopes of the Snowdon horseshoe, where Nick Furbank was suitably terrified and went on all fours along the narrow ridge of Cribgoch, but Alan strode on in dogged Turing fashion, as twenty years before, but with friends at last.

  Down from the mountain tops, it was time to pack up. The suitcase with the parts for the zeta-function machine was still in his room, along with the star globe and Christopher’s picture. He kept as souvenirs some of the gear wheels that had been cut, but gave the rest to Peter Matthews to sell for scrap. Alan was rather disappointed with the price they fetched.