Alan Turing: The Enigma (99 page)

I have just had my house broken into, and am still every few hours finding some fresh thing missing. Fortunately I am insured, and little has gone that is really irreplaceable. But the whole thing has had a very disturbing effect, especially as it followed shortly on a theft from me at the University. I go about expecting a brick to fall on my head or something disagreeable and unexpected anywhere.

A rather pathetic collection of oddments was missing – a shirt, some fish-knives, a pair of trousers, some shoes and shavers and a compass – even an opened bottle of sherry. He assessed it at a total value of £50. He reported the burglary to the police, and two CID officers came to take fingerprints in the house. Yet even while he did this, he suspected that there might be some connection with Arnold. He consulted a solicitor recommended to him by his neighbour Roy Webb, and on his advice wrote to Arnold on 1 February, reviving the question of the money missing from his wallet, saying that whatever the truth of the matter it had come between them, and that it would be best if they did not see each other again. He added in a somewhat schoolmasterish tone that it was Arnold’s duty to repay the £7. He also said that if Arnold came to his house again he would not be admitted.

But when Arnold reacted to this letter by calling at Hollymeade on the Saturday evening, 2 February, he found himself admitted after all. Again he angrily protested his innocence, and in a moment of emotion said he could go to the police and tell
everything
. Alan challenged him to ‘do his worst’ –
but it was an empty threat, for Arnold soon admitted he could do nothing against a man in Alans distinguished position. The anger was discharged, and a different mood prevailed. Giving Arnold a drink, Alan mentioned the burglary, and to this question Arnold immediately supplied an answer. He did not know the burglary had taken place, but did know exactly who might have done it. For he had mentioned Alan to an acquaintance called Harry, a twenty-year-old unemployed youth recently discharged from National Service in the Navy, while they were talking in the Oxford Street milk bar. They had been speaking rather boastfully of their respective successes. Harry had suggested a robbery, and although Arnold had refused to join in, he knew it had been planned.

The result was to re-establish a friendly, and indeed an erotic relationship. Arnold once again slept with Alan, although during the night Alan found himself in two minds, at one point going downstairs to put away the glass with Arnold’s fingerprints on it, in the hope that he could compare them with those left by the burglars. Next morning they went together into Wilmslow town, and Arnold waited outside the police station while Alan went in to pass on the information about the likely culprits, fabricating a story to explain how he had come by it. He had allowed the game of Presents to be taken a fair way without making a fuss; but to let it go unchecked would, in his view, be tantamount to giving in to blackmail.

Arnold went off having offered to do his best to track down the stolen goods, and indeed he was able to write to Alan with a report a few days later. But by that time everything had changed. One change had been marked by the Manchester bells, ringing not this time for victory, but for the death of George VI. On Thursday the new Queen Elizabeth flew back from Kenya and Winston Churchill, Prime Minister again, welcomed her at the airport. And it was on that very evening, as the new Elizabethan age began, that the detectives paid a call on Alan Turing.
No man is an Island, entire of itself
. Now he was in the soup.

^*
Newman had written
²
to von Neumann on 17 June 1946 that he was ‘at present grappling with Turing’s report, which I find a good deal less readable than yours.’ He had also spent a term at Princeton in late 1946, and discussed computers with von Neumann.

^*
In 1949, on a visit to the United States, Williams scandalised the employees of IBM, whose corporate motto think was everywhere in evidence, with this analysis of how he had succeeded where they had not.

^†
Yet already the end of the vacuum tube was in sight. A final letter from Jack Good to ‘Prof’, dated 3 October 1948, continued their discussion of the brain, and also asked: ‘Have you heard of the transistor (or Transitor)? It is a small crystal alleged to perform “nearly all the functions of a vacuum tube”. It might easily be the biggest thing since the war. Is England going to get a look-in?’

^*
This invention, later known as an ‘index register’, was of considerable importance in the future development of computer hardware design.

^*
The arrangement was now that he could receive a fellowship stipend only for a quarter in which he had spent twenty-five nights in residence – a condition fulfilled during August.

^†
Running in a team with Christopher Chataway.

^*
The next one was out of range at p = 521, as was discovered by computer search in 1952.

^*
This promise was not entirely fulfilled, since it turned out that the tracks had been too closely packed, and often were unusable.

^*
The opportunities open to him were rich, and his neglect of them very striking. He could, for instance, have used his knowledge of the ‘recursive function’ to develop a far more powerful and interesting treatment of the ‘sub-routine’. Church’s lambda-calculus, and all the hitherto abstruse and ‘useless’ work he had done on such problems as the ‘dots and brackets’ in mathematical logic, were now relevant to the devising of practical programming languages. The knowledge of probability and statistics that he had employed in Enigma work could equally profitably have been applied to the theory of programming. Experience with searching, sorting, and the ‘trees’ involved in his chess-playing ideas, were all particularly relevant to the data-processing problems it was now possible to attack on computers. He could have done much to set standards for the new engineering discipline, if only because he could so easily rise above the technicalities of any particular installation, and could have set his weight against the often absurd and debilitating separation of university mathematics from the developing field of computer applications. But with few exceptions, one of them being an insistence on program checking procedures which reflected his more abstract, rigorous, background, he abandoned this line of development.

^*
Alan Turing himself always sought to play down any such comparison, which in his view was irrelevant to the essential thesis that the brain could be regarded as a discrete state machine. Thus in the 1948 report for the NPL he had written: ‘We could produce fairly accurate electrical models to copy the behaviour of nerves, but there seems very little point in doing so. It would be rather like putting a lot of work into cars which walked on legs instead of continuing to use wheels.’

^*
The Mersenne prime problem was a highly artificial, if ingenious, application of the growing Manchester computer. Only from the autumn of 1949 could it be applied to ‘regular’ problems. Besides those of Alan Turing himself, as later described, it was used for optical calculations, tracing rays through systems of lenses, and for some mathematical work in connection with guided missiles.

^*
X-ray measurements give only the
amplitudes
of the different frequency components in the diffracted X-rays, and not the
phases
. The analysis depends upon guessing the phases, the criterion of a correct guess being that when the amplitudes and phases are put together, they lead to a picture of the crystal which is in accord with physical reality, with the right number of atoms and a positive electron density. This is exactly the same idea as guessing a key, given a piece of cipher-text: the criterion of a correct guess being that it give a sensible message.

The analogy with cryptanalysis is even closer in that the crystallographer attacks the problem, at first sight too enormous for contemplation, by making a hypothesis about the structure of the crystal. Thus Watson and Crick pursued the DNA analysis, as did Pauling, by making good guesses about the helix structure, and thus getting closer and closer to the solution. This is essentially the same idea as the ‘Probable Word’ method, which also effects a drastic reduction in the number of possible keys – so that with the Enigma, for instance, they were left only with a small number of Bombe ‘stops’ to try out for sensible German plain-text.

It is not surprising that Alan Turing could see how to quantify the idea of information required for a guess to be possible: this was very close to the quantification of ‘weight of evidence’ which constituted his major conceptual advance at Bletchley.

^*
Shannon was sceptical about this programme of work, and Shannon had a good point. By 1977 computer calculations would show that among the first
seven million
zeroes of the zeta function, there is not one that lies off the special line. This was a case where a brute force attack could yield only a negative result.

^*
If the operations are represented by letters, then such a sequence is represented by a ‘word’ – hence the name of the problem. For a
finite
group, there would of course exist such a definite method, namely the crude one of working through all the possibilities. The problem arises for
infinite
groups.

^†
A ‘semi-group’ is the abstract version of a set of operations which meets half of the conditions required for a ‘group’: the operations cannot necessarily be reversed.

^‡
A ‘cancellation semi-group’ is a semi-group with a property which makes it closer to being a group: if AC = BC then it must be that A = B.

^*
Polanyi rejected this argument, saying that a machine was a machine, a human mind was a human mind, and no amount of evidence could change this
a priori
fact.

^*
This oscillation between the two concepts of computability found its way into his
Programmers’ Handbook
: on the first page the programmer was greeted by the assertion that ‘There is also a part of the machine called the control which corresponds to the [human] computer himself. If his possible behaviour were very accurately represented this would have to be a formidably complicated circuit. However we really only require him to be able to obey the written instructions and these can be made so explicit that the control can be quite simple.’

^*
Thus a contemporary review article
⁴⁴
stated ‘The importance of the principle of patterned ‘field’ activities in the determination of embryonic systems has been generally recognised. … Yet their nature and mode of operation are still among the greatest puzzles of modern biology.’

^*
More strictly, all but the Manchester and Toronto machines were of a slightly modified design, the Mark I
^*
.

^*
Meanwhile D.G. Prinz, who worked for Ferranti, quite independently programmed the Manchester computer to solve two-move chess problems. But this would have been of minimal interest to Alan; given that a solution exists it is simply a matter of patience to run through every possibility until it is found. Unless it gave an idea of how the brain did it, or some feel of ‘pitting one’s wits’ against the machine, the problem of doing the programming, however ingenious, would have had little appeal for him. As in 1941, he was not interested in chess in itself, but as a model for thought.

^*
In this game three piles of matches are laid out, and two players take turns to remove as many matches as they please from any one pile. The player who removes the last match is the winner.

^*
Until his death in 1975, Christopher Strachey was to be a central innovative figure in British computing, and his draughts-playing program seminal in the study of ‘machine intelligence’.

^*
Alan was introduced to Angus Wilson at Cambridge by Robin Gandy. Although Angus Wilson had worked at Bletchley, he had not met Alan there.

^†
Was this plain-text or cipher-text? This is a good example of where the meaning of a word depends upon its social embodiment. At least since the 1930s it had been in general use among homosexual men as a code word with a plain meaning – in America. Thus D.W. Cory’s pioneer work
The Homosexual in America
, which appeared in 1951, explained: ‘Needed for years was an ordinary, everyday, matter-of-fact word, that could express the concept of homosexuality without glorification or condemnation. It must have no odium of the effeminate stereotype about it. Such a word has long been in existence, and in recent years has grown in popularity. The word
is gay.’
Alan Turing would usually use ‘homosexual’ or, among his friends, the word ‘queer’. But he could have known the American usage, and would entirely have approved of D.W. Cory’s rationale of it. For this reason the word will be used from here onwards; any anachronistic or transatlantic effect thus introduced will reflect quite properly the difficulty that Alan Turing had in communicating his attitudes in the Britain of the early 1950s. As with the ‘computer’ he was ahead of his time.