There was also another machine he had in mind, but this had nothing to do with Germany, except in the very different sense that it came out of the work of Riemann. Its purpose was to calculate the Riemann zeta-function. Apparently he had decided that the Riemann hypothesis was probably false, if only because such great efforts had failed to prove it. Its falsity would mean that the zeta-function did take the value zero at some point which was off the special line, in which case this point could be located by brute force, just by calculating enough values of the zeta-function.
This programme had already been started; indeed Riemann himself had located the first few zeroes and checked that they all lay on the special line. In 1935-6, the Oxford mathematician E. C. Titchmarsh had used the punched-card equipment which was then used for the calculation of astronomical predictions to show that (in a certain precise sense) the first 104 zeroes of the zeta-function did all lie on the line. Alan’s idea was essentially to examine the next few thousand or so in the hope of finding one off the line.
There were two aspects to the problem. Riemann’s zeta-function was defined as the sum of an infinite number of terms, and although this sum could be re-expressed in many different ways, any attempt to evaluate it would in some way involve making an approximation. It was for the mathematician to find a good approximation, and to prove that it was good: that the error involved was sufficiently small. Such work did not involve computation with numbers, but required highly technical work with the calculus of complex numbers. Titchmarsh had employed a certain approximation which – rather romantically – had been exhumed from Riemann’s own papers at Göttingen where it had lain for seventy years. But for extending the calculation to thousands of new zeroes a fresh approximation was required; and this Alan set out to find and to justify.
The second problem, quite different, was the ‘dull and elementary’ one of actually doing the computation, with numbers substituted into the approximate formula, and worked out for thousands of different entries. It so happened that the formula was rather like those which occurred in plotting the positions of the planets, because it was of the form of a sum of circular functions with different frequencies. It was for this reason that Titchmarsh had contrived to have the dull repetitive work of addition, multiplication, and of looking up of entries in cosine tables done by the same punched-card methods that were used in planetary astronomy. But it occurred to Alan that the problem was very similar to another kind of computation which was also done on a large practical scale – that of tide prediction. Tides could also be regarded as the sum of a number of waves of different periods: daily, monthly, yearly oscillations of rise and fall. At Liverpool there was a machine23 which performed the summation automatically, by generating circular motions of the right frequencies and adding them up. It was a simple analogue machine; that is, it created a physical analogue of the mathematical function that had to be calculated. This was a quite different idea from that of the Turing machine, which would work away on a finite, discrete, set of symbols. This tide-predicting machine, like a slide rule, depended not on symbols, but on the measurement of lengths. Such a machine, Alan had realised, could be used on the zeta-function calculation, to save the dreary work of adding, multiplying, and looking up of cosines.
Alan must have described this idea to Titchmarsh, for a letter24 from him dated 1 December 1937 approved of this programme of extending the calculation, and mentioned: ‘I have seen the tide-predicting machine at Liverpool, but it did not occur to me to use it in this way.’
There were some diversions. The hockey playing continued, although without Francis Price and Shaun Wylie the team had lost its sparkle. Alan found himself involved in making the arrangements. He also played a good deal of squash. At Thanksgiving he drove north to visit Jack and Mary Crawford for a second time. (‘I am getting more competent with the car.’) Before Christmas, Alan took up an invitation from his friend Venable Martin to go and stay with him. He came from a small town in South Carolina.
We drove down from here in two days and then I stayed there for two or three days before I came back to Virginia to stay with Mrs Welbourne. It was quite as far south as I had ever been – about 34°. The people seem to be all very poor down there still, even though it is so long since the civil war.
Mrs Welbourne was ‘a mysterious woman in Virginia’ who had a habit of inviting English students from the Graduate College for Christmas. ‘I didn’t make much conversational progress with any of them,’ Alan had to confess of her family. Alan and Will Jones organised another treasure hunt, although it lacked the élan of the previous year; one of the clues was in his collected Shaw. And in April he and Will made a trip to visit St John’s College, Annapolis, and Washington. ‘We also went and listened to the Senate for a time. They seemed very informal. There were only six or eight of them present and few of them seemed to be attending.’ They looked down from the gallery and saw Jim Farley, Roosevelt’s party boss. It was another world.
The main business of the year was the completion of his PhD thesis,25 investigating whether there was any way of escaping the force of Gödel’s theorem. The fundamental idea was to add further axioms to the system, in such a way that the ‘true but unprovable’ statements could be proved. But arithmetic, looked at in this way, had a distinctly hydra-headed nature. It was easy enough to add an axiom so that one of Gödel’s peculiar statements could be proved. But then Gödel’s theorem could be applied to the enlarged set of axioms, producing yet another ‘true but unprovable’ assertion. It could not be enough to add a finite number of axioms; it was necessary to discuss adding infinitely many.
This was just the beginning, for as mathematicians well knew, there were many possible ways of doing ‘infinitely many’ things in order. Cantor had seen this when investigating the notion of ordering the integers. Suppose, for example, that the integers were ordered in the following way: first all the even numbers, in ascending order, and then all the odd numbers. In a precise sense, this listing of the integers would be ‘twice as long’ as the usual one. It could be made three times as long, or indeed infinitely many times as long, by taking first the even numbers, then remaining multiples of 3, then remaining multiples of 5, then remaining multiples of 7, and so on. Indeed, there was no limit to the ‘length’ of such lists. In the same way, extending the axioms of arithmetic could be done by one infinite list of axioms, or by two, or by infinitely many infinite lists – there was again no limit. The question was whether any of this would overcome the Gödel effect.
Cantor had described his different orderings of the integers by ‘ordinal numbers’, and Alan described his different extensions of the axioms of arithmetic as ‘ordinal logics’. In one sense it was clear that no ‘ordinal logic’ could be ‘complete’, in Hilbert’s technical sense. For if there were infinitely many axioms, they could not all be written out. There would have to be some finite rule for generating them. But in that case, the whole system would still be based on a finite number of rules, so Gödel’s theorem would still apply to show that there were still unprovable assertions.
However, there was a more subtle question. In his ‘ordinal logics’, the rule for generating the axioms was given in terms of substituting an ‘ordinal formula’ into a certain expression. This was itself a ‘mechanical process’. But it was not a ‘mechanical process’ to decide whether a given formula was an ordinal formula. What he asked was whether all the incompleteness of arithmetic could be concentrated in one place, namely into the unsolvable problem of deciding which formulae were ‘ordinal formulae’. If this could be done, then there would be a sense in which arithmetic was complete; everything could be proved from the axioms, although there would be no mechanical way of saying what the axioms were.
He likened the job of deciding whether a formula was an ordinal formula to ‘intuition’. In a ‘complete ordinal logic’, any theorem in arithmetic could be proved by a mixture of mechanical reasoning, and steps of ‘intuition’. In this way, he hoped to bring
the Gödel incompleteness under some kind of control. But he regarded his results as disappointingly negative. ‘Complete logics’ did exist, but they suffered from the defect that one could not count the number of ‘intuitive’ steps that were necessary to prove any particular theorem. There was no way of measuring how ‘deep’ a theorem was, in his sense; no way of pinning down exactly what was going on.
One nice touch on the side was his idea of an ‘oracle’ Turing machine, one which would have the property of being able to answer one particular unsolvable problem (like recognising an ordinal formula). This introduced the idea of relative computability, or relative unsolvability, which opened up a new field of enquiry in mathematical logic. Alan might have been thinking of the ‘oracle’ in Back to Methuselah, through whose mouth Bernard Shaw answered the unsolvable problems of the politicians with ‘Go home, poor fool’!
Less clear from his remarks in the paper was to what extent he regarded such ‘intuition’, the ability to recognise a true but unprovable statement, as corresponding to anything in the human mind. He wrote that:
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. (We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions.) The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning….
and he claimed that his ideas on ‘ordinal logics’ represented one way of formalising this distinction. But it was not established that ‘intuition’ had anything to do with the incompleteness of finitely defined formal systems. After all, no one knew of this incompleteness until 1931, while intuition was a good deal older. It was the same ambiguity as in Computable Numbers, which mechanised mind, yet pointed out something beyond mechanisation. Did this have a significance for human minds? His views were not clear at this stage.
As for the future, his intention was to return to King’s, provided that, as expected, they renewed his Fellowship which was, in March 1938, at the end of its first three years. On the other hand, his father wrote advising him (not very patriotically, perhaps) to find an appointment in the United States. For some reason King’s College was slow in notifying him that the extension of his fellowship had been made. Alan wrote to Philip Hall on 30 March:
I am writing a thesis for a Ph.D, which is proving rather intractable, and I am always rewriting parts of it….
I am rather worried about the fact that I have heard nothing about re-election to my Fellowship. The most plausible explanation is simply that there has been no re-election, but [I] prefer to think there is some other reason. If you would make some cautious enquiries and send me a postcard I should be very grateful.
I hope Hitler will not have invaded England before I come back.
After the union with Austria on 13 March everyone was beginning to take Germany seriously. Meanwhile, Alan did dutifully go to Eisenhart and ask him ‘about possible jobs over here; mostly for Daddy’s information, as I think it is unlikely I shall take one unless you are actually at war before July. He didn’t know of one at present, but said he would bear it all in mind.’ But then a job materialised. Von Neumann himself offered a research assistantship at the IAS.
This might have meant a certain priority being given to von Neumann’s research areas – at that time in the mathematics associated with quantum mechanics and other areas of theoretical physics, and not including logic or the theory of numbers. On the other hand, a position with von Neumann would be the ideal start to the American academic career, which, presumably, Alan’s father thought wise. Competition was intense, and the market, already in depression, was flooded by European exiles. The stamp of approval from von Neumann would carry great weight.
In professional terms, this was a big decision. But all Alan wrote of the opportunity to Philip Hall on 26 April was: ‘Eventually a possibility of a job here turned up’, and to Mrs Turing on 17 May ‘I had an offer of a job here as von Neumann’s assistant at $1500 a year but decided not to take it.’ For he had cabled King’s to check that the fellowship had been renewed, and since it had, the decision was clear.
Despite himself, he had made his name in the Emerald City. It was not entirely necessary to have a reputation in order to be listened to. By this time, von Neumann was aware of Computable Numbers, even if he had not been a year earlier. For when he travelled with Ulam to Europe in this summer of 1938, he proposed26 a game of ‘writing down on a piece of paper as big a number as we could, defining it by a method which indeed has something to do with some schemata of Turing’s.’* But whatever the attractions, remunerations and compliments, the real issue was much simpler. He wanted to go home to King’s.
The thesis, which in October he had hoped to finish by Christmas, was delayed. ‘Church made a number of suggestions which resulted in the thesis being expanded to an appalling length.’ A clumsy typist himself, he engaged a professional, who in turn made a mess of it. It was eventually submitted on 17 May. There was an oral examination on 31 May, conducted by Church, Lefschetz and H. F. Bohnenblust. ‘The candidate passed an excellent examination, not only in the special field of mathematical logic, but also in other fields.’ There was a quick test in scientific French and German as well. It was vaguely absurd, to be examined in this way while at the same time he was refereeing the PhD thesis of a Cambridge candidate. This, as it happened, he had to reject. (To Philip Hall on 26 April: ‘I hope my remarks don’t encourage the man to go and rewrite the thing. The difficulty with these people is to find out a really good way of being blunt. However I think I’ve given him something to keep him quiet a long time if he really is going to rewrite it.’) The PhD was granted on 21 June. He made little use of the title, which had no application at Cambridge, and which elsewhere was liable to prompt people to retail their ailments.
His departure from the land of Oz was rather different from that in the fable. The Wizard was not a phoney, and had asked him to stay. While Dorothy had disposed of the Wicked Witch of the West, in his case it was the other way round. Though Princeton was fairly secluded from the orthodox, Teutonic side of America, it shared in a kind of conformity that made him ill at ease. And his problems remained unresolved. He was inwardly confident – but as in the Murder in the Cathedral which he saw performed in March (‘very much impressed’) he was living and partly living.
In one way, however, he resembled Dorothy. For all the time, there was something that he could do, and which was just waiting for the opportunity to emerge. On 18 July Alan disembarked at Southampton from the Normandie, with the electric multiplier mounted on its breadboard and wrapped up securely with brown paper. ‘Will be seeing you in the middle of July’, he had written to Philip Hall, ‘I also expect to find the back lawn criss-crossed with 8ft. trenches.’ It had not come to that, but there were more discreet preparations, in which he could take part himself.
Alan was right in thinking that HM Government was concerned with codes and ciphers.* It maintained a department to do the technical work. In 1938 its structure was still a legacy of the Great War, a continuation of the organisation discreetly known as Room 40 that the Admiralty had set up.
After the initial break of a captured German code book, passed to the Admiralty by Russia in 1914, a great variety of wireless and cable signals had been deciphered by a mostly civilian staff, recruited from universities and schools. The arrangement had the peculiar feature that the director, Admiral Hall, enjoyed control over diplomatic messages (for instance the famous Zimmermann telegram). Hall was no stranger to the exercise of power.27 It was he who showed Casement’s diary to the press, and there were more important instances of his28 ‘acting on intelligence independently of other departments in matters of policy that lay beyond the concerns of the Admiralty.’ The organisation survived the armistice,
but in 1922 the Foreign Office succeeded in detaching it from the Admiralty. By then it had been renamed as the ‘Government Code and Cypher School’, and was supposed to study29 ‘the methods of cypher communication used by foreign powers’ and to ‘advise on the security of British codes and cyphers.’ It now came technically under the control of the head of the secret service,* himself nominally responsible to the Foreign Secretary.
The director of GC and CS, Commander Alastair Denniston, was allowed by the Treasury to employ thirty civilian Assistants,30 as the high-level staff were called, and about fifty clerks and typists. For technical civil-service reasons, there were fifteen Senior and fifteen Junior Assistants. The Senior Assistants had all served in Room 40, except perhaps Feterlain, an exile from Russia who became head of the Russian section. There was Oliver Strachey, who was brother of Lytton Strachey and husband of Ray Strachey, the well-known feminist, and there was Dillwyn Knox, the classical scholar and Fellow of King’s until the Great War. Strachey and Knox had both been members of the Keynesian circle at its Edwardian peak. The Junior Assistants had been recruited as the department expanded a little in the 1920s; the most recently appointed of them, A. M. Kendrick, had joined in 1932.
The work of GC and CS had played an important part in the politics of the 1920s. Russian intercepts leaked to the press helped to bring down the Labour government in 1924. But in protecting the British Empire from a revived Germany, the Code and Cypher School was less vigorous. There was a good deal of sucess in reading the communications of Italy and Japan, but the official history31 was to describe it as ‘unfortunate’ that ‘despite the growing effort applied at GC and CS to military work after 1936, so little attention was devoted to the German problem.’
Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game Page 25