Alan Turing: The Enigma (31 page)

Meanwhile
the possibility had arisen of staying at Princeton for a second year. Alan wrote home on 22 February:

I went to the Eisenharts regular Sunday tea yesterday and they took me in relays to try and persuade me to stay another year. Mrs. Eisenhart mostly put forward social or semi-moral semi-sociological reasons why it would be a good thing to have a second year. The Dean weighed in with hints that the Procter Fellowship was mine for the asking (this is worth $2000 p.a.) I said I thought King’s would probably prefer that I return, but gave some vague promise that I would sound them on the matter. The people I know here will all be leaving, and I don’t much care about the idea of spending a long summer in this country. I should like to know if you have any opinions on the subject. I think it is most likely I shall come back to England.

Dean Eisenhart was an old-fashioned figure, who in his lectures would apologise for using the modern abstract group, but very kind. He and his wife made noble efforts to entertain the students at their tea-parties. Whatever his parents thought, Philip Hall had sent Alan the notice of vacancies for Cambridge lectureships, and this Alan would much have preferred if he could gain one. A lectureship would in effect mean a permanent home at Cambridge, which was the only possible resolution of his problems in life, as well as being due recognition of his achievements. Alan wrote back to him on 4 April:

I am putting in for it, but offering fairly heavy odds against getting it.

He also wrote to his mother, who was just setting off on a pilgrimage to Palestine:

Maurice and I are both putting in for it, though I don’t suppose either of us will get it: I think it is a good thing to start putting in for these things early, so as to get one’s existence recognised. It’s a thing I am rather liable to neglect. Maurice is much more conscious of what are the right things to do to help his career. He makes great social efforts with the mathematical big-wigs.

As he forecast, he failed to gain a Cambridge appointment. Ingham wrote from King’s, encouraging him to stay for another year, and this made up his mind. He wrote on 19 May:

I
have just made up my mind to spend another year here, but I shall be going back to England for most of the summer in accordance with previous programme. Thank you very much for your offer of help with this: I shall not need it, for if I have this Procter as the Dean suggests I shall be a rich man, and otherwise I shall go back to Cambridge. Another year here on the same terms would be rather an extravagance. …

My boat sails June 23. I might possibly do a little travelling here before the boat goes, as there will be very little doing here during the next month and it’s not a fearfully good time of year for work. More likely I shall not as I don’t usually travel for the sake of travelling.

I am sorry Maurice won’t be here next year. He has been very good company.

I am glad the Royal Family are resisting the Cabinet in their attempt to keep Edward VIII’s marriage quiet.

Since he was staying another year, he decided he should take a PhD degree, as Maurice had done. For his thesis, Church had suggested a topic that had come up in his lecture course, relating to the implications of Gödel’s theorem. Alan had written in March that he was ‘working out some new ideas in logic. Not so good as the computable numbers, but quite hopeful.’ These ideas would see him through.

As for the Procter Fellowship, it did indeed fall into his lap. It was for the Vice-Chancellor of Cambridge University to nominate the Fellow, so there were letters of recommendation sent to him. One of these was from the Wizard himself, who wrote:
¹²

June 1, 1937

Sir,

Mr A.M. Turing has informed me that he is applying for a Proctor [
sic
] Visiting Fellowship to Princeton University from Cambridge for the academic year 1937–1938. I should like to support his application and to inform you that I know Mr Turing very well from previous years: during the last term of 1935, when I was a visiting professor in Cambridge, and during 1936–1937, which year Mr Turing has spent in Princeton, I had opportunity to observe his scientific work. He has done good work in branches of mathematics in which I am interested, namely: theory of almost periodic functions, and theory of continuous groups.

I think that he is a most deserving candidate for the Proctor Fellowship, and I should be very glad if you should find it possible to award one to him.

I am, Respectfully, John von Neumann

Von Neumann would have been asked to write the letter, because his name carried such weight. But why did he make no mention of
Computable Numbers
, a far more substantial piece of work than the papers to which he referred? Had Alan failed to make him aware of it, even after the paper had been printed, and reprints sent round? If Alan had an
entrée
with von Neumann, the first thing he should have done was to exploit it to help bring
Computable Numbers
to attention. It would be typical of what was perceived as his lack of worldly sense, if he had been too shy to push his work upon the ‘mathematical bigwig’.

Against
Alan’s prediction, and perhaps to his mild chagrin, the redoubtable Maurice Pryce had been appointed to a Cambridge lectureship, as had Ray Lyttleton, the current Procter Fellow. And Alan did after all spend some time in travel, for Maurice Pryce sold him his car, a 1931 V8 Ford, which had taken him all over the continent on the tour that as a Commonwealth Fellow he had been obliged to make in summer 1936. Maurice taught him to drive, which was not an easy task, for Alan was ham-handed and not good with machines. Once he nearly reversed into the Carnegie Lake and drowned them both. Then on about 10 June they took off together for a Turing family visit, which no doubt Mrs Turing had long been urging upon Alan. It was to a cousin on her mocher’s side who had emigrated from Ireland. Jack Crawford now nearly seventy, was the retired Rector of Wakefield, Rhode Island.

The visit was not quite the expected grim chore of conventional politeness, for Alan approved of Jack Crawford, who in his youth had studied at the then Royal College of Science in Dublin:

I enjoyed the time I spent at Cousin Jack’s. He is an energetic old bird. He has a little observatory with a telescope that he made for himself. He told me all about the grinding of mirrors.… I think he comes into competition with Aunt Sybil for the Relations Merit Diploma. Cousin Mary is a little bit of a thing you could pick up and put in your pocket. She is very hospitable and rather timid: she worships Cousin Jack.

They were ordinary people, who made Alan feel more at home than did the grand Princeton figures. In their old-fashioned country way, they put Alan and Maurice in the same double bed.

The compartments of life were fractured. Maurice was amazed – he had not had the slightest suspicion. Alan apologised and withdrew at once. Then he blazed out, not with a trace of shame, but with
anger
, with a story of how his parents had been away in India so long, and of his years in boarding schools. It had all been said before, in
The Loom of Youth:

Then Jeffries’ wild anger, the anger that had made him so brilliant an athlete, burst out: ‘Unfair? Yes, that’s the right word; it is unfair. Who made me what I am but Fernhurst? … And now Fernhurst, that has made me what I am, turns round and says, “You are not fit to be a member of this great school!” and I have to go. …’

The deeply embarrassing moment brought to light a vein of self-pity that he otherwise never showed, as well as an analysis which he must have known to be facile. It would not do. It was time to look forward, not backward – but to what? How was he to continue? Maurice accepted the explanation, and they never spoke of it again. Alan boarded the
Queen Mary
on his twenty-fifth birthday, and on 28 June disembarked at Southampton. He missed the Fourth of July softball match at the Graduate College, between the British Empire and the Revolting Colonies.

Back
for three months in the mild Cambridge summer of 1937, there were three major projects on hand. First there was some tidying up of
Computable Numbers
. Bernays, in Zurich, had perhaps rather annoyingly found some errors
¹³
in his proof that the Hilbert decision problem, in its precise form, was unsolvable, and these had to be put right by a correction note in the LMS
Proceedings
. He also completed a formal demonstration
¹⁴
that his own ‘computability’ coincided exactly with Church’s ‘effective calculability’. By now there existed yet a third definition of the same sort of idea. This was the ‘recursive function’, which was a way of making absolutely precise the notion of defining a mathematical function in terms of other more elementary functions; Gödel had suggested it, and it had been taken up by Kleene. It was implicit in Gödel’s proof of the incompleteness of arithmetic. For when Gödel showed that the concept of proof by chess-like rules was an ‘arithmetical’ concept, one as ‘arithmetical’ as finding a highest common factor or something of the kind, he was really saying it could be done by a ‘definite method’. This idea, when formalised and extended somewhat, led to the definition of the ‘recursive function’. And now it had turned out that the general recursive function was exactly equivalent to the computable function. So Church’s lambda-calculus, and Gödel’s way of defining arithmetical functions, both turned out to be equivalent to the Turing machine. Gödel himself later acknowledged
¹⁵
the Turing machine construction as the most satisfactory definition of a ‘mechanical procedure’. At the time, it was a very striking and surprising fact, that three independent approaches to the idea of doing something in a definite way, had converged upon equivalent concepts.

The second project concerned the ‘new ideas in logic’ for a doctoral thesis. The basic idea was to see if there was any way in which to escape the force of Gödel’s result that there would always be true but unprovable assertions within arithmetic. This was not a new question, for Rosser, now at Cornell, had produced a paper
¹⁶
in March 1937 which took it up. But Alan planned a rather more general attack on the question.

His third project was a very ambitious one, for he had decided to try his strength on the central problem of the theory of numbers. It was not a new interest, for he had possessed Ingham’s book on the subject since 1933. But in 1937 Ingham sent him some recent papers,
¹⁷
on learning that he wished to make an attack himself. It was ambitious because the question he chose was one that had long absorbed and defeated the greatest pure mathematicians.

Although
the prime numbers were such ordinary things, it was easy to pose quite baffling questions about them in a few words. One question had been solved very early on. Euclid had been able to show that there were infinitely many prime numbers, so that although in 1937 the number 2
¹²⁷
– 1 = 170141183460469231731687303715884105727 was the largest known prime, it was also known that they continued for ever. But another property that was easy to guess, but very hard to prove, was that the primes would always thin out, at first almost every other number being prime, but near 100 only one in four, near 1000 only one in seven, and near 10,000,000,000 only one in 23. There had to be a reason for it.
¹⁸

In about 1793, the fifteen-year-old Gauss noticed that there was a regular pattern to the thinning-out. The spacing of the primes near a number
n
was proportional to the number of digits in the number n; more precisely, it increased as the natural logarithm of
n
. Throughout his life Gauss, who apparently liked doing this sort of thing, gave idle leisure hours to identifying all the primes less than three million, verifying his observation as far as he could go.

Little advance was made until 1859, when Riemann developed a new theoretical framework in which to consider the question. It was his discovery that the calculus of the complex numbers
^*
could be used as a bridge between the fixed, discrete, determinate prime numbers on the one hand, and smooth functions like the logarithm – continuous, averaged-out quantities – on the other. He thereby arrived at a certain formula for the density of the primes, a refinement of the logarithm law that Gauss had noticed. Even so, his formula was not exact, and nor was it proved.

Riemann’s formula ignored certain terms which he was unable to estimate. These error terms were only in 1896 proved to be small enough not to interfere with the main result, which now became the Prime Number Theorem, and which stated in a precise way that the primes thinned out like the logarithm – not just as a matter of observation, but proved to be so for ever and ever. But the story did not end here. As far as the tables went it could be seen that the primes followed this logarithmic law quite amazingly closely. The error terms were not only small compared with the general logarithmic pattern; they were
very
small. But was this also true for the whole infinite range of numbers, beyond the reach of computation? – and if so what was the reason for it?

Riemann’s
work had put this question into a quite different form. He had defined a certain function of the complex numbers, the ‘zeta-function’. It could be shown that the assertion that the error terms remained so very small, was essentially equivalent to the assertion that this Riemann zeta-function took the value zero only at points which all lay on a certain line in the plane. This assertion had become known as the Riemann Hypothesis. Riemann had thought it Very likely’ to be true, and so had many others, but no proof had been discovered. In 1900 Hilbert had made it his Fourth Problem for twentieth century mathematics, and at other times called it ‘the most important in mathematics, absolutely the most important’. Hardy had bitten on it unsuccessfully for thirty years.