Alan Turing: The Enigma (29 page)

He
enjoyed the play of
ideas
, and in the same letter he let slip a hint of ideas in which Bernard Shaw himself might have found a plot:

You have often asked me about possible applications of various branches of mathematics. I have just discovered a possible application of the kind of thing I am working on at present. It answers the question ‘What is the most general kind of code or cipher possible’, and at the same time (rather naturally) enables me to construct a lot of particular and interesting codes. One of them is pretty well impossible to decode without the key, and very quick to encode. I expect I could sell them to H.M. Government for quite a substantial sum, but am rather doubtful about the morality of such things. What do you think?

Ciphering would be a very good example of a ‘definite method’ applied to symbols, something that could be done by a Turing machine. It would be essential to the nature of a cipher that the encipherer behave like a machine, in accordance with whatever rules had been fixed in advance with the receiver of the message.

As for a ‘most general code or cipher possible’, in a sense any Turing machine could be regarded as encoding what it read on its tape, into what it wrote on the tape. However, to be useful there would have to be an
inverse
machine, which could reconstruct the original tape. His result, whatever it was, must have started on these lines. But as for the ‘particular and interesting codes’ he offered no further clue.

Nor did he touch again on the conflict indicated by the word ‘morality’: what was he to do? Mrs Turing, of course, was a Stoney; she assumed that science existed for the sake of useful applications, and she was not the person to doubt the moral authority of His Majesty’s Government. But the intellectual tradition to which Alan belonged was quite different. It was not only for the detachment of Cambridge, but for a very significant section of modern mathematical opinion that G.H. Hardy spoke when he wrote:
³

The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholely ‘useless’ (and this is true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work. … The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill.

In making explicit his response to the growing separation of mathematics from applied science, Hardy attacked the shallowness of the current ‘left-wing’ Lancelot Hogben interpretation of mathematics in terms of social and economic utility, an interpretation based on the ‘dull and elementary’ aspects of the subject. Hardy spoke more for himself, however, in holding that ‘useful’ mathematics had in any case worked more for ill than for good, being preponderantly military in application. He held the total uselessness of his own work in the theory of numbers to be a positive virtue, rather than a matter for apology:

No-one
has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.

Hardy’s own near-pacifist convictions stemmed from before the Great War, but no one touched by the Anti-War movements of the 1930s could fail to be unaware of a view that military applications were to be shunned. If Alan had now discovered something like a ‘warlike purpose’ in the play of symbols, he was faced, at least in embryo, with a mathematician’s dilemma. Behind the off-hand, teasing words to his mother, there lay a serious question.

Meanwhile the English students were brightening the Graduate College life with amusements of their own:

One of the Commonwealth Fellows, Francis Price (not to be confused with Maurice Pryce …) arranged a hockey match the other day between the Graduate College and Vassar, a women’s college (amer.)/university (engl.) some 130 miles away. He got up a team of which only half had ever played before. We had a couple of practice games and went to Vassar in cars on Sunday. It was raining slightly when we arrived, and what was our horror when we were told the ground was not fit for play. However, we persuaded them to let us play a pseudo-hockey match in their gymn. at wh[ich] we defeated them 11–3. Francis is trying to arrange a return match, which will certainly take place on a field.

The amateurism was deceptive, since Shaun Wylie, the topologist, and Francis Price, the physicist, both from New College, Oxford, were players of a national standard. Alan was hardly in the same class (even if he was not now ‘watching the daisies grow’), but enjoyed the games. Soon they were playing three times a week amongst themselves, and sometimes against local girls’ schools.

The effete English playing a women’s game might well have amazed the native Princeton students, but within the establishment there was a somewhat embarrassing anglophilia, in that all the most stuffy and mannered aspects of the English system were admired. In the summer of 1936, the Princeton chapel had been packed for a memorial service for George V. There was a professor in the Graduate College who harped upon his admiration for the royalty in a way that to educated English ears seemed only vulgar. As for George V’s successor, the revelations of Edward VIII’s Mediterranean cruise and Mrs Simpson created a particular sensation at Princeton. Alan wrote to his mother on 22 November:

I am sending you some cuttings about Mrs Simpson as representative sample of what we get over here on this subject. I don’t suppose you have even heard of her, but some days it has been ‘front page stuff’ here.

Indeed,
the British newspapers maintained their silence until 1 December, when the bishop of Bradford remarked that the King stood in need of God’s grace, and Baldwin showed his hand. On 3 December, Alan wrote:

I am horrified at the way people are trying to interfere with the King’s marriage. It may be that the King should not marry Mrs Simpson, but it is his private concern. I should tolerate no interference by bishops myself, and I don’t see that the King need either.

But the King’s marriage was not a private matter, but one that reflected upon the British state. It was a prophetic episode for Alan, ‘horrified’ at government interference with an individual life. For his class, the horror was rather that the King himself had betrayed King and Country, a logical paradox more upsetting than any that Russell or Gödel had found.

On 11 December the Windsors went into their butterfly life of exile, and the reign of George VI began. Alan wrote that day:

I suppose this business of the King’s abdication has come as rather a shock to you. I gather practically nothing was known of Mrs Simpson in England till about ten days ago. I am rather divided in my opinion of the whole matter. At first I was wholly in favour of the King retaining the throne and marrying Mrs Simpson, and if this were the only issue it would still be my opinion. However I have heard tales recently which seem to alter it rather. It appears that the King was extremely lax about state documents, leaving them about and letting Mrs Simpson and friends see them. There had been distressing leakages. Also one or two other things of same character, but this is the one I mind about most. However, I respect David Windsor for his attitude.

Alan’s respect extended to the acquisition of a gramophone record of the abdication speech. He further wrote on 1 January:

I am sorry that Edward VIII has been bounced into abdicating. I believe the Government wanted to get rid of him and found Mrs Simpson a good opportunity. Whether they were wise to try to get rid of him is another matter. I respect Edward for his courage. As for the Archbishop of Canterbury I consider his behaviour disgraceful. He waited until Edward was safely out of the way arid then unloaded a whole lot of quite uncalled-for abuse. He didn’t dare do it whilst Edward was King. Further he had no objections to the King having Mrs Simpson as a mistress, but marry her, that wouldn’t do at all. I don’t see how you can say that Edward was guilty of wasting his ministers’ time and wits at a critical moment. It was Baldwin who opened the subject.

The archbishop’s broadcast, of 13 December, had denounced the King for abandoning his duties for a mere ‘craving for private happiness’; the pursuit of happiness had never been accorded a high priority by the British rulers. Alan’s views on marriage and morals were those of a modernist; in a discussion at King’s with his theological contemporary Christopher Stead he had said that people should let their natural feelings take their course – and as for bishops, a class of person particularly dear to Mrs Turing, they epitomised for him the
ancien régime
. He talked to Venable Martin, his American friend from Church’s logic class, about the ‘very shabby way’ in which the King had been treated.

As
for work, on 22 November he wrote to Philip Hall:

I have not made any very startling discoveries over here, but I shall probably be publishing two or three small papers: just bits and pieces. One of them will be a proof of Hilbert’s inequality if it really turns out to be new, and another on groups which I did about a year ago and Baer thinks is worth publishing. I shall write these things up and then have another go at the Math[ematical] logic.

I find that ‘Go’ is only played very little here now, but I have had two or three games.

Princeton is suiting me very well. Beyond the way they speak there is only one – no two! – feature[s] of American life which I find really tiresome, the impossibility of getting a bath in the ordinary sense, and their ideas on room temperature.

By ‘the way they speak’, Alan meant such complaints as:
⁴

These Americans have various peculiarities in conversation which catch the ear somehow. Whenever you thank them for anything, they say ‘You’re welcome’. I rather liked it at first, thinking I was welcome, but now I find it comes back like a ball thrown against a wall, and become positively apprehensive. Another habit they have is to make the sound described by authors as ‘Aha’. They use it when they have no suitable reply to a remark, but think that silence could be rude.

The proofs of
Computable Numbers
had been sent to him at Princeton just after he had arrived, so that publication of the paper was imminent. Meanwhile, Alonzo Church had suggested that Alan might be able to give one of the regular seminars, to launch his discovery into the mainstream of Princeton mathematics. On 3 November he had written home:

Church has just suggested to me that I should give a lecture to the Mathematical Club here on my Computable Numbers. I hope I shall be able to get an opportunity to do this, as it will bring the thing to people’s attention a bit. I don’t expect the lecture will come off for some time yet.

In fact he only had to wait a month, but then there was a disappointment:

There was rather bad attendance at the Maths Club for my lecture on Dec. 2. One should have a reputation if one hopes to be listened to. The week following my lecture G.D. Birkhoff came down. He has a very good reputation and the room was packed. But his lecture wasn’t up to the standard at all. In fact everyone was just laughing about it afterwards.

It was also disappointing that when in January 1937
Computable Numbers
at last appeared in print, there was so little reaction. Church reviewed it for the
Journal of Symbolic Logic
, and thereby put the words ‘Turing machine’ into published form. But only two people asked for offprints: Richard Braithwaite back at King’s and Heinrich Scholz,
⁵
the almost lone representative of logic left in Germany, who wrote back saying that he had given a seminar on it at Münster, and begged almost plaintively for two copies of any future papers, explaining how difficult it was for him to keep abreast of developments otherwise. The world was rather less of a single country for mathematics now. Alan wrote home on 22 February:

I
have had two letters asking for reprints. … They seemed very much interested in the paper. I think possibly it is making a certain amount of impression. I was disappointed by its reception here. I expected Weyl who had done some work connected quite closely with it some years ago, at least to have made a few remarks about it.

He might also have expected John von Neumann to have made a few remarks about it. Here was a truly powerful Wizard playing against Alan’s version of the innocent Dorothy. Like Weyl, he had been very interested in the Hilbert programme and had once hoped to fulfil it, although his active interest in mathematical logic had ended with Gödel’s theorem. He once claimed
⁶
that after 1931 he never read another paper in logic, but this was at most a half-truth, for he was a prodigious reader, working long before anyone else got up in the morning, and covering the whole gamut of mathematical literature. Yet there was not a word about him at this point in Alan’s letters to his mother or Philip Hall.

As for the general readership of the LMS
Proceedings
, there were a number of reasons why Alan’s paper was unlikely to make an impression upon them. Mathematical logic seemed to be a marginal area of research, which many mathematicians would consider either as tidying up what was obvious anyway, or as creating difficulties where none really existed. The paper started attractively, but soon plunged (in typical Turing manner) into a thicket of obscure German Gothic type in order to develop his instruction tables for the universal machine. The last people to give it a glance would be the applied mathematicians who had to resort to practical computation in some field such as astrophysics or fluid dynamics, where the equations did not allow an explicit solution. There was little encouragement offered to them to do so.
Computable Numbers
made no concession to practical design, not even for the limited range of logical problems to which the machines were applied in the paper itself. For instance, he had made a convention that the machines should print out the ‘computable numbers’ on alternate squares of the ‘tape’, and use the intervening squares as working space. But it would have been much easier if he had made a more generous allowance of working space. So there was little about his work to attract anyone from outside the narrow circle of mathematical logic – with the possible exception of pure mathematicians who would be interested in the distinction between the computable numbers and the real numbers. It had nothing obvious to do with what Lancelot Hogben called ‘the world’s work’.