This article was written for a conference on Mathematics and War, Karlskrona, Sweden, August 2002. The editor may yet suggest changes for the published version.
Pages 1 | 2 | 3 | References
There is a common joke about Military Intelligence being a contradiction in terms, and the concept of the 'military use of mathematics' has a similar problem, for mathematics is not just used: it has to be created by awkward, non-standard, individual mathematicians. A subordinate cannot simply be ordered to do something extraordinary, unexpected and innovative, such as breaking a cipher system that is supposed to be unbreakable. Anyone approaching such work in the spirit of carrying out an order, is unlikely to be inspired. Turing exercised a wilful initiative and went against prevailing wisdom to attack the naval Enigma systems. Such an unorthodox success, however welcome in the desperate situation of 1940, clearly signified a certain danger for government control: if people act from self-willed initiative, rather than following orders and duty, they may progress to some other self-willed initiative less to the taste of the State. As later became well known, some of those who decided for themselves to fight Germany, also chose with equal force to assist the USSR. In the postwar period the emergent idea of 'security' had to take this into account, and after 1948 Turing was bound to find that the wartime personal networks, class-based trust and easy acceptance of his eccentricity no longer applied in state affairs.
What makes this particularly interesting and piquant in discussing Turing's biography, is that the interplay of rule-following Duty and creative Will was central to Turing's own theory of mind and machine. In his 1948 report on machine intelligence he described the problems to be addressed in terms of 'discipline' and 'initiative,' the latter being necessary to anything that could genuinely be called 'intelligence'. In his discussion of machine intelligence, a dialogue between mechanical and creative, obedience and surprise, runs throughout. It is a striking fact that he came down so strongly on the side of saying computers would be able to show intelligence, when he of all people knew the significance of creative originality. But this dialogue of compliance and rebellion — and resolving it by standing aside, withdrawing — ran through his life. In his 1950 paper, Turing made a joking reference to Casabianca, the boy on the burning deck who carries out his orders relentlessly as a computer, oblivious to common sense, and this image of military duty went back to his childhood learning of the poem at preparatory school, still within a First World War world. Even as a child he had known the limitations of rule-following, and had found the precepts of his class largely incomprehensible.
Yet Alan Turing had not been a rebel against his class or his school. In his upper-middle-class environment, dominated by the suffocating ideology of the Public School, he had neither complied nor rebelled; he had largely withdrawn into a scientific world of his own. There was, however, a moment in 1933 when the Anti-War student movement at Cambridge articulated the change that had taken place since the world of Duty plunged to disaster in 1914, and at twenty-one, Turing joined it and placed himself clearly on the modernist side. His contemporaries were insisting on deciding for themselves what to fight for, and the emergent left-liberal side decided not to fight for that old chant of Duty: 'King and Country.' 'Blatant militarist propaganda', Turing called a cinema recruiting film called 'Our Fighting Navy.' Ten years later, Alan Turing was to be seen by his WRNS assistants 'prancing' in his Bletchley Park hut at the news of the sinking of the Scharnhorst by that very Navy. Of course, the 1933 enlightenment had coincided with the one development that made war justifiable to the enlightened: the transformation of Europe's strongest industrial power into the aggressive engine of murderous fascism.
The young Alan Turing was well aware of the shock of Nazi Germany, and it was obvious where his sympathies lay. When visiting Germany in 1934, Turing's travelling companion was surprised to see how naturally a German socialist seemed to confide in him. Later, he was immediate in his response to the 1938 wave of persecution, and sponsored a young Jewish refugee. But in the intellectual context of the 1930s, dominated by the question of Communist party policy, Alan Turing's 1933 political engagement was short-lived; he was regarded as 'not a political person.' Alister Watson, who introduced him to Wittgenstein, was a Communist party member, as was also the young Robin Gandy who later became his student and closest friend. Others of Alan Turing's friends were fully engaged with political and economic issues. But for Alan Turing, it was the 'phoney' that attracted his scorn rather than political opposition; the compromises and alliances of political action were alien to him.
In temperament he was closer to his first lover, James Atkins, another mathematics student of his year, who was and remained a pacifist while becoming a professional musician. But Alan Turing found in the music of mathematics the means to undermine Nazi Germany. In 1936, just after he had completed his famous paper on computable numbers, and had arrived in Princeton, Alan Turing wrote to his mother, who was apt to ask him 'but what use is it':
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? (Turing 1936)
What theory of a 'most general' cipher he had in mind, what was his theory of cipher security, and and what were the interesting codes, remain unknown. Nor, in Turing's reference to the question of 'morality of such things', is it clear whether he means the morality of selling his work, or the morality of doing military work. However, whatever moral consideration Turing found most perplexing, the fact is that at Princeton in 1937 he proceeded with a cryptographic idea embodied in an electromagnetic relay device, and saw it specifically as relevant to looming war with Germany. It is very possible that this device was indeed offered to the British government, and was the reason why he became the first scientific officer at the Government Code and Cypher School. He received a salary, and later some special payments, but what was more important, he paid a moral price: he sacrificed the freedom of his mind.
Freedom, to Alan Turing, was more important than questions of money. He waxed more eloquent over the Abdication crisis than over any other issue, expressing, perhaps naively, a support for the King's freedom to marry, and deploring the 'hypocrisy' of the Archbishop of Canterbury. But tellingly, he deprecated the indiscretion of the ex-King in allowing state papers to be seen by Mrs Simpson. (Hodges 1983, p. 122) Soon afterwards, he had to promise to keep State secrets himself. But no doubt he found it exciting to be let into government's innermost secrets, and perhaps too the technical challenge of the Enigma problem became immediately and addictively fascinating.
The Abdication issue was an anticipation of his own quandary after his arrest as a homosexual in 1952. For in 1952 he showed himself the most devoted to personal freedom, but at the same time, as his friends did not know, he was the most tightly bound by secrecy at the highest level in the Anglo-American alliance. He was ahead of his time, as with all things, in an open insistence on his sexual identity, and his response to his trial and punishment (with hormone injections) in England was to travel over Europe. News of the Forbundet af 1948 organisation in Denmark and Norway, essentially the first open European gay movement, was a particular attraction.
It has yet to be disclosed what the security officials of Britain and the United States made of his priorities, but at a time when American commentators tended to place homosexuality on a level with communism as a danger to American interests it is not surprising that Turing was watched closely in 1953 when a young Norwegian tried to visit him. He was probably naive on such questions, for homosexuality had not been in itself a well-defined 'security' issue in 1939-45. As Donald Michie has emphasised, there were gay men at Bletchley Park even more open than Alan Turing. The change towards explicit 'vetting' for homosexuality, and the explicit exclusion of homosexuals, came only after 1948, and possibly Turing was unaware of the position he had placed himself. So in 1952 he was shocked to be excluded from secret work, and in 1953 highly indignant to be the object of police surveillance. It was this 'security' development which for society generally, as individually for Turing, created a change of consciousness, politicising a hitherto 'personal' issue.
In assessing Alan Turing's place in mathematics and war, we cannot overlook the culture of mathematics itself, of which he was part. It was and is a reticent and quiet culture. The introduction to this article mentioned a double secrecy surrounding Alan Turing's life: the official secrecy surrounding cryptology, and the social taboo about his sexuality. But it is a general reality for mathematicians that the very nature of their subject attracts almost as great an enforced silence. Very recently, mathematical culture has dipped its toe in the business of fortune and celebrity, but in the long aftermath of the Second World War it had no such profile. The contrast with the prestige of physics after the open and visible atomic bomb is particularly notable. The Second World War lesson was well learnt, and the National Security Agency and its British counterpart GCHQ became major employers of mathematical graduates, so that the postwar flowering of mathematics has to some extent rested upon the demands of the most secret government work. But the near-invisibility of mathematics combines easily with the total secrecy to make this alliance one incapable of arousing public interest.
Gauss called mathematics the queen of the sciences, who sometimes condescends to serve. The priorities of self-effacing mathematical culture are well illustrated by Newman, who was faced with the very difficult task of writing a Biographical Memoir of Turing immediately after his dramatic suicide in June 1954. Turing had been elected a Fellow of the Royal Society in 1951 and as such called for such a detailed Memoir, which appeared as (Newman 1955).
Alan Turing had died at the height of his powers in 1954, wrote Newman, who interestingly did not say that his apex lay either before or in the Second World War. Because of its secrecy, Newman could say nothing of significance about the wartime work, but even allowing for this constraint, he severely understated Turing's contribution with bland expressions such as 'a mild routine' and 'congenial set of fellow-workers.' Newman also understated Turing's role in the emergence of the computer. He stated that Turing's theory of the universal machine was not known by the designers of post-war digital computers, and omitted Turing's own design (Turing 1946) from his list of Turing's works. Newman gave far more attention in his Memoir to Turing's very abstract and difficult work on ordinal logics (Turing 1939). Above all, he portrayed the Second World War not as allowing Turing to turn the logical theory into practical application, but as interrupting Turing in his work on the logic of the uncomputable and the Riemann Hypothesis, preventing him from settling into a 'serious' problem. Thus subtly, perhaps unconsciously, he put an Anti-War impetus into his assessment — discounting those subjects that the war had accelerated, emphasising those that it had interrupted. His priorities were essentially the reverse of those that modern computer science would expect.
Newman was perfectly aware of the importance of Turing's work to worldly affairs. Newman knew it at first hand, and in no way dissented from the common agreement that Turing was the leading figure in the wartime work. In 1946 Newman wanted to reject the formal decoration he was offered by the government because he considered the rank of OBE that had been awarded to Turing to be so absurd an undervaluation. But Newman had been repelled from the emergent world of the computer, and it seems that by 1955 he considered both war and computation, however important politically and economically, to be secondary, transitory, matters compared with the timeless mathematics that it was his task to assess. Perhaps he was right: the queen of the sciences sometimes sees far ahead of her subjects. It is a point of view that Turing would have respected: in his last period he probably drew as much strength as he could from the Platonic qualities of the mathematical sciences, as counter to the harsh ironies of the world.
In 1953/4 Turing wrote a last semi-popular article on undecidable problems (Turing 1954a) and studied the axioms of quantum mechanics: topics that owed nothing to his Second World War experience. It seems that he wanted to go back to what he had learnt in1932 from von Neumann's axioms of quantum mechanics, and to think out for himself a new quantum mechanical theory, focussing on the problem of when and how wave-function 'reduction' is supposed to take place (Gandy 1954). Since the 1980s, Roger Penrose's views on logic, complex analysis and physics (Penrose 1989, 1994) have made this late development particularly intriguing. Penrose's views stand as the most radical contradiction of Turing's on the possibility of machine intelligence, but are based on the same materialist ground and focus on the very topics that perplexed Turing most: the interpretation of Gödel's theorem and the reality of a quantum-mechanical infrastructure to the brain. In particular (Penrose 1994) offers a detailed mathematical critique of Turing's 'mistakes' argument, as well as concentrating on the puzzle of the 'reduction' process. Had Turing followed his late interests further, and combined them with his knowledge of computability, he might have seen something much deeper connecting logic, space-time and quantum mechanics, than anything he did for computers. We cannot know.
The Second World War obviously stimulated progress in significant areas of mathematics and science. But maybe it stunted the growth of more subtle things that might have been. It is perhaps too soon to assess the balance. Many mathematicians might feel, though would perhaps not say openly, that the principle of the computer is a trivial matter compared with the serious problems of mathematics. Newman probably took this view in 1955; it is hard to imagine what Turing really thought, for he was no typical mathematician and had a vision that only partially overlapped with that of the classical discipline.
At the end, Alan Turing wrote in March 1954 a cryptic joke about quantum mechanics to his friend Robin Gandy, on a postcard (Turing 1954b) headed 'Messages from the Unseen World':
The exclusion principle is laid down purely for benefit of the electrons themselves, who might be corrupted (and become dragons or demons) if allowed to associate too freely.Free association was, of course, just what the State had forbidden to him. But this dialogue of acquiescence and dissent; of seriousness and humour; of self-assertion and withdrawal; abstraction and concreteness — these last words were typical of Alan Turing in mathematics and in war.
Pages 1 | 2 | 3 | References