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Alan Turing — a Cambridge Scientific Mind by Andrew Hodges for Cambridge Scientific Minds  (Cambridge University Press, 2002). For a guide to this website go to the Alan Turing Home Page

Isolation and Universality

This creates an isolation both for mathematical culture, and for individual mathematicians. There are escape routes: one is a parochial tunnel vision, or a sort of mathematical camp, making a joke of everything. But there is a heavier burden for those who see their mathematics as the foundation stone of certainty. Turing expressed this from the beginning. An isolated schoolboy in his philistine public-school setting, Turing read Einstein and Eddington, and wrote as if they were his friends. From them, perhaps the first modern media figures in science, and perhaps also from Bertrand Russell, he acquired the confidence to speak from the most shy and solitary position of universal themes. This confidence did not come from his family status; the Turings were anxious clingers to upper-middle-class status. It came from the Republic of Numbers.

Thus a young Turing announced the universal machine in 1936; the backroom-boffin proposed an electronic computer design in 1946; the unknown Manchester mathematician announced the prospect of artificial intelligence in 1950. But when he spoke to this effect on BBC radio in 1951, there was something the listeners could not know: unlike Eddington, Einstein, and Russell this unknown theorist had been stupendously useful.

The Uses of Numeracy

Hardy's sharpest reproof was to utilitarian — in his time marxist-influenced — talk of mathematics for planning and order. (It reads oddly now. Today's emphasis would be on short-term profit, which never entered into Hardy's dialectic.) It is now a commonplace remark that Hardy was wrong about relativity and quantum mechanics having no military use, as was soon to be shown. Hardy would have been surprised, I think, that advanced number theory is now of great significance for commercial cryptography. Though allowing the possibility of such an application, Hardy still ended ringingly 'I have done nothing useful.' Turing made no such predictions of purity and issued no soul-searching apology.

But Hardy's point was essentially a moral one about utility being irrelevant to value: the aesthetics of Wilde. He denied the justification by utility because it was untrue: not the real motivation for real mathematical thought. In this awkward truth-telling, Turing also shared Hardy's integrity of integers. 'Phoney' was one of his favourite words of opprobrium, along with 'Politicians, Charlatans, Salesmen.' Turing's motivations, even his practical work, were as unrelated to public or private profit as Hardy's.

It is curious to think whether Turing's obscure and unfashionable work of 1936 would receive a grant by modern criteria, which encourage scientists to make their work show immediate commercial relevance. Turing did in fact describe his ideas to his economist friend David Champernowne, who was appropriately sceptical of its practical value.

But Turing had also another mathematical magic, one that Hardy only touched upon as a secondary characteristic. Within the gamut of mathematics lies its strange special harmony with the physical world; hence follows a prophetic role of mathematics as the vanguard of science. Like Newton, Gauss, Riemann, Hamilton, and Hilbert — doing uneconomic work that took centuries to develop, differential geometry before relativity, complex vector spaces before quantum mechanics — the universal Turing machine came before electronic computers.

Turing had this prophetic gift, and his work moved from the pure and abstruse to world-conquering utility. Turing's 1936 work took the logical puzzles of self-reference, developed by Gödel from the work of Cantor and Russell, and by expressing it in the language of calculation, found both the absolute limits of computability, and the concept of the universal machine. His paper spoke to a small subset of pure mathematicians. Yet within ten years Turing turned the universal machine into the practical invention of the computer in its modern sense. Turing codified operations on numbers by numbers, and thereby both solved a deep problem in mathematics, and identified an idea essential to the modern world. Computer programs, operating on data, are themselves data; and Turing was eager to put this idea into application as soon as it could be embodied in electronics.

That application required the war. And war means loss, for mathematics as for everything else. Hardy's 'useless' relativity and quantum mechanics left a Cold War legacy of escalating arsenals; so did Turing's logic.

Matter and spirit

I have described him elsewhere as natural philosopher rather than mathematician, for Turing's inspirations had a first base in chemistry, the physics of mind and matter, in which the mind of Eddington and the matter of his first love, Christopher Morcom, combined. Cambridge mathematics was the right vehicle for this train of thought; and Newman, topologist and true Cambridge mind, was the catalyst who brought Turing to a frontier in the Entscheidungsproblem.  But it was not Cambridge that taught Turing to treat logic as applied mathematics; that came from something deeper. As Newman wrote in 1955, his introducing paper tape into logic came as a shock; perhaps partly because it was a mundane technological image sullying high mathematics. But unlike Hardy, Turing did not distinguish classes of mathematical work. In 1937, while absorbed in his most abstruse work, the development of 'ordinal logics' to probe and classify the uncomputable, Turing was also thinking about applying logic to ciphers, and building a cipher machine with electromagnetic relays.

Turing applied unsuccessfully for a Cambridge University lectureship in 1938; the University never employed him.

Die Zweite Heimat

But there was 'real' mathematics too: in group theory and probability, Turing brought the power of mathematics to a pre-scientific world (where only the Polish algebraists had gone before) and created his own information theory. To break the Enigma, Turing and Welchman put their astonishing logic into the electromagnetic Bombes. At the same time Turing took on Naval Enigma alone disregarding the discouragement of his superior, Commander Denniston: 'You know, the Germans don't mean you to read their stuff, and I don't expect you ever will.' And why did he? His colleague Hugh Alexander's report, released from secrecy 55 years later, reveals that Turing said they had  to be broken because the intellectual challenge was so great. Broken they duly were. Hardy-like, he refused to say he had done it from the call of Duty; the truth was that he had found it fascinating. I have always found the greatest drama in this, that the innocence of deeply unworldly mathematics, met the call of the greatest world crisis, and met it at its very centre.

Turing was fully aware of international events; he had even sponsored a refugee. But his Cambridge scientific mind made the war a chess game, one we now know to have been a duel with the young German logician Gisbert Hasenjaeger, entrusted with the keys of the Reich. After losing that war Hasenjaeger returned to logic, and when eventually interviewed on television, had outlived his victor by forty-five years. In 1936-7 Turing portrayed cryptography as something that would flow from his logic, something that would be a game against Germany, and something that meant an essentially moral choice, a sacrifice of purity. He chose in 1938, as Snow White bit on the apple. Did he sense even then, as he signed the Official Secrets Act, that he was killing truthfulness? It is strange irony that Turing's magical design for the Bombe turned upon the concept of following through the proliferating implications of false hypotheses.

The Bombe required a novel synthesis of ideas and engineering; and an ingenuity of logic that few at Bletchley Park understood; it was a miracle that Denniston was persuaded to invest so much in a great gamble. The first Bombe to be delivered was named Agnus  by Turing: a joke that atheist Hardy might have made (though Hardy at that moment was making his unapologetic apology, trying to avoid thought of the war). Further irony showed in the names they used: Turingismus  as if this ingenious statistical analysis of the Lorenz machine ciphers were some Hegelian philosophy, ROMSing,  as if the placing of long paper tapes against each other were the Resources of Modern Science  of marxist planners. 'Cillies'  were silly German Enigma operators' errors; but also perhaps the silliness of spending scientific talent on the crimes and folly of mankind. (Now, exiles from mathematical physics who enter the ephemera of futures markets, computer operating systems and e-commerce could echo the sentiment.) Newman, at Bletchley after 1942, lamented that it could not have been a pure-mathematical research group.

But it was science; unlike what was shown in television's Station X,  it was the power of scientific method; the production line of information with Turing uniquely placed to see electronics making program handling practical, and so able to embody the universal machine. Turing saw the future far clearer than practical people, having borrowed from it to defeat Nazi Germany.

Private plans and publication

Turing never clearly specified or published his claim to the stored program concept. Untroubled by Research Assessment citation counts, he left that problem to historians of the computer. Newman probably never thought of the computer as real mathematics, and might well not have thought its central principle worth publishing. It is more of a puzzle that Turing, quite prepared to pursue trivial mathematics, did not elucidate the way that the principle of 1936 had turned to the practice of 1946.

But Turing's computer was neither the useless reality of Hardy nor the useful reality of the atomic bomb; it was to do with simulating the brain, the arena for experiment on his thesis that the operation of the mind is computable. Even in his practical report for the NPL in 1946, this priority had shone through. As I have argued elsewhere, it was probably in 1941, in what must have seemed a revelation, perhaps enhanced by seeing machines overtake aspects of human guessing, that he decided intuition and originality must also be computable processes, explicable by programs allowed to change through experience into forms not originally written down. By 1945, Turing had concluded that the uncomputable was not needed to account for intuition or creativity, and thereafter argued vigorously to this effect. It is remarkable that Turing who showed such originality, denied originality its apparent meaning; he claimed intuition was all learning by experience. He was an anti-Socrates, though his life led to a Socratic end.

Turing set forth a major scientific paradigm as a result, by opening the arena of the discrete state machine; this at least he did not throw away; and it has given him his lasting status as a philosopher as well as mathematician. He transformed the ancient mind-body question by connecting it with computability, and turned it into a new experimental science. He was not saying that the brain is like a computer in its architecture; rather, as he often explained, that if the operation of the brain was computable, then no matter how complex, then it could be simulated by a program on a computer. For this was the well-proved property of the universal machine. And this was the theme that drew out his famous 1950 publication.

It was overconfident in its fifty-year prediction, as in 2000 we know, but we see now an over-confidence in all the cybernetic pioneers, the optimism perhaps essential to all inspired research, depression being its inevitable corollary. But this still leaves a puzzling aspect to Alan Turing: he failed to follow up or publish his neural net sketches, the learning machines whose potential was fundamental to his vision of intelligent machinery. There was a self-defeating element; a dark side to the vision. His ideas remained unknown when the American school of thought developed after 1956. Likewise he was lost to the history of practical computer development, never publishing his programming ideas.

Lying with Men

Another possible answer lies in the Wilde side: the demoralising burden of trying to live an honest criminal life. For each man kills the thing he loved,  he told Joan Clarke in 1941 when he rejected the temptation to live a lie, rather more dramatically than indicated in the staged and televised Breaking the Code.  His gay identity gained more support and confidence after the war, and what he had seen as a curse in 1930s, he began to see as a frontier of exploratory consciousness. It was the wrong time in the wrong place for that. Once arrested, Turing was turned into class traitor and cold-war risk in the moral panics and national security crisis of 1952. Amidst this, for him the issues of truth and trust were paramount. Turing believes machines think, Turing lies with men, Therefore machines do not think,  he wrote, camp humour combined with mathematical camp, but with truth at the centre. It was his honest way of dealing with the intensely dishonest world of the worldly, though with another irony, Turing was found out through his telling a lie to the police. Cambridge was little help to him in this; he was too open and democratic to be a Cambridge mind. But Newman was his character witness and called Turing 'particularly honest and truthful.' After the initial lie, Turing acquitted himself with Hardy's honesty: because, as Hardy said of the primality of 317, it is so.

Unlike Hardy, Alan Turing was on the receiving end of war through his usefulness; it made his brain a repository of the most secret knowledge of the West, and a security risk when he took that brain to Norway or Greece. Such provocative adventure was not the same as Hardy's escape route, the cricket field. But both must have known the same soul-destroying burden of being counted as less than human: as unbearable perhaps as trying to implement mathematics that no-one understood, or knowing that salvation had come from that which society hates and fears and concerning which one must remain completely silent.

Newman was described as 'a professor of pure mathematics' in the local newspaper's criminal reports. And so he was: just another professor. Newman and Turing, like Hardy, had an entirely realistic view of their worldly status. Richard Feynman, who nearly overlapped with Turing at those stifling Princeton tea-parties, had the fortune to live into times where such conflicts could be expressed more easily. Why should he care what other people think  could be asked as well of Turing as of Feynman, and again like Hardy, who in C. P. Snow's summary 'just didn't give a damn.'

Turing, like Feynman, could be accused of a higher selfishness along with high mathematics. In his last days Turing insisted on a social ideal of free association quite incompatible with his secret state status, rather as Feynman felt himself 'the only free man' who could speak up for his Pasadena low-life. But Turing was not a free man. His last postcards, in March 1954 referred mysteriously to electrons not allowed to associate too freely; another maths-camp joke, given meaning by the dark rider, that it was laid down for the good of the electrons themselves. The Agnus Dei, in Christian myth, is sacrificed by his own side.

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