Undecidables

Stuart Hampshire

  • Alan Turing: The Enigma by Andrew Hodges
    Burnett, 587 pp, £18.00, October 1983, ISBN 0 09 152130 0

This is a very long biography, and before it appeared Alan Turing was not very well-known; his genius was of a kind that is not likely to be spread abroad. An immense amount of work has gone into this book, which expresses profound, and sometimes almost obsessional, admiration. It is not hagiography, but rather a study of a hero, an intellectual hero. I found it continuously readable and interesting, and it will, I think, be found moving and unforgettable by those who are ready to enter into the cryptographical and mathematical technicalities. The author quite often steps forward and gives the reader a piece of his mind on public issues, and his manner of presentation and style are as unlike those of an assured professional biographer as they could possibly be. But the style matches the subject. Alan Turing evidently was proud to be an odd-man-out: he insisted on informality in all circumstances, and even among his mathematical colleagues he seems to have cultivated an air of amateurishness. Mr Hodges gives a most convincing picture of this side of his character.

The story of an intellectually adventurous life begins with an immensely normal, typically English, middle-class family and an ordinary public school education: no disturbing precocity as a schoolboy, no outward brilliance marking him out, as Russell and F.P. Ramsey were marked out in their earlier years. Mr Hodges includes photographs of seaside family holidays, redolent of their period and typical of a social class. He includes letters between Turing and a school friend to whom Turing was passionately attached, who died prematurely and who was always to be remembered with emotion. At Sherborne, and before Cambridge, Turing showed his independence and the gift for solitary thinking which he never lost: but he did not display any overwhelming mastery as a mathematician. Mr Hodges remarks that Turing did not at any time revolt against his school and family background in the supposedly traditional manner of intellectuals. A ‘dowdy, Spartan amateurism, in which possessions and consumption played a small role’, was a not unusual feature of middle-class public schools at that time. At Cambridge he rowed in the boat and played bridge with his contemporaries at King’s. Throughout his life he enjoyed communal games and simple amusements. But while he was still an undergraduate, he was invited to read a paper to the Moral Sciences Club, an unusual honour, and his subject was ‘Mathematics and Logic’. In 1935, at the age of 22, he was elected to a research fellowship at King’s, also an unusual recognition of his potentialities. His early achievements in logic and mathematics became known to the British mathematician Max Newman, and to the great John von Neumann in Princeton. He already speculated simultaneously on the foundations of mathematics, in Hilbert’s and Gödel’s sense, and on the conceivable capacities and limits of computing machines, and this combination of interests was his genius. He liked to think like an engineer, and at the same time to think in the broadest logical abstractions. Mr Hodges describes very well this two-faced talent, which set him apart from the more single-minded mathematicians and logicians of his time.

The paper that has preserved the name ‘Turing’ in perpetuity, at least for logicians, was published in 1937 with the title ‘Computable Numbers’. It was the first substantial response to the incompleteness theorem established by the Austrian logician living in the USA, Kurt Gödel. Gödel has been plausibly described as the greatest logician since Aristotle, because his incompleteness proof, formal and final, showed that any mathematical system of an interesting complexity must generate propositions which cannot be shown within the system to be true or shown to be false. The proof blocked the programme of the so-called formalist mathematicians, led by Hilbert, who had wanted to exhibit mathematical systems as derivable without remainder, and without paradox and contradiction, from the basic axioms of logic. The decision problem was left as a challenge to logicians, and Turing threw new light on it. He introduced the concept of computability by machine, the machine being an abstract entity described in mathematical terms and supposed capable of repeating its basic numerical operations indefinitely, without limit of time or space. Being computable by Turing machine was a test of the decidability of any predicate applicable to numbers. This brilliant result drew the attention of John von Neumann, the Hungarian physicist and mathematician who was probably the most powerful and inventive thinker of his time, a Leibniz of the 20th century, who had left Budapest and Germany for Princeton. His interest had been in quantum theory and he was the principal inventor of the theory of games. Turing went to Princeton, but in spite of von Neumann, he did not wish to remain in the USA and returned to his fellowship at King’s in Cambridge, his natural environment. After these few darkening pre-war years he was to make his second contribution to history, his second claim to be remembered.

Drawing on F.H. Hinsley’s magisterial history of wartime intelligence, and on other histories of the Ultra material, Mr Hodges describes Turing’s famous years at the Government Code and Cipher establishment at Bletchley, where he devised the machine which, reviewing alternative possibilities with sufficient speed, could just keep ahead, most of the time, with changes of key in the setting of the German naval ciphers on the Enigma machine. This success was an essential condition of winning the U-boat war in the Atlantic, which was an essential condition of preserving Britain as the launching-pad for the destruction of Germany from the West. Luck and genius came together at the right time, and Mr Hodges rightly sees that the informality of recruitment within the small old-boy network, based on Cambridge and (in this case less) Oxford, was totally effective. GC and CS, Bletchley Park, was a ramshackle and improvisatory organisation, unhierarchical and casually egalitarian, and its successes were successes of improvisation and of individual talent. Churchill understood its power, and there is an amusing story, told again here, of the direct appeal to him by cryptographers for facilities which had been denied to them by their thick-headed superior officers. Turing was evidently of the type of supremely talented, nonconforming Englishman, impatient, obstinate, independent, who despises generals, admirals, university deans, and all the trappings of authority. He had the temperament, and many of the attitudes, of George Orwell, and his family background was similar. ‘Phony’ was a favourite word of his, freely applied to anyone whose status or fame exceeded their proven abilities. Wittgenstein, the mathematician Max Newman, the economist David Champernowne, von Neumann, Robin Gandy the logician, were men with whom he could be at ease, free of the social patina and the trappings of respectibility which repelled him.

There are records of Turing’s participation in pre-war Cambridge, in Wittgenstein’s class on the philosophy of mathematics. Why should mathematicians make a fuss about paradoxes and contradictions revealed in their deductive systems? Did the contradictions do any harm? Is there any connection between a proof, in the mathematicians’ sense, and a claim to truth? These were Wittgenstein’s questions. At all stages of his short life Turing’s own work was to require philosophical scepticism, and his achievements emerged from a radical questioning of received opinions. Against Wittgenstein he stood by an orthodox account of mathematical truth, as vindicated by the application of mathematics: if we changed our arithmetic, our bridges would collapse. Mr Hodges sees that Wittgenstein and Turing were alike, first in being solitary thinkers, and, secondly, in being drawn to questions that are fundamental and simple: simple in the sense that technical ingenuity and elaboration are evidently insufficient for their solution. They call for a special kind of insight, which suddenly tilts the settled intellectual landscape to one side and illuminates it from a quite unexpected angle, obliterating many of the familiar lines of division within a discipline.

As the machines which Turing had designed at Bletchley, originally with Dilwyn Knox and Peter Twinn, were standardised and their development became a routine, he turned to the third area of his achievement: the conception of a general-purpose machine, more precisely of an automatic electronic digital computer with internal storage programme. In 1943-4, at a government research establishment, typically working with one assistant in a small hut, he turned away from the abstract general machine of 1936, designed for logic and notionally capable of calculations unlimited in extent, and formed the idea of a concrete but general machine, limited in the number and speed of its calculations only by the potentialities of electronic engineering. Mr Hodges explains that Turing’s genius included a flair for concrete and diagrammatic thinking. Like an 18th-century amateur designing scientific instruments, he liked by himself to imagine and to put together the tools that he would use. He therefore naturally saw that the new speeds of electronic switching, and new techniques of storage of information, must make a qualitative difference. A machine might soon be constructed that would come near to the human brain in the range of its adaptability to entirely different problems and to the receipt of entirely different information. A common form of thought, reproduced in the design of a universal thinker, would assimilate a great variety of entirely different subject-matters.

There is an old beguiling and utterly hopeless question about the fit between the thinker and his age, as also about the fit between the artist and his age. Von Neumann and Turing stand in the same relation to the conceptual innovation, and to the technology, of the succeeding age, our time, as did Galileo to the age following him. The straightforward historian will point out that it was not a change in the means of production which constituted the immediate stimulus to new thought, which was the accelerator, but rather the demands of government in war and then in the post-war competition in armaments. But at a less superficial level the evolution of the disciplines themselves – mathematics, logic, physics and engineering – favours in a definite decade a particular mutation of human intelligence. Von Neumann’s genius was versatility and economy. With extraordinary speed and certainty he seemed able to eliminate the noise in any communication he received and to disentangle the essential content. Turing seems to have had a contrasting habit of single-minded concentration: peripheral questions were totally ignored, and even his mathematical interests were narrow. But his thought was permanently to affect psychology and philosophy, no less than mathematics and logic, even though he was far from being a polymath like von Neumann.

The practical study of a general-purpose computer was in the usual way underfunded and largely neglected by the Government’s scientific advisers in England. With a few exceptions, conspicuously Professor Newman and Professor Flowers, they missed the point. The analogy between the human brain, and the manifest outcomes of its unknown processes, and the general-purpose computer more and more dominated Turing. So was born the study of artificial intelligence, the ‘AI’ now flourishing as a separate discipline in many universities and research institutes all over the world. In the philosophy department at Manchester University, where Max Newman was professor of mathematics, there was a discussion in October 1949 under the title ‘The Mind and the Computing Machine’, with Turing participating. His view then appeared in a celebrated article in the journal Mind for October 1950, under the title ‘Computing Machinery and Human Intelligence’. This article was his fourth important achievement and it was the first milestone on a now crowded highway of speculation about minds and machines. He argued that the function of thinking could be distinguished from other aspects of the human mind – for example, feeling and willing – and that the nature of thought was wholly revealed in its products, taken together with the inputs or stimuli stored in the mind. A machine which calculates and remembers, and thereby solves problems, is as much a thinker as persons are.

The disconcerting element in this argument was the claim that the experienced processes of thinking, often present in consciousness, are irrelevant to the characterisation of thought, and that the embodiment of thought in the human brain, which is also the vehicle of feeling and willing, is irrelevant to the proper characterisation of thought. Turing was aware that to overlook the link within the brain of the functions of thinking, abstractly defined, with the functions of feeling and willing would be to overlook some of the causes and effects of thinking, as usually defined, and therefore to overlook some of the features of thought, more broadly interpreted. It is a disability of machines that there should be ‘the difficulty of the same kind of friendliness occurring between man and machine as between white man and white man or black man and black man’. Organised knowledge advances by ruthless abstraction, and for Turing it was an incidental bonus if the abstraction necessary to scientific advance happened to offend conventional opinion and to puncture spiritual pretensions. After Turing it became a problem for psychologists, aided by philosophers, to look for possible links between functional descriptions of the brain and the distinguishable physical structures and processes within it. But Turing turned aside to problems in biology, looking for mathematical characterisations of some forms of growth in plants: yet another problem of generalised abstraction. His life ended in suicide in 1954.

It is to be hoped that there may be additions to Mr Hodges’s splendid book by some of his close friends mentioned here. I do not believe, on the evidence provided, all that Mr Hodges writes towards the end of the book, when he represents Anglo-American security precautions after the war as a great blow to Turing, to some degree spoiling his life by excluding him from secret government work. Mr Hodges’s own account, and the sources he cites, suggest that Turing’s strongest interests were not in service to governments. When Mrs Max Newman gave him War and Peace, it is not a surprise that, in her words, ‘he wrote to me expressing in moving terms his appreciation of Tolstoy’s understanding and thought.’ He had spent his short life in thinking about the nature of thought and he was evidently possessed by a sentiment of uncompromising integrity and self-respect, a Tolstoyan pride in his autonomy and independence, and in a certain moral isolation. He happened to be of a homosexual disposition and he loved a few men, sometimes happily, sometimes not. He was prosecuted for a homosexual offence, and received a comparatively mild but barbarous sentence: compulsory drug therapy. He was apparently calm in the face of this outrage and of the publicity that accompanied it. Not long afterwards he took poison. Mr Hodges makes Turing’s homosexuality a central theme of this book, and he puts a quotation from Whitman at the beginning of every chapter. It is not possible to guess whether this emphasis is true or false historically – true or false, that is, to Turing’s experience and feeling. But the reason for the emphasis is obvious, and will be respected.