Fred was very interested in an exchange of views and emotional experiences, feeling himself very puzzled by sex, and confronted by ex-public schoolboys much more conscious of homosexual attraction. He had taken advantage of the King’s freedom of discussion, and had been told by a Fellow that he ‘seemed a pretty normal bisexual male’. But it was not as simple as that, and nothing was ever simple for Fred Clayton.
Alan told his friend about how much he resented having been circumcised, and also of his earliest memories of playing with the gardener’s boy (presumably at the Wards’ house), which he thought had perhaps decided his sexual pattern. Rightly or wrongly, he gave Fred, and others too, the impression that public schools could be relied upon for sexual experiences – although more important perhaps was that schooldays continued to loom large in his consciousness of sexuality. Fred read Havelock Ellis and Freud, and also made discoveries in the classics which he would convey to his mathematical friend, not noted otherwise for an interest in Latin and Greek.
Puzzlement was an entirely reasonable reaction, in the conditions of 1933, when even in King’s there was so little to go on for those outside the most chromium-plated circles. These conversations were whispers in a crushing, deafening silence. It was not the effect of the law, whose prohibition of all male homosexual activity played but a tiny part in the Britain of the 1930s, in direct terms. It was more as J.S. Mill had written20 of heresy:
… the chief mischief of the legal penalties is that they strengthen the social stigma. It is that stigma which is really effective, and so effective is it, that the profession of opinions which are under the ban of society is much less common in England than is, in many other countries, the avowal of those which incur risk of judicial punishment.
Modern psychology had made a twentieth-century difference; the 1920s had given to the avant-garde the name of Freud to conjure with. But his ideas were used in practice to discuss what had ‘gone wrong’ with homosexual people, and such intellectual openings were outweighed by the continual efforts of the official world to render homosexuality invisible – a process in which the academic world played its part along with prosecutions and censorship. As for respectable middle class opinion, it was represented by the Sunday Express in 1928, greeting The Well of Loneliness with the words, ‘I had rather give a healthy boy or a healthy girl a phial of prussic acid than this novel.’ The general rule remained that of unmentionability above all else, leaving even the well-educated homosexual person with nothing more encouraging than the faint signals from the ancient world, the debris of the Wilde trials, and the rare exceptions to the rules supplied by the writings of Havelock Ellis and Edward Carpenter.
In a peculiar environment such as Cambridge, it might be a positive advantage to enjoy homosexual experience, simply in terms of the opportunity for physical release. The deprivation was not one of laws but of the spirit – a denial of identity. Heterosexual love, desire and marriage were hardly free from problems and anguish, but had all the novels and songs ever written to express them. The homosexual equivalents were relegated – if mentioned at all – to the comic, the criminal, the pathological, or the disgusting. To protect the self from these descriptions was hard enough, when they were embedded in the very words, the only words, that language offered. To keep the self a complete and consistent whole, rather than split into a facade of conformity, and a secret inner truth, was a miracle. To be able to develop the self, to increase its inner connections and to communicate with others – that was next to impossible.
Alan was at the one place that could support that development. Here, after all, was the circle round which Forster passed the manuscript of his novel Maurice which conveyed so much about being ‘an unmentionable of the Oscar Wilde sort’. How to complete the work, that was one problem. It had to have its own integrity of feeling, yet be credible as a story of the real world. There was a fundamental contradiction, which was not resolved by having his hero escape into the ‘greenwood’ of a happy ending.
There was another contradiction, in that this attempt at communication remained secret for fifty years.21 But here at least was the place where these contradictions were understood, and although Alan’s self-contained nature placed him on the edge of King’s society, he was protected from the harshness of the outside world.
If Alan enjoyed Back to Methuselah, it would also have been because Shaw dramatised his theory of the Life Force, which raised the same questions as the ‘spirit’. One of Shaw’s characters said ‘Unless this withered thing religion, and this dry thing science, come alive in our hands, alive and intensely interesting, we may just as well go out and dig the garden until it is time to dig our graves.’ This was Alan’s problem in 1933, although he could not accept Shaw’s easy solution. Bernard Shaw had no compunction about rewriting science if it did not agree with his ideas; determinism had to go, if it conflicted with a Life Force. Shaw fixed on Darwin’s theory of evolution, which he discussed as if it were an account of every kind of change, social and psychological change included, and rejected it as a ‘creed’: he wrote22 that
What damns Darwinian Natural Selection as a creed is that it takes hope out of evolution, and substitutes a paralysing fatalism which is utterly discouraging. As Butler put it, it ‘banishes Mind from the universe.’ The generation that felt nothing but exultant relief when it was delivered from the tyranny of an Almighty Busybody by a soulless Determinism has nearly passed away, leaving a vacuum which Nature abhors.
Science, for Shaw, had to be a Life Force, of which the super-intelligent Oracle of A.D. 3000 could say ‘Our physicists deal with it. Our mathematicians express its measurements in algebraic equations.’
But for Alan, science had to be true, rather than comforting. Nor did that mathematician and physicist John von Neumann have anything to say that lent credence to a Life Force. His Mathematische Grundlagen der Quantenmechanik had arrived in October 1932, but perhaps Alan had put off tackling it until the summer, when he also obtained books on quantum mechanics by Schrödinger and Heisenberg. On 16 October 1933 he wrote:
My prize book from Sherborne is turning out very interesting, and not at all difficult reading, although the applied mathematicians seem to find it rather strong.
Von Neumann’s account was very different from Eddington’s. In his formulation, the state of a physical system evolved perfectly deterministically; it was the observation of it that introduced an element of absolute randomness. But if this process of observation were itself observed from outside, it could be regarded as deterministic. There was no way of saying where the indeterminacy was; it was not localised in any particular place. Von Neumann was able to show that this strange logic of observations — quite unlike anything encountered with everyday objects – was consistent in itself, and agreed with known experiments. It left Alan sceptical about the interpretation of quantum mechanics, but certainly gave no support to the idea of the mind manipulating wave-functions in the brain.
Alan would not only have found von Neumann’s book ‘very interesting’ because it was tackling a subject of such philosophical importance to himself. It would also have been because of the way in which von Neumann approached his scientific subject as much as possible by logical thought. For science, to Alan Turing, was thinking for himself, and seeing for himself, and not a collection of facts. Science was doubting the axioms. He had the pure mathematician’s approach to the subject, allowing a free rein to thought, and seeing afterwards whether or not it had application to the physical world. He would often argue on these lines with Kenneth Harrison, who took the more traditional scientific view of experiments and theories and verification.
The ‘applied mathematicians’ would have found von Neumann’s study of quantum mechanics to be ‘rather strong’ because it required a considerable knowledge of recent pure-mathematical developments. He had taken the apparently different quantum theories of Schrödinger and Heisenberg, and by expressing their essential ideas in a much more abstract mathematical form, sh
own their equivalence. It was the logical consistency of the theory, not the experimental results, that von Neumann’s work treated. This suited Alan, who sought that kind of toughness, and it made a beautiful example of how the expansion of pure mathematics for its own sake had borne unexpected fruit in physics.
Before the war, Hilbert had developed a certain generalisation of Euclidean geometry, which involved considering a space with infinitely many dimensions. This ‘space’ had nothing to do with physical space. It was more like an imaginary graph on which could be plotted all musical sounds, by thinking of a flute, or violin, or piano tone as made up of so much of the fundamental, so much of the first harmonic, so much of the second harmonic, and so on – each kind of sound requiring (in principle) the specification of infinitely many ingredients.* A ‘point’ in such a ‘space’, a ‘Hilbert space’, would correspond to such a sound; then two points could be added (like adding sounds), and a point could be multiplied by a factor (like amplifying a sound).
Von Neumann had noticed that ‘Hilbert space’ was exactly what was needed to make precise the idea of the ‘state’ of a quantum-mechanical system, such as that of an electron in a hydrogen atom. One characteristic of such ‘states’ was that they could be added like sounds, and another was that there would generally be infinitely many possible states, rather like the infinite series of harmonics above a ground. Hilbert space could be used to define a rigorous theory of quantum mechanics, proceeding logically from clear-cut axioms.
The unforeseen application of ‘Hilbert space’ was just the kind of thing that Alan would produce to support his claim for pure mathematics. He had seen another vindication in 1932, when the positron was discovered. For Dirac had predicted it on the basis of an abstract mathematical theory, which depended upon combining the axioms of quantum mechanics with those of special relativity. But in arguing about the relationship between mathematics and science, Alan Turing found himself tackling a perplexing, subtle, and to him personally important aspect of modern thought.
The distinction between science and mathematics had only been clarified in the late nineteenth century. Until then it might be supposed that mathematics necessarily represented the relations of numbers and quantities appearing in the physical world, although this point of view had really been doomed as soon as such concepts as the ‘negative numbers’ were developed. The nineteenth century, however, had seen developments in many branches of mathematics towards an abstract point of view. Mathematical symbols became less and less obliged to correspond directly with physical entities.
In school algebra – eighteenth century algebra, in effect – letters would be used as symbols for numerical quantities. The rules for adding and multiplying them would follow from the assumption that they were ‘really’ interpreted in terms of numbers, but it would not be necessary nor indeed always appropriate to do so.
The point of such abstraction was that it liberated algebra, and indeed all mathematics, from the traditional sphere of counting and measurement. In modern mathematics, symbols might be used according to any rules whatever, and might be interpreted in ways far more general than in terms of numerical quantities, if indeed they bore any interpretation at all. Quantum mechanics presented a fine example of where the expansion and liberation of mathematics for its own sake had paid off in physics. It had proved necessary to create a theory not of numbers and quantities, but of ‘states’ – and ‘Hilbert space’ offered exactly the right symbolism for these. Another related development in pure mathematics, which quantum physicists was now busy exploiting, was that of the ‘abstract group’. It had come about through mathematicians putting the idea of ‘operation’ into a symbolic form, and treating the result as an abstract exercise.* The effect of abstraction had been to generalise, to unify, and to draw new analogies. It had been a creative and constructive movement, for by changing the rules of these abstract systems, new kinds of algebra with unforeseen applications had been invented.
On the other hand, the movement towards abstraction had created something of a crisis within pure mathematics. If it was to be thought of as a game, following arbitrary rules to govern the play of symbols, what had happened to the sense of absolute truth? In March 1933 Alan acquired Bertrand Russell’s Introduction to Mathematical Philosophy, which addressed itself to this central question.
The crisis23 had first appeared in the study of geometry. In the eighteenth century, it was possible to believe that geometry was a branch of science, being a system of truths about the world, which Euclid’s axioms boiled down into an essential kernel. But the nineteenth century saw the development of geometrical systems different from Euclid’s. It was also doubted whether the real universe was actually Euclidean. In the modern separation of mathematics from science, it became necessary to ask whether Euclidean geometry was, regarded as an abstract exercise, a complete and consistent whole.
It was not clear that Euclid’s axioms really did define a complete theory of geometry. It might be that some extra assumption was being smuggled into proofs, because of intuitive, implicit ideas about points and lines. From the modern point of view it was necessary to abstract the logical relationships of points and lines, to formulate them in terms of purely symbolic rules, to forget about their ‘meaning’ in terms of physical space, and to show that the resulting abstract game made sense in itself. Hilbert, who was always down-to-earth, liked to say: ‘One must always be able to say “tables, chairs, beer-mugs”, instead of “points, lines, planes”.’
In 1899, Hilbert succeeded in finding a system of axioms which he could prove would lead to all the theorems of Euclidean geometry, without any appeal to the nature of the physical world. However, his proof required the assumption that the theory of ‘real numbers’* was satisfactory. ‘Real numbers’ were what to the Greek mathematicians were the measurements of lengths, infinitely subdivisible, and for most purposes it could be assumed that the use of ‘real numbers’ was solidly grounded in the nature of physical space. But from Hilbert’s point of view this was not good enough.
Fortunately it was possible to describe ‘real numbers’ in an essentially different way. By the nineteenth century it was well understood that ‘real numbers’ could be represented as infinite decimals, writing the number π for instance as 3.14159265358979.… A precise meaning had been given to the idea that a ‘real number’ could be represented as accurately as desired by such a decimal – an infinite sequence of integers. But it was only in 1872 that the German mathematician Dedekind had shown exactly how to define ‘real numbers’ in terms of the integers, in such a way that no appeal was made to the concept of measurement. This step both unified the concepts of number and length, and had the effect of pushing Hilbert’s questions about geometry into the domain of the integers, or ‘arithmetic’, in its technical mathematical sense. As Hilbert said, all he had done was ‘to reduce everything to the question of consistency for the arithmetical axioms, which is left unanswered.’
At this point, different mathematicians adopted different attitudes. There was a point of view that it was absurd to speak of the axioms of arithmetic. Nothing could be more primitive than the integers. On the other hand, it could certainly be asked whether there existed a kernel of fundamental properties of the integers, from which all the others could be derived. Dedekind also tackled this question, and showed in 1888 that all arithmetic could be derived from three ideas: that there is a number 1, that every number has a successor, and that a principle of induction allows the formulation of statements about all numbers. These could be written out as abstract axioms in the spirit of the ‘tables, chairs and beer-mugs’ if one so chose, and the whole theory of numbers could be constructed from them without asking what the symbols such as ‘1’ and ‘+’ were supposed to mean. A year later, in 1889, the Italian mathematician G. Peano gave the axioms in what became the standard form.
In 1900 Hilbert greeted the new century by posing seventeen unsolved problems to the mathematical world. Of these, the second
was that of proving the consistency of the ‘Peano axioms’ on which, as he had shown, the rigour of mathematics depended. ‘Consistency’ was the crucial word. There were, for instance, theorems in arithmetic which took thousands of steps to prove – such as Gauss’s theorem that every integer could be expressed as the sum of four squares. How could anyone know for sure that there was not some equally long sequence of deductions which led to a contradictory result? What was the basis for credence in such propositions about all numbers, which could never be tested out? What was it about those abstract rules of Peano’s game, which treated ‘+’ and ‘1’ as meaningless symbols, that guaranteed this freedom from contradictions? Einstein doubted the laws of motion. Hilbert doubted even that two and two made four – or at least said that there had to be a reason.
One attack on this question had already been made in the work of G. Frege, starting with his 1884 Grundlage der Arithmetik. This was the logistic view of mathematics, in which arithmetic was derived from the logical relationships of the entities in the world, and its consistency guaranteed by a basis in reality. For Frege, the number ‘1’ clearly meant something, namely the property held in common by ‘one table’, ‘one chair’, ‘one beer-mug’. The statement ‘2 + 2 = 4’ had to correspond to the fact that if any two things were put together with any other two things, there would be four things. Frege’s task was to abstract the ideas of ‘any’, ‘thing’, ‘other’, and so forth, and to construct a theory that would derive arithmetic from the simplest possible ideas about existence.
Alan Turing: The Enigma The Centenary Edition Page 15
