Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game

by Andrew Hodges

  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’ numbers. But by the twentieth century, this view had been abandoned. A rule such as ‘x + y = y + x’ could be regarded as a rule for a game, as in chess, stating how the symbols could be moved around and combined legitimately. The rule might possibly be 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 were 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 ‘l’ 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 axiom
s’ 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 ‘l’ 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 Grundlagen 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.

  Frege’s work was, however, overtaken by Bertrand Russell, whose theory was on the same lines. Russell had made Frege’s ideas more concrete by introducing the idea of the ‘set’. His proposal was that a set which contained just one thing could be characterised by the feature that if an object were picked out of that set, it would always be the same object. This idea enabled one-ness to be defined in terms of same-ness, or equality. But then equality could be defined in terms of satisfying the same range of predicates. In this way the concept of number and the axioms of arithmetic could, it appeared, be rigorously derived from the most primitive notions of entities, predicates and propositions.

  Unfortunately it was not so simple. Russell wanted to define a set-with-one-element, without appealing to a concept of counting, by the idea of equality. Then he would define the number ‘one’ to be ‘the set of all sets-with-one-element’. But in 1901 Russell noticed that logical contradictions arose as soon as one tried to use ‘sets of all sets’.

  The difficulty arose through the possibility of self-referring, self-contradictory assertions, such as ‘this statement is a lie.’ One problem of this kind had emerged in the theory of the infinite developed by the German mathematician G. Cantor. Russell noticed that Cantor’s paradox had an analogy in the theory of sets. He divided the sets into two kinds, those that contained themselves, and those that did not. ‘Normally’, wrote Russell, ‘a class is not a member of itself. Mankind, for example, is not a man.’ But the set of abstract concepts, or the set of all sets, would contain itself. Russell then explained the resulting paradox in this way:

  Form now the assemblage of classes which are not members of themselves. This is a class: is it a member of itself or not? If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses – that it is, and that it is not, a member of itself – each implies its contradictory. This is a contradiction.

  This paradox could not be resolved by asking what, if anything, it really meant. Philosophers could argue about that as long as they liked, but it was irrelevant to what Frege and Russell were trying to do. The whole point of this theory was to derive arithmetic from the most primitive logical ideas in an automatic, watertight, depersonalised way, without any arguments en route. Regardless of what Russell’s paradox meant, it was a string of symbols which, according to the rules of the game, would lead inexorably to its own contradiction. And that spelt disaster. In any purely logical system there was no room for a single inconsistency. If one could ever arrive at ‘2 + 2 = 5’ then it would follow that ‘4 = 5’, and ‘0 = 1’, so that any number was equal to 0, and so that every proposition whatever was equivalent to ‘0 = 0’ and therefore true. Mathematics, regarded in this game-like way, had to be totally consistent or it was nothing.

  For ten years Russell and A. N. Whitehead laboured to remedy the defect. The essential difficulty was that it had proved self-contradictory to assume that any kind of lumping together of objects could be called ‘a set’. Some more refined definition was required. The Russell paradox was by no means the only problem with the theory of sets, but it alone consumed a large part of Principia Mathematica, the weighty volumes which in 1910 set out their derivation of mathematics from primitive logic. The solution that Russell and Whitehead found was to set up a hierarchy of different kinds of sets, called ‘types’. There were to be primitive objects, then sets of objects, then sets of sets, then sets of sets of sets, and so on. By segregating the different ‘types’ of set, it was made impossible for a set to contain itself. But this made the theory very complicated, much more difficult than the number system it was supposed to justify. It was not clear that this was the only possible way in which to think about sets and numbers, and by 1930 various alternative schemes had been developed, one of them by von Neumann.

  The innocuous-sounding demand that there should be some demonstration that mathematics formed a complete and consistent whole had opened a Pandora’s box of problems. In one sense, mathematical propositions still seemed as true as anything could possibly be true; in another, they appeared as no more than marks on paper, which led to mind-stretching paradoxes when one tried to elucidate what they meant.

  As in the Looking-Glass garden, an approach towards the heart of mathematics was liable to lead away into a forest of tangled technicalities. This lack of any simple connection between mathematical symbols and the world of actual objects fascinated Alan. Russell had ended his book saying, ‘As the above hasty survey must have made evident, there are innumerable unsolved problems in the subject, and much work needs to be done. If any student is led into a serious study of mathematical logic by this little book, it will have served the chief purpose for which it has been written.’ So the Introduction to Mathematical Philosophy did serve its purpose, for Alan thought seriously about the problem of ‘types’ – and more generally, faced Pilate’s question: What is truth?

  Kenneth Harrison was also acquainted with some of Russell’s ideas, and he and Alan would spend hours discussing them. Rather to Alan’s annoyance, however, he would ask ‘but what use is it?’ Alan would say quite happily that of course it was completely useless. But he must also have talked to more enthusiastic listeners, for in the autumn of 1933 he was invited to read a paper to the Moral Science Club. This was a rare honour for any undergraduate, especially one from outside the faculty of Moral Sciences, as philosophy and its allied disciplines were called at Cambridge. It would have been a quite unnerving experience, speaking in front of professional philosophers, but he wrote with customary sang froid to his mother:

  26/11/33

  … I am reading a paper to the Moral Science Club on Friday. Something by way of being Mathematical Philosophy. I hope they don’t know it all allready.

  The minutes24 of the Moral Science Club recorded that on Friday 1 December 1933:

  The sixth meeting of the Michaelmas term was held in Mr Turing’s rooms in King’s College. A. M. Turing read a paper on ‘Mathematics and logic’. He suggested that a purely logistic view of mathematics was inadequate; and that mathematical propositions possessed a variety of interpretations, of which the logistic was merely one. A discussion followed.

  R. B. Braithwaite (signed).

  Richard Braithwaite, the philosopher of science, was a young Fellow of King’s; and it might well have been through him that the invitation was made. Certainly, by the
end of 1933, Alan Turing had his teeth into two parallel problems of great depth. Both in quantum physics and in pure mathematics, the task was to relate the abstract and the physical, the symbolic and the real.

  *

  German mathematicians had been at the centre of this enquiry, as in all mathematics and science. But as 1933 closed, that centre was a gaping, jagged hole, with Hilbert’s Göttingen ruined. John von Neumann had left for America, never to return, and others had arrived in Cambridge. ‘There are several distinguished German Jews coming to Cambridge this year,’ wrote Alan on 16 October. ‘Two at least to the mathematical faculty, viz. Born and Courant.’ He might well have attended the lectures on quantum mechanics that Born gave that term, or those of Courant* on differential equations the next term. Born went on to Edinburgh, and Schrödinger to Oxford, but most exiled scientists found America more accommodating than Britain. The Institute for Advanced Study, at Princeton University, grew particularly quickly. When Einstein took up residence there in 1933, the physicist Langevin commented, ‘It is as important an event as would be the transfer of the Vatican from Rome to the New World. The Pope of physics has moved and the United States will become the centre of the natural sciences.’

  It was not Jewish ancestry alone that attracted the interference of Nazi officialdom, but scientific ideas themselves, even in the philosophy of mathematics:25

  A number of mathematicians met recently at Berlin University to consider the place of their science in the Third Reich. It was stated that German mathematics would remain those of the ‘Faustian man’, that logic alone was no sufficient basis for them, and that the Germanic intuition which had produced the concepts of infinity was superior to the logical equipment which the French and Italians had brought to bear on the subject. Mathematics was a heroic science which reduced chaos to order. National Socialism had the same task and demanded the same qualities. So the ‘spiritual connexion’ between them and the New Order was established – by a mixture of logic and intuition ….