WITTGENSTEIN: … Think of the case of the Liar. It is very queer in a way that this should have puzzled anyone – much more extraordinary than you might think…. Because the thing works like this: if a man says ‘I am lying’ we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn’t matter … it is just a useless language-game, and why should anybody be excited?
TURING: What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.
WITTGENSTEIN: Yes – and more: nothing has been done wrong…. where will the harm come?
TURING: The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort.
WITTGENSTEIN: … The question is: Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc., outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics? Turing says, ‘Because something may go wrong with the application.’ But nothing need go wrong. And if something does go wrong – if the bridge breaks down – then your mistake was of the kind of using a wrong natural law….
TURING: You cannot be confident about applying your calculus until you know that there is no hidden contradiction in it.
WITTGENSTEIN: There seems to me to be an enormous mistake there …. Suppose I convince Rhees of the paradox of the Liar, and he says, ‘I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 × 2 = 369.’ Well, we should not call this ‘multiplication’, that is all….
TURING: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere.
WITTGENSTEIN: But nothing has ever gone wrong that way yet ….
But Alan would not be convinced. For any pure mathematician, it would remain the beauty of the subject, that argue as one might about its meaning, the system stood serene, self-consistent, self-contained. Dear love of mathematics! Safe, secure world in which nothing could go wrong, no trouble arise, no bridges collapse! So different from the world of 1939.
He did not complete his research into the Skewes problem, which was left as an error-strewn manuscript40 and never taken up by him again. But he continued to pursue the more central problem, that of examining the zeroes of the Riemann zeta-function. The theoretical part, that of finding and justifying a new method of calculating the zeta-function, was finished at the beginning of March, and was submitted for publication.41 This left the computational part to be done. In this respect there had been a development. Malcolm MacPhail had written42 in connection with the electric multiplier:
How is your University fixed with storage batteries and lathes and so on which you can use for your machine? It’s such a pity that you will have to alter it. Hope you do not find it too much bunched together to be hard to work with. By the way if you are going to have time to work on it this fall and want some help don’t hesitate to ask my brother. I told him about the machine and how it worked. He’s very enthusiastic about your method of drawing wiring diagrams which rather surprised me. You know how conservative and old-fashioned engineers tend to be.
It so happened that his brother, Donald MacPhail, was a research student attached to King’s, studying mechanical engineering. The multiplier made no progress, but Donald MacPhail did now join in the zeta-function machine project.
Alan was not the only person to be thinking about mechanical computation in 1939. There were a number of ideas and initiatives, reflecting the growth of new electrical industries. Several projects were on hand in the United States. One of these was the ‘differential analyser’ that the American engineer Vannevar Bush had designed at the Massachusetts Institute of Technology in 1930. This could set up physical analogues of certain differential equations – the class of problem of most interest in physics and engineering. A similar machine had then been built by the British physicist D. R. Hartree out of Meccano components at Manchester University. This in turn had been followed by the commissioning of another differential analyser at Cambridge, where in 1937 the mathematical faculty had sanctioned a new Mathematical Laboratory to house it. One of Alan’s fellow ‘B-stars’ of 1934, the applied mathematician M. V. Wilkes, had been appointed as its junior member of staff.
Such a machine would have been useless for the zeta-function problem. Differential analysers could simulate only one special kind of mathematical system, and that only to a limited and very approximate extent. Similarly the Turing zeta-function machine would be entirely specific to the even more special problem on hand. It had no connection whatever with the Universal Turing Machine. It could hardly have been less universal. On 24 March he applied43 to the Royal Society for a grant to cover the cost of constructing it, and on their questionnaire wrote,
Apparatus would be of little permanent value. It could be added to for the purpose of carrying out similar calculations for a wider range of t* and might be used for some other investigations connected with the zeta-function. I cannot, think of any applications that would not be connected with the zeta-function.
Hardy and Titchmarsh were quoted as referees for the application, which won the requested £40. The idea was that although the machine could not perform the required calculation exactly, it could locate the places where the zeta-function took a value near zero, which could then be tackled by a more exact hand computation. Alan reckoned it would reduce the amount of work by a factor of fifty. Perhaps as important, it would be a good deal more fun.
The Liverpool tide-predicting machine made use of a system of strings and pulleys to create an analogue of the mathematical problem of adding a series of waves. The length of the string, as it wrapped itself round the pulleys, would be measured to obtain the total sum required. They started with the same idea for the zeta-function summation, but then changed to a different design. In this, a system of meshing gear wheels would rotate to simulate the circular functions required. The addition was to be done not by measuring length, but weight. There would in fact be thirty wave-like terms to be added, each simulated by the rotation of one gear wheel. Thirty weights were to be attached to the corresponding wheels, at a distance from their centres, and then the moment of the weights would vary in a wave-like way as the wheels rotated. The summation would be performed by balancing the combined effect of the weights by a single counterweight.
The frequencies of the thirty waves required ran through the logarithms of the integers up to 30. To represent these irrational quantities by gear wheels they had to be approximated by fractions. Thus for instance the frequency determined by the logarithm of 3 was represented in the machine by gears giving a ratio* of 34 × 31/57 × 35. This required four gear wheels, with 34, 31, 57 and 35 teeth respectively, to move each other so that one of them could act as the generator of the ‘wave’. Some of the wheels could be used two or three times over, so that about 80, rather than 120 gear wheels were required in all. These wheels were ingeniously arranged in meshing groups, and mounted on a central axis in such a way that the turning of a large handle would set them in simultaneous motion. The construction of the machine demanded a great deal of highly accurate gear-cutting to make this possible.
Donald MacPhail drew up a blueprint of the design,44 dated 17 July 1939. But Alan did not leave the engineering work to him. In fact his room, in the summer of 1939, was liable to be found with a sort of jigsaw puzzle of gear wheels across the floor. Kenneth Harrison, now a Fellow, was invited in for a drink and found it in this state. Alan tried and lamentably failed to explain what it was all for. It was certainly far from obvious that the motion of these wheels would say anything about the regularity with which the prime numbers thinned out, in their billions of billions out to infinity. A
lan made a start on doing the actual gear-cutting, humping the blanks along to the engineering department in a rucksack, and spurning an offer of help from a research student. Champ lent a hand in grinding some of the wheels, which were kept in a suitcase in Alan’s room, much impressing Bob when he came down from his school at Hale in August.
Kenneth Harrison had been much amazed, for he well knew from conversations with Alan that a pure mathematician worked in a symbolic world and not with things. The machine seemed to be a contradiction. It was particularly remarkable in England, where there existed no tradition of high status academic engineering, as there was in France and Germany and (as with Vannevar Bush) in the United States. Such a foray into the practical world was liable to be met with patronising jokes within the academic world. For Alan Turing personally, the machine was a symptom of something that could not be answered by mathematics alone. He was working within the central problems of classical number theory, and making a contribution to it, but this was not enough. The Turing machine, and the ordinal logics, formalising the workings of the mind; Wittgenstein’s enquiries; the electric multiplier and now this concatenation of gear wheels – they all spoke of making some connection between the abstract and the physical. It was not science, not ‘applied mathematics’, but a sort of applied logic, something that had no name.
By now he had edged a little further up the Cambridge structure, for in July the faculty asked that he should give his lectures on Foundations of Mathematics again in spring 1940, this time for the full fee of £50. In the normal course of events he could have expected fairly soon to be appointed to a university lectureship, and most likely to stay at Cambridge forever, as one of its creative workers in logic, number theory and other branches of pure mathematics. But this was not the direction in which his spirit moved.
Nor was it the direction of history. For there was to be no normal course of events. In March, the remains of Czecho-Slovakia slid into German control. On 31 March, the British government gave its guarantee to Poland, and committed itself to defending east European frontiers, while alienating the Soviet Union, already the world’s second industrial power. It was a device to deter Germany, not to aid Poland, there being no way in which Britain could render assistance to its new ally.
It might have seemed that there was, equally, no way in which Poland could help the United Kingdom. Yet there was. In 1938, the Polish secret service had dropped a hint that they held information on the Enigma. Dillwyn Knox had gone to negotiate for it, but returned empty-handed, complaining that the Poles were stupid and knew nothing. The alliance with Britain and France had changed the position. On 24 July, British and French representatives attended a conference in Warsaw and this time came away with what they wanted.
A month later everything changed again, the Anglo-Polish alliance becoming even more impractical than before. As far as Intelligence was concerned, the year had gained little for Britain. There was now a new wireless interception station at St Albans, replacing the old arrangement whereby the Metropolitan Police did the work at Grove Park. But there was still45 ‘a desperate shortage of receivers for wireless interception’, despite the pleas of GC and CS since 1932. The great exception was the fluke, handed over on a silver platter by the Poles.
The news-stands were announcing the Ribbentrop-Molotov pact as Alan set off from Cambridge for a week’s sailing holiday, together with Fred Clayton and the refugee boys. They went to Bosham, his usual holiday haunt, where he had hired a boat. Several anxieties lay beneath the quiet surface. The boys, who had not been sailing before, thought the two men incompetent, and altered their watches so that they would turn back in good time. ‘The lame leading the blind,’ was what Bob thought of it. Fred, however, was more worried about the emotional undertones of the holiday. Alan teased him a good deal, mocking the idea that after a couple of terms at Rossall a boy would be innocent of sexual experience.*
One day they sailed across to Hayling Island, and went ashore to look at the RAF planes lined up on the airfield. The boys were not very impressed with what they saw. The sun went down and the tide went out, and the boat was stuck in the mud. They had to leave it and wade across to the island to get back by bus, their legs encrusted with thick black mud. Karl said they looked like soldiers in long black boots.
It was at Bosham that King Cnut had shown his advisers that his powers did not extend to stemming the tide. The thin line of aircraft, charged with turning back the bombers, did not that August evening inspire much greater confidence. And who could have guessed that this shambling, graceless yachtsman, squelching bare-legged in the mud and grinning awkwardly at embarrassed Austrian boys, was to help Britannia rule the waves?
For now he would give no 1940 lectures. Nor indeed would he ever return to the safe world of pure mathematics. Donald MacPhail’s design would never be realised, and the brass gear wheels would lie packed away in their case. For other, more powerful wheels were turning: not only Enigma wheels, but tank wheels. The bluff was called, so the deterrent had failed to work. Yet Hitler had miscalculated, for this time British duty would be done. Parliament kept the government to its word, and there would be war with honour.
It was much as Back to Methuselah had prophesied in 1920:
And now we are waiting, with monster cannons trained on every city and seaport, and huge aeroplanes ready to spring into the air and drop bombs every one of which will obliterate a whole street, until one of you gentlemen rises in his helplessness to tell us, who are as helpless as himself, that we are at war again.
Yet they were not quite as helpless as they seemed. At eleven o’clock on 3 September, Alan was back at Cambridge, sitting in his room with Bob, when Chamberlain’s voice came over the wireless. His friend Maurice Pryce would soon be giving serious thought to the practical physics of chain reactions. But Alan had committed himself to the other, logical, secret. It would do nothing for Poland. But it would connect him with the world, to a degree surpassing the wildest dream.
* * *
* An abstract in French for the scientific journal Comptes Rendus. Mrs Turing helped with the French and the typing.
* The lambda-calculus represented an elegant and powerful symbolism for mathematical processes of abstraction and generalisation.
* He became bishop of Bath and Wells in 1937.
* The ‘complex’ number calculus exemplified the progress of mathematical abstraction. Originally, complex numbers had been introduced to combine ‘real’ numbers with the ‘imaginary’ square root of minus one, and mathematicians had agonised over the question of whether such things really ‘existed’. From the modern point of view, however, complex numbers were simply defined abstractly as pairs of numbers, and pictured as points in a plane. A simple rule for the definition of the ‘multiplication’ of two such pairs was then sufficient to generate an enormous theory. Riemann’s work in the nineteenth century had played a large part in its ‘pure’ development; but it was also found to be of great usefulness in the development of physical theory. Fourier analysis, treating the theory of vibrations, was an example of this. The quantum theory developed since the 1920s went even further in according complex numbers a place in fundamental physical concepts. None of these mathematical ideas are essential to what follows, although such connections between ‘pure’ and ‘applied’ were certainly relevant to a number of aspects of Alan Turing’s later work.
* 1034 is 10,000,000,000,000,000,000,000,000,000,000,000 – a number comparable with the number of elementary particles in a large building. But 101034 is far bigger: as 1 followed by 1034 zeroes it would require books with the mass of Jupiter to print it in decimal notation. It could be thought of as the number of possible man-made objects. Skewes’ number was much bigger again, as 1 followed by 101034 zeroes! In actual fact mathematicians had certainly thought about numbers far larger than these, here we have only gone through three stages of growth, but it is not difficult to make up a new notation to express the idea of going through ten such stages, or
1010, or 101010; or of regarding even these as just the first step in a process of super-growth, and then defining super-super-growth, and then …. Such definitions, indeed, had already played a role in the theory of ‘recursive functions’, one of the other approaches to the idea of ‘definite method’ which had been found equivalent to that of the Turing machine. But Skewes’ number was certainly remarkably large for a problem which could be expressed in such elementary terms.
* Certainly one attraction to Alan of the New Statesman would have been its exceptionally demanding puzzle column. In January 1937 he was delighted when his friend David Champernowne defeated such runners-up as M. H. A. Newman and J. D. Bernal in giving a witty solution, phrased in Carrollian language, to a problem set by Eddington called ‘Looking Glass Zoo’. (It depended upon a knowledge of the matrices used by Dirac in his theory of the electron.) But Alan’s comments on the Abdication, naive in idealism perhaps but certainly not ill-informed, indicate very clearly that his interest in the journal would not have been confined to this feature.
* Ulam writes further that ‘von Neumann had great admiration for him and mentioned his name and “brilliant ideas” to me already, I believe, in early 1939…. At any rate von Neumann mentioned to me Turing’s name several times in 1939 in conversations, concerning mechanical ways to develop formal mathematical systems.’
* In what follows, code refers to any conventional system of communicating text, whether secret or not. Cipher is used for communications designed to be incomprehensible to third parties. Cryptography is the art of writing in cipher; cryptanalysis that of deciphering what has been concealed in cipher. Cryptology covers both the devising and breaking of ciphers. At the time, these distinctions were not made, and Alan Turing himself referred to cryptanalysis as ‘cryptography’.
Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game Page 27
