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 1010; or of regarding even these as just the first step in a process of super-growth, and rhen 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 thet 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. Bermal 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 belive, in early 1939,. ... At any rate von Neumann mantioned to me Turings’s name several times in 1939 in conversations, concerning mechanical ways to develop formal mathematical systems.’<
br />
* 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’.
†The British spying organisation, variously entitled SIS, M16. Apart from this top-level administrative overlap it was and remained essentially distinct from the cryptanalytic department.
* David Champernowne also discussed the principle of the chain reaction with Alan after reading an article about it by J.B.S. Haldane in the Daily Worker.
* that is, for looking at even more zeroes of the zeta-function.
* Alan was wrong.
4
The Relay Race
Gliding o’er all, through all,
Through Nature, Time, and Space,
As a ship on the waters advancing,
The voyage of the soul – not life alone,
Death, many deaths I’ll sing.
Alan reported next day, 4 September, to the Government Code and Cypher School, which had been evacuated in August to a Victorian country mansion, Bletchley Park. Bletchley itself was a small town of ordinary dullness, a brick-built Urban District in the brickfields of Buckinghamshire. But it lay at the geometric centre of intellectual England, where the main railway from London to the north bisected the branch line from Oxford to Cambridge. Just to the north-west of the railway junction, on a slight hill graced by an ancient church, and overlooking the clay pits of the valley, stood Bletchley Park.
The trains were busy with the evacuation of 17,000 London children into Buckinghamshire, swelling Bletchley’s population by twenty-five per cent. ‘The few who returned,’ said an urban district councillor, ‘no one on earth would have billeted, and they did the wisest thing eventually to return to the hovels from whence they came.’ In these circumstances, the arrival of a few select gentlemen for the Government Code and Cypher School would have caused little stir, although it was said that when Professor Adcock first arrived at the station, a little boy shouted ‘I’ll read your secret writing, mister!’ in a most disconcerting way. Later on there were complaints by local residents about the do-nothings at Bletchley Park, and it was said that the MP had to be prevented from asking a question in Parliament. They had the pick of accommodation: the few hostelries of mid-Buckinghamshire. Alan was billeted at the Crown Inn at Shenley Brook End, a tiny hamlet three miles north of Bletchley Park, whither he cycled each day. His landlady, Mrs Ramshaw, was one of those who lamented that an able-bodied young man was not doing his bit. Sometimes he helped out in the bar.
The early days at Bletchley resembled the arrangements of a displaced senior common room, obliged through domestic catastrophe to dine with another college, but nobly doing its best not to complain. In particular there was a strong King’s flavour, with old-timers Knox, Adcock and Birch, and the younger Frank Lucas and Patrick Wilkinson as well as Alan. The shared background in Keynesian Cambridge was probably helpful for Alan. In particular it offered a link with Dillwyn Knox, a figure not generally noted for geniality or acessibility by Alan’s contemporaries. GC and CS was by no means a vast establishment. On 3 September, Denniston wrote1 to the Treasury:
Dear Wilson,
For some days now we have been obliged to recruit from our emergency list men of the Professor type who the Treasury agreed to pay at the rate of £600 a year. I attach herewith a list of these gentlemen already called up together with the dates of their joining.
Alan was not quite the first, for according to Denniston’s list there were nine of these ‘men of the Professor type’ at Bletchley by the time that he arrived with seven others the next day. Over the following year, about sixty more outsiders were brought in.
The ‘emergency in-take quadrupled the cryptanalytic staff of the Service sections and nearly doubled the total cryptanalytic staff.’ But only three of these first recruits came from the science side. Besides Alan, there were only W.G. Welchman and John Jeffries.* Gordon Welchman was the senior figure, lecturer in mathematics at Cambridge since 1929 and six years older than Alan. His field was algebraic geometry, a branch of mathematics then strongly represented at Cambridge, but one which never attracted Alan; their paths had not crossed before.
Welchman had not been involved with GC and CS before the outbreak of war as Alan had, and thus found himself, as a newcomer, relegated by Knox to the task of analysing the pattern of German call-signs, frequencies, and so forth. As it transpired, this was a job of immense significance, and his work rapidly brought such ‘traffic analysis’ to a quite new standard. It made possible the identification of the different Enigma key-systems, as was soon to prove so important, and opened GC and CS eyes to a much wider vision of what could be done. But no one could decipher the messages themselves. There was just a ‘small group which, headed by civilians and working on behalf of all three Services, struggled with the Enigma.’ This group consisted first of Knox, Jeffries, Peter Twinn and Alan. They established themselves in the stables building of the mansion, soon dubbed ‘the Cottage’, and developed the ideas that the Poles had supplied at the eleventh hour.
There was no glamour about ciphers. In 1939 the job of any cipher clerk, although not without skill, was dull and monotonous. But ciphering was the necessary consequence of radio* communication. Radio had to be used for aerial, naval, and mobile land warfare, and a radio message to one was a message to all. So messages had to be disguised, and not just this or that ‘secret message’, as with spies and smugglers, but the whole communication system. It meant mistakes, restrictions, and hours of laborious work on each message, but there was no choice.
The ciphers used in the 1930s did not depend on any great mathematical sophistication, but on the simple ideas of adding on and substituting. The ‘adding on’ idea was hardly new; Julius Caesar had concealed his communications from the Gauls by a process of adding on three to each letter, so that an A became D, a B became an E, and so on. More precisely, this kind of adding was what mathematicians called ‘modular’ addition, or addition without carrying, because it meant Y becoming B, Z becoming C, as though the letters were arranged around a circle.
Two thousand years later, the idea of modular addition by a fixed number would hardly be adequate, but there was nothing out-of-date about the general idea. One important type of cipher used the idea of ‘modular addition’, but instead of fixed number, it would be a varying sequence of numbers, forming a key, that would be added to the message.
In practice, the words of the message would first be encoded into numerals by means of a standard code-book. The job of the cipher clerk would then be to take this ‘plain-text’, say
6728 5630 8923,
and to take the ‘key’, say
9620 6745 2397,
and form the cipher-text
5348 1375 0210
by modular addition.
For this to be of any use, however, the legitimate receiver had to know what the key was, so that it could be subtracted and the ‘plain-text’ retrieved. There had to be some system, by which the ‘key’ was agreed in advance between sender and receiver.
One way of doing this was by means of the one-time principle. This was one of the few sound ideas of 1930s cryptography, as well as the simplest. It required the key to be written out explicitly, twice over, and one copy given to the sender, one to the receiver of the transmission. The argument for the security of this system was that provided the key were constructed by some genuinely completely random process, such as shuffling cards or throwing dice, there could be nothing for the enemy cryptanalyst to go on. Given cipher text ‘5673’, for instance, the analyst might guess that the plain-text was in fact ‘6743’ and the key therefore ‘9930’, or might guess that the plaintext was ‘8442
’ and the key ‘7231’, but there would be no way of verifying such a guess, nor reason to prefer one guess to another. The argument depended upon the key being absolutely patternless, and spread evenly over the possible digits, for otherwise the analyst would have reason to prefer one guess to another. Indeed, discerning a pattern in the apparently patternless was essentially the work of the cryptanalyst, as of the scientist.
In the British system, one-time pads were produced, to be used up one at a time. Provided the key was random, no page was used twice, and the pads were never compromised, the system was fool-proof. But it would involve the manufacture of a colossal quantity of key, equal in volume to the maximum that the particular communication link might require. Presumably this thankless task was undertaken by the ladies of the Construction Section of GC and CS, which on the outbreak of war was evacuated not to Bletchley but to Mansfield College, Oxford. As for the system in use, that was no joy either. Malcolm Muggeridge, who was employed in the secret service, found it2
a laborious business, and the kind of thing I have always been bad at. First, one had to subtract from the groups of numbers in the telegram corresponding groups from a so-called one-time pad; then to look up what the resultant groups signified in the code book. Any mistake in the subtraction, or, even worse, in the groups subtracted, threw the whole thing out. I toiled away at it, getting into terrible muddles and having to begin again…
Alan Turing: The Enigma The Centenary Edition Page 27