For one thing, he had agreed to take over M.H.A. Newman’s ‘Foundations of Mathematics’ lecture series at the university – despite not having a university post to go with his college one. As well as teaching logic, he was also arguing about it. Wittgenstein was also running a series of informal ‘lectures’ on the foundations of mathematics. Wittgenstein’s approach was less esoteric than Newman’s, and his discourses were down-to-earth, amusing and accessible. He would draw a smiley face and ask how this could be a representation of a well-known professor in the audience but not a representation of a mathematical concept. In fact, they weren’t so much lectures as facilitated discussions, with Wittgenstein engaging with a member of the audience and carrying on a staged debate, bringing in others as appropriate. During the first six months of 1939 there were 31 lectures, and Alan Turing attended almost all of them. From the very first lecture Wittgenstein used Alan as a foil for debating propositions. By lecture 19 they had got onto negation, contradictions and paradoxes; and a long discussion, spreading over four or five lectures, about the usefulness of contradictions ensued. ‘You might want to say,’ Wittgenstein suggested, ‘“logic and mathematics can’t reveal any truths if there are contradictions in it.”’ Of course, that wasn’t what Wittgenstein believed, and Alan Turing’s proof in Computable Numbers depended on contradiction to reveal truth.
While Alan was taking over Newman’s course, the Riemann zeta-function had been taking over Alan’s rooms. No doubt buoyed by his success in America with the relay multiplier, Alan worked up a design for a computing machine which would tackle the Riemann problem.
It is proposed to make calculations of the Riemann zeta-function on the critical line for 1,450 < t < 6,000 with a view to discovering whether all the zeros of the function in this range of t lie on the critical line. An investigation for 0 < t < 1,464 has already been made by Titchmarsh. The most laborious part of such calculations consists in the evaluation of certain trigonometrical sums.
In the present calculation it is intended to evaluate these sums approximately in most cases by the use of apparatus somewhat similar to what is used for tide prediction. I shall be working in collaboration with D. C. MacPhail, a research student who is an engineer. We propose to do most of the machine-shop work ourselves, and are therefore applying only for the cost of materials, and some preliminary computation.
D.C. (Donald) MacPhail was none other than the brother of Malcolm, with whom Alan had been messing about in the Physics Department machine shop at Princeton. Donald appeared at King’s in the autumn of 1938 to study for his Ph.D, and seems to have been adopted promptly by Alan to provide organisation and the dexterity to mill the finely machined pieces which the Riemann zeta-computer would need. Armed with a £40 grant from the Royal Society, and legitimate access to the Cambridge University Engineering Department workshops, Alan and Donald set to work. The concept was extraordinary. The machine would use fractions approximately equal to the logarithms needed in the calculations – Andrew Hodges gives the example of 34 x 31 / 57 x 35 ≈ log83 – and then gearwheels would be cut with a number of teeth corresponding to the factors in these ratios. Eighty wheels would be needed. If they were foolish enough to call, visitors to Alan’s rooms would find an orgy of grown-up Meccano, with the wheels laid out across the floor.
Neatness and order. Donald MacPhail’s neatly drawn schematic for the zeta-function machine belies the reality of gear-wheels strewn over Alan’s floor at King’s.
All I remember [said David Champernowne] is that the machine included a set of gear wheels the numbers of whose teeth were prime numbers, and I liked to fancy that as soon as the machine had found a root of the zeta function its centre of gravity would pass over the edge of a table and it would fall off uttering a swansong.
The intention was that the machine’s approximations would – even if they were inexact – show roughly where the zeros of the Riemann zeta-function lay, to enable the mathematicians to follow up more exactly with hand methods. Typical Alan Turing methodology: a combination of machinery, insight, and practical shortcuts. On 1 September 1939 Germany invaded Poland, forcing an indefinite postponement of work on the Riemann machine, but all of the talents Alan was using for the machine would characterise his work in unravelling Enigma at Bletchley Park.
Notes
1 Comptes Rendus, a French mathematical journal
1 Veronica Durrant, who was Isobel Morcom’s assistant
2 The Morcoms had had a memorial window installed in the local church
3 Christopher’s older brother
1 Dean of Rochester, later Bishop of Bath and Wells, and an acquaintance of Ethel Turing
6
PROF
KING’S COLLEGE, CAMBRIDGE, was a nest of spies. Not Kim Philby or Guy Burgess or Anthony Blunt or John Cairncross: they were at Trinity. Not Donald Maclean: he was at Trinity Hall. In any case, they were all spying for another side. The King’s College spies were an altogether subtler group. At least five members of King’s had been employed during World War One in a secret establishment forming part of the Naval Intelligence Division. In Room 40 of the Admiralty Old Building in Whitehall, this small elite had read the coded radio messages of the German Imperial Navy, thereby removing the threat of surprise from their sorties into the North Sea. In 1917 they achieved the greatest diplomatic coup to derive from a coded message when the content of the ‘Zimmerman telegram’ was revealed to the Americans, providing the final push which brought the United States into the war and sealed the fate of Imperial Germany. The members of Room 40 had not retired gracefully to the country in 1919. Some – notably Dilly Knox, a fellow of King’s – stayed on to transform Room 40 into the all-service cryptanalytical agency which, by 1939, had renamed itself as the Government Code & Cypher School (GC&CS). Others, like Professor Frank Adcock, went to the country, or rather his elegant set of rooms over the archway in front court at King’s, although his was a very active ‘retirement’. Prompted by Alastair Denniston, the Room 40 leader who was now the Director of GC&CS, Professor Adcock was retained to spot talent for the new organisation.
No fewer than 11 fellows of King’s (not counting Adcock himself, Dilly Knox and Frank Birch, another Room 40 veteran who had not retired to the country either) were recruited for GC&CS. Another eight alumni of King’s were also recruited. One of the fellows was Alan Turing, tapped on the shoulder by Adcock in 1938 in the manner we have seen. GC&CS had been preparing for war for a long time. By November 1938, Commander Denniston was able to write to the Foreign Office, which was responsible for his staffing needs. ‘I have been in touch with both Universities and have established direct contact through Dons who worked with us during the war, so that now we have a list of about 50 men earmarked for service under the Foreign Office in the event of war.’ On the list of available emergency staff were various ‘old members’, and over 20 others from Cambridge, of whom seven, including Alan Turing and Patrick Wilkinson, were from King’s. One, the literary critic F.L. Lucas, was so well known that his college is not specified and the list just says ‘see “Who’s Who”’. The Cambridge list also names M.H.A. Newman, but against his name is the word ‘no’. Newman was not to go to GC&CS. Not yet.
The talent agent. A caricature of Frank Adcock, from his days as a World War One codebreaker in Room 40.
Monstrous pile
In mid-March 1939 the German Army occupied Bohemia and Moravia, shredding the last scraps of credibility from the Munich settlement. But Denniston had already concluded that ‘a sufficient supply of professors is immediately available’ and on 2 August he informed the cryptanalysts that they would be moving to their ‘war site’ on 15 August 1939.
In order to carry out communication tests the war site will be manned a.m. 15th August by those detailed in G.C. & C.S. 1st Wave who are not on leave at the time.
All documents required at the war site are to be packed by 5.30 p.m. on 14th August. They will be moved during that night. As many as possible are to be placed
in small cupboards and filing cabinets which can be locked. Arrangements for labelling will be promulgated later. Stationery has already been sent down. Sufficient personal luggage should be taken for 15 days.
Those going by train should obtain a single ticket of the appropriate class to Bletchley. On arrival they should place their luggage in the cloak room from whence it will be collected after the allocation of billets etc. They should proceed on foot from the station to the war site enquiring if necessary for Bletchley Park, which is on the up (West) side of the railway. On leaving the station turn right up the hill and proceed through the second lodge gates. Suitable trains from Euston are the 8.37 a.m. and the 9.30 a.m. An advance for railway tickets can be obtained from Mr Travis. […]
The address for official correspondence and private letters will be Room 47 Foreign Office and the official telephone No. Whitehall 7947.
The staff are warned against any conversations regarding the work with other members of the staff whilst in their billets. If occasion should arise as to what you are doing the answer should be that you are part of the aerial defence of London. […]
This test is to be treated with the utmost secrecy by all members of this department.
Gas masks are to be taken.
Bletchley Park was not a glamorous place. Its main building was constructed over a 25-year period beginning in 1877, with improvements and innovations stacked one onto another without any concession to coherence or symmetry. It was described by Landis Gores, an American architect who was later stationed at Bletchley as a representative of United States Army Intelligence:
A maudlin and monstrous pile: the Mansion at Bletchley Park. The ground-floor room under the dome became Alastair Denniston’s office.
A maudlin and monstrous pile probably unsurpassed, though not for lack of competition, in the architectural gaucherie of the mid-Victorian era, built about 1860 in an undiscriminatingly imitative Tudor vocabulary out of an endemic dark red brick with beige coadestone trim, quoins, voussoirs and keystones, further hopelessly vulgarised by extensive porches and solaria as well as by batteries of tall casements in intermittent profusion, all of painted wood trim, mingling with what could only be termed incoherent abandon two-centre Gothic, three-centred Tudor, four-centred Perpendicular and ogival Flamboyant arches with English stick and French trefoil tracery. The profusion of top-story gables faced with cottage-style half-timbering, not to mention an overpowering copper-roofed octagonal-walled to onion-topped pleasure dome with finial immediately suggestive of the pseudo-orientalism of the Royal Pavilion at Brighton; oriels, turrets, bay windows and embrasures, all capped by myriad multi-potted chimneys in totally wanton location and configuration; altogether inchoate, unfocused and incomprehensible, not to say indigestible.
A building as exuberant as Mr Gores’s prose. Worse, its grounds were going to become a building site, as the GC&CS staff outgrew the mansion house, the outbuildings, the adjacent school, and numerous temporary wooden huts were thrown together quickly to deal with the overspill. The lawns and rose garden were ripped up to provide footings for the huts, and eventually the maze went too, when it was decided that brick buildings were needed to house the organisations which had outgrown the huts. By 1943, Bletchley Park had stopped looking like a park and taken on the appearance of a stolid and functional industrial centre, as indeed it had become. Only the lake and a fraction of the grounds visible from the Director’s ground-floor office remained in their pre-war state.
A week after Denniston was installed at Bletchley Park, the German foreign minister Joachim von Ribbentrop concluded his notorious pact with Vyacheslav Molotov, his Russian opposite number. The way was clear for Germany to occupy Poland and recover the territories lost under the Treaty of Versailles. Germans dressed as Poles staged a fake violation of Germany’s sovereign rights, and on 1 September Poland was invaded on two fronts. World War Two had begun. Denniston wrote to the Foreign Office:
Ref. No. 767.
3rd September, 1939.
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.
I will keep you informed at intervals of further recruitment.
Yours sincerely,
A.G. Denniston
Ref. No. 783.
7th September, 1939.
Dear Wilson,
In continuation of my No.767 of 3rd September:-
The following gentlemen have joined subsequently:- […]
Mr. A.M. Turing 4th September, 1939. […]
Yours ever,
A.G.D.
Although Mr Turing may have arrived on 4 September 1939, his actual mobilisation had taken place considerably earlier. Upon recruitment, he had been earmarked for Dilly Knox’s Research Section located in the Stableyard cottages, specifically to work on the breaking of messages encrypted on the German Enigma machine. The workings of the Enigma are now well-known: a typewriter-like keyboard had wiring leading to a lampboard, where the enciphered message would appear, joined by wiring which scrambled the message. In the standard Army and Air Force Enigma there were three coding wheels, each of which would substitute one letter for another, and which would rotate with every new letter of the coded message to create a new cipher; and a plugboard which would also switch some letters for others. Further complications were introduced by the rings on the coding wheels, which allowed for the internal wiring of each wheel to be rotated relative to its housing, and the existence of ‘turnover’ notches on the wheels which meant that more than one wheel might rotate at the same time. In all, the machine had 17,576 positions for the three chosen coding wheels, the same number of ring-settings, and 150 trillion plugboard settings, for a total of 158.9 million million million possible starting configurations, allowing for 10 plugboard cross-pairings and the choice of the three coding wheels for the day from a library of five. If Alan Turing’s imagined 100 Germans working eight hours a day on desk calculators were able to get through even a million possible settings every hour, it would take far, far longer than 100 years to pick the right setting by brute force. Even if each codebreaker could crank through 1014 permutations per hour it would take nearly 400 years to find the setting. Even allowing for German efficiency, let alone British, something rather cleverer than brute force would be needed to unlock the Enigma.
The early days. Cottages in the Stableyard where Alan Turing devised techniques for breaking Enigma.
Alan Turing had been working at Cambridge throughout 1939 on the mysteries of Enigma, and a major breakthrough had happened in July when the Polish Biuro Szyfrów shared its hard-won cryptanalytic secrets with Denniston, Dilly Knox, and their opposite numbers from French intelligence. The secrets included the internal wiring used by the German armed forces to connect the Enigma keyboard to the input plate feeding current into the coding wheels of the machine; also particulars of their electro-mechanical devices – a ‘cyclometer’ and an ominous-sounding ‘bomba’ – for finding the settings which gave the Enigma a new m-configuration, effectively enabling it to become a completely new cipher machine every day. In mid-August a Polish reconstruction of an Enigma machine was delivered to Colonel Stewart Menzies, the deputy head of the Secret Intelligence Service, at Victoria Station. Marian Rejewski, the chief Polish codebreaker, had used equations as well as machinery to tackle the Enigma. This was the kind of stuff which Alan Turing could get his teeth into. Although the purist Knox was always going to favour hand-based decryption methods, it was obvious to all that using a machine was the way to deal with the millions of permutations presented by the Enigma. Alan Turing went to see Knox at his house to be briefed on the Polish revelations. The questions now were what sort of a machine, how should it be designed, and who could build it.
The Enigma machine: the principal encipherment machine used by the Germans c
ould be set up 159 million million million different ways. After February 1942 the German Navy used a version which had 890 million million million permutations.
Bombe-ish boy
The Poles had described their machinery, and in it lay the foundations of the device which Alan designed to reveal the daily settings of the Enigma machine. The Poles were exploiting a precaution which the Germans were taking in transmitting their messages, and this precaution opened a tiny chink in the Enigma’s armour, a chink wide enough to insert the point of a crowbar and prise it wide open. More importantly, the chink was based on a principle which Alan Turing would exploit so as to enable all Army and Air Force Enigma key-nets to be broken open using a combination of mechanical and manual means.
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