Alan Turing: The Enigma The Centenary Edition

by Andrew Hodges

  It required a greater total number of valves than the Colossi because it stored long decimal numbers – the more so because of the primitive system the designers had employed whereby ten valves were allocated to each decimal digit required, a ‘9’ being represented by the ninth of those valves being ‘on’. In contrast the Colossi operated on single pulses which represented the logical ‘yes’ or ‘no’ of holes in telegraph tape.

  But this was a fairly superficial point of difference. Both alike demonstrated that thousands of valves, hitherto regarded as too unreliable for operation en masse, could be kept in simultaneous use.* And the ENIAC project was embodying the idea that Zuse, Aiken and Stibitz had missed. Like the Mark II Colossi, with their ability to automate acts of decision, the results of one counting operation automatically deciding what step was taken next, the ENIAC would have a form of conditional branching. It was designed so that it could be made to hop to and fro within the stock of instructions supplied to it, repeating sections as many times as the progress of the calculation showed was necessary, without the interference of human management. None of this went beyond what Babbage had envisaged – except that electronic components were so much faster, and that the ENIAC was (or was nearly) a reality.

  Like the Colossus, the ENIAC had been designed for a specific task, that of calculating artillery range tables. Essentially it was to simulate the trajectories of shells through varying conditions of air resistance and wind velocity, which involved the summation of thousands of little pieces of trajectory. It had external switches which would be set to store the constant parameters for a trajectory calculation, and further external devices for setting up the instructions on how to calculate the segments of motion. Then there would be valves to hold the intermediate working figures. In these arrangements it resembled the Colossi. But in both cases, people had quickly discovered the possibility for using the machines for a wider range of tasks than those for which they had been designed. The role of the original Colossus had been much extended by Donald Michie and Jack Good, and then the Mark II had once been set up to punch out a deciphered message, although this was done out of interest, not for the sake of efficiency. Even though a parasite on the German cipher machine, the flexibility offered by its instruction table facility was such that it could even be ‘almost’ set up to perform numerical multiplication. The ENIAC was flexible in a more serious way, and von Neumann had already discovered that it could be used, when ready, for Los Alamos problems.*

  But the ENIAC had not been conceived as a universal machine, and in one important respect the designers had departed from Babbage’s line of development. Babbage had been proud of the fact that the planned Analytical Engine would be able to ingest an unlimited number of instruction cards. The Aiken relay machine enjoyed the same feature, although cards had been replaced by a sort of pianola roll. But on the ENIAC the state of affairs was different. Its operations, being electronic, would be so fast as to make it impossible to supply cards or tape quickly enough. The engineers had had to find a way to make the instructions available to the machine in electronic times of a few millionths of a second.

  On the ENIAC they were arranging for this by a system of external devices which would set up the instructions for each job. It took the form of making connections with plugs as on a manual telephone exchange. (The Colossi had something very similar.) The advantage of this solution was that the instructions would in effect be available instantaneously, once the plugging work was done. The disadvantage was that the sequence of instructions was limited in length, and that it would take a day or so to do the plugging. It would be like building a new machine for each different task. Both ENIAC and Colossus were like kits out of which many slightly different machines could be made. Neither sought to embody the true universality of Babbage’s conception, in which the machinery would be entirely unchanged, and only the instruction cards rewritten.

  But even when von Neumann joined the ENIAC team as ‘adviser’ in late 1944, Eckert and Mauchly had perceived a quite different solution to their problem. This was to leave the hardware alone, and to make the instructions available at electronic speeds by storing them internally, in electronic form. The ENIAC was designed to store its arithmetical working internally, and the point of the original Colossus had been that it would hold Fish key-patterns internally. It was a quite different matter to consider holding the instructions to the machine internally. Instructions were naturally thought of as coming from the outside, to act upon an inside. But the ‘second electronic machine’ mentioned in von Neumann’s letter to Weaver was intended to incorporate this new idea.

  Every tradition of common sense and clear thinking would tend to suggest that ‘numbers’ were entirely different in kind from ‘instructions’. The obvious thing was to keep them apart: the data in one place, and the stock of instructions to operate on the data, in another place. It was obvious – but wrong. In March and April 1945, the ENIAC team had prepared a proposal, the Draft Report on the EDVAC. The EDVAC – the Electronic Discrete Variable Calculator – was the planned ‘second electronic machine’.

  The report was dated 30 June 1945, and signed by von Neumann. It was not his design, but the description of it bore the mark of his more mathematical mind rising above the technicalities.

  In particular, it articulated the very cautious, circumspect, but quite new idea, at which the ENIAC team had arrived in planning a better machine. It discussed the different kinds of storage that existing machines required: intermediate results, instructions, fixed constant parameters, statistical data, and then stated that

  The device requires a considerable memory. While it appeared, that various parts of this memory have to perform functions which differ somewhat in their nature and considerably in their purpose, it is nevertheless tempting to treat the entire memory as one organ.

  But such a proposal, that of the ‘one organ’, was equivalent to adopting the ‘one tape’ of the Universal Turing Machine, on which everything – instructions, data, and working – was to be stored. This was the new idea, different from anything in Babbage’s design, and one which marked a turning point in proposals for digital machines. For it threw all the emphasis on to a new place – the construction of a large, fast, effective, all-purpose electronic ‘memory’. And in its way it made everything much more simple, less cluttered in conception. Von Neumann might well have seen it as ‘tempting’, because it was almost too good an idea to be true. But it had been in Computable Numbers all the time.

  So the spring of 1945 saw the ENIAC team on the one hand, and Alan Turing on the other, arrive naturally at the idea of constructing a universal machine with a single ‘tape’. But they did so in rather different ways. The ENIAC, now already shown to be out-of-date in principle even before it was finished, had been something of a sledgehammer in cracking open the problem. And von Neumann had been obliged to hack his way through the jungle of every known approach to computation, assimilating all the current needs of military research and the capabilities of American industry. The result was something close to the Lancelot Hogben view of science: the political and economic needs of the day determining new ideas.

  But when Alan Turing spoke of ‘building a brain’, he was working and thinking alone in his spare time, pottering around in a British back garden shed with a few pieces of equipment grudgingly conceded by the secret service. He was not being asked to provide the solution to numerical problems such as those von Neumann was engaged upon; he had been thinking for himself. He had simply put together things that no one had put together before: his one-tape universal machine, the knowledge that large-scale electronic pulse technology could work, and the experience of turning cryptanalytic thought into ‘definite methods’ and ‘mechanical processes’. Since 1939 he had been concerned with little but symbols, states, and instruction tables – and with the problem of embodying these as effectively as possible in concrete forms. Now he could consummate it all.

  And now the war was over, his moti
ves were much nearer to those of G.H. Hardy than to the practicalities of the world’s work. They had more to do with the paradox of determinism and free will, than with the effecting of long calculations. Of course, no one was likely to pay for a ‘brain’ that had no useful application. And in this respect Hardy would have found justification of his views regarding the applications of mathematics. On 30 January 1945 von Neumann had written24 that the EDVAC was being designed for three-dimensional ‘aerodynamic and shock-wave problems …shell, bomb and rocket work …progress in the field of propellants and high explosives’. Such, in Churchill’s phrase, would be the ‘progress of mankind’. Alan Turing too would have to come a long way from the logic of Hilbert and Gödel if he was really to build a brain.

  The Draft Report on the EDVAC did carry a more theoretical burden (one reflecting von Neumann’s interests) in that it drew attention to the analogy between a computer, and the human nervous system. The use of the word ‘memory’ was an aspect of this. In its way, it was ‘building a brain’. However, its emphasis was placed not on an abstract thesis concerning ‘states of mind’, but on the similarities between input and output mechanisms, and afferent and efferent nerves respectively. It also drew upon the 1943 paper written by the Chicago neurologists W.S. McCulloch and W. Pitts, which analysed the activity of neurons in logical terms, and used their symbolism for describing the logical connections of electronic components.

  McCulloch and Pitts had been inspired by Computable Numbers, and so in a very indirect way, the EDVAC proposal owed something to the concept of the Turing machine. One of its curiosities, however, was that it made no mention of Computable Numbers, nor made precise the universal machine concept. Yet von Neumann had been familiar with it before the war, and must surely have recognised the connection when he liberated himself from the assumption that data and instructions had to be stored in different ways. According to S. Frankel,25 who worked at Los Alamos on the atomic bomb and was one of the first to use the ENIAC,

  in about 1943 or ‘44 von Neumann was well aware of the fundamental importance of Turing’s paper of 1936 ‘On computable numbers …’ …Von Neumann introduced me to that paper and at his urging I studied it with care.…he firmly emphasised to me, and to others I am sure, that the fundamental conception is owing to Turing – insofar as not anticipated by Babbage, Lovelace and others.

  So the Wizard might well have learnt something from Dorothy. However, the essential point about these two initiatives, American and British, was not the rather tenuous connection between them. It was their very marked independence.26

  Whatever ideas had flowed westwards, the Draft Report on the EDVAC was the first to put them together in writing. So once again, British originality had been pipped at the post by an American publication – and at a time when everyone was looking to the west. The Americans had won, and Alan was a sporting second. This time, however, American priority was nothing but an advantage to the Turing plans, for it provided the political and economic impetus that his own ideas alone could never have enjoyed.

  Indeed, it was probably only the existence of the ENIAC and the EDVAC idea that made possible the next stage of Alan Turing’s life. For in June he had a telephone call at Hanslope. It was from J. R. Womersley, Superintendent of the Mathematics Division at the National Physical Laboratory.

  Womersley was a new man in a new post in a new organisation. The National Physical Laboratory was not new; it had been set up in shabby, suburban Teddington in 1900, the British response to state-sponsored German scientific research. Its site was carved out of Bushy Park, itself largely given over to Supreme Headquarters, Allied Expeditionary Forces. It was the most extensive government laboratory in the United Kingdom, enjoying a high reputation within its traditional sphere, that of setting and maintaining physical standards for the benefit of British industry. Its current Director, installed in 1938, was Sir Charles Galton Darwin, grandson of the theorist of evolution and himself an eminent Cambridge applied mathematician. His major contribution had been in the field of X-ray crystallography, and like Humpty Dumpty, who was able to explain the Jabbervocky, he was regarded as the27 ‘interpreter of the new quantum theory to experimental physicists.’ Large, awesome, and remote, during the war he had spent a year as director of what became the British Central Scientific Mission in Washington, and had been the first scientific adviser to the British Army.

  The Mathematics Division, however, was new. Indeed, it was a computational equivalent of the planned welfare state, the product of a calculators’ Beveridge Report. In about March 1944 a proposal28 had been mooted for an independent Mathematical Station, and this suggestion, a fine example of wartime planning for peace, went to a large interdepartmental committee, itself a manifestation of cooperation and coordination unthinkable in peacetime days. The government accepted the principle of continuing the funding found necessary in war, and a centralised, rationalised institution was planned to take over the various ad hoc offices which had done the mostly dreary work of numerical computation for military purposes. Sir Charles Darwin had persuaded the committee to establish it as a division of the NPL.

  But the telephone call to Hanslope was not made at Darwin’s behest. It was made on the initiative of his subordinate, Womersley, who had been selected as head of the new Division on 27 September 1944. Womersley, a bulky Yorkshireman then attached to the Ministry of Supply and a member of the interdepartmental committee, was probably the nominee of D.R. Hartree, who in mathematical matters was a power behind the Darwin throne. Womersley had appeared as joint author with Hartree of a 1937 paper on the application of the differential analyser to partial differential equations.

  The official research programme for the new Division in October 1944 included ‘Investigation of the possible adaptation of automatic telephone equipment to scientific computing’ and ‘Development of electronic counting device suitable for rapid computing’. Behind these words there were more definite intentions to imitate the American developments. Hartree, with his differential analyser at Manchester University, already had an interest in computing machinery, and had a finger in many wartime scientific projects. In the high levels where he moved, some details of the secret Aiken and ENIAC machines filtered through. Such knowledge was reflected in Womersley’s report in December 1944, which, though it placed emphasis upon building a large differential analyser, remarked on the speed of electronics, and suggested that ‘a machine can be made to perform certain cycles of operations mechanically …the instructions to the machine [could] depend on the result of previous operations …the problem is already being tackled in the USA.’ The press release29 in April 1945, when the new Division was officially inaugurated, only made mention of ‘analytical engines, including the differential analyser and other machines both existing and awaiting invention …it is certain that this field is capable of great developments, but it is more difficult to predict in what directions they will lie.’ But it seemed that the direction in which to look was westwards, and in February 1945, Womersley had been packed off on a two-month tour of the computing installations of the United States, where on 12 March he was the first non-American to be allowed access to the ENIAC, and to be informed of the EDVAC report.

  By 15 May, Womersley was back at the NPL ‘revising his plans’. The American revelations were enough to give anyone pause for thought. But they held a particular meaning for Womersley, who held one very unusual card up his sleeve. Before the war, while employed at Woolwich Arsenal on practical computation, he had learnt of Turing machines. Even more remarkably, for a mathematician-in-the-street, he had not been daunted by the abstruse language of mathematical logic. According to his claim:30

  1937-38 Paper Computable Numbers seen by JRW and read. JRW met C.I. Norfolk, a telephone engineer who had specialised in totalisator design and discussed with him the planning of a ‘Turing Machine’ using automatic telephone equipment, Rough schematics prepared, and possibility of submitting a proposal to NPL discussed. It was decided that
machine would be too slow to be effective.

  June 1938 JRW purchased a uniselector and some relays on Petty Cash at RD Woolwich for spare-time experiments. Experiments abandoned owing to pressure of work on ballistics.

  On seeing Aiken’s machine at Harvard, Womersley had written home to his wife that he saw it as ‘Turing in hardware’. And thus it was that in June 1945, according to his account:

  JRW meets Professor M.H.A. Newman.* Tells Newman he wishes to meet Turing. Meets Turing same day and invites him home. JRW shows Turing the first report on the EDVAC and persuades him to join NPL staff, arranges interview and convinces Director and Secretary.

  Alan would be appointed a Temporary Senior Scientific Officer at £800 per annum. Don Bayley, when told of this, did not think much of the rank, but Alan told him that it was the highest to which they could recruit, and that he had been assured of a promotion within a few weeks. It was not quite ‘Fivepence farthing for one – twopence for two’, as of the eggs that the Sheep offered Alice in the Looking Glass shop, but at £600 for naval Enigma, and £800 for the digital computer, the British government had certainly acquired a bargain in Alan Turing. Alan claimed that Womersley had asked him whether he knew ‘the integral of cos x’ which as Don Bayley immediately said, was a ludicrously trivial question to ask any prospective SSO, let alone a B-star Wrangler. ‘Ah’, said Alan, in a joke against his own capacity for carelessness, ‘but what if I had got it wrong?

  On his side, Womersley expressed himself to colleagues as delighted with the capture of Alain Turing for his new department. And for Alan, who was careless about rank and terms of appointment, it was still the exciting prospect of having the British government itself support the realisation of a Universal Turing Machine. He had done his bit for them; now they could return the compliment. The NPL had been founded ‘to break down the barrier between theory and practice’, and this was exactly what Alan proposed to do. Whatever his doubts about the Civil Service, it offered him a chance. When he bid farewell to Joan Clarke and the others tidying up Hut 8, he talked excitedly of the future of automatic computers, and reassured them that mathematicians would not be put out of work.