This article was written for a conference on Mathematics and War, Karlskrona, Sweden, August 2002. The editor may yet suggest changes for the published version.
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In discussions of the origin of the computer there is an inevitable rivalry between the claims of mathematics and engineering. Engineers, often overlooked and accorded a low social status, do not take kindly to being treated as mere technical assistants when in fact they have contributed immensely skilled and creative solutions. Nevertheless, whether we speak of the computer, or of machines such as the Bombe or Colossus, no engineer originally conceived the nature and purpose of the machine: the conception was that of mathematicians. The point of the Bombe was to implement Turing's brilliant logic; the point of the Colossus was to implement ingenious developments of his statistical theory. The point of the modern computer is that it implements Turing's universal machine, on which instructions can be stored and manipulated in exactly the same way as data. It is usually said that von Neumann, in formulating the EDVAC plan in 1945, played the same mathematical role independently, but Martin Davis has recently argued that von Neumann could not have played this part without learning of Turing's work before the war (Davis 2000).
However, Turing was unusual in wishing to dominate every aspect of the computer design: not only its central principle and purpose, but the applications for which it would be used, and the details of its design. His report (Turing 1946) for the National Physical Laboratory, London, covered all of these. In every way it reflected the influence of his war work, but there was an irony. Official secrecy about the wartime success was total and prevented Turing from arguing from his extensive experience to establish some influence over the engineering side of the problem. So although his report alluded to important military applications in the solution of combinatorial problems, showed other influences of non-numerical Bletchley Park work, and claimed Foreign Office support, the NPL management continued to regard mathematics and engineering as belonging to two distinct planets and blocked Turing's efforts to argue otherwise.
Turing was particularly frustrated over the inability to command the necessary technology, because although vital for the trying out of a practical form of a universal machine, the actual form of the implementation was essentially a secondary matter. Turing correctly saw his ACE plan as necessarily obsolescent, leading the way to better machines built using faster control circuits and larger storage systems. Of far greater significance, in Turing's plans, was the scope of what a practical universal machine would be doing: namely, implementing an open-ended array of software or as he called it, instruction tables.
In that first report he wrote a number of key remarks about the potential of the universal machine:
It is intended that the setting up of the machine for new problems shall be virtually only a matter of paper work... There will be positively no internal alterations to be made even if we wish suddenly from calculating the energy levels of the neon atom to the enumeration of groups of order 720.
Instruction tables will have to be made up by mathematicians with computing experience and perhaps a certain puzzle-solving ability. There will probably be a great deal of work of this kind to be done, for every known process has got to be translated into instruction table form at some stage.
The process of constructing instruction tables should be every fascinating. There need be no real danger of it ever becoming a drudge, for any processes that are quite mechanical may be turned over to the machine itself.
In (Turing 1947) he elaborated on the potential for computer languages opened up by this last remark:
Actually one could communicate with these machines in any language provided it was an exact language, i.e. in principle one should be able to communicate in any symbolic logic, provided that the machine were given instruction tables which would allow it to interpret that logical system...
These and other observations mapped out the scope of the computing industry of the future (although Turing did remarkably little to implement his prophetic ideas for computer languages). But, at a more profound level still, even this software manifesto was also secondary. He was, he said, building a brain: embarking on an experimental programme for what he called 'intelligent machinery', or what would now be called 'artificial intelligence.'
This deeper scientific programme was also influenced by his war experience. Before the war Turing had, in his work on ordinal logics, apparently accepted the standpoint of Gödel in which the mathematician does something beyond the scope of rule-following when seeing the truth of a Gödel sentence. It has been argued by the author (Hodges 1997, 2002), that Turing must have had a change of mind, or at least a definite clarification of mind, in about 1941. By that time the spectacle of the Bombes and of the mechanisation of human guessing and judgment by Bayesian statistical methods had supplied a vivid argument to the effect that the concept of being 'merely' mechanical was misleading. To capture the spirit of the time, it is worth noting from a recently declassified document (Good et al. 1945) the vivid impression that the beginning of the electronic information technology age made on its beholders:
It is regretted that it is not possible to give an adequate idea of the fascination of a Colossus at work... the fantastic speed of thin paper tape eround the glittering pulleys... the wizardry of purely mechanical decoding letter by letter (one novice thought she was being hoaxed); the uncanny action of the typewriter in printing the correct scores without and beyond human aid...Turing, even though he perfectly understood what was going on, was doubtless also influenced by this magical power of the purely mechanical. Possibly, Turing was also encouraged by exposure to intellectual currents which suggested thinking of the nervous system in behaviourist terms.
In any case, it is clear that by 1945 Turing had decided that the brain did not actually perform any uncomputable operations of the kind suggested by Gödel's theorem. He had formulated his 'mistakes' argument: that in assessing mathematical proofs the brain will sometimes make mistakes, and that accordingly Gödel's theorem does not have any force. His guiding thought was that the brain must be effectively a finite machine and therefore performs operations which can, at least in principle, be run on a universal machine, the computer. A further report (Turing 1948) explored the question of how a machine could be led to perform actions which appear not to be of a 'mechanical' nature, as understood in common parlance. Turing discussed program self-modification, networks of logical elements as models of neuronal systems, the idea of 'genetical search.' His ideas for emulating human 'education' combined what would now be called top-down and bottom-up approaches to Artificial Intelligence. This report was the basis of his more famous philosophical paper (Turing 1950).
Turing spent a year at Cambridge to do this theoretical work in 1948, because the National Physical Laboratory was making such desultory progress with implementing his computer plan. While he was there, another opportunity appeared to open. The key figure was the topologist M. H. A. Newman, the Cambridge lecturer who had introduced Turing to the frontier of mathematical logic in 1935, the first reader of Turing's consequent logical discoveries, and indeed to some extent Turing's collaborator in logic. Newman took the idea of the universal machine with him when he went to Manchester University as professor of pure mathematics in 1945. He also took an acquaintance with electronics, since he had directed the development of the Colossus as applied to the Lorenz cipher problem at Bletchley. In fact, he was equipped with all the force of Turing's ideas except that he had none of Turing's interest in mastering electronics for himself.
Newman's fervent desire was that the concentration and investment of scientific resources that had been shown possible in the emergency of the war, should now be dedicated to pure science. He rapidly made a proposal to the Royal Society, and was successful in getting a large grant for a computer on which to do research in mathematics (including for instance the Four-Colour Theorem), on an ambitious and long-term scale. Newman was able to recruit for his project the leading electronic engineers who came to Manchester from wartime radar development work, and he imparted to them the essential stored-program ideas that had to be implemented.
Newman offered Turing the role of running this emergent Computing Laboratory. Turing accepted. But even before he arrived, the success in June 1948 of the engineers' prototype meant that it was demanded for the British atomic bomb development. The government contract went to the engineers, not to Newman; the Royal Society connection was abandoned. Turing was sidelined along with Newman himself, and had to relaunch himself with quite new work: his mathematical theory of morphogenesis. By 1950 it was the end of the road for the synthesis and collaboration achieved in the Second World War. What is now the engineering discipline of 'computer science' took on a life of its own and the connection with mathematics was largely lost, Turing's legacy being abandoned with it.
There is, however, an important caveat to be made here: we do not know what Turing did as secret work for GCHQ after 1948. This Cold War question is still as secret now as the Second World War operations were in 1970. And it opens an aspect of the subject of Mathematics and War where all is speculation, an unresolved enigma. Perhaps the most extraordinary aspect of Alan Turing's story is his role as a great innocent figure of pure science, yet at the heart of the most urgent world crisis. It is a magical picture, but this sequel to it is still incomplete and obscure.
Was he involved in the Venona problem, which involved tracking KGB messages to and from its western agents? If so, was he eager for participation in the work against Soviet influence? Or was he only responding to a strong personal plea from Hugh Alexander, who from being Turing's deputy in 1941 had risen to become director of GCHQ in 1952? Or was it more the challenge of another 'interesting and amusing' mathematical problem? What were the inner springs? We do not know. I turn now to those more general questions about the setting of Turing's mind in the world of war.
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