This preface was written to complement the material in the fourth volume ('Mathematical Logic') of the Collected Works. Its comments and allusions are in a form appropriate to this publication. In particular, note that this essay precedes a transcription of the letter that Robin Gandy wrote to Max Newman in June 1954. For notes on the Collected Works see the bibliography page on this website.
In this letter, Alan Turing's student, friend and colleague Robin Gandy tried to capture the intellectual currents that seemed most striking in the immediate aftermath of his sudden death. This note attempts to encapsulate the background to Turing's extraordinary exploration of new ideas in physics, as documented by the letter. It also indicates the context of Robin Gandy's knowledge of Turing's work. The reader will know from the Preface to this volume that the Collected Works project itself long rested on Robin Gandy's subsequent dedication, so that Robin's letter was both a last word on Turing's living communications and a first contribution to this necessary future undertaking.
Turing's first serious interest in mathematics came through reading Einstein and Eddington and he had a fine understanding of the principles of general relativity and quantum mechanics while still at school (ATTE, pages 33, 40). His first recorded serious scientific question was influenced by Eddington's 1928 book, The Nature of the Physical World: is there a quantum-mechanical basis to human will? (ATTE, page 63)
As a student, Turing ignored the conventional Cambridge division between 'pure' and 'applied' mathematics. He was seriously reading Russell, but also seriously considered research in mathematical physics (ATTE, page 95) and his Fellowship dissertation on the Central Limit Theorem was on a borderline between 'pure' and 'applied.' John Britton's comment [Volume 1 of the Collected Works ] on Turing's lasting interest in applying probability theory extends also to an interest in the physical world quite unlike anything that the Cambridge tradition of pure mathematics encouraged.
Indeed it could be said that Turing treated mathematical logic like an applied mathematician. Max Newman's Memoir [this volume of the Collected Works] called him 'at heart more of an applied than a pure mathematician' which must have surprised all those who thought they knew Turing as a logician. In 1937, Turing immediately turned to the embodiment of primitive logical operations in electromagnetic relays, and this fascination with connecting the abstract with the practical continued undiminished thereafter, as indeed Robin Gandy's letter pointed out in expressing his agreement with Newman.
One of the most fascinating questions about Turing's subsequent development concerns the extraordinary flux of his ideas in 1937-9, a period when he was busy in logic and analytic number theory, but also tackling the Enigma and arguing with Wittgenstein. At this time he developed his most abstract and advanced work, the ordinal logics. But even this he gave an extra-mathematical interpretation. The ordinal logics are motivated by Gödel's proof that seeing the truth of mathematical statements requires methods which cannot be mechanised. Turing decribed these non-mechanical steps as 'intuitive judgments.' Had he pursued his interest in an underlying physical viewpoint, what would he have said the brain was doing in such moments of intuition? The question might driven him to the something resembling Penrose's position, as it emerged in the 1980s [Penrose 1988], on the necessity for an uncomputable element in fundamental physics.
In his Memoir, Max Newman dwelt on the loss to science that arose because this period was terminated by the outbreak of war. Newman's choice of priorities, emphasising the ordinal logics and what Turing might have done had he continued to concentrate on mathematics, but regarding the wartime work and the building of electronic computers as unfortunate or unimportant interruptions, must have struck many as an incomprehensible distortion. In a longer term these judgments may acquire new force.
Leaving aside what Turing might have done, the reality of the Second World War gave immense stimulus to the physical embodiment of Turing's logical ideas, and developed his acquaintance first with electromagnetic and and then with electronic technology. In wartime years, no dividing line was drawn between Turing's logic and its physical implementation. A recently declassified paper of 1942, [Turing 1942] Turing's report on his visit to the factory where the American Bombes were built, is striking in its revelation of its author as reporting authoritatatively on engineering questions (e.g. the electronic testing of commutators) as well as on logical design.
It was in this context that Turing determined to outsmart American speech-scrambling with his own electronic system, the Delilah. Shortly afterwards, in 1944, Robin Gandy first began close contact with Turing, who was then soldering electronic components with his own hands, and enjoying the role of solving electrical engineering problems (starting, naturally, from Maxwell's equations.) One of many ironies in Gandy's story, which emerged in the many discussions I had with him while writing my Turing biography in 1977-83, was that he then saw himself a mathematical physicist, and so was ignorant of logic. Although in Turing's presence in 1945, the future logician and Turing disciple did not observe the emergence of the practical stored-program computer from the logic of the universal machine.
It is another curious fact that neither Robin Gandy nor anyone else no-one seems to have observed the important change in Turing's position regarding uncomputability and intelligence, which I have recently discussed in [TNP] as evident by 1945. During the war Turing had apparently come to the position that intelligence did not mean infallibility, so that undecidability and incompleteness were not relevant to understanding mental action. I now think this was a key point in Turing's development, in which he abandoned the significance he had attached to ordinal logics in 1938-9, and instead attached great importance to the ability of computers to modify their own programs and do what programmers could not have foreseen. I believe he decided that 'intuition' could be accounted for by learning processes and the implicit programming of neural networks. But this was apparently a dialogue about logic and physics in Turing's own mind alone.
Turing's knowledge and interest in applied physics did not end with the Second World War. Turing worked from first principles on the design of delay lines for his ACE plan, though the results did not convince professional engineers and his circuit designs were probably his least satisfactory work. In this he failed where his rival Maurice Wilkes, a classic Cambridge applied mathematician, brilliantly succeeded. It should also be said that Turing's knowledge of applied mathematical techniques continued to surprise those who classified him as a pure mathematician. Thus, at Manchester he impressed Alick Glennie, who did computational work for the British atomic bomb, with his current knowledge of hydrodynamics. In his own work, he effortlessly introduced inverse differential operators for handling partial differential equations in his morphegenetic theory.
In the more theoretical side of the physical basis of digital computing, Turing showed rather an inconsistent interest. He maintained a careful concern to point out that logical discrete systems are embodied in the physically continuous, but gave only slight references to actual physics. His 1948 work [Turing 1948, page 7] contains a thermodynamic calculation relevant to computer reliability (see also the story told by John Britton in the introduction to Volume 1) but he paid no attention to the quantum-mechanical basis of electronics.
Turing's underlying thesis, increasingly evident as his claims for mechanical intelligence grew in confidence, was that whatever it is the brain does, it must be a computable process. Considering the importance of this conviction, his references to underlying physical law, as discussed in [TNP], are somewhat thin and cavalier. It is surprising also that although Robin Gandy moved to logic under Turing's influence, and became his student with a thesis on the logical foundations of physics, they never seem to have discussed Turing's underlying assumptions about fundamental physics. Turing never considered quantum computation. Nor did Turing ever press questions about digital approximations to continuous systems. In the famous paper [Turing 1950] he gave a classic comment (page 440) on the 'butterfly effect' in physical systems (in terms of snowflakes and avalanches), pointing out the lack of analogy in discrete computation; and yet also in brief words (dealing with the 'Argument from Continuity in the Nervous System' on page 451) espoused faith in the discrete approximations used in applied mathematics. This was entirely consistent with his practical experience: he was pioneering the use of digital computers for exploring the evolution of critical effects in his morphogenetic theory. But it was not a serious examination of the relationship of computation to continuous physics, such as modern theoreticians of analogue computing now undertake.
This somewhat patchy nature of Turing's post-war physical interests means that there is little to prepare us for a sudden explosion of ideas about the fundamental physics of quantum mechanics and relativity in 1953-4. There is in fact just one link between Turing's major work in logic and computability, and his late interest in physics. In his radio talk [Turing 1951], Turing referred briefly to the unpredictability of quantum mechanics as implying that physical systems might not be amenable to simulation by the universal Turing machine. In [ATTE, page 441] I brought out the reference that Turing made here to Eddington's views, suggesting the connection with those early thoughts about physics and Mind. But I would now take Turing's question more seriously, noting that the unpredictability of quantum mechanics lies in its still-mysterious 'reduction' process. The philosopher B. J. Copeland [Copeland 1999] has also drawn attention to Turing's 1951 sentence, but in a context suggesting that Turing was connecting 'randomness' with 'oracle-machines.' This is unjustified: the point is that Turing was beginning to give active thought to the theory of wave-function reduction, as is described in Robin Gandy's letter.
Another irony confronts us here: Robin Gandy had by that time switched entirely to becoming Turing's successor in the field of mathematical logic where, in turn, Turing had abandoned an active interest. Hence the verbal explanations Turing gave to Robin, as mentioned in his letter, were probably mainly lost on him. Nevertheless Robin kept the postcards that Turing sent in 1954, and from which he quoted in his letter to Newman. (These survive in the King's College archive, also being reproduced in [ATTE, page 513].) These are only scraps, but enough to suggest serious new directions. The language, depending on cryptic comments to be decoded by Robin in the light of their shared good humour, disguises their seriousness.
In [ATTE, page 495] I referred to Turing taking up Dirac's theory of spinors; I have since learned from Sir Roger Penrose that Turing's tensor analysis notes [Turing 1954?] must be notes on Dirac's Cambridge lectures on quantum mechanics, which somehow he had found time to attend. The postcards also show Dirac's influence. One comment of Turing's, written as a sideline on one of the postcards, 'Does the gravitational constant decrease,' is indeed pure Dirac. But the other comments have greater originality.
But it is uncannily close to what Roger Penrose, just twenty-two at Turing's death, has since actually done. Turing's lines about 'boundary conditions' as opposed to 'differential equations' are clearly aimed at Eddington's mysticism in their reference to 'Religion.' Yet they can also be read seriously as a reference to the nature of physical law: physical science has so far rested on laws framed as differential equations but there is no absolute reason why this should remain the case. It is Penrose now who suggests that the union of gravity and quantum mechanics must involve some new kind of physical law, a boundary condition on space-time singularities which introduces asymmetry in time.
In 1983, in [ATTE, page 514], I only hinted at how the physical programme sketched here by Turing has since been realised by Penrose. Since then Roger Penrose has further suggested that an uncomputable element must enter into wave-function reduction in order to explain consciousness — we behold a mystery and close this Volume in natural wonder.
References[ATTE]: Andrew Hodges, Alan Turing, the Enigma: see the page on this website.
[TNP]: Andrew Hodges, Turing, a natural philosopher: see the page on this website.
B. J. Copeland (ed.) A lecture and two radio broadcasts on machine intelligence by Alan Turing, in F. Furukawa, D. Michie, S. Muggleton (eds.), Machine Intelligence 15, (Oxford: Oxford University Press, 1999)
Roger Penrose, On the Physics and Mathematics of Thought, in: The Universal Turing Machine, a Half-Century Survey, ed. Rolf Herken, (Verlag Kammerer & Unverzagt, Berlin, 1988). This paper preceded Penrose's better known The Emperor's New Mind (Oxford University Press, 1989).
Turing 1942: Visit to National Cash Register Corporation of Dayton, Ohio, a report by
Turing of December 1942, in 'Bombe Correspondence' (Crane Collection) CSNG LIB, Box 139, RG 38, Records of the Office of Naval Intelligence. I owe this to Lee A. Gladwin of the National Archives and Records Administration, Washington DC.
Turing 1948: Intelligent Machinery, see the bibliography on this website
Turing 1950: Computing Machinery and Intelligence, see the bibliography on this website
Turing 1951: Radio talk, see the bibliography on this website
Turing 1954?: Unpublished notes on tensors and spinors, see the bibliography on this website