The Nature of Turing and the Physical World a preface by Andrew Hodges
(2000) for The Collected Works of A. M. Turing
For a guide to this website go to the Alan Turing Home Page
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 that cannot be mechanized. Turing described 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, emphasizing 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 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 authoritatively 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 speechscrambling 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 storedprogram 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 morphogenetic 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 'oraclemachines.' 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.
The word 'fount' (i.e. a typographical 'font'), is probably Turing's own. It
refers to the convention of using different fonts for tensor indices related to
different symmetry groups. 'Particles are founts' implies a classification of
particles by underlying symmetry groups, a key idea of elementary particle
theory.
The statement about charge is essentially the idea of gauge theory, which in
1954 was taking on new life in the famous Yang-Mills paper of that year,
which discussed non-Abelian gauge groups.
The reference to the 'interior of the light cone of the Creation' is not trivial.
Light-ray-based descriptions were the key to the renaissance of General
Relativity in 1954, a field in which little had happened in fifteen years but was
soon transformed into a new arena both for differential geometry and
astrophysics. (Clear evidence for the Big Bang was, however, eleven years
away.)
The 'hyperboloids of wondrous light' are a mystery, but suggest some new
geometrical theory for the propagation of quantum-mechanical wave
functions.
Robin Gandy's letter then (in section 8) records Turing's dissatisfaction with
standard 'Copenhagen' quantum mechanics, essentially because it gives no
explanation of how or when the reduction of the wave function is supposed
to occur. The introduction of non-linearity (as mentioned in section 7) is
consistent with how later thinkers have addressed this still-mysterious
question.
To summarize: Turing was probably as much in touch with the problems of
fundamental physics in 1954 as he was with those of mathematical logic in
1935, and the stage was set for a major creative act. Newman saw Turing's
mind as still at 'the height of its power' in 1954. We can see these 'Messages
from the Unseen World' as showing, or shadowing, what Turing might have
done had he lived on.
But it is uncannily close to what Roger Penrose, just twenty-two at Turing's
Turing's lines about
'boundary conditions' as opposed to
'differential equations' are clearly aimed at Eddington's
death, has since actually done.
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.
This report may now be read on a page on this website.
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