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Home , Jack Copeland, Max Newman

TURING AND THE PHYSICS OF THE MIND

B.

31 August 2011

Introduction Turing's lecture 'Can Digital Think?' was broadcast on BBC Radio on 15th May 1951 (repeated on 3rd July). It was the second in a series of lectures entitled 'Automatic Calculating Machines'. Others contributors to the series included (like Turing from the University of Manchester), (University of Cambridge), Maurice Wilkes (Cambridge), and F.C. Williams (Manchester).1 In modern times, 'Can Digital Computers Think?' was virtually unknown until 1999, when I included it in a small collection of unpublished work by Turing ('A Lecture and Two Radio Broadcasts on Machine Intelligence by ', in Machine Intelligence 15) and again in The Essential Turing in 2004. The previously published text, reproduced here, is from Turing's own typescript and incorporates corrections made in his hand. In this broadcast Turing's overarching aim was to defend his view that 'it is not altogether unreasonable to describe digital computers as brains'. The broadcast contains a bouquet of fascinating arguments, and includes discussions of the Church-Turing thesis and of free will. There is a continuation of Turing’s discussion of 'Lady Lovelace's dictum' (which he had begun in 'Computing Machinery and Intelligence' the previous year), and a priceless analogy that likens the attempt to program a to act like a brain to trying to write a treatise about family life on Mars—and moreover with insufficient paper. The broadcast makes manifest Turing's real attitude to talk of machines thinking. In 'Computing Machinery and Intelligence', he famously said that the question 'Can machines think?' is 'too meaningless to deserve discussion',2 but in this broadcast he made liberal use of phrases such as 'programming a machine to think' and 'the attempt to make a thinking machine'. In one passage he said: our main problem [is] how to programme a machine to imitate the brain, or as we might say more briefly, if less accurately, to think. However, the feature of the broadcast that is of absolutely outstanding interest, and is the topic of this note, is Turing's brief discussion of the possibility that physical action is not always computable.3 attributed the opposite view to Turing. He said, 'It seems likely that he [Turing] viewed physical action in general—which would include the action of a human brain—to be always reducible to some kind of Turing-machine action'.4 Penrose even named this claim Turing's thesis. Yet Turing never endorsed this thesis. As 'Can Digital Computers Think?' makes clear, Turing was aware that the thesis might be false.

Physics and Uncomputability: A Brief History This overview mentions only a few of the important milestones in the history of the current debate about physics and uncomputability.5

1. Scarpellini In an article published in German in 1963, Scarpellini speculated that non-recursive (i.e. non Turing-machine-computable) processes might occur in nature. He wrote: [O]ne may ask whether it is possible to construct an analogue- computer which is in a position to generate functions f(x) for which the predicate ! f(x) cos nxdx > 0 is not decidable [by ] while the machine itself decides by direct measurement whether ! f(x) cos nxdx is greater than zero or not.6 Scarpellini's suggestion was de novo. He had no knowledge of Turing's 'Can Digital Computers Think?'. Working in isolation at the Battelle Research Institute in Geneva in 1961, he conceived the idea that natural processes describable by classical analysis might falsify the suggestion that every function effectively computable by machine is also computable by Turing machine. He recollects that the influence of Turing's work on his paper was twofold: 'Technically, in that Turing machines appear, explicitly or implicitly, at various points of my paper; and conceptually, in that his work caused me to perform for the continuum, i.e., analysis, constructions analogous to those that he introduced for discrete '.7 In comments on his 1963 paper made in 2003 Scarpellini said: '[I]t does not seem unreasonable to suggest that the brain may rely on analogue processes for certain types of computation and decision-making. Possible candidates which may give rise to

2 such processes are the axons of nerve cells. ... [I]t is conceivable that the mathematics of a collection of axons may lead to undecidable propositions.'8

2. Komar In 1964 Komar showed that the behaviour of a quantum system having an infinite number of degrees of freedom can be uncomputable, proving that, in general, the cannot determine whether or not two arbitrarily chosen states of such a system are macroscopically distinguishable. He wrote: '[T]here exists no [effective] procedure for determining whether two arbitrarily given physical states can be superposed to show interference effects characteristic of quantum systems. ... [I]t is rather curious … that the issue of the macroscopic distinguishability of quantum states should be among the undecidable questions.'9

3. Kreisel Kreisel emphasised that it is an open question whether there are uncomputable natural processes. He discussed this theme in relation to classical mechanics, classical electrodynamics and quantum mechanics, in remarks throughout a series of papers spanning three decades.10 He said in 1967: 'There is no evidence that even present day quantum theory is a mechanistic, i.e. recursive theory in the sense that a recursively described system has recursive behaviour'.11

4. Pour-el and Richards In 1979 Pour-El and Richards published their paper 'A computable ordinary differential equation which possesses no computable solution', followed in 1981 by their equally transparently titled 'The wave equation with computable initial data such that its unique solution is not computable'.12 They explained their general approach as follows: Our results are related to some remarks of Kreisel. In [1974], Kreisel concerns himself with the following question. Can one predict theoretically on the basis of some current physical theory—e.g. classical mechanics or quantum mechanics—the existence of a physical constant which is not a recursive real? Since physical theories are often expressed in terms of differential

3 equations, it is natural to ask the following question: Are the solutions of "' = F(x,"), "(0) = 0, computable when F is?13 Pour-El and Richards proved in their second paper that the behaviour of a certain system with computable initial conditions and evolving in accordance with the familiar three-dimensional wave equation is not computable. In a review of their two papers Kreisel wrote: The authors suggest, albeit briefly and in somewhat different terms, that they have described an analogue computer that—even theoretically—cannot be simulated by a Turing machine. Here 'analogue computer' refers to any physical system, possibly with a discrete output, such as bits of computer hardware realizing whole 'subroutines'. (Turing's idealized digital computer becomes an analogue computer once the physical systems are specified that realize—tacitly, according to physical theory—his basic operations.)14

5. Karp and Lipton In a conference presentation in 1980, Karp and Lipton discussed infinite families of digital circuits, for example circuits consisting of boolean gates or McCulloch-Pitts neurons.15 Each individual circuit in an infinite family is finite. A family may be regarded as a representation of a piece of hardware that grows over time, each stage of growth consisting of the addition of a finite number of new nodes (e.g. neurons). The behaviour of any given circuit in a family can be calculated by some Turing machine or other (since each circuit is finite), but there may be no single Turing machine able to do this for all circuits in the family. In the case of some families, that is to say, the function computed by the growing hardware—successive members of the family computing values of the function for increasingly large inputs—is uncomputable.

6. Doyle In 1982 Doyle suggested that the physical process of equilibriating—for example, a quantum system's settling into one of a discrete spectrum of states of equilibrium—is 'so

4 easily, reproducibly, and mindlessly accomplished' that it can be granted equal status alongside the operations usually termed effective. He wrote: My suspicion is that physics is easily rich enough so that ... the functions computable in principle, given Turing's operations and equilibriating, include non-recursive functions. For example, I think that chemistry may be rich enough that given a diophantine equation ... we plug values into [a] molecule as boundary conditions, and solve the equation iff the molecule finds an equilibrium.16

7. Rubel Rubel emphasised that aspects of brain function are analog in nature and suggested that the brain be modelled in terms of continuous mathematics, as against the discrete mathematics of what he called the 'binary model'. A proponent of analog computation, he noted that '[A]nalog computers, besides their versatility, are extremely fast at what they do ... In principle, they act instantaneously and in real time. ... Analog computers are still unrivalled when a large number of closely related differential equations must be solved.'17 He maintained that not all analog computers need be amenable to digital simulation, even in principle: One can easily envisage other kinds of black boxes of an input- output character that would lead to different kinds of analog computers. ... Whether digital simulation is possible for these 'extended' analog computers poses a rich and challenging set of research questions.18

8. Geroch and Hartle Geroch and Hartle argued that theories describing uncomputable physical systems 'should be no more unsettling to physics than has the existence of well-posed problems unsolvable by any algorithm been to mathematics', suggesting that such theories 'may be forced upon us' in the quantum domain.19 They drew an analogy between the process of deriving predictions from such a physical theory and the process of calculating approximations to some uncomputable real number. One algorithm may deliver the first n digits of the

5 decimal representation of the real number, another the next m digits, and so on. Discovering each algorithm is, as they put it, akin to finding the theory in the first place—a creative act. They said: To predict to within, say, 10%, one manipulates the mathematics of the theory for a while, arriving eventually at the predicted number. To predict to within 1%, it will be necessary to work much harder, and perhaps to invent a novel way to carry out certain mathematical steps. To predict to 0.1% would require still more new ideas. ... The theory certainly 'makes definite predictions', in the sense that predictions are always in principle available. It is just that ever increasing degrees of sophistication would be necessary to extract those predictions. The prediction process would never become routine.20 This sophisticated analogy might have appealed to Turing. In a little noticed passage in 'On Computable Numbers' Turing defined a certain infinite binary sequence #, which he showed to be uncomputable, and said: 'It is (so far as we know at present) possible that any assigned number of figures of # can be calculated, but not by a uniform process. When sufficiently many figures of # have been calculated, an essentially new method is necessary in order to obtain more figures.'21 In a wartime letter to Newman, Turing spoke at greater length about the necessity of introducing mathematical methods transcending any single uniform process, and he emphasised the connection between the necessity for 'essentially new methods' and the 'ordinal ' of his 1939 paper.22 Placing the incompleteness results in a different light—these are usually stated in terms of there being true mathematical statements that are not provable—Turing said: The straightforward unsolvability or incompleteness results about systems of logic amount to this $) One cannot expect to be able to solve the for a system %) One cannot expect that a system will cover all possible methods of proof.

6 [W]e ... make proofs ... by hitting on one and then checking up to see that it is right. ... When one takes %) into account one has to admit that not one but many methods of checking up are needed. In writing about ordinal logics I had this kind of idea in mind."23 The picture of uncomputable physical systems drawn by Geroch and Hartle, and their rejection of an implication from a physical system's uncomputability to its unpredictability, fit very comfortably with Turing's thinking about the foundations of mathematics. As we shall see, though, Turing's pioneering discussion of uncomputability in physics concerned only the straightforwardly correct converse implication, from a physical system's unpredictability (over arbitrarily long spans of behaviour) to its uncomputability.

9. Pitowsky In a conference presentation given in 1987, Pitowsky considered the question 'Can a physical machine compute a non-recursive function?'.24 Referring to the thesis that no non- recursive function is physically computable as Wolfram's thesis,25 Pitowsky said: The question of whether Wolfram's thesis is valid is a problem in the physical sciences, and the answer is still unknown. Yet there are very strong indications that Wolfram's thesis may be invalid.26 Pitowsky described notional physical devices, compatible with general relativity, that are able to compute functions that no Turing machine can compute. Pitowsky's proposals have been further developed by Hogarth, Shagrir, and others.27

10. Penrose In 1989 the speculation that physics—and in particular the physics of the mind—may not always be computable hit the headlines, with the publication of Penrose's book The Emperor's New Mind. Penrose suggested that 'non-algorithmic action' may 'have a role within the physical world of very considerable importance' and that 'this role is intimately bound up with ... "mind"'.28 In a précis of the book, Penrose wrote: I have tried to stress that the mere fact that something may be scientifically describable in a precise way does not imply that it is computable. It is quite on the cards that the physical activity underlying our conscious thinking may be governed by precise but

7 nonalgorithmic physical laws and our conscious thinking could indeed be the inward manifestation of some kind of nonalgorithmic physical activity.29

Turing on Physics and Uncomputability Turing's early observation that there might be uncomputable physical processes has not been widely noticed (as witnessed by Penrose's attribution to him of a thesis equivalent to Wolfram's, under the name 'Turing's Thesis'). Yet Turing was one of the first, perhaps the very first, to raise the question whether there are uncomputable physical processes, and he must be regarded as a founding father of the enquiry whose origins are being sketched here. In the course of his discussion in 'Can Digital Computers Think?', Turing considered the claim that if 'some particular machine can be described as a brain we have only to programme our digital computer to imitate it and it will also be a brain'. He observed that this 'can quite reasonably be challenged', pointing out that there is a problem if the machine's behaviour is not 'predictable by calculation'; and he drew attention to the view of physicist Arthur Eddington (expressed in Eddington's 1927 Gifford Lectures, 'The Nature of the Physical World') that in the case of the brain—and indeed the world more generally—'no such prediction is even theoretically possible' (as Turing summarized Eddington) on account of 'the indeterminacy principle in quantum mechanics'.30 Turing's casual observation that something about the physics of the brain might make it impossible for a digital computer to calculate the brain's behaviour may largely have passed over the heads of his BBC radio audience. Yet, with hindsight, this observation prefaced the discussion of physics and uncomputability that would gradually unfold over the following decades.

Uncomputability and Freewill Eddington's discussion of quantum indeterminacy was closely bound up with his discussion of the 'freedom of the human mind'.31 Turing, too, was much interested in the issue of freewill, and seems to have believed that the mind is a partially random machine.32 We have the word of one of Turing's closest associates, Newman, that Turing had a deep-seated conviction that the real brain has a 'roulette wheel' somewhere in it.33

8 Turing's principal aim in 'Can Digital Computers Think?' was not, though, to offer an analysis of freewill, but to answer affirmatively the question posed by his title. He wished to guard his view that appropriate digital computers can be described as brains against any objection along the following lines: if a brain's future states cannot be predicted by computation, and if this feature of the brain is (not a detail of minor importance but) the seat of our free will, then digital computers, with their deterministic action, must be a completely different sort of beast. Turing argued that his proposition 'If any machine can appropriately be described as a brain, then any digital computer can be so described' is entirely consistent with the possibility that the brain is the seat of free will: To behave like a brain seems to involve free will, but the behaviour of a digital computer, when it has been programmed, is completely determined. ... [I]t is certain that a machine which is to imitate a brain must appear to behave as if it had free will, and it may well be asked how this is to be achieved. One possibility is to make its behaviour depend on something like a roulette wheel or a supply of radium. ... It is, however, not really even necessary to do this. It is not difficult to design machines whose behaviour appears quite random to anyone who does not know the details of their construction. Turing called machines of the latter sort 'apparently partially random'. An example that he gave elsewhere is a Turing machine in which 'the digits of the number & [are] used to determine the choices',34 although the secret cryptographic machines that he had worked on during the war, such as the Lorenz SZ40/42 ('Tunny'), would have formed much better examples of deterministic machines whose behaviour can appear quite random—if only he could have mentioned them.35 (His involvement with all aspects of wartime codebreaking was subject to the British government's Official Secrets Act.) A genuinely partially random machine, on the other hand, is a discrete-state machine that contains a genuinely random element.36 Except in the case where (even under idealisation) the machine has only a finite number N of configurations, a partially random discrete-state machine cannot be simulated by any Turing machine. This is because, as

9 Church pointed out in 1939, if a sequence of integers a1, a2, ... an, ... is random, then there is 37 no function f(n) = an that is calculable by Turing machine. Randomness is an extreme form of uncomputability. Apparently partially random machines imitate partially random machines. As is well known, Turing advocated imitation as the basis of a test—the —that '[y]ou might call ... a test to see whether the machine thinks'.38 An appropriately programmed digital computer could give a convincing imitation of the behaviour produced by a human brain even if the brain is a partially random machine. The appearance that this deterministic machine gives of possessing free will is, Turing said, 'mere sham', but it is in his view nevertheless 'not altogether unreasonable' to describe a machine that successfully 'imitate[s] a brain' as itself being a brain. Turing's strategy for dealing with what can be termed the freewill objection to human-level AI is elegant and provocative.39

10 REFERENCES

Church, A. 1940. 'On the Concept of a Random Sequence' American Mathematical Society Bulletin, 46: 130-135.

Copeland, B.J. 1999. 'A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in K. Furukawa, D. Michie, and S. Muggleton (eds) Machine Intelligence 15. Oxford: .

Copeland, B.J. 2000. 'Narrow Versus Wide Mechanism' Journal of , 96: 5-32.

Copeland, B.J. (ed.) 2002. Hypercomputation. Part 1. Special issue of , vol. 12(4).

Copeland, B.J. 2002a. 'Hypercomputation', in Copeland 2002.

Copeland, B.J. (ed.) 2003. Hypercomputation. Part 2. Special issue of Minds and Machines, vol. 13(1).

Copeland, B.J. (ed.) 2004. The Essential Turing. Oxford: Oxford University Press.

Copeland, B.J. et al. 2006. Colossus: The Secrets of ’s Codebreaking Computers. Oxford: Oxford University Press.

Copeland, B.J., Shagrir, O. 2007. 'Physical Computation: How General are Gandy’s Principles for Mechanisms' Minds and Machines, 17: 217-231.

Copeland, B.J., Shagrir, O. 2011. 'Do Accelerating Turing Machines Compute the Uncomputable?' Minds and Machines, 21: 221-239.

Copeland, B.J., Shagrir, O. 2012 (forthcoming). 'Turing and Gödel on Computability and the Mind'.

Copeland, B.J., Sylvan, R. 1999. 'Beyond the Universal Turing Machine' Australasian Journal of Philosophy, 77: 46-66.

Doyle, J. 1982. 'What is Church’s Thesis?' Laboratory for Computer Science, MIT. Published in Copeland 2002.

Eddington, A.S. 1929. The Nature of the Physical World. Cambridge: Cambridge University Press.

Geroch, R., Hartle, J.B. 1986. 'Computability and Physical Theories' Foundations of Physics, 16: 533-550.

Hogarth, M.L. 1992. 'Does General Relativity Allow an Observer to View an Eternity in a Finite Time?' Foundations of Physics Letters, 5: 173-181.

11 Hogarth, M.L. 1994. 'Non-Turing Computers and Non-Turing Computability', Association 1994, 1: 126-38.

Komar, A. 1964. 'Undecidability of Macroscopically Distinguishable States in Quantum Field Theory' Physical Review, second series, 133B: 542-544.

Karp, R.M., Lipton, R.J. 1982. 'Turing Machines that Take Advice', in E. Engeler et al. (eds) 1982. Logic and Algorithmic. Genève: L'Enseignement Mathématique.

Kreisel, G. 1965. '', in T.L. Saaty (ed.) 1965. Lectures on Modern Mathematics, vol.3. New : John Wiley.

Kreisel, G. 1967. 'Mathematical Logic: What Has it Done For the Philosophy of Mathematics?', in R. Schoenman (ed.) 1967. : Philosopher of the Century. London: George Allen and Unwin.

Kreisel, G. 1970. 'Hilbert's Programme and the Search for Automatic Proof Procedures', in Laudet, M. et al. (eds) 1970. Symposium on Automatic Demonstration. Lecture Notes in Mathematics, vol. 125. Berlin: Springer.

Kreisel, G. 1971. 'Some Reasons for Generalising Recursion Theory', in R.O. Gandy, C.M.E. Yates (eds) 1971. Logic Colloquium '69. Amsterdam: North-Holland.

Kreisel, G. 1972. 'Which Number Theoretic Problems can be Solved in Recursive 1 Progressions on &1 'Paths Through 0?' Journal of Symbolic Logic, 37: 311-334.

Kreisel, G. 1974. 'A Notion of Mechanistic Theory' Synthese, 29: 11-26.

Kreisel, G. 1982. Review of Pour-El and Richards, Journal of Symbolic Logic, 47: 900-902.

Kreisel, G. 1987. 'Church's Thesis and the Ideal of Formal Rigour' Notre Dame Journal of Formal Logic, 28: 499-519.

Penrose, R. 1989. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford: Oxford University Press.

Penrose, R. 1990. Précis of The Emperor's New Mind. Behavioral and Brain Sciences, 13: 643-655, 692-705.

Penrose, R. 1994. Shadows of the Mind: A Search for the Missing Science of Consciousness. Oxford: Oxford University Press.

Pitowsky, I. 1990. 'The Physical Church Thesis and Physical Computational Complexity' Iyyun, 39: 81-99.

Pour-El, M.B., Richards, J.I. 1979. 'A Computable Ordinary Differential Equation Which Possesses No Computable Solution' Annals of Mathematical Logic, 17: 61-90.

12 Pour-El, M.B., Richards, J.I. 1981. 'The Wave Equation with Computable Initial Data such that its Unique Solution is not Computable' Advances in Mathematics, 39: 215-239.

Pour-El, M.B., Richards, J.I. 1989. Computability in Analysis and Physics. Berlin: Springer.

Proudfoot, D., Copeland, B.J. 2011 (forthcoming). 'Artificial Intelligence', in E. Margolis, R.I. Samuels, S.P. Stich (eds) The Oxford Handbook of Philosophy of Cognitive Science. New York: Oxford University Press.

Rubel, L.A. 1985. 'The Brain as an Analog Computer' Journal of Theoretical Neurobiology, 4: 73-81.

Rubel, L.A. 1989. 'Digital Simulation of Analog Computation and Church's Thesis' Journal of Symbolic Logic, 54: 1011-1017.

Scarpellini, B. 1963 'Zwei unentscheitbare Probleme der Analysis' Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 9: 265-289. Published in translation as 'Two Undecidable Problems of Analysis' (with a new commentary by Scarpellini) in Copeland 2003.

Shagrir, O., Pitowsky, I. 2003. 'Physical Hypercomputation and the Church-Turing Thesis', in Copeland 2003.

Turing, A.M. 1936. 'On Computable Numbers, with an Application to the Entscheidungsproblem' Proceedings of the London Mathematical Society, Series 2, 42 (1936-37): 230-265.

Turing, A.M. 1939. 'Systems of Logic Based on Ordinals' Proceedings of the London Mathematical Society, Series 2, 45: 161-228.

Turing, A.M. c. 1940. 'Letter from The Crown, Shenley Brook End', in Copeland 2004.

Turing, A.M. 1948. 'Intelligent Machinery'. National Physical Laboratory report. First published in a form reproducing the layout and wording of Turing's document in Copeland 2004.

Turing, A.M. 1950. 'Computing Machinery and Intelligence' Mind 59: 433-460.

Turing, A.M., Braithwaite, R.B., Jefferson, G., Newman, M.H.A. 1952. 'Can Automatic Calculating Machines Be Said To Think?', in Copeland 2004.

Wolfram, S. 1985. 'Undecidability and Intractability in Theoretical Physics' Physical Review Letters, 54: 735-738.

1 Letter from Maurice Wilkes to Copeland (9 July 1997). 2 Turing 1950.

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3 Aspects of Turing's broadcast not treated here are discussed in my 2004 (ch. 13), 2000, and introduction to 1999. 4 Penrose 1994: 21. 5 For more information, see Copeland 2002, 2002a, 2003, Copeland and Sylvan 1999. 6 Scarpellini 1963 (Scarpellini's translation is from Copeland 2003: 77). 7 Letter from Bruno Scarpellini to Copeland (3 August 2011). 8 Scarpellini in Copeland 2003:84-85. 9 Komar 1964: 543-544. 10 See, for example, Kreisel 1965, 1967, 1971, 1972, 1982, 1987, and especially 1970, 1974. 11 Kreisel 1967: 270. 12 See also the book Pour-El and Richards 1989. 13 Pour-El and Richards 1979: 63. 14 Kreisel 1982: 901. 15 Karp and Lipton 1982. 16 Doyle 1982 (pp. 519-520 in Copeland 2002). 17 Rubel 1985: 78-79. 18 Rubel 1989: 1011. 19 Geroch and Hartle 1986: 534, 549. 20 Geroch and Hartle 1986: 549. 21 Turing 1936: 253. 22 Turing c. 1940; Turing 1939. 23 Turing c. 1940: 212-213. See further Copeland and Shagrir 2012 (forthcoming). 24 Pitowsky 1990: 82. 25 See Wolfram 1985. 26 Pitowsky 1990: 86. 27 Hogarth 1992, 1994; Shagrir and Pitowsky 2003. See also Copeland and Shagrir 2007, 2011. 28 Penrose 1989: 557. See also Penrose 1994. 29 Penrose 1990: 653. 30 Eddington 1929, ch. 14. 31 See Eddington 1929: 310. 32 See further Copeland 2000. 33 Newman in interview with Christopher Evans. ('The Pioneers of Computing: An Oral History of Computing'. London: Science Museum.) 34 Turing 1948: 416. 35 The story of Tunny is told in Copeland 2006; see Appendix 6 for a detailed description of Turing's main contributions to the attack on Tunny. 36 Turing 1948: 416. 37 Church 1940. 38 Turing in Turing et al. 1952: 495. See also Turing 1950. 39 For more on Turing and human-level AI, see Proudfoot and Copeland 2011.

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