TURING AND THE PHYSICS OF THE MIND B. Jack Copeland 31 August 2011 Introduction Turing's lecture 'Can Digital Computers 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 Max Newman (like Turing from the University of Manchester), Douglas Hartree (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 was virtually unknown until 1999, when I included it in a small collection of unpublished work by Turing ('A Lecture and Two Radio Broadcasts on Machine Intelligence by Alan Turing', in Machine Intelligence 15) and again in The Essential Turing in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 computer 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 Roger Penrose 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 Turing machine] 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 mathematics'.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 universal Turing machine 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 logic 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.