What ' Gaps ' ?

Grush and Churchland (1995) attempt to address aspects of the proposal that we have been making concerning a possible physical mechanism underlying the phenomenon of consciousness. Unfortunately, they employ arguments that are highly misleading and, in some important respects, factually incorrect. Their article ‘Gaps in Penrose’s Toilings’ is addressed specifically at the writings of one of us (Penrose), but since the particular model they attack is one put forward by both of us (Hameroff and Penrose, 1995; 1996), it is appropriate that we both reply; but since our individual remarks refer to different aspects of their criticism we are commenting on their article separately. The logical arguments discussed by Grush and Churchland, and the related physics are answered in Part l by Penrose, largely by pointing out precisely where these arguments have already been treated in detail in Shadows of the Mind (Penrose, 1994). In Part 2, Hameroff replies to various points on the biological side, showing for example how they have seriously misunderstood what they refer to as ‘physiological evidence’ regarding to effects of the drug colchicine. The reply serves also to discuss aspects of our model ‘orchestrated objective reduction in brain microtubules – Orch OR’ which attempts to deal with the serious problems of consciousness more directly and completely than any previous theory. Part 1: The Relevance of Logic and Physics Logical arguments It has been argued in the books by one of us, The Emperor’s New Mind (Penrose, 1989 – henceforth Emperor) and Shadows of the Mind (Penrose, 1994 – henceforth Shadows) that Gödel’s theorem shows that there must be something non–computational involved in mathematical thinking. The Grush and Churchland (1995 – henceforth G&C) discussion attempts to dismiss this argument from Gödel’s theorem on certain grounds. However, the main points that they put forward are ones which have been amply addressed in Shadows. It is very hard to understand how G&C can make the claims that they do without giving any indication that virtually all their points are explicitly taken into account in Shadows. It might be the case that the arguments given in Shadows are in some respects inadequate, and it would have been interesting if G&C had provided a detailed commentary on these particular arguments, pointing out possible shortcomings where they occur. But it would seem from what G&C say that they have not even read, and certainly not understood, these arguments. A natural reaction to their commentary would be simply to say "go and read the book and come back when you have understood its arguments." However, it will be helpful to pinpoint the specific issues that they raise here, and to point out the places in Shadows where these issues are addressed. The main argument that they appear to be raising against Penrose’s (1989; 1994) use of Gödel’s theorem (to demonstrate non–computability in mathematical thinking) is that mathematical thinking contains errors. They give the impression that the possibility of errors by mathematicians is not even considered by Penrose. However, in §§3.2, 3.4, 3.17, 3.19, 3.20 and 3.21 of Shadows the question of possible errors in human or robot mathematical reasoning is explicitly addressed at length. (The words ‘errors’ and ‘erroneous’ even appear explicitly in the headings of two of those sections and it is hard to see why G&C make no reference to these parts of the book.) In addition, on page 16 of their commentary, G&C claim that ‘most of the technical machinery’ involved in Penrose’s arguments refer to what they call ‘Ala’ and ‘Alc’, on their page 15, which they choose not to dispute; whereas in fact by far the most difficult technical arguments given in Shadows are those which specifically address the possibility of errors in human or robot mathematical reasoning (these are given in §§ 3.19 and 3.20 of Shadows). It is difficult to understand why G&C fail to refer to this discussion, seeming to suggest (quite incorrectly) that Penrose has an in-built faith in the complete accuracy in the reasoning of mathematicians! G&C have a curious way of formalizing what they believe to be the ingredients of Penrose’s arguments. In particular, on page 16 they refer to ‘Penrose’s Premise A 1: Human thought, at least in some instances, perhaps in all, is sound, yet non-algorithmic’ (which they break down into A I a, . . . , A 1 e). Their ‘Premise A 1’ is nowhere to be found in Penrose’s writings. It is fully admitted by Penrose that actual human thinking can be unsound even when seeming to be carried out in the most rigorous fashion by mathematicians. It may well be that there is a genuine and deep misunderstanding implicit in what G&C are attempting to say, and it may be helpful to try to clarify the issue here. For the purposes of our present discussion (and for the essential discussion given in Shadows) it will be sufficient to restrict attention to a very specific class of mathematical statements, namely those referred to as "pi 1"–sentences. Such sentences are assertions that particular (Turing–machine) computations do not halt. There are some very famous examples of mathematical assertions which take the form of "pi 1"– sentences, the best known being the so–called ‘Fermat’s Last Theorem’. Other examples are ‘Goldbach’s conjecture’ (still unproved) that every even number greater than 2 is the sum of two primes, Lagrange’s Theorem that every natural number is the sum of four squares, and the famous 4–colour theorem. It is useful to concentrate one’s attention on "pi 1"–sentences because this is all one needs for application of the Gödel argument to the issue of computability in human mathematical thinking. There is no relevant issue of dispute between mathematicians as to the meaningfulness and objectivity of the truth of such sentences. (One might, however, worry about the ‘intuitionists’ or other constructivists in this context –and some reference to such viewpoints is given on p. 18 and footnote 30 on p. 20 of the G&C article. However, such constructivist viewpoints do not evade the Gödel argument and the use made of it in Shadows as is explicitly addressed in the discussion of Q9 on page 87 of Shadows, a discussion not even referred to by G&C.) As far as we can make out, G&C are not disputing the absolute (‘Platonic’) nature of the truth or falsity of explicit "pi 1"–sentences. The issue is the accessibility of the truth of "pi 1"–sentences by human reasoning and insight. We should make clear what is meant by a word such as ‘accessibility’ in this context, since there seem to be a great many misconceptions by philosophers and others as to how mathematical understanding actually operates. It is not a question of some kind of ‘mystical intuition’ that (some) mathematicians might have, and which is unavailable to ordinary mortals. What is being referred to by ‘access’ is simply the normal procedure of mathematical proof. It is not even a question of how some mathematician might have the inspiration to arrive at a proof. It is merely the question of the understanding which is involved in the ability to follow a proof in principle. (See, in particular, in the response to Q12 pp. 101 3 of Shadows.) However, it should be made clear in this context that the word ‘proof’ does not refer necessarily to a formalized argument within some pre-assigned logical scheme. For example, the arguments given by Andrew Wiles (as completed by Taylor and Wiles) to demonstrate the validity of Fermat’s last theorem were certainly not presented as formal arguments, within, say, the Zermelo–Fraenkel axiom system. The essential point about such arguments is that they have to be correct as mathematical reasoning. It is a secondary matter to try to find out within which formal mathematical systems such arguments can be formulated. Indeed, what the Gödel argument shows (and this is not in dispute) is that if the rules of some formal system, F, can be trusted as providing correct demonstrations of mathematical statements —and here we need restrict attention only to "pi 1"– sentences—then the particular "pi 1"–sentence G(F) must also be accepted as true even though it is not a consequence of the very rules provided by F. (Here the sentence G(F) is the Gödel proposition which asserts the consistency of the formal system F—assuming that F is sufficiently extensive. It can also be taken as the explicit statement Ck(k) exhibited on p. 75 of Shadows. What this shows is that mathematical understanding (i.e. mathematical proof-in the sense above) cannot be encapsulated in any humanly acceptable formal system. Here ‘acceptable’ means acceptable to mathematicians as a reliable means of obtaining mathematical truths, where attention may be restricted to the truth of "pi 1"–sentences. The notion of ‘proof’ that is being referred to above certainly raises profound issues. However, it would be unreasonable to dismiss it as something which is too ill–defined for scientific consideration or perhaps ‘mystical’. There is indeed something mysterious about the very nature of ‘understanding’ and this is what is involved here. But the notion of proof that is involved in mathematical understanding is extraordinarily precise and accurate. There is no other form of argument within science or philosophy which really bears comparison with it. Moreover, this notion transcends any individual mathematician. But it is what mathematicians individually strive for. If one mathematician claims to have an argument for demonstrating the validity of some assertion—say a "pi 1"–sentence—then it should in principle be possible to convince another mathematician that the argument, and hence the conclusion, is correct unless there is an error, in which case it is up to the mathematicians to locate this error. There is no question but that mathematicians do, not infrequently, make errors. This is not the point. The point is that it is possible for there actually to be an argument of some nature, to be found, which demonstrates the truth of the "pi 1"–sentence in question, and it is the mathematicians’ business to try to find such an argument, whether or not they make mistakes in the process of attempting to achieve this. If there is an argument accessible to human understanding, then ‘access’ to this particular "pi 1"– sentence is possible. The point about the Gödel argument in this context is that there is no way to encapsulate the means whereby this access is achieved, in terms of mathematically acceptable computational rules. It is no part of Penrose’s argument against computationalism that al! "pi 1"–sentences should be humanly accessible (although there is some discussion of this possibility in chapter 8 of Shadows). What is argued for in Shadows is the assertion that the class of "pi 1"–sentences which are in principle humanly accessible is not a class which is computationally accessible (technically; not a recursive set—it is certainly not a knowably recursive set). The issue of practical accessibility (as opposed to in principle accessibility) of "pi 1"–sentences by individual mathematicians, or perhaps by the mathematical community as a whole, is of course a somewhat separate matter, but these issues are discussed in considerable detail in Chapter 3 of Shadows (and also in the discussion of Q8 on pp. 83J9). No mention of this is made by G&C. Nor do they refer to the fact that in Shadows the Gödel argument is applied to computability both in the ‘in principle’ and the ‘in practice’ sense, according to context. A detailed commentary on all of the remaining misunderstandings of Penrose’s arguments displayed by G&C would take more space than is available to us here. However, it will be helpful to list some of the more important of these. At the end of the paragraph which finishes at the top of page 17, in a rather confused sentence, they seem to be saying that there are no sound procedures of mathematical deliberation, beyond, say, Zermelo–Fraenkel set theory (ZF). This shows that they do not understand Gödel. He demonstrated that if ZF is sound, then G(ZF) does indeed enable one to go beyond ZF (etc.). On page 18 they refer to the changing perceptions of mathematical rigour over the centuries, seeming to suggest that Cauchy and Cantor might have had different conceptions of what is unassailably true in mathematics. (Here, ‘unassailable’ refers merely to ‘correctly proved’ in the sense given above.) There is nothing wrong with the continual broadening of the kind of reasoning which can be used to obtain mathematical results, say, "pi 1"–sentences, and what Cantor did was to increase this breadth enormously over what had been achieved before. This led to delicate, and sometimes disputed issues, such as the Axiom of Choice. The possibility of differing viewpoints with regard to the Axiom of Choice is explicitly addressed in the discussion of Q11 on pp. 97-101 of Shadows (although G&C seem oblivious of this fact). The conclusion is that this does not significantly affect the non– computability argument. G&C make a number of points which seem to suggest that they think that Penrose is unaware of certain elementary logical points. This is particularly irritating. For example at the bottom of p.19 they point out that: ‘M does not believe that A is unassailably true’ does not entail that ‘M disbelieves A’. Of course it does not. This kind of distinction is essential to the arguments of Chapter 3 of Shadows.