Roger Penrose Consciousness Quotes

We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.

A scientific world-view which does not profoundly come to terms with the problem of conscious minds can have no serious pretensions of completeness.

Consciousness is the phenomenon whereby the universe's very existence is made known.

The final conclusion of all this is rather alarming. For it suggests that we must seek a non-computable physical theory that reaches beyond every computable level of oracle machines (and perhaps beyond).

I don't know if the universe has a purpose, but I would say that there is something more to it, in the sense that the presence of conscious beings is probably something deeper, not just not random. [quoting Roger Penrose]

It is in mathematics that our thinking processes have their purest form.

There are completely deterministic universe models, with clear-cut rules of evolution, that are impossible to simulate computationally.

What Godel and Rosser showed is that the consistency of a (sufficiently extensive) formal system is something that lies outside the power of the formal system itself to establish.

In some Platonic sense, the natural numbers seem to be things that have an absolute conceptual existence independent of ourselves.

I argue that the phenomenon of consciousness cannot be accommodated within the framework of present-day physical theory.

To me the world of perfect forms is primary (as was Plato's own belief)-its existence being almost a logical necessity-and both the other two worlds are its shadows.

Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules.

Somehow, consciousness is needed in order to handle situations where we have to form new judgements, and where the rules have not been laid down beforehand.

How is that perceiving beings can arise from out of the physical world, and how is that mentality is able seemingly to 'create' mathematical concepts out of some kind of mental model.

If we try to make general inferences about the theoretical possibility of a reliable computational model of the brain, we ought indeed to come to terms with the mysteries of quantum theory.

All I would myself ask for would be that our perceptive interrogator should really feel convinced, from the nature of the computer’s replies, that there is a conscious presence underlying these replies

What is particularly curious about quantum theory is that there can be actual physical effects arising from what philosophers refer to as counterfactuals-that is, things that might have happened, although they did not in fact happen.

Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules. Support for the Platonic viewpoint ...was an important part of Godel's initial motivations.

Either individually or in these larger arrays, microtubules are responsible for cellular and intra-cellular movements requiring intelligent spatiotemporal organization. Microtubules have a lattice structure comparable to computational systems. Could microtubules process information?

What is it that we can do with conscious thought that cannot be done unconsciously? The problem is made more elusive by the fact that anything that we do seem originally to require consciousness for appears also to be able to be learnt and then later carried out unconsciously (perhaps by the cerebellum

These are deep issues, and we are yet very far from explanations. I would argue that no clear answers will come forward unless the interrelating features of all these worlds are seen to come into play. No one of these issues will be resolved in isolation from the others. I have referred to three worlds and the mysteries that relate them one to another. No doubt there are not really three worlds but one, the true nature of which we do not even glimpse at present.

It has been said that, in scale, a human being is about halfway between an atom and a star. Interestingly, this is also the regime in which physics becomes most complicated; on the atomic scale, we have quantum mechanics, on the large scale, relativity. It is in between these two extremes where our lack of understanding of how to combine these theories becomes apparent. The Oxford scientist Roger Penrose has written convincingly of his belief that whatever it is that we are missing from our understanding of fundamental physics is also missing from our understanding of consciousness. These ideas are important when one considers what have become known as anthropic points of view, best summarized as the belief that the Universe must be the way it is in order to allow us to be here to observe it.

Awareness, I take to be one aspect-the passive aspect-of the phenomenon of consciousness. Consciousness has an active aspect also, namely the feeling of free will.

It would seem that it is in some kind of combination of top-down and bottom-up organization that we must expect to find the most successful AI systems.

do not see how natural selection, in itself, can evolve algorithms which could have the kind of conscious judgements of the validity of other algorithms that we seem to have.

I am not so much concerned, at this stage, with how individual mathematicians might differently approach a mathematical problem, but more with what is universal about our understandings and our mathematical perceptions.

The gravitational field itself contains energy, and this energy measurably contributes to the total energy (and therefore to the mass, by Einstein's E = mc^2) of a system. Yet it is a nebulous energy that inhabits empty space in a mysterious non-local way.

Thus, Godel appears to have taken it as evident that the physical brain must itself behave computationally, but that the mind is something beyond the brain, so that the mind's action is not constrained to behave according to the computational laws that he believed must control the physical brain's behavior.

It will be one of my purposes, in later arguments, to show that there is indeed an aspect of 'genuine understanding' that cannot be properly simulated in any computational way whatever. Consequently, there must indeed be a distinction between genuine intelligence and any attempt at a proper computational simulation of it.

Roger Penrose tried to conceive the unconscious as the space of superposition of thoughts, and the passage to consciousness as the collapse of wave oscillations into a single reality: “Could thoughts exist in some sort of quantum superposition on an unconscious level only to become conscious when there is a specific selection

Every one of our conscious brains is woven from subtle physical ingredients that somehow enable us to take advantage of the profound organization of our mathematically underpinned universe-so that we, in turn, are capable of some kind of direct access, through that Platonic quality of 'understanding', to the very ways in which our universe behaves at many different levels.

It is important to bear in mind that this is a property of single photons. Each individual photon must be considered to feel out both routes that are open to it, but it remains one photon; it does not split into two photons in the intermediate stage, but its location undergoes the strange kind of complex-number weighted co-existence of alternatives that is characteristic of quantum theory.

A point that should be emphasized is that the energy that defines the lifetime of the superposed state is an energy difference, and not the total, (mass-) energy that is involved in the situation as a whole. Thus, for a lump that is quite large but does not move very much-and supposing that it is also crystalline, so that its individual atoms do not get randomly displaced-quantum superpositions could be maintained for a long time. The lump could be much larger than the water droplets considered above. There could also be other very much larger masses in the vicinity, provided that they do not get significantly entangled with the superposed state we are concerned with. (These considerations would be important for solid-state devices, such as gravitational wave detectors, that use coherently oscillating solid-perhaps crystalline-bodies.)

The reason that I have concentrated on non-computability, in my arguments, rather than on complexity, is simply that it is only with the former that I have been able to see how to make the necessary strong statements. It may well be that in the working lives of most mathematicians, non-computability issues play, if anything, only a very small part. But that is not the point at issue. I am trying to show that (mathematical) understanding is something that lies beyond computation, and the Godel (-Turing) argument is one of the few handles that we have on that issue. It is quite probable that our mathematical insights and understandings are often used to achieve things that could in principle also be achieved computationally-but where blind computation without much insight may turn out to be so inefficient that it is unworkable (cf. 3.26). However, these matters are much more difficult to address than the non-computability issue.

Technology provides the potential, by use of well-produced books, film, television, and interactive computer-controlled systems of various kinds. These, and other developments, provide many opportunities for expanding our minds-or else for deadening them. The human mind is capable of vastly more than it is often given the chance to achieve. Sadly, these opportunities are all to frequently squandered, and the minds of neither young nor old are provided the openings that they undoubtedly deserve.

The perceiving of mathematical truth can be achieved in very many different ways. There can be little doubt that whatever detailed physical activity it is that takes place when a person perceives the truth of some mathematical statement, this physical activity must differ very substantially from individual to individual, even though they are perceiving precisely the same mathematical truth. Thus, if mathematicians just use computational algorithms to form their unassailable mathematical truth judgments, these very algorithms are likely to differ in their detailed construction, from individual to individual. Yet, in some clear sense, the algorithms would have to be equivalent to one another.

We cannot say, in familiar everyday terms, what it 'means' for an electron to be in a state of superposition of two places at once, with complex-number weighting factors w and z. We must, for the moment, simply accept that this is indeed the kind of description that we have to adopt for quantum-level systems. Such superpositions constitute an important part of the actual construction of our microworld, as has now been revealed to us by Nature. It is just a fact that we appear to find that the quantum-level world actually behaves in this unfamiliar and mysterious way. The descriptions are perfectly clear cut-and they provide us with a micro-world that evolves according to a description that is indeed mathematically precise and, moreover, completely deterministic!

The thrust of Godel's argument for our purposes is that it shows us how to go beyond any given set of computational rules that we believe to be sound, and obtain a further rule, not contained in those rules, that we must believe to be sound also, namely the rule asserting the consistency of the original rules. The essential point, for our purposes, is: belief in soundness implies belief in consistency. We have no right to use the rules of a formal system F, and to believe that the results that we derive from it are actually true, unless we also believe in the consistency of that formal system. (For example, if F were inconsistent, then we could deduce, as TRUE, the statement '1=2', which is certainly not true!) Thus, if we believe that we are actually doing mathematics when we use some formal system F, then we must also be prepared to accept reasoning that goes beyond the limitations of the system F, whatever that system F may be.

Of course none of this will stop us from wanting to know what it is that is really going on in consciousness and intelligence. I want to know too. Basically the arguments of this book are making the point that what is not going on is solely a great deal of computational activity-as is commonly believed these days-and what is going on will have no chance of being properly understood until we have a much more profound appreciation of the very nature of matter, time, space, and the laws that govern them. We shall need also to have much better knowledge of the detailed physiology of brains, particularly at the very tiny levels that have received little attention until recent years. We shall need to know more about the circumstances under which consciousness arises or disappears, about the curious matter of its timing, of what it is used for, and what are the specific advantages of its possession-in addition to many other issues where objective testing is possible. It is a very broad field indeed, in which progress in many different directions is surely to be anticipated.

If, as I believe, the Godel argument is consequently forcing us into an acceptance of some form of viewpoint C, the we shall also have to come to terms with some of its other implications. We shall find ourselves driven towards a Platonic viewpoint of things. According to Plato, mathematical concepts and mathematical truths inhabit an actual world of their own that is timeless and without physical location. Plato's world is an ideal world of perfect forms, distinct from the physical world, but in terms of which the physical world must be understood. It also lies beyond our imperfect mental constructions; yet, our minds do have some direct access to this Platonic realm through an 'awareness' of mathematical forms, and our ability to reason about them. We shall find that whilst our Platonic perceptions can be aided on occasion by computation, they are not limited by computation. It is this potential for the 'awareness' of mathematical concepts involved in this Platonic access that gives the mind a power beyond what can ever be achieved by a device dependent solely upon computation for its action.

But there is also some empirical evidence that sheds light on the relationship between quantum principles and consciousness. Anesthesiologist Stuart Hameroff claims to have found evidence that anesthesia arrests consciousness by hindering the motion of electrons in microtubules, minute tunnels of protein that serve as a kind of skeleton for cells. Hameroff speculates that microtubules could be a possible site for quantum effects in the brain,17 and his speculations have led mathematical physicist Roger Penrose to endorse the hypothesis.18 Attempts to develop models of consciousness based on quantum mechanics have also been made by neuroscientist John Eccles, and physicists Henry Stapp and Evan Harris Walker.o Walker and the experimental physicist Helmut Schmidt (the latter responsible for many of the micro-PK experiments described earlier) have also proposed mathematical theories of psi based on quantum mechanics.19 These theories rest upon two propositions that are now supported by experimental evidence: that mind can influence random quantum events, and that influence can occur instantaneously at a distance.p

It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other.

Specifically, the awareness that I claim is demonstrably non-computational is our understanding of the properties of natural numbers 0,1,2,3,4,....(One might even say that our concept of a natural number is, in a sense, a form of non-geometric 'visualization'.) We shall see in 2.5, by a readily accessible form of Godel's theorem (cf. response to query Q16), that this understanding is something that cannot be simulated computationally. From time to time one hears that some computer system has been 'trained' so as to 'understand' the concept of natural numbers. However, this cannot be true, as we shall see. It is our awareness of what a 'number' can actually mean that enables us to latch on to the correct concept. When we have this correct concept, we can-at least in principle-provide the correct answers to families of questions about numbers that are put to us, when no finite set of rules can do this. With only rules and no direct awareness, a computer-controlled robot (like Deep Thought) would be necessarily limited in ways in which we are not limited ourselves-although if we give the robot clever enough rules for its behaviour it may perform prodigious feats, some of which lie far beyond unaided human capabilities in specific narrowly enough defined areas, and it might be able to fool us, for some while, into thinking that it also possesses awareness.

Q5. Have not I merely shown that it is possible to outdo just a particular algorithmic procedure, A, by defeating it with the computation Cq(n)? Why does this show that I can do better than any A whatsoever? The argument certainly does show that we can do better than any algorithm. This is the whole point of a reductio ad absurdum argument of this kind that I have used here. I think that an analogy might be helpful here. Some readers will know of Euclid's argument that there is no largest prime number. This, also, is a reductio ad absurdum. Euclid's argument is as follows. Suppose, on the contrary, that there is a largest prime; call it p. Now consider the product N of all the primes up to p and add 1: N=2*3*5*...*p+1. N is certainly larger than p, but it cannot be divisible by any of the prime numbers 2,3,5...,p (since it leaves the remainder 1 on division); so either N is the required prime itself or it is composite-in which case it is divisible by a prime larger than p. Either way, there would have to be a prime larger than p, which contradicts the initial assumption that p is the largest prime. Hence there is no largest prime. The argument, being a reductio ad absurdum, does not merely show that a particular prime p can be defeated by finding a larger one; it shows that there cannot be any largest prime at all. Likewise, the Godel-Turing argument above does not merely show that a particular algorithm A can be defeated, it shows that there cannot be any (knowably sound) algorithm at all that is equivalent to the insights that we use to ascertain that certain computations do not stop.

Q7. The total output of all the mathematicians who have ever lived, together with the output of all the human mathematicians of the next (say) thousand years is finite and could be contained in the memory banks of an appropriate computer. Surely this particular computer could, therefore, simulate this output and thus behave (externally) in the same way as a human mathematician-whatever the Godel argument might appear to tell us to the contrary? While this is presumably true, it ignores the essential issue, which is how we (or computers) know which mathematical statements are true and which are false. (In any case, the mere storage of mathematical statements is something that could be achieved by a system much less sophisticated than a general purpose computer, e.g. photographically.) The way that the computer is being employed in Q7 totally ignores the critical issue of truth judgment. One could equally well envisage computers that contain nothing but lists of totally false mathematical 'theorems', or lists containing random jumbles of truths and falsehoods. How are we to tell which computer to trust? The arguments that I am trying to make here do not say that an effective simulation of the output of conscious human activity (here mathematics) is impossible, since purely by chance the computer might 'happen' to get it right-even without any understanding whatsoever. But the odds against this are absurdly enormous, and the issues that are being addressed here, namely how one decides which mathematical statements are true and which are false, are not even being touched by Q7. There is, on the other hand, a more serious point that is indeed being touched upon in Q7. This is the question as to whether discussions about infinite structures (e.g. all natural numbers or all computations) are really relevant to our considerations here, when the outputs of humans and computers are finite.

At this point, the cautious reader might wish to read over the whole argument again, as presented above, just to make sure that I have not indulged in any 'sleight of hand'! Admittedly there is an air of the conjuring trick about the argument, but it is perfectly legitimate, and it only gains in strength the more minutely it is examined. We have found a computation Ck(k) that we know does not stop; yet the given computational procedure A is not powerful enough to ascertain that facet. This is the Godel(-Turing) theorem in the form that I require. It applies to any computational procedure A whatever for ascertaining that computations do not stop, so long as we know it to be sound. We deduce that no knowably sound set of computational rules (such as A) can ever suffice for ascertaining that computations do not stop, since there are some non-stopping computations (such as Ck(k)) that must elude these rules. Moreover, since from the knowledge of A and of its soundness, we can actually construct a computation Ck(k) that we can see does not ever stop, we deduce that A cannot be a formalization of the procedures available to mathematicians for ascertaining that computations do not stop, no matter what A is. Hence: (G) Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth. It seems to me that this conclusion is inescapable. However, many people have tried to argue against it-bringing in objections like those summarized in the queries Q1-Q20 of 2.6 and 2.10 below-and certainly many would argue against the stronger deduction that there must be something fundamentally non-computational in our thought processes. The reader may indeed wonder what on earth mathematical reasoning like this, concerning the abstract nature of computations, can have to say about the workings of the human mind. What, after all, does any of this have to do with the issue of conscious awareness? The answer is that the argument indeed says something very significant about the mental quality of understanding-in relation to the general issue of computation-and, as was argued in 1.12, the quality of understanding is something dependent upon conscious awareness. It is true that, for the most part, the foregoing reasoning has been presented as just a piece of mathematics, but there is the essential point that the algorithm A enters the argument at two quite different levels. At the one level, it is being treated as just some algorithm that has certain properties, but at the other, we attempt to regard A as being actually 'the algorithm that we ourselves use' in coming to believe that a computation will not stop. The argument is not simply about computations. It is also about how we use our conscious understanding in order to infer the validity of some mathematical claim-here the non-stopping character of Ck(k). It is the interplay between the two different levels at which the algorithm A is being considered-as a putative instance of conscious activity and as a computation itself-that allows us to arrive at a conclusion expressing a fundamental conflict between such conscious activity and mere computation.

In order for A to apply to computations generally, we shall need a way of coding all the different computations C(n) so that A can use this coding for its action. All the possible different computations C can in fact be listed, say as C0, C1, C2, C3, C4, C5,..., and we can refer to Cq as the qth computation. When such a computation is applied to a particular number n, we shall write C0(n), C1(n), C2(n), C3(n), C4(n), C5(n),.... We can take this ordering as being given, say, as some kind of numerical ordering of computer programs. (To be explicit, we could, if desired, take this ordering as being provided by the Turing-machine numbering given in ENM, so that then the computation Cq(n) is the action of the qth Turing machine Tq acting on n.) One technical thing that is important here is that this listing is computable, i.e. there is a single computation Cx that gives us Cq when it is presented with q, or, more precisely, the computation Cx acts on the pair of numbers q, n (i.e. q followed by n) to give Cq(n). The procedure A can now be thought of as a particular computation that, when presented with the pair of numbers q,n, tries to ascertain that the computation Cq(n) will never ultimately halt. Thus, when the computation A terminates, we shall have a demonstration that Cq(n) does not halt. Although, as stated earlier, we are shortly going to try to imagine that A might be a formalization of all the procedures that are available to human mathematicians for validly deciding that computations never will halt, it is not at all necessary for us to think of A in this way just now. A is just any sound set of computational rules for ascertaining that some computations Cq(n) do not ever halt. Being dependent upon the two numbers q and n, the computation that A performs can be written A(q,n), and we have: (H) If A(q,n) stops, then Cq(n) does not stop. Now let us consider the particular statements (H) for which q is put equal to n. This may seem an odd thing to do, but it is perfectly legitimate. (This is the first step in the powerful 'diagonal slash', a procedure discovered by the highly original and influential nineteenth-century Danish/Russian/German mathematician Georg Cantor, central to the arguments of both Godel and Turing.) With q equal to n, we now have: (I) If A(n,n) stops, then Cn(n) does not stop. We now notice that A(n,n) depends upon just one number n, not two, so it must be one of the computations C0,C1,C2,C3,...(as applied to n), since this was supposed to be a listing of all the computations that can be performed on a single natural number n. Let us suppose that it is in fact Ck, so we have: (J) A(n,n) = Ck(n) Now examine the particular value n=k. (This is the second part of Cantor's diagonal slash!) We have, from (J), (K) A(k,k) = Ck(k) and, from (I), with n=k: (L) If A(k,k) stops, then Ck(k) does not stop. Substituting (K) in (L), we find: (M) If Ck(k) stops, then Ck(k) does not stop. From this, we must deduce that the computation Ck(k) does not in fact stop. (For if it did then it does not, according to (M)! But A(k,k) cannot stop either, since by (K), it is the same as Ck(k). Thus, our procedure A is incapable of ascertaining that this particular computation Ck(k) does not stop even though it does not. Moreover, if we know that A is sound, then we know that Ck(k) does not stop. Thus, we know something that A is unable to ascertain. It follows that A cannot encapsulate our understanding.

Think of the superposed state of a lump in two different locations as being like an unstable nucleus that decays, after a characteristic 'half-life' timescale, into something else more unstable. In the case of the superposed lump locations we likewise think of an unstable quantum state which decays, after a characteristic lifetime (given, roughly on average, by the reciprocal of the gravitational energy of separation), to a state where the lump is in one location or the other-representing two possible decay modes.

I believe that the problem of quantum measurement should be faced and solved well before we can expect to make any real headway with the issue of consciousness in terms of physical action-and that the measurement problem must be solved in entirely physical terms. Once we are in possession of a satisfactory solution, then we may be in a better position to move towards some kind of answer to the question of consciousness. It is my view that solving the quantum measurement problem is a prerequisite for an understanding of mind and not at all that they are the same problem. The problem of mind is a much more difficult problem than the measurement problem!

The point is that we really have no conception of how to consider linear superpositions of states when the states themselves involve different space-time geometries. A fundamental difficulty with 'standard theory' is that when the geometries become significantly different from each other, we have no absolute means of identifying a point in one geometry with any particular point in the other-the two geometries are strictly separate spaces-so the very idea that one could form a superposition of the matter states within these two separate spaces becomes profoundly obscure. Now, we should ask when are two geometries to be considered as actually 'significantly different' from one another? It is here, in effect, that the Planck scale of 10^-33 cm comes in. The argument would roughly be that the scale of the difference between these geometries has to be, in an appropriate sense, something like 10^-33 cm or more for reduction to take place. We might, for example, attempt to imagine (Fig. 6.5) that these two geometries are trying to be forced into coincidence, but when the measure of the difference becomes too large, on this kind of scale, reduction R takes place-so, rather than the superposition involved in U being maintained, Nature must choose one geometry or the other. What kind of scale of mass or of distance moved would such a tiny change in geometry correspond to? In fact, owing to the smallness of gravitational effects, this turns out to be quite large, and not at all unreasonable as a demarcation line between the quantum and classical levels. In order to get a feeling for such matters, it will be useful to say something about absolute (or Planckian) units.

In general, when we consider an object in a superposition of two spatially displaced states, we simply ask for the energy that it would take to effect this displacement, considering only the gravitational interaction between the two. The reciprocal of this energy measures a kind of 'half-life' for the superposed state. The larger this energy, the shorter would be the time that the superposed state could persist.

Once there is a sufficient disturbance in the environment, according to the present ideas, reduction will rapidly actually take place in that environment-and it would be accompanied by reduction in any 'measuring apparatus' with which that environment is entangled. Nothing could reverse that reduction and enable the original entangled state to be resurrected, even imagining enormous advances in technology. Accordingly, there is no contradiction with the measuring apparatus actually registering either YES or NO-as in the present picture it would indeed do.

A point that should be emphasized is that the energy that the energy that defines the lifetime of the superposed state is an energy difference, and not the total, (mass-) energy that is involved in the situation as a whole. Thus, for a lump that is quite large but does not move very much-and supposing that it is also crystalline, so that its individual atoms do not get randomly displaced-quantum superpositions could be maintained for a long time. The lump could be much larger than the water droplets considered above. There could also be other very much larger masses in the vicinity, provided that they do not get significantly entangled with the superposed state we are concerned with. (These considerations would be important for solid-state devices, such as gravitational wave detectors, that use coherently oscillating solid-perhaps crystalline-bodies.)

In the GRW scheme, however, an object as large as a cat, which would involve some 10^27 nuclear particles, would almost instantaneously have one of its particles 'hit' by a Gaussian function (as in Fig. 6.2), and since this particle's state would be entangled with the other particles in the cat, the reduction of that particle would 'drag' the others with it, causing the entire cat to find itself in the state of either life or death. In this way, the X-mystery of Schrodinger's cat-and of the measurement problem in general-is resolved.

There are strong reasons for suspecting that the modification of quantum theory that will be needed, if some form of R is to be made into a real physical process, must involve the effects of gravity in a serious way. Some of these reasons have to do with the fact that the very framework of standard quantum theory fits most uncomfortably with the curved-space notions that Einstein's theory of gravity demands. Even such concepts as energy and time-basic to the very procedures of quantum theory-cannot, in a completely general gravitational context, be precisely defined consistently with the normal requirements of standard quantum theory. Recall, also, the light-cone 'tilting' effect (4.4) that is unique the physical phenomenon of gravity. One might expect, accordingly, that some modification of the basic principles of quantum theory might arise as a feature of its (eventual) appropriate union with Einstein's general relativity. Yet most physicists seem reluctant to accept the possibility that it might be the quantum theory that requires modification for such a union to be successful. Instead, they argue, Einstein's theory itself should be modified. They may point, quite correctly, to the fact that classical general relativity has its own problems, since it leads to space-time singularities, such as are encountered in black holes and the big bang, where curvatures mount to infinity and the very notions of space and time cease to have validity (see ENM, Chapter 7). I do not myself doubt that general relativity must itself be modified when it is appropriately unified with quantum theory. And this will indeed be important for the understanding of what actually takes place in those regions that we presently describe as 'singularities'. But it does not absolve quantum theory from a need for change. We saw in 4.5 taht general relativity is an extraordinarily accurate theory-no less accurate than is quantum theory itself. Most of the physical insights that underlie Einstein's theory will surely survive, not less than will most of those of quantum theory, when the appropriate union that moulds these two great theories together is finally found.

Applying the standard U-procedures of quantum mechanics, we find that the photon's state, after it has encountered the mirror, would consist of two parts in two very different locations. One of these parts then becomes entangled with the device and finally with the lump, so we have a quantum state which involves a linear superposition of two quite different positions for the lump. Now the lump will have its gravitational field, which must also be involved in this superposition. Thus, the state involves a superposition of two different gravitational fields. According to Einstein's theory, this implies that we have two different space-time geometries superposed! The question is: is there a point at which the two geometries become sufficiently different from each other that the rules of quantum mechanics must change, and rather than forcing the different geometries into superposition, Nature chooses between one or the other of them and actually effects some kind of reduction procedure resembling R?

It is a famous theorem first proved by the great (Italian-) French mathematician Joseph L. Lagrange in 1770 that every number is, indeed, the sum of four squares.

Another example of a class of well-defined mathematical problems that have no algorithmic solution is the tiling problem. This is formulated as follows: given a set of polygonal shapes, decide whether those shapes will tile the plane; that is, is it possible to cover the entire Euclidean plane using only these particular shapes, without gaps or overlaps?

The mathematical proofs that Hilbert's tenth problem and the tiling problem are not soluble by computational means are difficult, and I shall certainly not attempt to give the argument here. The central point of each argument is to show, in effect, how any Turing-machine action can be coded into a Diophantine or tiling problem. This reduces the issue to one that Turing actually addressed in his original discussion: the computational insolubility of the halting problem-the problem of deciding those situations in which a Turing-machine action fails ever to come to a halt. In 2.3, various explicit computations that do not ever halt will be given; and in 2.5 a relatively simple argument will be presented-based essentially on Turing's original one-that shows, amongst other things, that the halting problem is indeed computationally insoluble.

This book will not supply an answer to these deep issues , but I believe that it may open the door to them by a crack-albeit only by a crack. It will not tells us that there need necessarily be a 'self' whose actions are not attributable to external cause, but it will tell us to broaden our view as to the very nature of what a 'cause' might be. A 'cause' could be something that cannot be computed in practice or in principle. I shall argue that when a 'cause' is the effect of our conscious actions, then it must be something very subtle, certainly beyond computation, beyond chaos, and also beyond any purely random influences. Whether such a concept of 'cause' could lead us any closer to an understanding of the profound issue (or the 'illusion'?) of our free wills is a matter for the future.

Whereas I reject mysticism in its negation of scientific criteria for the furtherance of knowledge, I believe that within an expanded science and mathematics there will be found sufficient mystery ultimately to accommodate even the mystery of mind.

I suppose that this viewpoint-that physical systems are to be regarded as merely computational entities-stems partly from the powerful and increasing role that computational simulations play in modern twentieth-century science, and also partly from a belief that physical objects are themselves merely 'patterns of information', in some sense, that are subject to computational mathematical laws. Most of the material of our bodies and brains, after all, is being continuously replaced, and it is just its pattern that persists. Moreover, matter itself seems to have merely a transient existence since it can be converted from one form into another. Even the mass of a material body, which provides a precise physical measure of the quantity of matter that the body contains, can in appropriate circumstances be converted into pure energy (according to Einstein's famous E=mc^2)-so even material substance seems to be able to convert itself into something with a theoretical mathematical actuality. Furthermore, quantum theory seemst o tell us that material particles are merely 'waves' of information. (We shall examine these issues more thoroughly in Part II.) Thus, matter itself is nebulous and transient; and it is not at all unreasonable to suppose that the persistence of 'self' might have more to do with the preservation of patterns than of actual material particles.

According to C, the problem of conscious awareness is indeed a scientific one, even if the appropriate science may not yet be at hand. I strongly support this viewpoint; I believe that it must indeed be by the methods of science-albeit appropriately extended in ways that we can perhaps only barely glimpse at present-that we must seek our answers. That is the key difference between C and D, whatever similarities there may seem to be in the corresponding opinions as to what present-day science is capable of achieving.

There is even a view, not uncommonly expressed, that might best be regarded as a combination of A and D (or perhaps B and D)-a possibility that will actually feature significantly in our later deliberations. According to this view, the brain's action is indeed that of a computer, but it is a computer of such wonderful complexity that its imitation is beyond the wit of man and science, being necessarily a divine creation of God-the 'best programmer in the business'!

Perhaps it is conceivable that, in the future, some different kind of 'computer' might be introduced, that makes critical use of continuous physical parameters-albeit within the standard theoretical framework of today's physics-enabling it to behave in a way that is essentially different from a digital computer.

Turing was able to show that there are certain classes of problem that do not have any algorithmic solution (in particular the 'halting problem' that I shall describe shortly). However, Hilbert's actual tenth problem had to wait until 1970 before the Russian mathematician Yuri Matiyasevich-providing proofs that completed certain arguments that had been earlier put forward by the Americans Julia Robinson, Martin Davis, and Hilary Putnam-showed that there can be no computer program (algorithm) which decides yes/no systematically to the question of whether a system of Diophantine equations has a solution. It may be remarked that whenever the answer happens to be 'yes', then that fact can, in principle, be ascertained by the particular computer program that just slavishly tries all sets of integers one after the other. It is the answer 'no', on the other hand, that eludes any systematic treatment. Various sets of rules for correctly giving the answer 'no' can be provided-like the argument using even and odd numbers that rules out solutions to the second system given above-but Matisyasevich's theorem showed that these can never be exhaustive.

Of course I have not defined any of the terms 'intelligence', 'understanding', or 'awareness'. I think that it would be most unwise to attempt to give full definitions here. We shall need to rely, to some extent, on our intuitive perceptions as to what these words actually mean. If our intuitive concept of 'understanding' is that it is something that is necessary for 'intelligence', then an argument which establishes the non-computational nature of 'understanding' will also establish the non-computational nature of 'intelligence'. Moreover, if 'awareness' is something that is needed for 'understanding', then a non-computational physical basis for the phenomenon of awareness might account for such a non-computational nature for 'understanding'. Thus, my own use of these terms (and, I maintain, common usage also) entails the implications: (a) 'intelligence' requires 'understanding' and (b) 'understanding' requires 'awareness

The foregoing remarks illustrate the fact that the 'tilting' of light cones, i.e. the distortion of causality, due to gravity, is not only a subtle phenomenon, but a real phenomenon, and it cannot be explained away by a residual or 'emergent' property that arises when conglomerations of matter get large enough. Gravity has its own unique character among physical processes, not directly discernible at the level of the forces that are important for fundamental particles, but nevertheless it is there all the time. Nothing in known physics other than gravity can tilt the light cones, so gravity is something that is simply different from all other known forces and physical influences, in this very basic respect. According to classical general relativity theory, there must indeed be an absolutely minute amount of light-cone tilting resulting from the material in the tiniest speck of dust. Even individual electrons must tilt the light cones. But the amount of tilting in such objects is far too ridiculously tiny to have any directly noticeable effect whatsoever.

It is only with very large masses indeed that light-cone tilting can be directly observed; whereas its actual presence in very tiny amounts in bodies as small as specks of dust is a clear-cut implication of Einstein's theory.

Yet despite the fact that gravity is different from other physical forces, there is a profound harmony integrating gravity with all of the rest of physics. Einstein's theory is not something foreign to the other laws, but it presents them in a different light. (This is particularly so for the laws of conservation of energy, momentum, and angular momentum.) This integration of Einstein's gravity with the rest of physics goes some way to explaining the irony that Newton's gravity had provided a paradigm for the rest of physics despite the fact, as Einstein later showed, that gravity is actually different from the rest of physics! Above all, Einstein taught us not to get too complacent in believing, at any stage of our understanding, that we have, as yet, necessarily found the appropriate physical viewpoint.

May we expect that there is something corresponding to be learnt with regard to the phenomenon of consciousness? If so, it would not be mass that would need to be large for the phenomenon to become apparent-at least not only mass-but some kind of delicate physical organization. According to the arguments put forward in Part I, such organization would have to have found a way of making use of some hidden non-computational ingredient already present in the behaviour of ordinary matter-an ingredient that, like the light-cone tilting of general relativity, would have totally escaped attention had that attention been confined to the study of the behaviour of tiny particles.

However, in other circumstances, such as with PSR 1913 + 16, the situation is very different, and gravitational radiation from the system indeed has a significant role to play. Here, Einstein's theory provides a firm prediction of the detailed nature of the gravitational radiation that the system ought to be emitting, and of the energy that should be carried away. This loss of energy should result in a slow spiralling inwards of the two neutron stars, and a corresponding speeding up of their orbital rotation period. Joseph Taylor and Russell Hulse first observed this binary pulsar at the enormous Aricebo radio telescope in Puerto Rico in 1974. Since that time, the rotation period has been closely monitored by Taylor and his colleagues, and the speed-up is in precise agreement with the expectations of general relativity (cf. Fig. 4.11). For this work, Hulse and Taylor were awarded the 1993 Nobel Prize for Physics. In fact, as the years have rolled by, the accumulation of data from this system has provided a stronger and stronger confirmation of Einstein's theory. Indeed, if we now take the system as a whole and compare it with the behaviour that is computed from Einstein's theory as a whole-from the Newtonian aspects of the orbits, through the corrections to these orbits from standard general relativity effects, right up to the effects on the orbits due to loss of energy in gravitational radiation-we find that the theory is confirmed overall to an error of no more than about 10^-14. This makes Einstein's general relativity, in this particular sense, the most accurately tested theory known to science!

Perhaps Cardano's curious combination of a mystical and a scientifically rational personality allowed him to catch these first glimmerings of what developed to be one of the most powerful of mathematical conceptions. In later years, through the work of Bombelli, Coates, Euler, Wessel, Argand, Gauss, Cauchy, Weierstrass, Riemann, Levi, Lewy, and many others, the theory of complex numbers has flowered into one of the most elegant and universally applicable of mathematical structures. But not until the advent of the quantum theory, in the first quarter of this century, was a strange and all-pervasive role for complex numbers revealed at the very foundational structure of the actual physical world in which we live-nor had their profound link with probabilities been perceived before this. Even Cardano could have had no inkling of a mysterious underlying connection between his two greatest contributions to mathematics-a link that forms the very basis of the material universe at its smallest scales.

G* No individual mathematician ascertains mathematical truth solely by means of an algorithm that he or she knows to be sound.

Perhaps different mathematicians do actually have inherently different perceptions as to the truth of statements that relate to non-constructively infinite sets. It is certainly true that they often profess to having such different perceptions. But I think that such differences are basically similar to the differences in expectations that different mathematicians might have with regard to the truth of ordinary mathematical propositions. These expectations are merely provisional opinions. So long as a convincing demonstration or refutation hs not been found, the mathematicians may disagree amongst themselves as to what they expect, or guess is true, but the possession of such a demonstration by one of the mathematicians would (in principle) enable the others also to become convinced. With regard to foundational issues, such demonstrations are indeed lacking. It might be the case that convincing demonstrations will never be found. Perhaps they cannot be found because such demonstrations do not exist, and it is simply the case that there are different equally valid viewpoints with regard to these foundational issues.

The simplest type of computational loop occurs when the system, at some stage, arrives back in exactly the same state as it had been in on a previous occasion. With no additional input it would then simply repeat the same computation endlessly. It would not be hard to devise a system that, in principle (though perhaps very inefficiently), would guarantee to get out of loops of this kind whenever they occur (by, say, keeping a list of all the states that it had been in previously, and checking at each stage to see whether that state has occurred before). However, there are many more sophisticated types of 'looping' that are possible. basically, the loop problem is the one that the whole discussion of Chapter 2 (particularly 2.1-2.6) was all about; for a computation that loops is simply one that does not stop. An assertion that some computation actually loops is precisely what we mean by a Pi-1 sentence (cf. 2.10, response to Q10). Now, as part of the discussion of 2.5, we saw that there is no entirely algorithmic way of deciding whether a computation will fail to stop-i.e. whether it will loop. Moreover, we conclude from the discussion above that the procedures that are available to human mathematicians for ascertaining that certain computations do loop-i.e. for ascertaining the truth of Pi1-sentences-lie outside algorithmic action. Thus we conclude that indeed some kind of 'non-computational intelligence' is needed if we wish to incorporate all humanly possible ways of ascertaining for certain that some computation is indeed looping. It might have been thought that loops could be avoided by having some mechanism that gauges how long a computation has been going on for, and it 'jumps out' if it judges that the computation has indeed been at it for too long and it has no chance of stopping. But this will not do, if we assume that the mechanism whereby it makes such decisions is something computational, for then there must be the cases where the mechanism will fail, either by erroneously coming to the conclusion that some computation is looping when indeed it is not, or else by not coming to any conclusion at all (so that the entire mechanism itself is looping). One way of understanding this comes from the fact that the entire system is something computational, so it will be subject to the loop problem itself, and one cannot be sure that the system as a whole, if it does not come to erroneous conclusions, will not itself loop.

As the arguments of this book have shown, mathematical understanding is something different from computation and cannot be completely supplanted by it. Computation can supply extremely valuable aid to understanding, but it never supplies actual understanding itself. However, mathematical understanding is often directed towards the finding of algorithmic procedures for solving problems. In this way, algorithmic procedures can take over and leave the mind free to address other issues. A good notation is something of this nature, such as is supplied by the differential calculus, or the ordinary 'decimal' notation for numbers. Once the algorithm for multiplying numbers together has been mastered, for example, the operations can be performed in an entirely mindless algorithmic way, rather than 'understanding' having to be invoked as to why those particular algorithmic rules are being adopted, rather than something else.

One thing that we conclude from all this is that the 'learning robot' procedure for doing mathematics is not the procedure that actually underlies human understanding of mathematics. In any case, such bottom-up-dominated procedure would appear to be hopelessly bad for any practical proposal for the construction of a mathematics-performing robot, even one having no pretensions whatever for simulating the actual understandings possessed by a human mathematician. As stated earlier, bottom-up learning procedures by themselves are not effective for the unassailable establishing of mathematical truths. If one is to envisage some computational system for producing unassailable mathematical results, it would be far more efficient to have the system constructed according to top-down principles (at least as regards the 'unassailable' aspects of its assertions; for exploratory purposes, bottom-up procedures might well be appropriate). The soundness and effectiveness of these top-down procedures would have to be part of the initial human input, where human understanding an insight provide the necesssary additional ingredients that pure computation is unable to achieve. In fact, computers are not infrequently employed in mathematical arguments, nowadays, in this kind of way. The most famous example was the computer-assisted proof, by Kenneth Appel and Wolfgang Haken, of the four-colour theorem, as referred to above. The role of the computer, in this case, was to carry out a clearly specified computation that ran through a very large but finite number of alternative possibilities, the elimination of which had been shown (by the human mathematicians) to lead to a general proof of the needed result. There are other examples of such computer-assisted proofs and nowadays complicated algebra, in addition to numerical computation, is frequently carried out by computer. Again it is human understanding that has supplied the rules and it is a strictly top-down action that governs the computer's activity.

There is one are of work that should be mentioned here, referred to as 'automatic theorem proving'. One set of procedures that would come under this heading consists of fixing some formal system H, and trying to derive theorems within this system. We recall, from 2.9, that it would be an entirely computational matter to provide proofs of all the theorems of H one after the other. This kind of thing can be automated, but if done without further thought or insight, such an operation would be likely to be immensely inefficient. However, with the employment of such insight in the setting up of the computational procedures, some quite impressive results have been obtained. In one of these schemes (Chou 1988), the rules of Euclidean geometry have been translated into a very effective system for proving (and sometimes discovering) geometrical theorems. As an example of one of these, a geometrical proposition known as V. Thebault's conjecture, which had been proposed in 1938 (and only rather recently proved, by K.B. Taylor in 1983), was presented to the system and solved in 44 hours' computing time. More closely analogous to the procedures discussed in the previous sections are attempts by various people over the past 10 years or so to provide 'artificial intelligence' procedures for mathematical 'understanding'. I hope it is clear from the arguments that I have given, that whatever these systems do achieve, what they do not do is obtain any actual mathematical understanding! Somewhat related to this are attempts to find automatic theorem-generating systems, where the system is set up to find theorems that are regarded as 'interesting'-according to certain criteria that the computational system is provided with. I do think that it would be generally accepted that nothing of very great actual mathematical interest has yet come out of these attempts. Of course, it would be argued that these are early days yet, and perhaps one may expect something much more exciting to come out of them in the future. However, it should be clear to anyone who has read this far, that I myself regard the entire enterprise as unlikely to lead to much that is genuinely positive, except to emphasize what such systems do not achieve.

Thus the robot is incapable of knowing that it was constructed according to the mechanisms M. Since we are aware-or at least can be made aware-that the robot was so constructed, this seems to tell us that we have access to mathematical truths, e.g. Omega (Q(M)), that are beyond the robot's capabilities, despite the fact that the robot's abilities are supposed to be equal of (or in excess of) human capabilities.

The aim of the next nine sections will be to present careful arguments to show that none of the loopholes (a), (b), and (c) can provide a plausible way to evade the contradiction of the robot. Accordingly, it, and we also, are driven to the unpalatable (d), if we are still insistent that mathematical understanding can be reduced to computation. I am sure that those concerned with artificial intelligence would find (d) to be as unpalatable as I find it to be. It provides perhaps a conceivable standpoint-essentially the A/D suggestion, referred to at the end of 1.3, whereby divine intervention is required for the implanting of an unknowable algorithm into each of our computer brains (by 'the best programmer in the business'). In any case, the conclusion 'unknowable'-for the very mechanisms that are ultimately responsible for our intelligence-would not be a very happy conclusion for those hoping actually to construct a genuinely artificially intelligent robot! It would not be a particularly happy conclusion, either, for those of us who hope to understand, in principle and in a scientific way, how human intelligence has actually arisen, in accordance with comprehensible scientific laws, such as those of physics, chemistry, biology, and natural selection-irrespective of any desire to reproduce such intelligence in a robot device. In my own opinion, such a pessimistic conclusion is not warranted, for the very reason that 'scientific comprehensibility' is a very different thing from 'computability'. The conclusion should be not that the underlying laws are incomprehensible, but that they are non-computable.

The upshot is that no mathematically aware conscious being-that is, no being that is capable of genuine mathematical understanding-can operate according to any set of mechanisms that it is able to appreciate, irrespective of whether it actually knows that those mechanisms are supposed to be the ones governing its own routes to unassailable mathematical truth. (We recall, also, that its 'unassailable mathematical truth' just means what it can mathematically establish-which means by means of 'mathematical proof' though not necessarily 'formal' proof.)

More precisely, we are driven, by the foregoing reasoning, to conclude that there is no robot-knowable set of computational mechanisms, free of genuinely random ingredients, that the robot could accept as being even a possibility for underlying its mathematical belief system-provided that the robot is prepared to accept that the specific procedure that I have been suggesting for constructing the formal system Q(M) from the mechanisms M actually does encapsulate the totality of Pi1-sentences that it believes in unassailably-and, correspondingly, that the formal system Qm(M) encapsulates the totality of Pi-1-sentences that it unassailably believes would follow from the hypothesis M. Moreover, there is the further point that genuinely random ingredients might have to be included into the mechanisms M if the robot is to achieve a potentially consistent mathematical belief system.

Some AI proponents might argue that in order for an AI system to gain any 'actual' understanding, it would need to be programmed in a way that involves bottom-up procedures in a much more basic way than is usual for chess-playing computers. Accordingly, its 'understandings' would develop gradually by its building up a wealth of 'experience', rather than having specific top-down algorithmic rules built into it. Top-down rules that are simple enough for us to appreciate easily could not, by themselves, provide a computational basis for actual understanding-for we can use our very understandings of these rules to realize their fundamental limitations.

Although it might well be possible for a sufficiently cleverly constructed such system to preserve an illusion, for some considerable time (as with Deep Thought), that it possesses some understanding, I shall maintain that a computer system's actual lack of understanding should-in principle, at least-eventually reveal itself.

Any complicated activity, which may be mathematical calculations, or playing a game of chess, or commonplace actions-if they have been understood in terms of clear-cut computational rules-are the things that modern computers are good at; but the very understanding that underlies these computational rules is something that is itself beyond computation.

Why do I claim that this 'awareness', whatever it is, must be something non-computational, so that no robot, controlled by a computer, based merely on the standard logical ideas of a Turing machine (or equivalent)-whether top-down or bottom-up-can achieve or even simulate it? It is here that the Godelian argument plays its crucial role. It is hard to say much at the present time about our 'awareness' of, for example, the colour red; but there is something definite that we can say concerning our awareness of the infinitude of natural numbers. It is 'awareness' that allows a child to 'know' what it means for this sequence to go on for ever, when only absurdly limited, and seemingly almost irrelevant, kinds of descriptions in terms of a few oranges and bananas have been given. The concept of 'three' can indeed be abstracted, by a child, from such limited examples; and, moreover, the child can also latch on to the fact that this concept is but one of the unending sequence of similar concepts ('four', 'five', 'six', etc.). In some Platonic sense, the child already 'knows' what the natural numbers are.

Yet, how is it that descriptions of numbers in terms of apples or bananas can allow a child to know what 'three days' means, that same abstract concept of 'three' being involved as with 'three oranges'? Of course, this appreciation may well not come at once, and the child may get it wrong at first, but that is not the point. The point is that this kind of realization is possible at all. The abstract concept of 'three', and of this concept as being one of an infinite sequence of corresponding concepts-the natural numbers themselves-is something that can indeed be understood, but, I claim, only through the use of one's awareness.

A computational procedure is said to have a top-down organization if it has been constructed according to some well-defined and clearly understood fixed computational procedure (which may include some preassigned store of knowledge), where this procedure specifically provides a clear-cut solution to some problem at hand. (Euclid's algorithm for finding the highest common factor of two natural numbers, as described in ENM, p. 31, is a simple example of a top-down algorithm.) This is to be contrasted with a bottom-up organization, where such clearly defined rules of operation and knowledge store are not specified in advance, but instead there is a procedure laid down for the way that the system is to 'learn' and to improve its performance according to its 'experience'. Thus, with a bottom-up system, these rules of operation are subject to continual modification. One must allow that the system is to be run many times, performing its actions upon a continuing input of data. On each run, an assessment is made-perhaps by the system itself-and it modifies its operations, in the lifht of this assessment, with a view to improving this quality of output. For example, the input data for the system might be a number of photographs of human faces, appropriately digitized, and the system's task is to decide which photographs represent the same individuals and which do not. After each run, the system's performance is compared with the correct answers. Its rules of operation are then modified in such a way as to lead to a probable improvement in its performance on the next run.

I think it would be fair to say that only with certain instances of top-down (or primarily top down) organization have computers exhibited a significant superiority over humans. The most obvious example is in straightforward numerical calculation, where computers would now win hands down-and also in 'computational' games, such as chess or draughts (checkers), and where there may be only a very few human players able to beat the best machines. With bottom-up (artificial neural network) organization, the computers can, in a few limited instances, reach about the level of ordinary well-trained humans.

One important issue that we must address is the possibility that there might be numerous quite different, perhaps inequivalent, algorithms that are responsible for the different modes of mathematical understanding that pertain to different individuals. Indeed, one thing is certainly clear from the start, and that is that even amongst practising mathematicians, different individuals often perceive mathematics in quite different ways from one another. To some, visual images are supremely important, whereas to others, it might be precise logical structure, subtle conceptual argument, or perhaps detailed analytic reasoning, or plain algebraic manipulation. In connection with this, it is worth remarking that, for example, geometrical and analytical thinking are believed to take place largely on opposite sides-right and left, respectively-of the brain. Yet the same mathematical truth may often be perceived in either of these ways. On the algorithmic view, it might seem, at first, that there should be a profound inequivalence between the different mathematical algorithms that each individual might possess. But, despite the very differing images that different mathematicians (or other people) may form in order to understand or to communicate mathematical ideas, a very striking fact about mathematicians' perceptions is that when they finally settle upon what they believe to be unassailably true, mathematicians will not disagree, except in such circumstances when a disagreement can be traced to an actual recognizable (correctable) error in on or the other's reasoning-or possibly to their having differences with respect to a very small number of fundamental issues;

Nevertheless, the Geroch-Hartle work does indicate the clear possibility that non-computability may have a genuine role in whatever quantum gravity theory finally emerges as being physically correct.

In a particular approach to quantum gravity, Robert Geroch and James Hartle (1986) found themselves confronted with a computationally unsolvable problem, namely the topological equivalence problem for 4-manifolds. Basically, their approach involved the questions of deciding when two four-dimensional spaces are 'the same', from the topological point of view (i.e. when it is possible to deform one of them continuously until it coincides with the other, where the deformation does not allow tearing or gluing the spaces in any way). In Fig. 7.14, this is illustrated in the two-dimensional case, where we see that the surface of a ball is different. In two dimensions, the topological equivalence problem is computationally solvable, but it was shown by A.A. Markov in 1958 that there is not algorithm for solving this problem in the four-dimensional case. In fact, what is shown effectively demonstrates that if there were such an algorithm, then one could convert that algorithm into another algorithm which could solve the halting problem, i.e. it could decide whether or not a Turing-machine action will stop. Since, as we have seen in 2.5, there is no such algorithm, it follows that there cannot be any algorithm for solving the equivalence problem for 4-manifolds either.

Though it indeed seems reasonable to rule out space-time geometries with closed timelike lines as descriptions of the classical universe, a case can be made that they should not be ruled out as potential occurrences that could be involved in a quantum superposition.

Does awareness play some kind of role as a 'bridge' to a world of Platonic absolutes.

Support for the Platonic viewpoint (as opposed to the formalist one) was an important part of Godel's initial motivations. On the other hand, the arguments from Godel's theorem serve to illustrate the deeply mysterious nature of our mathematical perceptions. We do not just 'calculate' in order to form these perceptions, but something else is profoundly involved-something that would be impossible without the very conscious awareness that is, after all, what the world of perceptions is all about.

In the present chapter, we tried to pinpoint the place in the brain where quantum action might be important to classical behaviour, and have apparently been driven to consider that it is through the cytoskeletal control of synaptic connections that this quantum/classical interface exerts its fundamental influence on the brain's behaviour.

There is another point that should be made, however, and this is that it need not be the case that human mathematical understanding is in principle as powerful as any oracle machine at all. As noted above, the conclusion G does not necessarily imply that human insight is powerful enough, in principle, to solve each instance of the halting problem. Thus, we need not necessarily conclude that the physical laws that we seek reach, in principle, beyond every computable level of of oracle machine (or even reach the first order). We need only seek something that is not equivalent to any specific oracle machine (including also the zeroth-order machines, which are Turing machines). Physical laws could perhaps lead to something that is just different.

If we now consider what it means to perform a quantum computation in such a situation, we apparently come to the conclusion that non-computable operations can be performed! This arises from the fact that in the space-time geometries with closed timelike lines, a Turing-machine operation can feed on to its own output, running around indefinitely, if necessary, so that the answer to the question 'does that computation ever stop' has an actual influence on the final result of the quantum computation. Deutsch comes to the conclusion that in his quantum gravity scheme, quantum oracle machines are possible. As far as I can make out, his arguments would apply just as well to higher-order oracle machines also. Of course, many readers may feel that all this should be taken with an appropriate amount of salt. Indeed, there is no real suggestion that the scheme provides us with a consistent (or even plausible) theory of quantum gravity. Nonetheless, the ideas are logical within their own framework and are suggestively interesting-and it seems quite reasonable to me that when the appropriate scheme for quantum gravity is eventually found, then some important vestiges of Deutsch's proposal will indeed survive.

One appears to conclude from these experiments that: (i) the conscious act of 'free will' is a pure illusion, having been, in some sense, already preprogrammed in the preceding unconscious activity of the brain; or (ii) there is a possible 'last-minute' role for the will, so that it can sometimes (but not usually) reverse the decision that had been unconsciously building up for a second or so before; or (iii) the subject actually consciously wills the finger-flexing at the earlier time of a second or so before the flexing takes place, but mistakenly perceives, in a consistent way, that the conscious act occurs at the much later time, just before the finger is indeed flexed.