Roger Penrose 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.

... Even an aardvarks think their offspring are beautiful

No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.

Roger Penrose and I showed that Einstein’s general theory of relativity implied that the universe must have a beginning and, possibly, an end.

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]

Objective mathematical notions must be thought of as timeless entities and are not to be regarded as being conjured into existence at the moment that they are first humanly perceived.

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

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

There are considerable mysteries surrounding the strange values that Nature's actual particles have for their mass and charge. For example, there is the unexplained 'fine structure constant' ... governing the strength of electromagnetic interactions, ....

To make this condition mathematically clearer, it is convenient to assert it in the form that the space-time can be continued smoothly, as a conformal manifold, a little way prior to the hypersurface . To before the Big Bang? Surely not: the Big Bang is supposed to represent the beginning of all things, so there can be no ‘before’. Never fear—this is just a mathematical trick. The extension is not supposed to have any physical meaning! Or might it …?

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

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).

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.

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.

What the Second Law indeed states, roughly speaking, is that things are getting more ‘random’ all the time.

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

The algorithm has some kind of disembodied ‘existence’ which is quite apart from any realization of that algorithm in physical terms.

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.

If you come from mathematics, as I do, you realize that there are many problems, even classical problems, which cannot be solved by computation alone

associates and collaborators were Roger Penrose, Robert

associates and collaborators were Roger Penrose,

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.

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.

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

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.

The more deeply we probe the fundamentals of physical behaviour, the more that it is very precisely controlled by mathematics.

It is always the case, with mathematics, that a little direct experience of thinking over things on your own can provide a much deeper understanding than merely reading about them.

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.

In order to decide whether or not an algorithm will actually work, one needs insights, not just another algorithm.

People think of these eureka moments and my feeling is that they tend to be little things, a little realization and then a little realization built on that.

is not easy to ascertain what an algorithm actually is, simply by examining its output.

My own view is that to understand quantum nonlocality we shall require a radical new theory. This new theory will not just be a slight modification of quantum mechanics but something as different from standard quantum mechanics as general relativity is from Newtonian gravity. It would have to be something which has a completely different conceptual framework.

The loop approach to quantum gravity is now a thriving field of research. Many of the older ideas, such as supergravity and the study of quantum black holes, have been incorporated into it. Connections have been discovered to other approaches to quantum gravity, such as Alain Connes's non-commutative approach to geometry, Roger Penrose's twistor theory and string theory.

Roger Penrose of Oxford University has calculated that the odds of the low entropy conditions present in the Big Bang having come about as a result of chance are around one chance in 1010(123).9 The number 1010(123) is so huge a number that even if a zero were inscribed on every subatomic particle in the entire universe, one could not even approach writing down this number.

The second was the chapter in Roger Penrose’s book The Emperor’s New Mind in which he discusses entropy. He points out that there’s a sense in which it’s incorrect to say we eat food because we need the energy it contains. The conservation of energy means that it is neither created nor destroyed; we are radiating energy constantly, at pretty much the same rate that we absorb it. The difference is that the heat energy we radiate is a high-entropy form of energy, meaning it’s disordered. The chemical energy we absorb is a low-entropy form of energy, meaning it’s ordered. In effect, we are consuming order and generating disorder; we live by increasing the disorder of the universe. It’s only because the universe started in a highly ordered state that we are able to exist at all.

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.

I can at least state that my point of view entails that it is our present lack of understanding of the fundamental laws of physics that prevents us from coming to grips with the concept of ‘mind’ in physical or logical terms

According to strong AI, it is simply the algorithm that counts. It makes no difference whether that algorithm is being effected by a brain, an electronic computer, an entire country of Indians, a mechanical device of wheels and

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.

The ultimate behaviour of these matter distributions, taking the form of massless radiation (in accordance with CCC’s §3.2 requirements), can then leave its signature on the crossover 3-surface, and then perhaps be readable in subtle irregularities in the CMB.

As scientists would discover after Einstein’s death, Schwarzschild’s odd theory was right. Stars could collapse and create such a phenomenon, and in fact they often did. In the 1960s, physicists such as Stephen Hawking, Roger Penrose, John Wheeler, Freeman Dyson, and Kip Thorne showed that this was indeed a feature of Einstein’s general theory of relativity, one that was very real. Wheeler dubbed them “black holes,” and they have been a feature of cosmology, as well as Star Trek episodes, ever since.3 Black holes have now been discovered all over the universe, including one at the center of our galaxy that is a few million times more massive than our sun. “Black holes are not rare, and they are not an accidental embellishment of our universe,” says Dyson. “They are the only places in the universe where Einstein’s theory of relativity shows its full power and glory. Here, and nowhere else, space and time lose their individuality and merge together in a sharply curved four-dimensional structure precisely delineated by Einstein’s equations.”4 Einstein

Computer scientist and author Douglas R. Hofstadter phrased this succinctly in his fantastic book Godel, Escher, Bach: An Eternal Golden Braid: "Provability is a weaker notion than truth." In this sense, there will never be a formal method of determining for every mathematical proposition whether it is absolutely true, any more than there is a way to determine whether a theory in physics is absolutely true. Oxford's mathematical physicist Roger Penrose is among those who believe that Godel's theorems argue powerfully for the very existence of a Platonic mathematical world.

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.

However, in 1930 (published in 1931), Godel produced his bombshell, which eventually showed that the formalists' dream was unattainable! He demonstrated that there could be no formal system F, whatever, that is both consistent (in a certain 'strong' sense that I shall describe in the next section) and complete-so long as F is taken to be powerful enough to contain a formulation of the statements of ordinary arithmetic together with standard logic. Thus, Godel's theorem would apply to systems F for which arithmetical statements such as Lagrange's theorem and Goldbach's conjecture, as described in 2.3, could be formulated as mathematical statements.

there seems to be something non-algorithmic about our conscious thinking. In particular, a conclusion from the argument in Chapter 4, particularly concerning Gödel’s theorem, was that, at least in mathematics, conscious contemplation can sometimes enable one to ascertain the truth of a statement in a way that no algorithm could.

Moreover, the slightest ‘mutation’ of an algorithm (say a slight change in a Turing machine specification, or in its input tape) would tend to render it totally useless, and it is hard to see how actual improvements in algorithms could ever arise in this random way. (Even deliberate improvements are difficult without ‘meanings’ being available.

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

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.

The big question in cosmology in the early 1960s was did the universe have a beginning? Many scientists were instinctively opposed to the idea, because they felt that a point of creation would be a place where science broke down. One would have to appeal to religion and the hand of God to determine how the universe would start off. This was clearly a fundamental question, and it was just what I needed to complete my PhD thesis. Roger Penrose had shown that once a dying star had contracted to a certain radius, there would inevitably be a singularity, that is a point where space and time came to an end. Surely, I thought, we already knew that nothing could prevent a massive cold star from collapsing under its own gravity until it reached a singularity of infinite density. I realised that similar arguments could be applied to the expansion of the universe. In this case, I could prove there were singularities where space–time had a beginning. A eureka moment came in 1970, a few days after the birth of my daughter, Lucy. While getting into bed one evening, which my disability made a slow process, I realised that I could apply to black holes the casual structure theory I had developed for singularity theorems. If general relativity is correct and the energy density is positive, the surface area of the event horizon—the boundary of a black hole—has the property that it always increases when additional matter or radiation falls into it. Moreover, if two black holes collide and merge to form a single black hole, the area of the event horizon around the resulting black hole is greater than the sum of the areas of the event horizons around the original black holes.

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.

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

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.

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.