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]

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

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).

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

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.

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.

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 …?

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.

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

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

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.

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

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.

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

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

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.

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.

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

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.

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.

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.

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

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.

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.

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.

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.

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.

During one seminar offered by this group, I heard a statement that shocked me. The speaker walked us through what the chances were that our universe would come into existence and concluded that the odds that our universe would form in the way that it did, with a big bang at high energies, were nearly zero! In fact, it was possible to calculate the odds, which the eminent British mathematician and theoretical physicist Roger Penrose (later a Nobel Prize laureate) had done in the late 1970s. When Penrose calculated the likelihood of our universe spontaneously forming, he got a staggering number: 1 in 10^10^123. Less than a one in a googolplex chance.

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.

The judgement-forming that I am claiming is the hallmark of consciousness is itself something that the AI people would have no concept of how to program on a computer.

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 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.

Embracing the subjective interpretation quickly leads us to assertions that are "patently absurd," underscoring the independence of mathematical knowledge of any human activity" Just take me there...

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 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.

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.

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;

Roger Penrose discovered that if you launch the object at a clever angle and make it split into two pieces as figure 6.4 illustrates, then you can arrange for only one piece to get eaten while the other escapes the black hole with more energy than you started with. In other words, you’ve successfully converted some of the rotational energy of the black hole into useful energy that you can put to work. By repeating this process many times, you can milk the black hole of all its rotational energy so that it stops spinning and its ergosphere disappears.

the quanta inside microtubules in the brain create human consciousness, this is in fact a complete fabrication I made up from the words of the very real theory of quantum mind. The actual quantum mind theory is a totally different thing and exceedingly difficult, which means I can’t understand a word of it. For those of you who are interested in the subject, a man called Roger Penrose has written a book on the topic, so please read that. And then please break it down and quietly explain it to me. (LOL)

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.

When I participated in small discussion meetings in Roger Penrose’s office, I felt that the Copernican principle was being violated; I was manifestly in a preferred frame of reference at the very centre of the universe.

The dream of Strong Artificial Intelligence—and more specifically the growing interest in the idea that a computer can become conscious and have first-person subjective experiences—has led to a cultural shift. Prophets like Kurzweil believe that we are much closer to cyberconsciousness and superintelligence than most observers acknowledge, while skeptics argue that current AI systems are still extremely primitive and that hopes of conscious machines are pipedreams. Who is right? This book does not attempt to address this question, but points out some philosophical problems and asks some philosophical questions about machine consciousness. One fundamental problem is that we do not understand human consciousness. Many in science and artificial intelligence assume that human consciousness is based on information or computations. Several writers have tried to tackle this assumption, most notably the British physicist Roger Penrose, whose controversial theory suggests that consciousness is based upon noncomputable quantum states in some of the tiniest structures in the brain, called microtubules. Other, perhaps less esoteric thinkers, like Duke’s Miguel Nicolelis and Harvard’s Leonid Perlovsky, are beginning to challenge the idea that the brain is computable. These scientists lead their fields in man-machine interfacing and computer science. The assumption of a computable brain allows artificial intelligence researchers to believe they will create artificial minds. However, despite assuming that the brain is a computational system—what philosopher Riccardo Manzotti calls “the computational stance”—neuroscience is still discovering that human consciousness is nothing like we think it is. For me this is where LSD enters the picture. It turns out that human consciousness is likely itself a form of hallucination. As I have said, it is a very useful hallucination, but a hallucination nonetheless. LSD and psychedelics may help reveal our normal everyday experience for the hallucination that it is. This insight has been argued about for centuries in philosophy in various forms. Immanuel Kant may have been first to articulate it in modern form when he called our perception of the world “synthetic.” The fundamental idea is that we do not have direct knowledge of the external world. This idea will be repeated often in this book, and you will have to get used to it. We only have knowledge of our brain’s creation of that world for us. In other words, what we see, hear, and subsequently think are like movies that our brain plays for us after the fact. These movies are based on perceptions that come into our senses from the external world, but they are still fictions of our brain’s creation. In fact, you might put the disclaimer “based on a true story” in front of each experience you have. I do not wish to imply that I believe in the homunculus argument—what philosopher Daniel Dennett describes as the “Cartesian Theater”—the hypothetical place in the mind where the self becomes aware of the world. I only wish to employ the metaphor to illustrate the idea that there is no direct relationship between the external world and your perception of it.

Perhaps consciousness is, after all, merely a spectator who experiences nothing but an ‘action replay’ of the whole drama.

though much of what is actually involved in mental activity might do so.

One can argue that a universe governed by laws that do not allow consciousness is no universe at all.

Furthermore, there are weighty voices within science that are not as enthusiastic about the multiverse. Prominent among them is that of Sir Roger Penrose, Hawking’s former collaborator, who shared with him the prestigious Wolf Prize. Of Hawking’s use of the multiverse in The Grand Design Penrose said: “It’s overused, and this is a place where it is overused. It’s an excuse for not having a good theory.”44 Penrose does not, in fact, like the term “multiverse”, because he thinks it is inaccurate: “For although this viewpoint is currently expressed as a belief in the parallel co-existence of different alternative worlds, this is misleading. The alternative worlds do not really ‘exist’ separately, in this view; only the vast particular superposition…is taken as real.”45

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.