Shadows of the Mind Quotes

“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).”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“A scientific world-view which does not profoundly come to terms with the problem of conscious minds can have no serious pretensions of completeness.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.)”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“It is in mathematics that our thinking processes have their purest form.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“There are completely deterministic universe models, with clear-cut rules of evolution, that are impossible to simulate computationally.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“In some Platonic sense, the natural numbers seem to be things that have an absolute conceptual existence independent of ourselves.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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!”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“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.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness