Shadows of the Mind Quotes

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

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

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

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

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

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

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

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

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

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

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

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

“G* No individual mathematician ascertains mathematical truth solely by means of an algorithm that he or she knows to be sound.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

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

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

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

“Any complicated activity, which may be mathematical calculations, or playing a game of chess, or commonplace actions-if they have been understood in terms of clear-cut computational rules-are the things that modern computers are good at; but the very understanding that underlies these computational rules is something that is itself beyond computation.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

“Although it might well be possible for a sufficiently cleverly constructed such system to preserve an illusion, for some considerable time (as with Deep Thought), that it possesses some understanding, I shall maintain that a computer system's actual lack of understanding should-in principle, at least-eventually reveal itself.”
― Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness

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

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

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

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

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

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

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

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

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

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

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

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