Arkady Plotnitsky

Literature Program
Duke University
aplotnit@acpub.duke.edu

 

Roger Penrose, The Large, the Small, and the Human Mind (with Abner Shimony, Nancy Cartwright, and Steven Hawking), Cambridge: Cambridge University Press, 1997; with a glance back at The Emperor’s New Mind, Shadows of the Mind, and The Nature of Space and Time.

 

At 4 p.m. on May 11, 1997, “the truly impossible occurred,” as Newsweek reported (May 26, 1997, 84). The computer Deep Blue defeated the world chess champion and one of the greatest chess players of all time, Garry Kasparov. In the process, his confidence, bolstered by his impressive previous victories over computers, was shattered along with the confidence and hopes of much of the chess world. Indeed, the defeat was taken by some, including by Kasparov, as a humiliation. It appears to have been especially humiliating because it was inflicted on a great chess mind, capable of the most complex tactical and strategic thinking, by the raw power of computation. A great chess mind was defeated by crude number-crunching–a much inferior form of thinking or, in this case, not even thinking. Such minds have always been seen as able to circumvent and transcend protracted computational routines, and as entities whose own workings are themselves inaccessible to computational analysis.

 

In the last installment of his ongoing argument against the possibility of artificial intelligence, The Large, the Small, and the Human Mind, Roger Penrose uses chess to illustrate the difference between the computational approach used by digital computers and non-computational thinking, which, according to him, fundamentally defines the human mind and gives it ultimate superiority over computers or computer-like computational intelligence (103-5). Penrose gives examples of two chess problems that are easily solved by even mediocre human players but have defeated a computer (in this case Deep Thought, the forerunner of Deep Blue). Penrose’s lectures (given in 1995 and published earlier this year) preceded the latest chapter of the story of computer chess just described, and he may even have been inspired in part by Kasparov’s previous decisive victories over computers and by the seemingly unquestionable ultimate superiority of chess players or, one might say, chess thinkers over chess computers.

 

This latest episode of this, by now long, history does not prove the opposite. Nor does it prove the ultimate superiority of any one form of thinking over another, or of (human) thinking over (computer-like or other) non-thinking. That is, assuming that human thinking is indeed non-computational–a claim that, however appealing or likely, remains hypothetical, as Penrose admits. Everyone acknowledges that Deep Blue cannot think–leaving aside here the complexity, if not impossibility, of the latter concept itself. Nor does the episode prove that artificial intelligence is any more likely now than it was before. What it does prove is that computational thinking or (even more humiliatingly for its opponents) computation without thinking, number-crunching, can be taken to a level high enough to supersede non-computational thinking in certain specific cases. The fact that this is possible in some cases is in part (there are more general conceptual reasons) what drives the idea–which is more or less the idea of artificial intelligence–that it may be possible in any given case. Much else, it is further extrapolated, would be possible as well for computational devices. It may, for example, be possible for machines to perform any given task that human thinking can perform, or conceivably even to think or have consciousness or even to feel, just as humans do. Or–a rarely, if ever, discussed but interesting and radical possibility–it may be possible to perform even the most complex tasks better than humans without thinking and, thus, in a certain sense without intelligence. Or, as I said, it might also be possible to assume that at bottom the human mind is itself only a computational device, a number-crunching machine. These are the types of possibilities that are pursued in the field of artificial intelligence and related endeavors and that are argued against by, among others, Penrose.

 

At a certain level, at stake here is still the power of creative thinking (seen as ultimately non-computational) against computational or calculating thinking (seen as ultimately uncreative) or against the nonthinking of machines. Along and interactively with other classical oppositions and hierarchies of that type, this opposition and this hierarchy has, from Plato (or even the pre-Socratics) to the present, defined the history of thinking about human thinking and its superiority over other animals, on the one hand, and machines, on the other.

 

This theme is among several that link Penrose’s books to what have become known as “postmodern” problematics (using this term with caution here), within which the nature and structure, or deconstruction, of both of these oppositions–the human and the animal, and the human and the machine–and their interactions have been explored at great length. The general question of artificial intelligence is, of course, another such theme, although it is indissociable from the oppositions just described. This question is fundamentally connected by Penrose to a number of key questions raised by such revolutionary developments in twentieth century science and technology as Einsteinian relativity and quantum physics, post-Gödelian mathematical logic, modern biology, and computer technology; and these questions have in turn been central to postmodernist thinking. Such issues as the nature of causality and reality of the physical world, of truth and certainty in mathematics and mathematical logic, the nature of scientific explanations necessary for understanding the human brain (or indeed mind), all considered by Penrose, are part of the postmodernist intellectual scene and have been hotly debated recently, or, again, throughout modern (or earlier) intellectual history.

 

Penrose’s ideas have themselves been the subject of considerable debate. The Large, the Small and the Human Mind is the last installment of his “trilogy” on “computers, minds, and the laws of physics,” initiated by The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics (1989), and continued, in part in reply to questions posed and debates provoked by the first volume, with Shadows of the Mind: A Search for the Missing Science of Consciousness (1994). There is, moreover, another companion (technical) volume, containing a debate with Stephen Hawking, The Nature of Space and Time (1996), which is more specialized and concerned more exclusively with physics, but with many echoes of and significant connections to Penrose’s books.

 

Beyond and as part of an investigation into the question of the human mind and artificial intelligence, Penrose’s trilogy contains major semi-popular or, more accurately, semi-technical expositions of key revolutionary mathematical and scientific theories defining twentieth-century science–post-Gödelian mathematical logic, relativity, and quantum physics–and his controversial forays into the biology of the brain. Derived from a 1995 series of lectures, The Large, the Small, and the Human Mind is a summary presentation (with some updating) of the two previous volumes and a sample of the debate provoked by them. It includes contributions by Abner Shimony, Nancy Cartwright, and Stephen Hawking, and Penrose’s responses to their arguments. Both earlier volumes contain elegant extended expositions of the mathematical and scientific theories just mentioned. While, in principle, these expositions do not require a specialized knowledge of mathematics and science, it is doubtful that, in practice, they are sufficiently accessible to general readers to be read in depth by most of them. By contrast, the latest volume can be read as a more general introduction to Penrose’s main ideas and is more readable for general readers than the two previous volumes or, especially, The Nature of Space and Time. The latter work, however, contains arguably the most interesting discussions of general relativity and cosmology, parts of which can be read and productively assimilated by non-specialists, albeit with considerable (and, in my view, well-deserving) effort. Some of Penrose’s elegant mathematical thinking and presentation is found in the last volume as well, in particular in his discussions of relativity and non-Euclidean geometry in Chapter 1, and the reader will be especially rewarded here. Quantum mechanics, mathematical logic, and biology are, as will be seen, more complex cases in this respect.

 

Penrose’s overall argument can be summarized as follows. He wants to argue definitively for the impossibility for artificial intelligence (at least that based on computation, such as that carried on by digital computers) to reach the level of human intelligence and, more generally, consciousness (both based, he believes, on non-computational processes and themselves effects of physical systems whose design is non-computable). This definitive argument is to be rigorously grounded in mathematically and scientifically ascertainable facts about (physical) nature and (mathematical) mind. Or, more accurately–and this nuance is crucial–Penrose contends that there is, at this point, enough scientific data to hope that one will be able to offer a definitive, rigorous argument of that type in the future (it is, Penrose acknowledges, speculative at present) from the new physics of the world and/as the new physics of the brain. Penrose also offers a specific argument for, or more accurately, a vision of what kind of physics it would and should be. Penrose’s argument, however, is fundamentally based on his interpretations of these theories and data, in particular quantum physics and post-Gödelian mathematical logic, two crucial ingredients of his overall argument. These interpretations are not uncontestable (and have been contested), as Penrose admits. This point is especially significant precisely because Penrose offers a vision of specific future theories of non-computational physical processes modifying existing theories in physics (and a program for developing such theories), and his case against artificial intelligence is fundamentally tied to the possibility and plausibility of such theories. Most especially at stake is a particular form of “quantum gravity” theory, a theory that joins quantum physics and general relativity, the Einsteinian theory of gravitation. Theories envisioned by Penrose will also account for a non-computable physics of the human brain and the non-computational mathematics of the human mind, and specifically for the phenomenon of consciousness. That is, these will be theories of physics that will lead to the physics of the brain that makes thinking and consciousness possible, along the way bridging the classical (macro) level–“the Large”–and the quantum (micro) level–“the Small”–so far resisting such a bridging in physical theory. In Penrose’s own words,

 

We look for the non-computability in physics which bridges the quantum and the classical levels. This is quite a tall order. I am saying that not only do we need new physics, but we also need new physics which is relevant to the action of the brain. (103)

 

One needs the non-computability in physics, which will, first, bridge the hitherto unbridged classical and quantum physics, and then will bridge two other hitherto unbridged territories–the physics and biology of the human brain. A tall order indeed, and Penrose acknowledges that his vision is speculative and that it reflects certain particular “prejudices” of a philosophical nature (97). There are, as will be seen, also “prejudices” that Penrose does not quite see, either by (overtly) seeing some of them as non-prejudicial arguments or by (unconsciously) missing, being blind to, the presence of others in his argument.

 

Before commenting on the nature or, one might say, the structure of Penrose’s speculation, I would like to discuss Penrose’s presentation of key scientific theories involved. As I have indicated, Chapter 1, “Space-Time and Cosmology,” which deals with “the Large,” may well be the most rewarding chapter of the book. It also offers a discussion of the kind of theory (modelled on and extending Einstein’s general relativity) that conceptually grounds Penrose’s vision of what a physical theory should be, whether as a theory of the large, the small, or of the human brain/mind, or, as is Penrose’s ultimate desideratum, of all three together. One finds in this chapter a beautiful, highly informative, and reasonably accessible exposition of, among other things, special and general relativity, relativistic cosmology, and non-Euclidean geometry (the mathematical basis of general relativity). The overall exposition is itself mostly geometrical, which, as will be seen, also signals a significant philosophical point, as an example of something that is more likely to be non-computable. Penrose’s readers, especially non-scientists, will learn much about the conceptual richness of modern mathematics and science, and might be motivated to read or reread Penrose’s discussion of these subjects in his earlier books, or even in the more technical exposition of The Nature of Space and Time, his debate with Hawking.

 

While the arguments of that debate may appear only tangential to the main ostensible concern (“the human mind”) of Penrose’s project, these arguments are in fact crucial, as is clear from the extension of this debate in The Large, The Small, and the Human Mind. The Nature of Space and Time provides a more comprehensive and more sustained picture of the experimental evidence (or at least testability) and theoretical argument as to what kind of theory “quantum gravity” could or should be. Penrose and Hawking disagree on that issue, as well as in their overall philosophical positions–the Platonism of Penrose versus the positivism of Hawking, or what they see as such.

 

The specific shape of the evidence and arguments for (or against) a theory of quantum gravity is, as I said, crucial to Penrose’s argument concerning the human mind as something that, in contrast to computational digital computers, is based on non-computational thinking, mathematical or other. The debate with Hawking shows, however, how deeply speculative and at times problematic (at least at certain points and along certain lines) are Penrose’s ideas concerning quantum gravity even on scientific grounds. These complexities would remain, regardless of whether one agrees or disagrees with Hawking’s own vision of this theory, which is not without some speculation either; or regardless of how one can use currently available theories in approaching some of the problems at issue (Hawking’s primary concern).

 

Obviously, such complexities in themselves offer no grounds for a criticism of Penrose’s ideas concerning modern physics, especially relativity, which have elicited much admiration, as have Hawking’s ideas. One might want to be more cautious in evaluating certain aspects of Penrose’s presentation of his argument as limiting the scope of the debate in modern physics concerning key scientific and philosophical questions at issue, on which I shall further comment below. My point at the moment is double. First, general readers of Penrose’s work should be aware of the complexities of the scientific (rather than only philosophical) nature of Penrose’s argument concerning relativity and cosmology, and even more so of other scientific theories and questions he considers. Second, The Nature of Space and Time is especially indicative of this situation and is especially significant in exposing such complexities, more than is The Large, the Small, and the Human Mind.

 

For this and other reasons, of all the books mentioned here, The Nature of Space and Time may well be the most interesting and exciting one for scientists and, I would even argue, for general readers as well. The latter argument can, I think, be made in spite of the more technical and difficult, and at points prohibitive, nature of the book, but, to a degree, also because of that nature. A nonscientist may be unrecoverably frustrated even by such (by the book’s standards) benign elaborations as: “The No-Boundary Proposal (Hartle and Hawking): The path integral for quantum gravity should be taken over all compact Euclidean metrics. One can paraphrase this as ‘The Boundary Condition of the Universe Is That It Has No Boundary'” (The Nature of Space and Time, 79). The primary interest for general readers may be the very scene of the debate between two great scientific minds, rather than the scientific or even philosophical substance at stake, and the book is framed (not altogether justifiably) as a continuation of the Bohr-Einstein debate concerning quantum mechanics. An attempt to penetrate more deeply into the philosophical substance of the debate may, however, bring considerable rewards. Michael Atiyah (a great mathematician in his own right) describes the situation well in his foreword:

 

Although some of the presentation requires a technical understanding of the mathematics and physics, much of the argument is conducted at a higher (or deeper) level that will interest a broader audience. The reader will at least get an indication of the scope and subtlety of the ideas being discussed and of the enormous challenge of producing a coherent picture of the universe that takes full account of both gravitation and quantum physics. (viii)

 

First, then, by reading The Nature of Space and Timeone gets an indication of the scope and subtlety of the ideas being discussed, and, I would add, of their conceptual and metaphorical richness. This richness is one of the great philosophical or, one might even say, poetic achievements of modern mathematics and science. Penrose’s and Hawking’s ideas and their ways of thinking offer some superb examples here. I am thinking in particular of Hawking’s ideas concerning the “gluing” of spaces of different non-Euclidean geometries and curvatures in constructing his model or rather theory of the universe, or his ideas concerning more radical (than even in conventional quantum physics) non-causalities in quantum gravity (59-60, 103), or of Penrose’s own rich geometrical ideas, which inform and shape his books, and in many ways define his mathematical imagination.

 

It is, one could argue, this richness that connects modern science and modern–and postmodern–discourses in the humanities in the most interesting and significant ways. It is easy to understand from this perspective why, for example, Gilles Deleuze appeals to Riemann’s mathematical ideas at key junctures of his work. We are, however, far from having really approached what Riemann’s work and subsequent mathematics and science have to offer us by way of new concepts, metaphors, ways of thinking, and so forth. Arguably the most interesting and important challenge for the interdisciplinary studies involving science is to convey or indeed present this richness, on the one hand, and meaningfully to absorb as much of this richness as possible, on the other. Obviously, this traffic can proceed in both directions, and Penrose’s philosophical arguments could benefit from absorbing certain key recent developments in the humanities. One also could and should, at least at certain points, expect in the theoretical discourses in the humanities a level of complexity and even a certain technical specificity, and hence also difficulty, comparable to those of mathematics and science. Philosophical (rather than mathematical and scientific) complexity is sometimes lacking in Penrose’s books. Their main value in this context is instead in the possibilities they offer for this traffic by their presentation of mathematical and scientific ideas.

 

Equally important is Atiyah’s remark concerning “the enormous challenge of producing a coherent picture of the universe that takes full account of both gravitation and quantum physics.” First of all, this remark is, again, indicative of the speculative and tentative nature of Penrose’s ideas (and other ideas he considers) even as concerns relativity and, even more so, quantum physics, or, to a still greater degree, those concerning the biology of the brain, even in the scientific context. These scientific ideas, moreover, are the subject of much controversy and debate in these fields themselves. Secondly, equally significantly, not only key scientific but key philosophical ideas–such as those concerning physical reality and its mathematical nature, determinism, the question of truth in mathematics and mathematical logic, and so forth–are as much part of the debate within the scientific community itself as of the debate in the humanities (or the social sciences), or of the debate between both communities.

 

As we progress with Penrose from the classical world, including relativity, the world of the large (although it is no longer quite clear how classical this world really is), to quantum physics, to Gödel’s theorem, to the biology of the brain, or (all the more so) as we travel between them, the level of complexity, ambiguity, speculation, debate, and so forth increases. This is all the more so because, as I indicated at the outset, Penrose’s arguments, including his ultimate argument for the non-computability of the human mind and against artificial intelligence, depend not only on complex aspects of mathematical and physical theories themselves but on his interpretations of these theories. These interpretations are far from being broadly accepted (which Penrose acknowledges) and some of their aspects are rather idiosyncratic, and sometimes problematic. Penrose’s non-computability argument, however, irreducibly depends on these interpretations. While, to his credit, Penrose acknowledges such complexities and complications and considers some of them in his books, these books, I would argue, do not fully reveal to their readers the extent of these complexities and complications, and of the debates concerning them. There are, that is, levels of complexity surrounding Penrose’s ideas that are hidden (I am not saying deliberately) from an unprepared reader, and the debates incorporated in The Large, the Small, and the Human Mind or even in The Nature of Space and Time help only partially in this regard. Science itself appears to provide no conclusive evidence for most, if any, of Penrose’s key ideas, and indeed certain of his claims concerning scientific theories are questionable. One can argue that modern mathematics and science, especially quantum physics or post-Gödelian mathematical logic, provide more evidence against classical or traditional thinking (based on the concepts of reality, determinism, truth, knowledge, and so forth) than for it. Nor is there indeed any measurable consensus of opinion on these issues among the scientific or philosophical communities themselves. Whether one speaks of mathematics, physics, or biology (or indeed of consciousness and the mind), classical ideas and ideals are put into question by science itself, including even by what is seen as classical science. Modern science questions some of the same ideas as does some of the most radical postmodernist work in the humanities, and questions them just as radically.

 

To illustrate the argument just offered, I would like to consider one of Penrose’s key comments on quantum mechanics in The Large, the Small, and the Human Mind:

 

One of the things which people say about quantum mechanics is that it is fuzzy and indeterministic, but this is not true. So long as we remain at this level [of the quantum, small-scale behavior], quantum theory is deterministic and precise. In its most familiar form, quantum mechanics involves use of the equation known as Schrödinger's Equation which governs the behavior of the physical state of a quantum system--called its quantum state--and this is a deterministic equation.... Indeterminacy in quantum mechanics only arises when you perform what is called "making a measurement" and that involves magnifying an event from the quantum level to the classical level. (8)

 

One can indeed say that there is nothing fuzzy or imprecise about quantum mechanics in the sense that it is as precise and effective as any mathematical theory in the history of physics. The claim that it is deterministic is far more complicated, however, and may indeed be unacceptable, at least in this strong form–“this is not true.” There is certainly more disagreement with the view advocated by Penrose than this statement or Penrose’s overall treatment of the subject would suggest, even though he, again, indicates that his overall view of quantum physics is not widely accepted. One might argue that there is a degree of consensus that Schrödinger’s equation itself is mathematically deterministic. There is, however, hardly any consensus at all as to what, if anything, it is deterministic about. At best it may be deterministic about indeterminism–that is, in gauging the distribution of the randomness in quantum behavior, which behavior, it is true, is manifest only at the macro level of measurement. One cannot, however, infer from this fact, in the way Penrose appears to do, that the micro–quantum–behavior is physically deterministic on the basis of Schrödinger’s equation alone. This is a (metaphysical) assumption, not a logical inference on the basis of the available data of quantum physics. In Max Born’s elegant formulation: “The motion of particles follows the probability law but the probability itself propagates according to the law of causality” (cited in Pais, 258). Probabilities can be gauged in a reasonably deterministic manner, for example, by using Schrödinger’s equation. The process itself, however, is never fully predictable, and is constrained by Heisenberg’s uncertainty relations, which are inherent in Schrödinger’s equation as well. Indeed in any given case just about anything can happen. In this, quantum physics is very much like life, or chess. To cite Hawking’s comments in The Nature of Space and Time: “Einstein was wrong when he said, ‘God does not play dice.’ Consideration of black holes suggests, not only that God does play dice, but that he sometimes confuses us by throwing them where they can’t be seen” (26), and speaking of further indeterminacy that gravity may introduce: “It means the end of the hope of scientific determinism, that we could predict the future with certainty. It seems God still has a few tricks up his sleeve” (60).

 

A number of other examples of the kind just considered can be given here, in particular (still in his discussion of the quantum world) certain aspects of his interpretation of the Einstein-Podolsky-Rosen argument and Bell’s theorem, or some aspects of his interpretation of Gödel’s and Turing’s findings in mathematical logic. As with Penrose’s claim concerning quantum determinism, these examples are not random. They occur at crucial junctures of his overall argument concerning the human mind and artificial intelligence. I mention these examples even though my space does not allow me to consider them in the detail necessary to offer a fully rigorous critical argument. My aim, however, is not so much to criticize Penrose, but to indicate the broader (than Penrose himself suggests) scope of the hypothetical and the “prejudicial” in the landscape surveyed by his books.

 

I borrow the characterization “prejudicial” from Penrose himself, but give it a broader philosophical rather than negative meaning, as Penrose perhaps does as well. Penrose organizes his key philosophical “prejudices”–“that the entire physical world can in principle be described in terms of mathematics”; “that there are not mental objects floating around out there that are not based in physicality”; and “that, in our understanding of mathematics, in principle at least, any individual item in the Platonic world is accessible to our mentality, in some sense”–into a Penrose triangle of the Platonic, Physical, and Mental Worlds (96-97, 137-39). The Penrose triangle is arguably the most famous object which can be drawn so as to appear physically possible, but which cannot actually exist, and as such it was an inspiration for Escher’s famous drawings, which are often in turn used by Penrose. One finds a picture of the Penrose triangle in The Large, the Small, and The Human Mind (138). The title itself suggests (I think deliberately) a triangle and a Penrose triangle, similar to that of Platonic, Physical, and Mental Worlds. Both these triangles are in fact multiply connected into a kind of complex and perhaps ultimately impossible network. As I have pointed out, one of the main questions of the book (of all the books at issue here) is that of the possibility of bridging the hitherto unbridgeable; and the Penrose triangle is of course a very fitting figure in this context. While Penrose ultimately aims at, at least, some bridging, the metaphor itself inevitably suggests that at best one can only achieve an illusion of bridging, but can never actually implement it. Penrose is obviously aware of this, but I think that the broader space of the hypothetical and the prejudicial, as here considered, not only makes the figure of the Penrose triangle even more pertinent and poignant here, but also suggests a different implication of its use by Penrose. It suggests that each of the entities Penrose wants to bridge–whether large or small, human or inhuman–are themselves networks of real and Penrose triangles, or of much more complex figures or unfigures and networks of that type. This irreducible multiplicity might give us a better sense of the figures and of the unfigurability of the large, the small, and the human mind, and of the possible or impossible interconnections between them.

 

By the same token, however, this richer and more complex conceptual geometry also suggests that we may connect things that Penrose (perhaps) wants to separate, for example, the computable and the noncomputable, or the human and the nonhuman. As I have pointed out earlier, the “prejudice” against computational thinking has a very long history which extends from the pre-Socratics to Heidegger and beyond. I can only consider here one early event in this history, in which it is, fittingly, geometry that (as against both arithmetic and logic) appears to have been especially associated with non-computational and/as creative thinking–the thinking of mathematical and perhaps (at least for Plato) all philosophical discovery. It appears that ultimately Penrose takes a similar view as well, although he does, of course, argue for the ultimate non-computability of arithmetic as well in view of Gödel’s findings. My example is all the more fitting here since it has to do with the diagonal of the square. Just as it was the square where numerical computation was defeated by the Greeks, it was the square–now that of the chess board–where the latest defeat of the non-computational, the mind of Garry Kasparov, took place.

 

The diagonal of the square was both a great glory and a great problem, almost a scandal, in Greek mathematics and philosophy. For the diagonal and the side of a square were proved to be incommensurable, a discovery often attributed to Plato’s student Theaetetus. Their “ratio” is irrational, that is, it cannot be represented as a ratio of two whole numbers, and hence is not a rational number. This was the first example of such a number–what we now call the square root of, for example, two–a number that was proved to be unrepresentable as a ratio of two positive integers. It was an extraordinary and, at the time, shocking discovery, which was in part responsible for a crucial shift from arithmetics to geometry in mathematics and philosophy, since the diagonal is well within the limits of geometrical representation but outside those of arithmetical representation–as the Greeks conceived of it. To cite Maurice Blanchot:

 

The Greek experience, as we reconstitute it, accords special value to the "limit" and reemphasizes the long-recognized scandalousness of the irrational: the indecency of that which, in measurement, is immeasurable. (He who first discovered the incommensurability of the diagonal of the square perished; he drowned in a shipwreck, for he had met with a strange and utterly foreign death, in the nonplace bounded by absent frontiers). (103)

 

The Greeks, then, might have been more ambivalent about the relationships between geometrical and arithmetical, or logical, thinking (and their relation to computation and the non-computable) than is commonly thought, even though Plato or Socrates might have seen geometry as the greatest model of mathematical or even philosophical discovery. In closing his book Penrose relates (a bit too loosely) his philosophical triangle to the so-called cohomology theory, which is part of the field of algebraic topology:

 

You may ask, "Where is the impossibility [of the Penrose triangle]?" Can you locate it?....You cannot say that the impossibility is at any specific place in the picture--the impossibility is a feature of the whole structure. Nevertheless, there are precise mathematical ways in which you can talk about such things. This can be done in terms of breaking it apart, glueing it together and extracting certain abstract mathematical ideas from the detailed total pattern of glueings. The notion of cohomology is the appropriate notion in this case. This notion provides us with a means of calculating the degree of impossibility of this figure. (137-39, emphasis added)

 

The appeal to calculation in the end of a book that celebrates the incalculable and the non-computable could delight an early deconstructionist a couple of decades ago, and one finds the deconstruction of oppositions of that very type in the works of Derrida and de Man, among others. Penrose’s comment, however, can hardly be conceived as unselfconscious here. We must of course also be aware of the difference between calculation and computability. (Penrose, it should be noted, does not deny the significance of either). My point is that, by associating algebraic structures with topological ones, cohomology theory connects the often incalculable or even inconceivable geometry and topology (or indeed inconceivable algebra) to arithmetical and algebraic calculations and makes it possible to know something about the noncomputable and the (geometrically or otherwise) inconceivable. Mathematics may suggest to us a better model than we might be able to offer to mathematics. This model may be simultaneously both that of computation and that of noncomputability, or even of that which is neither one nor the other, and a sign of intelligence that is neither artificial (or otherwise inhuman) nor human, nor divine.

Works Cited

 

• Blanchot, Maurice. The Writing of the Disaster. Trans. Ann Smock. Lincoln: U of Nebraska P, 1983.
• Hawking, Stephen. A Brief History of Time: From the Big Bang Black Holes. New York: Bantam, 1988.
• Hawking, Stephen and Roger Penrose. The Nature of Space Time. Princeton, NJ.: Princeton UP, 1996.
• Pais, Abraham. Inward Bound: Of Matter and Forces in Physical World. Oxford: Oxford UP, 1986.
• Penrose, Roger. The Emperor’s New Mind: Concerning Minds and the Laws of Physics. Oxford: Oxford UP, 1989.
• —. The Large, the Small and the Human Mind. Cambridge UP, 1997.
• —. Shadows of the Mind: A Search for the Science of Consciousness of Consciousness. Oxford: Oxford UP, 1994.