STEPHEN W. HAWKING: Although I'm regarded as a dangerous radical by particle physicists for proposing that there may be loss of quantum coherence, I'm definitely a conservative compared to Roger. I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation. I think Roger is a Platonist at heart but he must answer for himself. [pp.3-4]ROGER PENROSE: At the beginning of this debate Stephen said that he thinks that he is a positivist, whereas I am a Platonist. I am happy with him being a positivist, but I think that the crucial point here is, rather, that I am a realist. Also, if one compares this debate with the famous debate of Bohr and Einstein, some seventy years ago, I should think that Stephen plays the role of Bohr, whereas I play Einstein's role! For Einstein argued that there should exist something like a real world, not necessarily represented by a wave function, whereas Bohr stressed that the wave function doesn't describe a "real" microworld but only "knowledge" that is useful for making predictions. [pp.134-135]

Stephen Hawking and Roger Penrose,

The Nature of Space and Time[Princeton University Press, 1996].

*The Emperor's New Mind* by Roger Penrose may be the best book about modern science yet written. The range of issues addressed by Penrose is vast, from Relativity and quantum mechanics, to many questions about mathematics, and ultimately to important questions about Artificial Intelligence; and Penrose's authority as one of the greatest living mathematicians to address these things is unique. Penrose's thesis, therefore, that Artificial Intelligence through computers, as presently constructed, cannot in principle duplicate the workings of the human brain, and his argument that Einstein was not wrong to express grave philosophical doubts about quantum mechanics, are given a weight that they would not have if merely some philosopher, or anyone else less intimately involved with the mathematical and theoretical underpinnings of physics, had expressed similar views.

The Emperor's "new clothes," of course, were *no* clothes. The Emperor's "New Mind," we then suspect, is nothing of the sort as well. That computers as presently constructed cannot possibly duplicate the workings of the brain is argued by Penrose in these terms: that all digital computers now operate according to *algorithms*, rules which the computer follows step by step. However, there are plenty of things in mathematics that *cannot* be calculated algorithmically. We can discover them and know them to be true, but clearly we are using some devices of calculation ("insight") that are not algorithmic and that are so far not well understood -- certainly not well enough understood to have computers do them instead. This simple argument is devastating.

Along the way, Penrose has an interesting take on Gödel's Proof of the incompleteness of mathematics. He notes (pp. 105-108) that if David Hilbert were right and all of mathematics could be completely reduced to a formal syntactic system, then mathematics need have no *meaning* -- "true" and "false" would simply mean "derivable" and "non-derivable" in the formalism of the system. Hilbert himself recognized this and had said that mathematical terms could mean "beer steins, sausages, and tables" instead of what they are interpreted to mean mathematically (obviously, Hilbert spent some time in German beer gardens). Instead, Gödel demonstrated that in any formal syntactic system there will be propositions that are *true* but not through formal derivation from the axioms of the system. Thus, Penrose notes, they are true because of their *meaning*, not because of their syntax relation to an axiomatic system. This reinforces the thesis of Jerrold Katz, that syntactic simples are not semantic simples, and so some truths will depend on semantic contents that cannot be exhaustively expressed as syntax.

Philosophical implications are also evident in Penrose's evaluation of the difficulties of quantum mechanics. The principal conceptual difficulty is that reality existing in a unique and determined state always applies to the observer and the observer's instruments but only applies to other external objects *after* they have been observed. Thus, Schrödinger's Cat is both alive and dead at the same time, until the box is opened and the cat is observed. This was Schrödinger's own *reductio ad absurdum* of quantum mechanics, a feature not always noted in triumphalist treatments of the subject, since it raises the question *who counts* as an observer. A cat would seem to be a sophisticated enough being to count as an observer, or does it? And if it doesn't, why do we? And if a cat does, how about a mouse? A grasshopper? A bacterium? What is really the principle for making the distinction? It is clear that there isn't one (except just between "us" and "them"), and Penrose examines different possibilities, none of which seems entirely satisfactory.

Reality as it exists apart from observation is the deterministic expression of Schrödinger's equation for the Wave Function. But at the same time, the Wave Function is interpreted by Heisenberg and Bohr as the sum of all possible states (Richard Feynman's all possible histories) of the physical system. Observation collapses the wave function into one actual and unique state, but the transition is indeterministic -- any one of the possibilities may become actual. This duality between possibilities and observation, however, sounds like nothing so much as Kant's distinction between things-in-themselves apart from experience and the act of synthesis that brings sensible intuition into consciousness. Kantian synthesis is, indeed, the act by which a conscious observer experiences the world. Kant never thought of things-in-themselves as the sum of all possible histories, but that is because he didn't know how we could think of them at all. Quantum mechanics serendipitously supplies a clue.

At the same time, Kant's theory that the mind applies the forms of space and time to intuition in the process of bringing it into consciousness *does* seem to have a parallel in quantum mechanics. Thanks to John Bell, the Einstein-Podolsky-Paradox has been demonstrated to be true: that the Wave Function collapses *instantaneously*, even across cosmological distances, which violates the postulate of Special Relativity that nothing can travel faster than the velocity of light. Quantum mechanical effects are therefore "non-local," i.e. they ignore space, but only, of course, *until* the Wave Function collapses: after that, we are back in Einstein's universe again. The only theory in the history either of philosophy or science that would explain non-locality is Kant's theory of the transcendental ideality of space.

Penrose, however, does not notice that possibility. Instead, he seems to labor under traditional misunderstandings of Kant's theory of space and geometry. Thus, Penrose's mention of Kant's theory on page 158 leaves something to be desired. Penrose says that if the postulates of Euclid are not *logically* "self-evident" tautologies, then what does it mean to say that they are *a priori*? It means, of course, that they are true independent of experience. Kant holds that space and time are *a priori* "forms of intuition" and that the postulates of Euclid are based on them. This means that while non-Euclidean geometries can be *conceived*, they cannot be *imagined* or *visualized*. The difference between the abstract *conception* of something, which is always possible if logically consistent, and the imaginative *visualization* of something, is a point that trips up the reading of Kant by many, whether by philosophers, scientists, or others.

A key traditional misunderstanding is that the very *existence* of non-Euclidean geometry refutes Kant's theory of mathematics. What is often seen stated, e.g. by the great French mathematician Poincaré, is that non-Euclidean geometry is *impossible* if the postulates of geometry are synthetic *a priori* propositions. However, it was recognized by Leonard Nelson that Kant's theory in fact allows for a *prediction* of the existence of non-Euclidean geometry. That is a big difference. The confusion occurs because people forget the basic *definition* of "synthetic": that *any* synthetic proposition can be denied without contradiction and thus that the *contradictory* of any synthetic proposition is *conceivable*, just as Hume would have said that the contradictory of any "matter of fact" is conceivable. That is true of *any* synthetic proposition, whether *a priori* or *a posteriori*. But, if the postulates of Euclid are axiomatically independent, and if the contradictories of the postulates of Euclid are conceivable and involve no contradiction, then a non-Euclidean geometry built with them would be just as consistent as that of Euclid. The construction of non-Euclidean geometries thus *vindicated* Kant rather than refuted him. It was Hume and Hegel, who thought that the postulates were analytic, who were refuted.

In a personal communication with the author, Penrose himself contended that what can be imagined or visualized in geometry is just a matter of experience and practice. This did not mean that Penrose could actually visualize a proper Lobachevskian space or a four-dimensional Euclidean space, just that he expected that this would be possible as we get used to these things in the future. But the matter does not seem to be so subjective. It is easy for us to deceive ourselves that we imagine non-traditional geometries when we are really only *conceiving* of them and in fact are visualizing *projections* or *models* of them in three-dimensional Euclidean space. Escher's "depiction" of Lobachevskian space in Penrose's book, or any such depiction of a non-Euclidean space, seems to involve the use of curved Euclidean lines to represent "straight" non-Euclidean lines (as in the triangle on page 156) or results in uniform or congruent non-Euclidean figures becoming distorted in shape or in size. None of these models tells us how we are to draw a fourth Cartesian coordinate that meets at 90 degrees with all of the other three, or how we are to extend truly straight lines that start off parallel and then will run into each other (as on the surface of a sphere, or in a Riemannian space).

All we need is one counterexample to prove Penrose's claim, but it sounds as though he *expects* one, not that he *has* one. Penrose cites [in private communication again] Tom Banchoff's computer graphics in the fascinating video production *Not Knot* [Geometry Center, University of Minnesota, distributed by Jones and Bartlett Publishers, Inc. 20 Park Plaza, suite 1435, Boston, MA 02116] as examples of the visualization of non-Euclidean geometry. Piet Hutt, of the Institute for Advanced Study in Princeton, made a similar statement to the author. But, as Penrose admits himself, the Lobachevskian, non-Euclidean space of *Not Knot* is represented in *projection* on a two-dimensional screen. Whether one can become familiar with those projections and then make the leap to the real thing is just the issue. I have a little model in my office, a 3 dimensional model projection of a hyper-cube, which would be the 4 dimensional analogue of a cube; and I occasionally spend some time looking at the thing, which would unfold into eight cubes in 4 dimensions. It can give me an eerie feeling, but I still haven't popped those extra cubes out into the fourth dimension (as happens in Robert Heinlein's short story, "There Was a Crooked House"). Since various people have been trying to do this for over a century, I begin to wonder if my, and other's, inability is a mere limitation in our experience. It sounds like even new-born infants have more of a sense of space than we would expect if their spatial sense was learned. In fact, all the images we use from our retinas are two-dimensional: It would be nice if we could see solid objects from all directions at once. We seem to have trouble even imagining that.

Some interesting statements in this vein occur in a book called *The Matter Myth*, by Paul Davies (a physicist in Australia) and John Gribbin (a writer who was trained in astrophysics at Cambridge). For instance, Davies and Gribbin say on pages 110-111:

I believe that the reality exposed by modern physics is fundamentally alien to the human mind, and defies all power of direct visualization...The realization that not everything that is so in the world can be grasped by the human imagination is tremendously liberating...

Eddington's implicit boast of being the only person other than Einstein able to understand the general theory of relativity did not mean, I believe, that he and Einstein alone could visualize the revolutionary new concepts such as curved spacetime. But he may well have been among the first physicists to appreciate that in this subject true understanding comes only by

relinquishingthe need to visualize.

If the reality grasped by modern physics is "fundamentally alien to the human mind," then it is not clear how "true understanding" could ever be possible! However, Davies and Gribbin seem to be doing the very thing that Kant had allowed for: that something could be abstractly *understood* through reason (Euclidean postulates can be denied without creating a contradiction) without our being able to supply an *object* from our imagination that would correspond to it.

Adapting Kant to quantum mechanics and Relativity requires a couple of modifications: First, that things-in-themselves be seen in terms of the Wave Function, as the sum of all possible histories that would exist apart from observation; and second, that the real physical space of phenomenal objects is not necessarily the space that we are able to imaginatively visualize (as discussed in The Ontology and Cosmology of Non-Euclidean Geometry). Making both of these modifications together would require that space neither exists among things-in-themselves as such nor is merely something imposed idiosyncratically by the human brain. This would require that the nature of consciousness, with its attendant phenomenal objects, be part of the structure of reality and not just a psychologistic adaptation in the human species. The nature of Kant's metaphysics is certainly open to that possibility, even if it was a question that he himself did not approach asking. Friesian theory is a bit closer, although the Friesian theory of the *knowledge* contained in human reason is commonly misunderstood, even by Karl Popper, as merely psychologistic, which of course it is not. But Friesian theory itself might need to make a distinction between the true knowledge *that* space exists in phenomenal objects, and a merely *human* limitation in how that space can be visualized.

Although Penrose's *The Emperor's New Mind* does not get us so far down the road philosophically as to apply Kant to things like quantum mechanics, his exploration is so much more serious philosophically than almost anything that has been done since Einstein and Schrödinger that the next steps seem clearly suggested. No more need be asked from any book about science and mathematics.