David Deutch - The Fabric of Reality

So explanation does, after all, play the same paramount role in pure mathematics as it does in science. Explaining and understanding the world — the physical world and the world of mathematical abstractions — is in both cases the object of the exercise. Proof and observation are merely means by which we check our explanations. Roger Penrose has drawn a further, radical and very Platonic lesson from Gödel’s results. Like Plato, Penrose is fascinated by the ability of the human mind to grasp the abstract certainties of mathematics. Unlike Plato, Penrose does not believe in the supernatural, and takes it for granted that the brain is part of, and has access only to, the natural world. So the problem is even more acute for him than it was for Plato: how can the fuzzy, unreliable physical world deliver mathematical certainties to a fuzzy, unreliable part of itself such as a mathematician? In particular, Penrose wonders how we can possibly perceive the infallibility of new, valid forms of proof, of which Gödel assures us there is an unlimited supply.

Penrose is still working on a detailed answer, but he does claim that the very existence of this sort of open-ended mathematical intuition is fundamentally incompatible with the existing structure of physics, and in particular that it is incompatible with the Turing principle. His argument, in summary, runs as follows. If the Turing principle is true, then we can consider the brain (like any other object) to be a computer executing a particular program. The brain’s interactions with the environment constitute the inputs and outputs of the program. Now consider a mathematician in the act of deciding whether some newly proposed type of proof is valid or not. Making such a decision is tantamount to executing a proof-validating computer program within the mathematician’s brain. Such a program embodies a set of Hilbertian rules of inference which, according to Gödel’s theorem, cannot possibly be complete. Moreover, as I have said, Gödel provides a way of constructing, and proving, a true proposition which those rules can never recognize as proven. Therefore the mathematician, whose mind is effectively a computer applying those rules, can never recognize the proposition as proven either. Penrose then proposes to show the proposition, and Gödel’s method of proving it to be true, to that very mathematician. The mathematician understands the proof. It is, after all, self-evidently valid, so the mathematician can presumably see that it is valid. But that would contradict Gödel’s theorem. Therefore there must be a false assumption somewhere in the argument, and Penrose thinks that the false assumption is the Turing principle.

Most computer scientists do not agree with Penrose that the Turing principle is the weakest link in his story. They would say that the mathematician in the story would indeed be unable to recognize the Gödelian proposition as proven. It may seem odd that a mathematician should suddenly become unable to comprehend a self-evident proof. But look at this proposition:

David Deutsch cannot consistently judge this statement to be true.

I am trying as hard as I can, but I cannot consistently judge it to be true. For if I did, I would be judging that I cannot judge it to be true, and would be contradicting myself. But you can see that it is true, can’t you? This shows it is at least possible for a proposition to be unfathomable to one person yet self-evidently true to everyone else.

Anyway, Penrose hopes for a new, fundamental theory of physics replacing both quantum theory and the general theory of relativity. It would make new, testable predictions, though it would of course agree with quantum theory and relativity for all existing observations. (There are no known experimental counter-examples to those theories.) However, Penrose’s world is fundamentally very different from what existing physics describes. Its basic fabric of reality is what we call the world of mathematical abstractions. In this respect Penrose, whose reality includes all mathematical abstractions, but perhaps not all abstractions (like honour and justice), is somewhere between Plato and Pythagoras. What we call the physical world is, to him, fully real (another difference from Plato), but is somehow part of, or emergent from, mathematics itself. Furthermore, there is no universality; in particular, there is no machine that can render all possible human thought processes. Nevertheless, the world (especially, of course, its mathematical substrate) is still comprehensible. Its comprehensibility is ensured not by the universality of computation, but by a phenomenon quite new to physics (though not to Plato): mathematical entities impinge directly on the human brain, via physical processes yet to be discovered. In this way the brain, according to Penrose, does not do mathematics solely by reference to what we currently call the physical world. It has direct access to a Platonic reality of mathematical Forms, and can perceive mathematical truths there with (blunders aside) absolute certainty.

It is often suggested that the brain may be a quantum computer, and that its intuitions, consciousness and problem-solving abilities might depend on quantum computations. This could be so, but I know of no evidence and no convincing argument that it is so. My bet is that the brain, considered as a computer, is a classical one. But that issue is independent of Penrose’s ideas. He is not arguing that the brain is a new sort of universal computer, differing from the universal quantum computer by having a larger repertoire of computations made possible by new, post-quantum physics. He is arguing for a new physics that will not support computational universality, so that under his new theory it will not be possible to construe some of the actions of the brain as computations at all.

I must admit that I cannot conceive of such a theory. However, fundamental breakthroughs do tend to be hard to conceive of before they occur. Naturally, it is hard to judge Penrose’s theory before he succeeds in formulating it fully. If a theory with the properties he hopes for does eventually supersede quantum theory or general relativity, or both, whether through experimental testing or by providing a deeper level of explanation, then every reasonable person would want to adopt it. And then we would embark on the adventure of comprehending the new world-view that the theory’s explanatory structures would compel us to adopt. It is likely that this would be a very different world-view from the one I am presenting in this book. However, even if all this came to pass, I am nevertheless at a loss to see how the theory’s original motivation, that of explaining our ability to grasp new mathematical proofs, could possibly be satisfied. The fact would remain that, now and throughout history, great mathematicians have had different, conflicting intuitions about the validity of various methods of proof. So even if it is true that an absolute, physico-mathematical reality feeds its truths directly into our brains to create mathematical intuitions, mathematicians are not always able to distinguish those intuitions from other, mistaken intuitions and ideas. There is, unfortunately, no bell that rings, or light that flashes, when we are comprehending a truly valid proof. We might sometimes feel such a flash, at a ‘eureka’ moment — and nevertheless be mistaken. And even if the theory predicted that there is some previously unnoticed physical indicator accompanying true intuitions (this is getting extremely implausible now), we should certainly find it useful, but that would still not amount to a proof that the indicator works. Nothing could prove that an even better physical theory would not one day supersede Penrose’s, and reveal that the supposed indicator was unreliable after all, and some other indicator was better. Thus, even if we make every possible concession to Penrose’s proposal, if we imagine it is true and view the world entirely in its terms, it still does not help us to explain the alleged certainty of the knowledge that we acquire by doing mathematics.