
Not knowably sound

Sir Roger Penrose is unique in offering something close to a proof in formal logic that minds are not merely computers. There is a kind of piquant appeal in an argument against the power of formal symbolic systems which is itself clothed largely in formal symbolic terms. Although it is this 'mathematical' argument, based on the famous proof by Gödel of the incompleteness of arithmetic, which has attracted the greatest attention, an important part of Penrose's theory is provided by positive speculations about how consciousness might really work. He thinks that consciousness may depend on a new kind of quantum physics which we don't, as yet, have a theory for, and suggests that the microtubules within brain cells might be the place where the crucial events take place. I think it must be admitted that his negative case against computationalism is much stronger than these positive theories.
Besides the direct arguments about consciousness, Penrose's two books on the subject feature excellent and highly readable passages on fractals, tiling the plane, and many other topics. At times, it must be admitted, the relevance of some of these digressions is not obvious  I'm still not convinced that the Mandelbrot Set has anything to do with consciousness, for example  but they are all fascinating and remarkably lucid pieces in their own right. 'The Emperor's New Mind' is particularly wideranging, and would be well worth reading even if you weren't especially interested in consciousness, while a large part of 'Shadows of the Mind' is somewhat harder going, and focuses on a particular argument which purports to establish that "Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth". 

I like the books myself, mostly, but I don't find them convincing. Of course, people find a lengthy formal argument intimidating, especially from someone of Penrose's acknowledged eminence. But does anyone seriously think this kind of highly abstract reasoning can tell us anything real about how things actually work?


You don't think maths tells us anything about the real world then? Well, let's start with the Gödelian argument, anyway. Gödel proved the incompleteness of arithmetic, that is, that there are true statements in arithmetic which can never be proved arithmetically. Actually, the proof goes much wider than that. He provides a way of generating a statement, in any formal algebraic system, which we can see is true, but which cannot be proved within the system. Penrose's point is that any mechanical, algorithmic, process is based on a formal system of some kind. So there will always be some truths that computers can't prove  but which human beings can see are true! So human thought can't be just the running of an algorithm. 

These unprovable truths are completely uninteresting ones, of course: the sort of thing Gödel produces are arid selfreferential statements of no wider relevance. But in any case, the doctrine that people can always see the truth of any such Gödel statement is a mere assertion. In the simple cases Penrose considers, of course human beings can see the truth of the statements, but there's no proof that the same goes for more complex ones. If we actually defined the formal system which brains are running on, I believe we might well find that the Gödel statement for that system really was beyond the power of brains to grasp. 

I don't think that that could ever happen  it just doesn't work like that. The complexity of the system in question isn't really a factor. And in any case, brains are not 'running on' formal systems! 

Oh, but they have to be! I'm not suggesting the 'program' for any given brain is simple, but I can see three ways we could in principle construct it.
One. If we list all the sensory impressions and all the instructions to act that go into or out of a brain during a lifetime, we can treat them as inputs and outputs. Now there just must be some function, some algorithm, which produces exactly those outputs for those inputs. If nothing simpler is available (I'm sure it would be) there is always the algorithm which just lists the inputs to date and says 'given these inputs, give this output'.
Two. If you don't like that approach, I reckon the way neurons work is sufficiently clear for us to construct a complete neuronal model of a brain (in principle  I'm not saying it's a practical proposition); and then that would clearly represent an implementation of a complex function for the person in question.
Three. As a last resort, we just model the whole brain in excruciating detail. It's a physical object, and obeys the normal laws of physics, so we can construct a mechanical description of how it works.
Any of these will do. The algorithms we come up with might well be huge and unwieldy, but they exist, which is all that matters. So we must be able to apply Gödel to people, too. 

Nonsense! For a start, I don't believe 'inputs' and 'outputs' to human beings can be defined in those terms  reality is not digital. But the whole notion of a person's own algorithm is absurd! The point about computers is that their algorithms are defined by a programmer and kept in a recognised place, clearly distinguished from data, inputs, and hardware, so it's easy to say what they are in advance. With a brain, there is nothing you can point to in advance as the 'brain algorithm'. If you insist on interpreting the brain as running an algorithm, you just have to wait and see which bits of the brain and which bits of the rest of the person and their environment turn out to be relevant to their 'outputs' in what ways and then construct the algorithm to suit. We can never know what the total algorithm is until all the inputs and outputs have been dealt with. In short, it turns out not to be surprising that a person can't see the truth of their own Gödel statement, because they have to dead before anyone can even decide what it is! 

Alright, well look at it this way. We're only talking about things that can't be proved within a particular formal system. Humans can see the truth of these statements, and even prove them, because they go outside the formal system to do so. There's no real reason why a computer can't do the same. It may operate one algorithm to begin with, but it can learn and develop more comprehensive algorithms for itself as it goes. Why not? 

That's the whole point! Human beings can always find a new way of looking at something, but an algorithm can't. You can't have an algorithm which generates new algorithms for itself, because if it did, the new bits would by definition be part of the original algorithm. 

I think it must be clear to anyone by now that you're just playing with words. I still say that all this is simply too esoteric to have any bearing on what is essentially a practical computing problem. If I understand them correctly, both Dennett and your friend Searle agree with me (in their different ways). The algorithms in practical AI applications aren't about mathematical proof, they're about doing stuff. 

I was puzzled by Dennett's argument in 'Darwin's dangerous idea' in particular. He's quite dismissive about the whole thing, but what he seems to say is this. The narrow set of algorithms picked out by Penrose may not be able to provide an arithmetical proof, but what about all the others which Penrose has excluded from consideration? This is strange, because the ones excluded from consideration, according to Dennett, are: algorithms which don't do anything at all; algorithms which aren't interesting; algorithms which aren't about arithmetic; algorithms which don't produce proofs; and algorithms which aren't consistent! Can we reasonably expect proofs from any of these? Maybe not, says Dennett, but some of them might play a good game of chess... This seems to miss the point to me.
What I fear is that this kind of reasoning leads to what I call the Roboteer's argument (I've seen it put forward by people like Kevin Warwick and Rodney Brooks). The Roboteer says, OK, so computers will never work the way the human brain works. So what? That doesn't mean they can't be intelligent and it doesn't mean they can't be conscious. Planes don't fly the way birds do, but we don't say it isn't proper flight because of that... 

Personally, I don't see anything wrong with that argument. What about this quantum malarkey? You're not going to tell me you go along with that? There is absolutely no reason to think quantum physics has anything to do with this. It may be hard to understand, but it's just as calculable and deterministic as any other kind of physics. All there really is to this is that both consciousness and quantum physics seem a bit spooky. 

It isn't conventional, established quantum physics we're talking about. Having established that human thought goes beyond the algorithmic, Penrose needs to find a noncomputable process which can account for it; but he doesn't see anything in normal physics which fits the bill. He wants the explanation to be part of physics  you ought to sympathise with that  so it has to be in a new physical theory, and new quantum physics is the best candidate. Further strength is given to the case by the ideas Stuart Hameroff and he have come up with about how it might actually work, using the microtubules which are present in the structure of nerve cells. 

They're present in most other kinds of cell, too, if I understand correctly. Microtubules have perfectly ordinary jobs to do within cells which have nothing to do with thinking. We don't understand the brain completely, but surely we know by now that neurons are the things that do the basic work. 

It isn't quite as clear as that. There has been a tendency, right since the famous McCulloch and Pitts paper of 1947, to see neurons as simple switches, but the more we know about them the less plausible that seems. Actually there is some highly complex chemistry involved. Personally, I would also say that the way neurons are organised looks very much like the sort of thing you might construct if you wanted to catch and amplify the effects of very smallscale events. One molecule  in the eye, one quantum, as Penrose points out  can make a neuron fire, and that can lead to a whole chain of other firings. 

At the end of the day, the problem is that quantum physics just doesn't help. It doesn't give us any explanatory resources we couldn't get from normal physics. 

That's too sweeping. There are actually several reasons, in my view, to think that quantum physics might be relevant to consciousness (although these are not Penrose's reasons). One is that the way two different states of affairs can apparently be held in suspense resembles the way two different courses of action can be suspended in the mind during the act of choice. A related point is the possibility that exploiting this kind of suspension could give us spectacularly fast computing, which might explain some of the remarkable properties of the brain. Another is the special role of observation  becoming conscious of things  in causing the collapse of the wavefunction. A third is that quantum physics puts some limits on how precisely we can specify the details of the world, which seems to militate against the kind of argument you were making earlier, about modelling the brain in total detail. I know all of these are open to strong objections: the real reason, as I've already said, is just that quantum physics is the most likely place to find the kind of new science which Penrose thinks is needed. 

I don't see it. It seems to me inevitable that any new physics that may come along is going to be amenable to simulation on a computer  if it wasn't, it hardly seems possible it could be clear enough to be a reasonable theory. 

In other words, your mind is closed to any possibility except computationalism. Consciousness seems to me to be such an important phenomenon that I simply cannot believe it is something just 'accidentally' conjured up by a complicated computation... 

Read: 
"The Emperor's New Mind" An essential text on the consciousness question, and contains a great deal of fascinating stuff. The foreword is by the late great Martin Gardner, and fans of his will find a similar combination of deep thought with clear and entertaining exposition here. 
"Shadows of the Mind " Similar ground is covered here, although in this case the details of Penrose's argument about the nonalgorithmic nature of mathematical thought is uppermost, and hence the going is just a little harder. Still a very readable treatment of subjects which inevitably require a degree of thought and concentration to assimilate. 

Some Links: 
Biographical details  from the University of St Andrews site 
Critical paper and response  a critical paper by Patricia Churchland with a response from Penrose and Hameroff. 
Psyche  Symposium on 'Shadows of the Mind' in the online journal. 


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