An Expedient Mind: ARGUMENTS - PENROSE (The Emperor's New Mind)

Tartan Hen Publications : Home | more books | Contact : feedback@tartanhen.co.uk

PENROSE’S IMPERIAL CLOTHING

Reference:

(ENM) The Emperor’s New Mind - Roger Penrose (Oxford Univ. Press 1989)

(SOTM) Shadows of the Mind - Rogor Penrose (Vintage 1995)

Professor Roger Penrose is a very erudite man - Rouse Ball Professor of Mathematics at Oxford and an established expert of international renown on many esoteric branches of the subject. As he reminds us several times throughout the book, he worked with Stephen Hawking on black holes and other exotic cosmological concepts. So it is, I guess, a brave, or a very foolish person, who would do battle with this intellectual Goliath. I am not sure into which category I should place myself.
According to the foreword, Penrose’s book The Emperor’s New Mind is destined to become a classic. It is also - “... the most powerful attack yet written on strong AI. Objections have been raised in past centuries to the reductionist claim that a mind is a machine operated by the laws of physics, but Penrose’s offensive is more persuasive because it draws on information not available to earlier writers.”

According to the Times, Shadows of the Mind is “One of the most important works of the second half of the 20th Century”.

I will tackle ENM first.
I confess that I found the book both fascinating and objectionable. It is fascinating because in it, Penrose provides a kind of Cook’s Tour of some very obscure mathematical and scientific ideas, all of them explained with an admirable authority.
However, the book is also objectionable, and for several reasons.
Most of the complex ideas discussed, have nothing at all to do with main subject. They serve another, more devious purpose. They overwhelm the reader with Penrose’s prodigious academic erudition. And they pad out the text so that the crucial parts, which are relevant to the main subject, are spaced well apart.
The reader, who wants to understand in detail the logic of his argument, is forced to follow a large number of cross references. Those who take the trouble do so, will, I fear, be disappointed. Most of these references turn out to be chained, with one reference leading to another and another and another. And like an expedition to find one of the magnetic poles, one finds that the compass, which points the way when one still has some distance to go, becomes useless as the crucial spot is approached.
The thrust of Penrose’s attack is this - he wants to prove that the human mind is able to do things which are beyond the reach of any artificial machine. The particular mental activity which he has chosen to serve as an example, is what he calls “mathematical insight”. The reason why he takes us through that extended tour of obscure mathematical concepts, is to prove, beyond doubt, that Penrose himself has lots and lots of this mathematical insight. The point, however, is this - can he prove that mathematical insight is not algorithmic?
The strategy Penrose adopts is one which has become commonplace in logic. If you cannot prove directly that X is true, then try to prove that NOT(X) is impossible. It is a perfectly valid approach, and had Penrose been able to do that, quite a large amount of academic work in the AI field would have come to a sudden halt. However, he did not succeed. What he did manage to do, was to muddy the waters sufficiently to convince the unwary. The task I have set myself, is the clarification of those waters. To do that, I must unpick his argument, piece by piece, cross reference by cross reference, and show, that at its heart, his apparent “proof” is no more than an assertion of opinion. At least in one case, it is also based on an unwarranted assumption.

ON ALGORITHMS
Godel’s Theorem is one of the most abstruse bits of mathematical logic ever devised. Even so, the end result of Godel’s theorem, is not too inaccessible to the non-expert, once the basic idea of a logical system has been grasped.
A logical system consists of a set of axioms (statements which are defined to be absolutely true) and a set of rules of inference, which indicate what can be deduced from true statements. Anything which can be deduced from the axioms, using those rules of inference, is also true and is called “a theorem” of the system.

That is a rather loose description which uses the term "true" in a colloquial sense. But in formal logic, there is a subtle distinction to be made between what is "true" and what is "provable". Strict logic does not allow us to assume that these two concepts are identical. There are actually quite complicated theorems which prove that a statement which is provable, is also true (in most logical systems). But the question which Godel's theorem addresses is the converse - is it the case that what is true will always be provably true. And the rather surprising answer is NO.
The axioms and the rules of inference are arbitrary. The axioms consist of nothing more than strings of meaningless symbols. The rules of inference are re-write rules with the form - “If you have a string, or a set of strings with this pattern, you may re-write it as that pattern”. The techniques of proof are, therefore, divorced from any kind of reality.
And that, it can be claimed, is the strength of logic. It means that when we are proving that something is true, we are concerned only with the form of argument, not its content. Emotional attachment to some result cannot influence the outcome. When we are dealing with every day problems, we can assign meanings or interpretations to the symbols, in order to find out what the implications are in reality.

The rationale is similar to the way we can prove geometrical relationships concerning such arbitrary shapes as triangles and circles without bothering ourselves what these shapes actually represent. The results can later be applied in many engineering projects. That circle might be a wheel or a circular building but the relationships worked out on the arbitrary shapes, will hold in all cases.
Another important thing about a logical system, is that it is a closed system. Once the axioms and rules have been defined, no other extraneous facts or methods of deduction are allowed to creep in unnoticed.
We can talk about a logical system from two standpoints. We can stand outside the system and discuss its form, what axioms we might have included, the kind of thing which cannot be proven by the rules of inference, and so on. But if we step inside the system, then the only form of discussion, which is permissible, has to be couched in terms of those axioms and those rules of inference. Nothing else. If you drag additional facts, or ways of thinking into the argument, then you have broken the rules and the results obtained cannot then be applied with confidence in any real-life context.
Anyway, what Godel did, was to prove that it is possible to make a statement WITHIN a logical system, which cannot be proven true WITHIN the system, while at the same time we know that statement is TRUE for reasons which lie OUTSIDE the system. “True, but not provably true” would be one way to summarise the situation.
The kind of statement, which Godel used to make this proof, was a complicated self-referent statement. In a rather round-about way, the statement asked a question about the proof of its own truth-value. We will call that kind of statement a “Godel-statement” (abbreviated to “GS”). What it says is (in effect)

GS: “There is no way to prove the truth of this statement within this system.”

Those final words “within this system” are very important.
If what that statement (GS) says is false, then there must actually be a way to prove it that what GS says is true. But it cannot be true and false at the same time. Therefore, the statement cannot be false. It must, therefore, be the case that there is indeed no way to prove that what the statement says is true WITHIN THE SYSTEM.
If you ask “In that case, how come we have just proven it to be true?”
The answer is that we did not do that WITHIN THE SYSTEM. For the purposes of that proof we stepped OUT OF THE SYSTEM. Not only did we make use of axioms which were not available to us when we were inside the system but, because we were outside the system, we escaped from the self-reference loop, which generated that logical contradiction.
The significance of Godel’s theorem is usually lost on non-mathematicians but it is widely regarded by mathematicians as fundamental. It pulls the rug out from underneath some of the fondly held beliefs which mathematicians once held about the nature of their discipline.

MATHEMATICAL INSIGHT
Penrose, however, has introduced Godel’s Theorem into his argument for another reason. It is an established fact, that any proof of a theorem which operates WITHIN a logical system, must be algorithmic. It consists entirely of a sequence of those re-write rules applied to the axioms and theorems of the system. However, the other proof, the one used by Godel to prove the truth of GS, comes from OUTSIDE the system. It makes use of unspecified axioms. It just “seems” to be okay. It seems to depend upon “intuition”, or, as Penrose puts it - it depends on “mathematical insight”. So the question which arises is this - is this “mathematical insight” the non-algorithmic process which Penrose seeks? Penrose actually goes a bit further than that. What he is talking about is the process by which we devised this OUTSIDE proof method and saw that it is a valid way to proceed.
The short answer is, that we just do not know. I have a feeling that if I was able to re-produce any of these OUTSIDE methods as an algorithm, that Penrose would reply “but how were you able to think of that”, and thus take the argument a step further into the obscure. For the present, all we know is that the other proof methods, the ones which are INSIDE the logic system, are algorithmic. What lies OUTSIDE, lies in the realm of the unknown.
But what we most definitely CANNOT say, is that the process which lies outside the system, MUST be non-algorithmic. That would be an unwarranted assumption. In logic, it is a schoolboy howler. Let me call it “H” (for “howler”).

H: If we cannot prove X then NOT(X) must be true.

Pleased note too, that H is a direct contradiction of Godel’s Theorem.
But Penrose does not state his case in those terms. As though Godel’s Theorem is not sufficiently complicated for his purposes, he launches into a convoluted discussion about how the logic system can be extended to include an appropriate GS as an axiom. This expanded logic system has its own GS statement and the same situation, then arises. Once again we cannot prove the truth of the new GS from within, but we can step outside the system and see (by insight) that it must be true. This process can be repeated indefinitely to produce an infinite family of Godelean systems.
But always there will remain within the system, as it is defined, a GS which cannot be proven true (from within the system) but which is known to be true for reasons which lie outside the defined system. Where does this get us exactly? Penrose asks the same question.

“Does this ever end? In a sense, no; but it leads us into some difficult mathematical considerations that cannot be gone into detail here.”
After making reference to some earlier work by Turing and Feferman, Penrose continues thus ...

“However, this does to some degree beg the question of how we actually DECIDE whether a proposition is true or false. The critical issue, at each stage, is to see how to code the adjoining of an infinite family of Godel propositions into providing a single additional axiom (or finite number of axioms). This requires that our infinite family can be systematized in some algorithmic way. To be sure that such a systematization correctly does what it is supposed to do, we shall need to employ INSIGHTS from outside the system ..... It is these insights which cannot be systematized - and indeed must lie outside ANY algorithmic action!”

Cannot? Must?

With that last sentence, having stirred the mixture to the point of confusion, Penrose slips in that important, but unwarranted assumption. It is a disguised form of that schoolboy howler which I called “H”. According to Penrose, if this INSIGHT is not inside the system (and therefore known to be algorithmic) it must be non-algorithmic. After that he continues with growing confidence.

“... it seems to me that it is a clear consequence of the Godel argument that the concept of mathematical truth cannot be encapsulated in any formalistic scheme. Mathematical truth is something which goes beyond mere formalism.”

and then with certainty.

“The notion of mathematical truth goes beyond the whole concept of formalism. There is something ‘God-given’ about mathematical truth.”

And

“Real mathematical truth goes beyond mere man-made constructions.”

All of these quotations come from chapter 4. In chapters 5,6,7,8 and 9 we are given a dissertation on - Gallileo, Newtonian physics, Hamiltonian mechanics, Maxwell's electromagnetic theory, computability, the wave equation, Lorentz equation, Einstein's theory of general relativity, quantum mechanics, Hilbert space, Riemann spheres of state, photon spin, Schrodinger's equation, Dirac's equation, quantum field theory, cosmology, time, entropy, the big bang, black holes, space-time singularities, quantum gravity, Weyl curvature hypothesis and Hawking's box.

In chapter 10, after having beaten his reader into intellectual submission, he returns to the subject of Godel’s Theorem and God-given mathematical insight.

“It has been an underlying theme of the earlier chapters that there seems to be something non-algorithmic about our conscious thinking. In particular, a conclusion to the argument in Chapter 4, particularly concerning Godel’s Theorem, was that, at least in mathematics, conscious contemplation can sometimes enable one to ascertain the truth of a statement in a way no algorithm could.”

The emphasis is mine. I would like to point out the way that Penrose mixes caution with certainty. In one sentence he says that something “seems” to be the case. A few sentences later he asserts the same proposition as if it had been proven beyond doubt. In that quotation above, compare the use of the word “seems” in the first sentence with the apparent certainty expressed by the words “... in a way no algorithm could”.
He has by this time moved from talking about mathematics to the more general issue of consciousness which he believes is the central property of human intelligence. (I do not disagree with that idea. I disagree only with the way he tries to “prove” that cannot be the property of an algorithmic physical system). Penrose continues ...

“... a good part of the reason for believing that consciousness is able to influence judgements in a non-algorithmic way stems from consideration of Godel’s theorem. If we can see that the role of consciousness is non-algorithmic when forming mathematical judgements, ... then surely we may be persuaded that such a non-algorithmic ingredient could also be crucial for the role of consciousness in more general (non-mathematical) circumstances.”
Let us recall the arguments given in Chapter 4 establishing Godel’s theorem and its relation to computability. It was shown there that whatever (sufficiently extensive) algorithm a mathematician might use to establish mathematical truth - or, what amounts to the same thing, whatever formal system he might adopt as proving the criterion of truth - there will always be mathematical propositions, such as the explicit Godel proposition Pk(k) of the system, that his algorithm cannot provide an answer for. If the workings of the mathematician’s mind are entirely algorithmic, then the algorithm (or formal system) that he uses to form his judgements is not capable of dealing with the proposition Pk(k) constructed from his personal algorithm. [my emphasis] Nevertheless, WE can (in principle) see that Pk(k) is actually true! This would seem to provide HIM with a contradiction, since HE ought to be able to see that also. Perhaps this indicates that the mathematician was NOT using an algorithm at all.”

This argument is devious. It is indeed the case that the Godel statement (which he denotes Pk(k) and which I denoted GS) could not be proved true by a proof algorithm within the system to which GS belonged. But somehow those words “within the system” have vanished. Penrose is trying to get us to accept the idea that the reason for the failure of the proof method, is that it is an algorithm. But that is not why it failed. It failed because the GS statement is self-referring. It refers not only to itself but to the system within which it is embedded. To be able to prove the truth of GS we had to get out of the system in order to escape from that self-referent loop. There is nothing there which shows that the successful method (operating from the outside the system) is not also an algorithm.

We can see this quite readily by removing the restriction about the proof being within the system. This modified GS now reads -

Modified-GS = “This statement cannot be proved correct by any method whatsoever.”

There is now no way we can step out of the system to escape the self-referent loop.

(1) If we suppose that Mod-GS is false then there must in fact be a way to prove it is true (hence a contradiction).

(2) If we say that it cannot be false (because of that contradiction) then it must be true, then we have just proved it true which means it must be false (hence another contradiction). It is therefore neither true nor false. It is undecidable. And that will be the case whether we method we use is an algorithm, or what Penrose regards as a non-algorithmic “insight”.

And that is one of my main irritations with this book. The whole issue could have been dispensed with very quickly without the reader getting trapped like Laocoon in that serpentine argument about Godel’s theorem. Penrose reminds me of a taxi driver in Brussells who once drove me about three times round the city in darkness to deliver me to a hotel which was round the corner from my starting place. In that case however, I was at least, delivered to the promised destination.

SHADOWS OF THE MIND
Penrose's book, "Shadows of the Mind" was published some years later in response to numerous objections which were raised to his earlier text. Once again, in this text, Penrose plunges headlong in to Godel’s Incompleteness Theorem. It is interesting to note that in the ENM, he quoted Feferman, as an authority on that subject. It is doubly interesting therefore that Feferman has seen fit to publish on the internet a refutation of some of Penrose’s exposition of the Godelean argument.

see http://psyche.cs.monash.edu.au/v2/psuche-2-07-feferman.html

Feferman, it should be noted, agrees with Penrose that emulation of human mental processes lies beyond the reach of artificial systems but feels that the Godelean argument is irrelevant. He has however, provided us with five and a half pages of detailed corrections to Penrose’s treatment ending with these words ...

“I have not detailed all occurrences of technical errors that Penrose makes in connection with Godel’s incompleteness theorems in Ch 2, many of which propagate through Ch 3. Given the weight which Penrose attaches to his Godelean argument, all of these errors should give one pause. One has here lots more of the “slapdash scholarship” that Martin Davis complained about in his commentary on ENM (1993, p116), and they suggest that Penrose may stretch that scholarship perilously thin in areas distant from his own expertise.” [section 3.12]

In his general discussion of what follows from Godel’s theorem, Feferman comments on the way the proof sequence of a formal system differs from mathematical insight.

“On the algorithmic model, ...... one starts with the ‘statement’ possibly to be established and plugs away mechanically following the algorithm that determines A in the hopes that it will end by ‘proving it’. .... it would be ridiculous to think that anything like that such a search of proofs takes place in the activity of working mathematicians. How it is that they arrive at proof is through a marvelous combination of heuristic reasoning, insight and inspiration (building of course, on prior knowledge and experience) for which there are no general rules, though some patterns have been discerned by Polya and others: there no formula for mathematical success. It is only when one finally arrives at a proof that one can check (mechanically in principle, but not in practice) that does indeed establish the theorem in question.” [section 4.2]

While I bow to Feferman’s expertise on the subject of Godel’s incompleteness theorems, I do think that, with that comment, he exhibits the same misconception as Penrose on the subject of algorithms. Perhaps that is a reflection of the mathematician’s view of things. While the mechanical operation of a proof sequence in the context of a formal system is indeed an algorithm, so to is heuristic reasoning. “Insight”, I suggest, makes use of various stratagems such as analogy and metaphorical relationships, pattern recognition, concept formation and many others. All of these can take the form of an algorithm. I believe too that there is no other way in which these methods of thought can be expressed. What is quite evident is that Penrose has not proved otherwise, although he struggles with convoluted arguments, stretched to very great length, to give the impression that he has done so.
And what of “inspiration”? I suggest that is just a poetic name for original applications of these algorithmic procedures.

In a forthright article entitled PENROSE IS WRONG, Drew McDermott a well known expert in the field of Artificial Intelligence, takes Penrose to task for his argument based on Godel's Theorem, and for his rather vague claims about the excellence of "mathematical Insight". McDermott points out that there have been well documented occasions when the world community of mathematicians have been fooled by an apparently sound (and insightful) argument on some mathematical topic, only to have that insight disproved at a later time, by a formal argument. This destroys Penrose's argument which is based on the unfailing reliability of mathematical insight. It suggests instead that mathematical insight is nothing more than a collection of heuristic techniques, which often produce the correct result, but can sometimes be wrong. A heuristic is an algorithm albeit often a very complicated one, and in the case of human heuristics probably not a single entity or one that is readily knowable. McDermott's article can be found at http://psyche.cs.monash.edu.au/v2/psyche-2-17-mcdermott.html

Early in the book Penrose itemises four different view points on the issue of artificial and human minds. These are ..

(A) All thinking is computational; in particular, feelings of conscious awareness are evoked merely by carrying out appropriate computations.

(B) Awareness is a feature of the brain’s physical action; and whereas any physical action can be simulated computationally, computational simulation cannot by itself evoke awareness

(C) Appropriate physical action of the brain evokes awareness, but this physical action cannot even be properly simulated computationally.

(D) Awareness cannot be explained by physical, computational or any other scientific terms.

Penrose declares his own preference for option (C). He also assumes that option (A) is characteristic of the position adopted by those who support the idea of artificial intelligence. I have to point out, however, that while there may be some who hold to option (A) (in the terms he describes), the position which I defend and which is often called the “brain-mind identity” theory, is that the phenomenon of feeling and self-awareness is not EVOKED by a physical mechanism. My position is that the physical mechanism IS the mechanism of thinking, feeling and conscious awareness. The difference may seem slight, but it is in fact crucial and fundamental to the thesis I have developed in this book.

Tartan Hen Publications : Home | more books | Contact : feedback@tartanhen.co.uk