Roger Penrose - Shadows Of Mind

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1. General Remarks

1.1 I am glad to have this opportunity to address some of the criticisms that have been aimed at

arguments in my book Shadows of the Mind (henceforth Shadows). I hope that in the following

remarks I am able to remove some of the confusions and misunderstandings that still surround the

arguments that I tried to make in that book - and also that we may be able to move forward from

there.

1.2 In the accompanying PSYCHE articles, the great majority of the commentators' specific

criticisms have been concerned with the purely logical arguments given in Part 1 of Shadows, with

comparatively little reference being made to the physical arguments given in Part 2 - and virtually

none at all to the biological ones.<1> This is not unreasonable if it is regarded that the entire

rationale for my physical and biological arguments stands or falls with my purely logical

arguments. Although I do not entirely agree with this position - since I believe that there are strong

motivations from other directions for the kinds of physical and biological action that I have been

promoting in Shadows - I am prepared to go along with it for the moment. Thus, most of my

remarks here will be concerned with the implications of Gödel's theorem, and with the claims made

by many of my critics that my arguments do not actually establish that there must be a

noncomputational ingredient in human conscious thinking.

1.3 In replying to these arguments, I should first point out that, very surprisingly, almost none of the

commentators actually addresses what I had regarded as the central (new) core argument against the

computational modelling of mathematical understanding! Only Chalmers actually draws attention

to it, and comments in detail on this argument, remarking that "most commentators seem to have

missed it".<2> Chalmers also remarks that "it is unfortunate that this argument was so deeply

buried". I apologize if this appears to have been the case; but I am also very puzzled, since its

essentials are summarized in the final arguments of "Reductio ad absurdum - a fantasy dialogue",

which is the section of Shadows (namely Section 3.23) that readers are particularly directed

towards. This section is referred to also by McDermott and by Moravec, but neither of these

commentators actually addresses this central argument explicitly, and nor do any of the other

commentators. This is particularly surprising in the case of McCullough, as he is concerned with

some of the subtleties of the logic involved, and also of Feferman, in view of his very carefully

considered logical discussion.

1.4 It would appear, therefore, that I have an easy solution to the problem of replying to all nine

commentators. All I need do is show why the ingenious argument put forward by Chalmers (based

partly on McCullough's very general considerations) as a counter to my central argument is in fact

(subtly) invalid! However, I am sure that this mode of procedure would satisfy none of the other

commentators, and many of them also have interesting other points to make which need

commenting upon. Accordingly, in the following remarks, I shall attempt to address all the serious

points that they do bring up. My reply to this main argument of Chalmers (partly dependent upon

that of McCullough) will be given in Section 3, but it will be helpful first to precede this by

addressing, in Section 2, the significant logical points that are raised by Feferman in his careful

commentary.

2. Some Technical Slips in Shadows

2.1 Feferman quite correctly draws attention to some inaccuracies in Shadows with regard to certain

logical technicalities. The most significant of these (in fact, the only really significant one for my

actual arguments) concerns a misunderstanding on my part with regard to the assertion of omega-

consistency of a formal system F, which I had chosen to denote by the symbols Omega(F), and its

relation to Gödel's first incompleteness theorem. (As it happens, two others before Feferman had

also pointed out this particular error to me.) As Feferman says, the assertion that some particular

formal system is "omega-consistent" is certainly not of the form of a PI_1-sentence (i.e. not of the

form of an assertion: "such-and-such a Turing computation never halts" - I call these "P-sentences"

from here on). This much I should have been (and essentially was) aware of, despite the fact that in

the first two printings of Shadows, p.96 I made the assertion that Omega(F) is a P-sentence. The

fact of the matter was that I had somehow (erroneously) picked up the belief that the statement that

Gödel originally exhibited in his famous first incompleteness theorem was equivalent to the omega-

consistency of the formal system in question, not that it merely followed from this omega-

consistency. Accordingly, I had imagined that for some technical reason I did not know of, this

omega-consistency must actually be equivalent (for sufficiently extensive systems F) to the

particular assertion "C_k(k)" that I had exhibited in Section 2.5, when the rules of the formal

system F are translated into the algorithm A. Accordingly, I had mistakenly believed that Omega(F)

must, for some subtle reason (unknown to me), be equivalent to the P-sentence C_k(k) (at least for

sufficiently extensive systems F).

2.2 This error affects none of the essential arguments of the book but it is unfortunate that in

various parts of Chapter 3, and most particularly in the "fantasy dialogue" in Section 3.23, the

notation "Omega(F)" is used in circumstances where I had intended this to stand for the actual P-

sentence C_k(k). In later printings of Shadows, this error has been corrected: I use the Gödel

sentence G(F) (which asserts the consistency of F and is a P-sentence) in place of Omega(F). It is in

any case much more appropriate to use G(F) in the arguments of Chapter 3, rather than Omega(F),

and I agree with Feferman that the introduction of "Omega(F)" was essentially a red herring. In

fact, the presentation in Shadows would have usefully simplified if omega-consistency had not even

been mentioned.

2.3 The next most significant point of inaccuracy - or rather imprecision - in Shadows that Feferman

brings up is that there is a discrepancy between different notions of the term "sound" that I allude to

in different parts of the book. (This is actually quite an important issue, in relation to some of the

discussion to follow, and I shall need to return to it later in Section 3.) His point is, essentially, that

in some places I need make use of the soundness of a formal system only in the limited sense of its

capacity to assert the truth of certain P-sentences, whereas in other places I am actually referring to

soundness in a more comprehensive sense, where it applies to other types of assertion as well. I

agree that I should have been more careful about such distinctions. In fact, it is the weaker notion of

soundness that would be sufficient for all the "Gödelian" arguments that I actually use in Part 1 of

Shadows, though for some of the more philosophical discussions, I had in mind soundness in a

stronger sense. (This stronger sense is not needed on pp. 90-92 if omega-consistency is dropped;

nor is it needed on p.112, the weaker notion of soundness now being equivalent to consistency.)

2.4 Basically, I am happy to agree with all the technical criticisms and corrections that Feferman

refers to in his section discussing my treatment of the logical facts". (I should attempt a point of

clarification concerning his puzzlement as to why I should make the "strange" and "trivial"

assertions he refers to on p.112. No doubt I expressed myself badly. The point that I was attempting

to make concerned the issue of the relationship between the formal string of symbols that constitute

"G(F)" and "Omega(F)" and the actual meanings that these strings are supposed to represent. I was

merely trying to argue that meanings are essential - a point with which Feferman strongly concurs,

in his commentary.) It should be made clear that none of these corrections affects the arguments of

Chapter 3 in any way (so long as Omega(F) is replaced by G(F) throughout), as Feferman himself

appears to affirm in his last paragraph of the aforementioned section.

2.5 I find it unfortunate, however, that he does not offer any critique of the arguments of Chapter 3.

I would have found it very valuable to have had the comments of a first-rate logician such as

himself on some of the specifics of the discussions in Chapter 3. Feferman seems to be led to

having some unease about the arguments presented there, not because of specific errors that he has

detected, but merely because my "slapdash scholarship" may be "stretched perilously thin in areas

different from [my] own expertise". A related point is made by McCarthy, McDermott and Baars in

connection with my evidently inadequate referencing of the literature on AI, and on other theories

that relate to consciousness, either in its computational, biological, or psychological respects.

2.6 I think that a few words of explanation, from my own vantage point, are necessary here. An

ability to search thoroughly through the literature has never been one of my strong points, even in

my own subject (whatever that might be!). My method of working has tended to be that I would

gather some key points from the work of others and then spend most of my time working entirely

on my own. Only at a much later stage would I return to the literature to see how my evolved views

might relate to those of others, and in what respects I had been anticipated or perhaps contradicted.

Inevitably I shall miss things and get some things wrong. The most likely source of error tends to be

with second-hand information, where I might misunderstand what someone else tells me when

reporting on the work of a third person. Gradually these things sort themselves out, but it takes

time.

2.7 My reason for mentioning this is to emphasize that errors of the nature of those pointed out by

Feferman are concerned essentially with this link of communication with the outside (scientific,

philosophical, mathematical, etc.) world, and not with the internal reasonings that constitute the

essential Gödelian arguments of Shadows. Most specifically, the main parts of Chapter 3

(particularly 3.2, 3.3 and 3.5-3.24) are entirely arguments that I thought through on my own, and

are therefore independent of however "slapdash" my scholarship might happen to be! I trust that

these arguments will be judged entirely on their intrinsic merits.

3. The Central New Argument of Shadows

3.1 Chalmers provides a succinct summary of the central new argument that I presented in Shadows

(Section 3.16, and also 3.23 and 3.24 - but recall that my Omega(F) should be replaced by G(F)

throughout Section 3.16 and 3.23). Let me repeat the essentials of Chalmers's presentation here -

but with one important distinction, the significance of which I shall explain in a moment.

3.2 We try to suppose that the totality of methods of (unassailable) mathematical reasoning that are

in principle humanly accessible can be encapsulated in some (not necessarily computational) sound

formal system F. A human mathematician, if presented with F, could argue as follows (bearing in

mind that the phrase "I am F" is merely a shorthand for "F encapsulates all the humanly accessible

methods of mathematical proof"):

(A) "Though I don't know that I necessarily am F, I conclude that if I were, then the system F would

have to be sound and, more to the point, F' would have to be sound, where F' is F supplemented by

the further assertion "I am F". I perceive that it follows from the assumption that I am F that the

Gödel statement G(F') would have to be true and, furthermore, that it would not be a consequence

of F'. But I have just perceived that "if I happened to be F, then G(F') would have to be true", and

perceptions of this nature would be precisely what F' is supposed to achieve. Since I am therefore

capable of perceiving something beyond the powers of F', I deduce that, I cannot be F after all.

Moreover, this applies to any other (Gödelizable) system, in place of F."

3.3 (Of course, one might worry about how an assertion like "I am F" might be made use of in a

logical formal system. In effect, this is discussed with some care in Shadows, Sections 3.16 and

3.24, in relation to the Sections leading up to 3.16, although the mode of presentation there is

somewhat different from that given above, and less succinct.)

3.4 The essential distinction between the above presentation and that of Chalmers is that he makes

use (in the first (2) of his Section 2) of the stronger conditional assumption "I know that I am F",

rather than merely "I am F", the latter being all that I need for the above. Thus, if we accept the

validity of the above argument, the conclusion is considerably stronger than the "strong" conclusion

that Chalmers draws ("threatening to the prospects of AI") to the effect that it "would rule out even

the possibility that we could empirically discover that we were identical to some system F".

3.5 In fact, it was this stronger version (A) that I presented in Shadows, from which we would

conclude that we cannot be identical to any knowable (Gödelizable) system F whatever, whether we

might empirically come to believe in it or not! I am sure that this stronger conclusion would provide

an even greater motivation for people (whether AI supporters or not) to find a flaw in the argument.

So let me address the particular objection that Chalmers (and, in effect, also McCullough) raises

against it.

3.6 The problem, according to Chalmers, is that it is contradictory to "know that we are sound".

Accordingly, he argues, it would be invalid to deduce the soundness of F, let alone that of F', from

the assumption "I am F". On the face of it, to a mathematician, this seems an unlikely let-out, since

in all the above discussions we are referring simply to the notion of mathematical proof. Moreover,

the "I" in the above discussion refers to an idealized human mathematician. (The problems that this

notion raises, such as those referred to by McDermott, are not my concern at the moment. I shall

return to such matters later; cf. Section 6.) Suppose that F indeed represents the totality of the

procedures of mathematical proof that are in principle humanly accessible. Suppose, also, that we

happen to come across F and actually entertain this possibility that we might "be" F, in this sense

(without actually knowing, for sure, whether or not we are indeed F). Then, under the assumption

that it is F that encapsulates all the procedures of valid mathematical proof, we must surely

conclude that F is sound. The whole point of the procedures of mathematical proof is that they instil

belief. And the whole point of the Gödel argument, as I have been employing it, is that a belief in

the conclusions that can be obtained using some system H entails, also, a belief in the soundness

and consistency of that system, together with a belief (for a Gödelizable H) that this consistency

cannot be derived using H alone.

3.7 This notwithstanding, Chalmers and McCullough argue for an inconsistency of the very notion

of a "belief system" (which, as I have pointed out above, simply means a system of procedures for

mathematical proof) which can believe in itself (which means that mathematicians actually trust

their proof procedures). In fact, this conclusion of inconsistency is far too drastic, as I shall show in

a moment. The key issue is not that belief systems are inconsistent, or incapable of trusting

themselves, but that they must be restricted as to what kind of assertion they are competent to

address.

3.8 To show that "a belief system which believes in itself" need not be inconsistent, consider the

following. We shall be concerned just with P-sentences (which, we recall, are assertions that

specified Turing machine actions do not halt). The belief system B, in question, is simply the one

which "believes" (and is prepared to assert as "unassailably perceived") a P-sentence S if and only

if S happens to be true. B is not allowed to "output" anything other than a decision as to whether or

not a suggested P-sentence is true or false - or else it may prattle on, as is its whim, generating P-

sentences together with their correct truth values. However, as part of its internal musings, it is

allowed to contemplate other kinds of thing, such as the fact that it does, indeed, produce only

truths in its decisions about P-sentences. Of course, B is not a computational system - it is a Turing

oracle system, as far as its output is concerned - but that should not matter to the argument. Is there

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Roger Penrose - Shadows Of Mind

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