From newshub.ccs.yorku.ca!torn!cs.utexas.edu!uunet!tcsi.com!iat.holonet.net!uupsi!psinntp!scylla!daryl Wed Oct 14 14:58:25 EDT 1992
Article 7190 of comp.ai.philosophy:
Newsgroups: comp.ai.philosophy
Path: newshub.ccs.yorku.ca!torn!cs.utexas.edu!uunet!tcsi.com!iat.holonet.net!uupsi!psinntp!scylla!daryl
>From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Human intelligence vs. Machine intelligence
Message-ID: <1992Oct9.003020.7551@oracorp.com>
Organization: ORA Corporation
Date: Fri, 9 Oct 1992 00:30:20 GMT
Lines: 54

About Penrose' argument that human intelligence cannot be algorithmic:

His argument is summed up on pages 416-417 (paperback edition) of
_The Emperor's New Mind_:

    Let us recall the arguments given in Chapter 4 establishing Godel's
    theorem and its relation to computability. It was shown 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 providing his criterion of truth---there will
    always be mathematical propositions, such as the explicit Godel proposition
    P_k(k) of the system (cf. p. 140), 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 actually
    uses to form his judgements is not capable of dealing with the proposition
    P_k(k) constructed from his personal algorithm. Nevertheless, *we* can
    (in principle) see that P_k(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!

What Godel's theorem tells us is that for any formalizable theory T
powerful enough to do arithmetic, there is a sentence G of
arithmetic---Penrose'construction calls it P_k(k)---such that if T is
consistent (it never proves a contradiction), then (1) G is true, and
(2) T is unable to prove G. From this if follows that if we know that
T is consistent (where we know something if it is true, and we believe
it), then we will also know that G is true, and so we know something
that T is unable to prove. Therefore, if we know that T is consistent,
then T is not powerful enough to capture all of human reasoning.

That much is true. Where Penrose jumps off the deep end is in
concluding that therefore no theory can formalize human reasoning. The
most that can be concluded is that *if* there is a theory that
formalizes all of human reasoning, then it must be a theory so complex
that we can't know whether it is consistent or not.

For example, let's examine whether it is possible for all human
reasoning about arithmetic to be formalized in the theory NF (Quine's
New Foundations, a variant of set theory which is different from ZFC
but is not known to be consistent relative to ZFC). We can do Penrose'
trick of coming up with a sentence G such that *if* NF is consistent,
then (1) G is true, and (2) NF doesn't prove G. Do we know that G is
true? No, because G is only true if NF is consistent, and we don't
know whether NF is consistent. Therefore, Penrose' trick fails to come
up with a sentence which we know to be true and which NF can't prove.
In other words, Penrose is mistaken.

Daryl McCullough
ORA Corp.
Ithaca, NY





