From: chalmers@bronze.ucs.indiana.edu (David Chalmers)
Subject: Re: Human intelligence vs. Machine intelligence
Organization: Indiana University
Date: Sun, 11 Oct 1992 21:07:35 GMT

Aaron Sloman writes:

>I suspect he must be referring to my discussion of an error that I
>claimed was common to Penrose and many of his critics. I.e. Penrose
>and crtics both think there is some way of discovering that Godel's
>"undecidable" sentence (G(F)) says something *true* about the formal
>system F that allegedly provides a basis for arithmetic.

What is F?  If F is something like Peano arithmetic, say, then
we can certainly see the truth of G(F) -- it's straightforwardly
equivalent to the consistency of PA, which is no less clear than
the truth of most other mathematical propositions.  If F is the
humongous system that represents our joint mathematical competence,
then it's entirely unclear that we could see the truth of G(F)
when it was presented to us -- even presented to us in a "nice"
way that made explicit its relation to F.  Presumably, determining
the consistency of F would be beyond us -- it would be just a step
beyond the limits of our mathematical competence.

Certainly: we can see that if F is a consistent formulation of our
joint mathematical competence, then G(F) is true.  But to say this
is not the same as to say that: if F is a consistent formulation of
our joint mathematical competence, then we can see that G(F) is
true (note the shift in the "we can see" operator).  The latter is
presumably false if we are formalizable, and Penrose doesn't
provide any reason to suppose that it is true.

>My claim was that since the sentence was undecidable, adding either it
>or its negation to F would produce a new consistent system F1 or F2,
>and since both F1 and F2 are consistent both would have models, with
>G(F) true in one set of models and false in the other set (the
>socalled "non-standard" models). Hence nobody can rightly claim simply
>to see that G(F) is true.
>
>If they say they can because they know *which* model they are thinking
>about, then they must explain *how* they identify that model. Godel's
>result shows that it cannot be by means of a formal system. If there
>are some other means, then they apparently support Penrose who is
>claiming exactly that!

Certainly we can do this if F is a system like Peano arithmetic
All the argument shows is that the correct formalization of our
competence must be more complex than F.  (I suspect that those who
have disagreed with you have had in mind the case where F is a
simple system, like PA or whatever, whose consistency we can indeed
see.  Penrose appeals to this case in his argument, for example,
saying that when we "see" the truth of the Godel sentence for PA,
we are doing something non-algorithmic.  This conclusion is
manifestly unsupported.)  If F is the ridiculous formalization of
our competence, then again there's not the slightest reason to
suppose that we can see that G(F) is true.

Taking a look at your AI article, you say that you mean F to be
any system rich enough to express the arithmetic of natural
numbers -- i.e. something more like the PA case than the humongous
case.  In that case, it seems that I am disagreeing with you above,
and that your argument in the second paragraph quoted above is
fallacious ("Godel's results" only show that seeing G(F) can't
be by means of *that* formal system F, not that it can't be
by means of *any* formal system).  Maybe you really meant to be
talking about the humongous F, however.

-- 
Dave Chalmers                            (dave@cogsci.indiana.edu)      
Center for Research on Concepts and Cognition, Indiana University.
"It is not the least charm of a theory that it is refutable."
Xref: sparky comp.ai:3864 comp.ai.neural-nets:3876 alt.consciousness:585 comp.ai.philosophy:6373 sci.psychology:4085