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Article 5064 of comp.ai.philosophy:
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>From: daryl@oracorp.com (Daryl McCullough)
Subject: goedel and ai - correct version!!
Message-ID: <1992Apr10.000621.24586@oracorp.com>
Organization: ORA Corporation
Date: Fri, 10 Apr 1992 00:06:21 GMT
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atten@phil.ruu.nl (Mark van Atten) writes:

> II.2 Penrose's argument

> Let F be a formal system, and G(F) an undecidable formula in F.
> (e.g., Con(F))
>
> Then Penrose's argument is this: The ***deduction*** of G(F) from F
> is true and valid, we can ***see*** that. The important thing is that
> it is the deduction is seen to be  valid, while it is not formalizable
> (is that correct English?).

Mark, we already did Penrose in this newsgroup. Sigh.

Penrose's arguments are, in my opinion, completely wrong. First of
all, to say that we deduce Con(F) from F is mistaken. The statement
Con(F) is simply a formalization of the statement "It is impossible to
prove a contradiction from the axioms of F". If we believe that F is
consistent, then we believe the formalization of this, Con(F). Where
is there some non-formalizable deduction in any of this?

Penrose claims that our ability of "reflection" is somehow not
formalizable, but this is an empty claim. His proof doesn't prove it,
any more than Lucas' proof did. As a matter of fact, Penrose' proof is
simply Lucas' proof, recycled.

> Perhaps mathematical intuition cannot see ALL of true math. (Goedel
> thinks it can, however), but that doesn't matter for this argument:
> there is at least one math. truth that can not be formalized and
> hence, is not algorithmic.

I agree that mathematical truth is not algorithmic. But the little
pieces that we have been able to grasp certainly are.

> It must be borne in mind that this is a
> question of principle. Probably no one will ever be able to compute
> the 10^10^10^10 digit of pi, but in ***principle***, it is possible.

> Again: it is the fact that we see the validity of Goedel's proof, not
> the truth of G(f); that is the difference with Lucas.

Godel's proof is a *theorem* of Peano arithmetic. It is completely
formalizable, and therefore does not prove that there is some kind of
reasoning that we do that is not formalizable.

Daryl McCullough
ORA Corp.
Ithaca, NY






