From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.ecf!utgpu!cs.utexas.edu!uunet!mcsun!sun4nl!alchemy!plato!atten Thu Apr 16 11:33:53 EDT 1992
Article 5031 of comp.ai.philosophy:
Path: newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.ecf!utgpu!cs.utexas.edu!uunet!mcsun!sun4nl!alchemy!plato!atten
>From: atten@phil.ruu.nl (Mark van Atten)
Newsgroups: comp.ai.philosophy
Subject: Re: goedel and ai - correct version!!
Keywords: ai,goedel
Message-ID: <atten.702902833@groucho.phil.ruu.nl>
Date: 10 Apr 92 10:47:13 GMT
References: <atten.702555787@groucho.phil.ruu.nl> <centaur.702598337@cc.gatech.edu>
Sender: news@phil.ruu.nl
Organization: Department of Philosophy, University of Utrecht, The Netherlands
Lines: 52
Nntp-Posting-Host: groucho.phil.ruu.nl

centaur@terminus.gatech.edu (Anthony G. Francis) writes:

>atten@phil.ruu.nl (Mark van Atten) writes:
>>There is an ongoing discussion right now on mathematical realism in the ai
>>group. Some people question the relevance of this to ai. In this article,
>>I want to discuss
>>1 The objective existence of mathematical objects
>>2 The significance of this for ai

>>II THE SIGNIFICANCE OF GOEDEL'S THEOREM FOR AI
>>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?).
>>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.
>>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.

>The deduction of G(F) is not formalizable from _within_ F, but that
>does not mean that it is not formalizable at all. It is possible to 
>devise a new formal system F', in which it is possible to prove truths 
>_about_ F. From within F', it is possible to derive that G(F) is an 
>undecideable formula within F, and that G(F) is true and valid. That is,
>the validity of the deduction of G(F) from F can be determined in a 
>formalizable way, even though this cannot be determined from within F.

This is not a valid argument. To see why, let's compare it with the proof
that there are infinite many prime numbers. It starts with the assumption
that there is a largest prime number; let's call it n. Then a larger prime
is constructed, thus proving that n cannot be the largest one. However, since
no assumptions were made about n (except its being the alleged largest prime),
the argument obviously succeeds for any proposed candidate; therefore, it is
concluded that there is no largest prime at all.
The analogy is obvious. No one would claim that, since we called the largest
prime n, we've only proven that there is some  prime number for which a larger prime
exists. It's a proof of principle. So is Penrose's argument. He does not claim
that for any given consistent formal system, we can see the truth of its
Goedel sentence; that is not obvious at all, as Hofstadter points out in his
refutation of Lucas. Penrose argues that, given any consistent formal system,
we are able to see the validity of the deduction of its Goedel sentence. In a
way, we are always a step ahead of the next formal system.

Mark.


