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From: jqb@netcom.com (Jim Balter)
Subject: Re: Putnam reviews Penrose.
Message-ID: <jqbDBI63F.LFn@netcom.com>
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References: <3t0tn4$p32@netnews.upenn.edu> <jqbDBD193.Iv5@netcom.com> <95Jul9.214427edt.6061@neat.cs.toronto.edu> <3tqldi$bvb@saba.info.ucla.edu>
Date: Mon, 10 Jul 1995 13:42:50 GMT
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In article <3tqldi$bvb@saba.info.ucla.edu>,
Michael Zeleny <zeleny@oak.math.ucla.edu> wrote:
>cbo@cs.toronto.edu (Calvin Bruce Ostrum) writes:
>>Jim Balter <jqb@netcom.com> wrote:
>
>>> So much gibberish, so little time.
>
>>So skip the gibberish and get down to what counts.
>
>In a Goedelian discussion one would be well advised to exploit
>fixpoints.  This sort of advice would be best applied at home.
>
>>>Goedel-limited robots could "arrive" at such statements, they could "see"
>>>that they are true, they could "believe" that they are true, they could
>>>"know" that they are true, they just couldn't *prove* that they are true.
>>>Just like human mathematicians.
>
>>Very good.  This is the first time I have seen anyone put so simply
>>and clearly the basic fallacy in all of the "Goedel proves man exceeds
>>machine" stuff.  Goedel's theorem is a *theorem*.  No reasoning system
>>whatsoever can escape it, and it's that simple.  No reasoning system
>>whatsoever can exhibit a proof of a Goedel sentence for a formal system
>>F in that system.  There *is* no such proof, after all.  
>
>Only if your conception of proof is sufficiently impoverished to
>exclude the sort of informal reasoning that enables us to see the
>Goedel sentence as expressing a truth.

But informal reasoning by robots is not excluded by any Goedelian proof.  So
you are begging the question.  This alleged impoverishment is *Goedel's*; it
is he who formalized the notion of reasoning to which his limitation applies.
Of course, if we drop the formal requirements on what constitutes reasoning,
we can prove the Goedel sentence.  I'm perfectly willing to entertain informal
reasoning, both for myself and for my robot.  Your imputation of an
impoverished conception of proof seems here a bit dishonest.

>>The whole fallacy as is normally presented relies on an equivocation
>>between "provide a formal proof of" and "see to be true" (along with
>>those other locutions).
>
>Being able to provide a formal proof is clearly not in question here.

It is clearly the demand placed upon computation by penrose et. al.

>By restricting your consideration to techniques available within any
>given formal system, you are begging the question of what constitutes
>valid mathematical reasoning.

The question begging has to do with where the distinction between human
reasoning and robot reasoning lies, if not in demanding that the robot reason
formally.  The robot could, for example, use probabilistic methods and accept
as true propositions that have not been contradicted after a large number of
trials.  This could lead it to conclude as true the Goedel proposition for its
formal system, without violating any Goedelian limitation.  Of the robot can
simply do exactly what humans do.  Since humans are using informal, empirical,
and synthetic methods to reach their conclusions, robots can do the same,
without risking the ire of Goedel.

>>One may want to argue that machines can't "see" anything to be true
>>unless they have provided a formal proof, but this has to be argued and
>>it seems to have little to do with Goedel's theorem.
>
>The quality of this discussion would be immensely improved if its
>participants were to consult Goedel's own statements on this subject,
>which are to be found in the recently published third volume of his
>_Collected Works_.  A good secondary source on the opposite side is
>Judson Webb's book on mechanism, mentalism, and metamathematics.

Perhaps, although this sounds like an argument from authority.  We trust
Goedel's formal rigor; his subjective statements are more controversial.
But you might enlighten us with some apporpriate quotations.
-- 
<J Q B>

