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From: jqb@netcom.com (Jim Balter)
Subject: Re: SRS Theory - Evolution anf Mathematics
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Date: Sat, 28 Sep 1996 02:30:58 GMT
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In article <324B6499.3AB6@dircon.co.uk>,
Chris Gordon-Smith  <gsmith@dircon.co.uk> wrote:
>I spent some time last year trying to figure out what the flaw is. I think 
>its that he believes that you can't do mathematics using an unsound 
>algorithm. I read the book and was convinced that you can't do mathematics 
>with a sound algorithm, but the argument against using an unsound algorithm 
>seemed to be just that this was 'implausible'.

Actually there are many flaws, but I think the clearest one is seen in the
exchange between Penrose and Daryl McCullough in Psyche.  McCullough points
out that Penrose depends upon mathematicians being able to *guarantee* that a
proposition is true without providing a *formal* proof of it.  Penrose
concedes that this is so, but holds that there are such guarantees.  This
reduces his argument to a dependency upon a claim that appears to be
semantically absurd, and in any case is unsupported and unsupportable.

Penrose's idea of guarantees is based upon his notion of unassailable truth,
which boils down to a Sorites fallacy, with the typical refutation
("unassailability" is not a yes/no proposition; a 4 trillion page proof is
somewhat less unassailably true than a one liner).  And his idea of an ideal
mathematical community, by which he attempts to avoid the sorts of cumulative
errors endemic to Sorites arguments, is tantamount to imagining an ideal
Turing machine community that can avoid Godel limitations.  Merely imagining
it doesn't mean that it is realizable, and Penrose does not show (and
certainly cannot *guarantee*) that such an idealization is realizable.

Basically, it is foolish to depend upon thought experiments and
logico-philosophical arguments; they are almost always flawed.  Only after
they have been through the fires of examination and controversy, come out
unscathed, and been incorporated into the ideal community's corpus, should
they be taken as pillars for further developments.  But given that Penrose's
argument is not accepted Chalmers, Dennett, Putnam, McCullough, Feferman,
McDermott, McCarthy, Minsky, and Churchland, to name just a few, it is
anything but unassailable, and people who themselves have not been able to
find a flaw in it should not be so arrogant as to think that that fact carries
much weight.
-- 
<J Q B>

