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>From: ian@cambridge.oracorp.com (Ian Sutherland)
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan14.194748.14755@cambridge.oracorp.com>
Organization: ORA Corp, 675 Mass Ave, Cambridge, MA 02139
References: <1992Jan13.165429.27512@oracorp.com> <1992Jan13.192559.7488@husc3.harvard.edu>
Date: Tue, 14 Jan 92 19:47:48 GMT

In article <1992Jan13.192559.7488@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>>DMC is Daryl McCullough
>>MZ is Mikhail Zeleny

>DMC:
>>Look, Mikhail, the question of a Turing machine halting on an input
>>involves only existential quantification over integers. There are no
>>quantifications over *sets* of integers, and so, therefore, it is a
>>first-order statement, and not a second-order statement. If you insist
>>that even *mentioning* integers makes it a second-order statement,
>>then by that criterion, any mathematical definition is second-order.
>
>In view of non-categoricity of first-order PA, any mathematical definition
>that involves quantification over the integers, whether in object language
>or in meta-language (e.g. by appealing to the conventional notion of a
>proof as a *finite* sequence of propositions) is ipso facto second-order.
>Is this so hard to understand?

It's not at all hard to understand the facts you state, it's just
hard to understand why you think they imply that human reasoning is
nonalgorithmic.  Your argument boils down to saying "first order
theories can't completely capture finiteness, and we can, so we're
not first order".  This is the same kind of erroneous argument that
Penrose gives.  It reduces believing something that seems moderately
plausible to something which seems probably false.  What leads you to
believe that human beings can completely capture the notion of
finiteness in their own minds?
-- 
Ian Sutherland                          ian@cambridge.oracorp.com

Sans peur


