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Article 2736 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan15.115855.7592@husc3.harvard.edu>
Date: 15 Jan 92 16:58:53 GMT
References: <1992Jan13.165429.27512@oracorp.com> <1992Jan13.192559.7488@husc3.harvard.edu> <1992Jan14.194748.14755@cambridge.oracorp.com>
Organization: Dept. of Math, Harvard Univ.
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Nntp-Posting-Host: zariski.harvard.edu

In article <1992Jan14.194748.14755@cambridge.oracorp.com> 
ian@cambridge.oracorp.com (Ian Sutherland) writes:

>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.

MZ:
>>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?

IS:
>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?

In philosophy, the only erroneous argument is an invalid argument.
Penrose's arguments are valid, and so are mine.  If they fail to convince
you, it's because your intuitions differ.  My intuition is that my mind can
completely capture the notion of finiteness.  You are welcome to your own
intuitions.  End of this discourse.

>-- 
>Ian Sutherland                          ian@cambridge.oracorp.com
>
>Sans peur


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