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Article 2596 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan9.110732.7279@husc3.harvard.edu>
Date: 9 Jan 92 16:07:30 GMT
References: <1992Jan9.131829.15232@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
Lines: 64
Nntp-Posting-Host: zariski.harvard.edu

In article <1992Jan9.131829.15232@oracorp.com> 
daryl@oracorp.com writes:

>Mikhail Zeleny writes:

MZ:
>> Try representing in a first-order language set-theoretic concepts like
>> *countable set* and *finite set*, or topological concepts like *open set*
>> and *continuous function*, or analytic concepts like *set of measure 0*, or
>> probabilistic concepts like *random variable*.  It seems to me that our
>> success in discovering and manipulating such concepts amounts to prima
>> facie evidence of our ability to grasp non-recursive abstract entities.

DMC:
>All of these concepts *have* been represented in first-order language!

Not true.

DMC:
>They all can be expressed within the first-order language of
>set-theory, ZFC. The concepts cannot be characterized completely in a
>first-order language, but they can be characterized sufficiently well
>to accommodate human reasoning about them. If you think otherwise,
>point out a result in any of these fields that is not in fact a
>theorem of ZFC.

Our understanding of the results of first-order ZFC is related to our grasp
of its standard model.  I invite you to meditate on the difference between
the latter and its countable submodels before you emit any further
exclamations.  While you are at it, feel free to characterize "sufficiently
well" the predicate "... is finite" in a first-order language of your
choice.  I eagerly await the results.

MZ:
>> Now, if you wish to explain this evidence away, it is incumbent upon you to
>> demonstrate that our ability can indeed be represented in an accounted for
>> by a first-order language used formalistically.

DMC:
>There is no evidence that humans can go beyond formalized reasoning, so
>it is not clear what you are demanding be explained.

On the contrary, any semantic consideration goes beyond formalistic, purely
syntactical symbol manipulation.  But we've been through all that before in
our discussion of the reflection principles.  Unless you have something new
to add, let's agree to disagree.

>Daryl McCullough
>ORA Corp.
>301A Harris B. Dates Dr.
>Ithaca, NY 14850-1313


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