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Article 2604 of comp.ai.philosophy:
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>From: gene@nynexst.com (Gene Miller)
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan9.190644.331@nynexst.com>
Sender: news@nynexst.com (For News purposes)
Organization: Nynex Science and Technology
References: <1992Jan9.131829.15232@oracorp.com> <1992Jan9.110732.7279@husc3.harvard.edu>
Date: Thu, 9 Jan 92 19:06:44 GMT

In article <1992Jan9.110732.7279@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>daryl@oracorp.com 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:
>>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.

MZ:
>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.

Is anything gained by viewing our visual, and kinesthetic imagination
as a kind of "analogue computer" used in mathematical reasoning?

    1) When I think of 5 pennies, I may "see" a picture of them in my mind,
    perhaps as the vertices of a "W".

    2) When I think of aleph_0 pennies, I may "see" them lined up ahead
    of me, like the ties of a railroad track, going off to the horizon,
    with no visible end. The projective geometry that allows me to
    "see" the infinitude of the set is unconscious, requiring no
    symbolic reasoning.

    3) When I think of open sets of the real line, I may "feel" the line as
    a "necklace", with the points as little hard beads on a string. When the
    string is broken, the bead at which it is broken terminates one of
    necklace segments, and the other segment is left unterminated (open).

Do not these visual and kinesthetic images serve as aids in reasoning
about these objects?

Similarly, could not all mathematical reasoning (including the
consistency of arithmetic) be based on built-in or learned
perceptual and motor capabilities intended to manipulate things
in the physical world (occasionally aided by symbol manipulation).

And could not this alleged "analogue computer" be simulated on a
digital computer?

Note: More rigorous phrasing of these ideas are welcomed.
-- 
Gene Miller		Phone 914 644 2834
gene@nynexst.com	Fax 914 644 2260


