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Article 2605 of comp.ai.philosophy:
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>From: kubo@zariski.harvard.edu (Tal Kubo)
Newsgroups: comp.ai.philosophy
Subject: Re: Penrose on Man vs. Machine
Keywords: human reasoning transcends formalization
Message-ID: <1992Jan9.151511.7300@husc3.harvard.edu>
Date: 9 Jan 92 20:15:09 GMT
References: <1992Jan9.131829.15232@oracorp.com>
Sender: Tal Kubo
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Organization: Dept. of Math, Harvard Univ.
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In article <1992Jan9.131829.15232@oracorp.com> daryl@oracorp.com writes:
>Mikhail Zeleny writes:
>
>> 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.
>
>All of these concepts *have* been represented in first-order language!
>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.
>
>> 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.
>
>There is no evidence that humans can go beyond formalized reasoning, so
>it is not clear what you are demanding be explained.
>
>Daryl McCullough
>ORA Corp.
>301A Harris B. Dates Dr.
>Ithaca, NY 14850-1313

"There are more things in heaven and earth, Horatio,
 Than are dreamt of in your philosophy."

Humans perform reasoning whose formalization, if possible, defies my
imagination.  For examples of a *few* such forms of reasoning, I suggest
you that peruse the literature on subjects like quantum groups,
motivic cohomology, or string theory.  In each of these areas, people
reason informally about ill-defined objects to produce palpable, even
provable, results and conjectures on well-defined ones.

Even if human reasoning were described in some formal language
equipped with some trans-recursive oracle, this would fail to capture the
essence and the mystery of human understanding: that we can discover and
apprehend truths sans preuve, even lacking the means for a proof of
reasonable length.  No Turing machine, turbocharged with any
oracles you like, could do better than finding all the true statements and
writing down their proofs.  It would lack the abilities accessible
to human intuition: to discern meaningful and important statements, and to
get at their truth or falsity, guided by means other than complete proof or
refutation.

That we can meaningfully (well, let the reader judge) discuss abstractions
such as reasoning and formalization speaks to our essential transcendence
of formalized reasoning.  Whereas I can imagine the limits of computer
"reasoning", I see no such limit to human understanding.

-- Tal Kubo   kubo@zariski.harvard.edu


