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Article 2009 of comp.ai.philosophy:
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>From: smoliar@hilbert.iss.nus.sg (stephen smoliar)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: Existence
Message-ID: <1991Dec10.132011.11290@nuscc.nus.sg>
Date: 10 Dec 91 13:20:11 GMT
References: <1991Dec8.103340.6300@husc3.harvard.edu> <1991Dec8.175957.2225@arizona.edu> <1991Dec8.181146.2226@arizona.edu>
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Organization: Institute of Systems Science, NUS, Singapore
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In article <1991Dec8.181146.2226@arizona.edu> bill@NSMA.AriZonA.EdU (Bill
Skaggs) writes:
>
>  The crucial thing is to distinguish between the mathematical entity
>'7' and the psychological concept '7'.  The psychological concept
>of number arises from a small set of preverbal notions, including
>place, movement, object constancy, grouping, and symmetry:  roughly,
>two groups of things have the same number if they can be rearranged
>so that their locations form a symmetric array.  Number, defined in
>this way, gives rise to an equivalence relation between groups of
>things (the relation of "having the same number").  In short, the
>psychological concept '7' is the property that all groups of 7
>things have in common.
>
Actually, I suspect there are a variety of different ways in which "7" can be
perceived psychologically.  Indeed, even Searle tried to appeal to this
approach during a seminar he gave at UCLA several years ago.  Unfortunately,
he lacked Bill's acuity in identifying that what he was trying to talk about
was just this idea of a PSYCHOLOGICAL perception of "7" and, instead, went back
to his usual cant about what is means to "really understand 7."  (I believe
that is pretty close to a direct quote.)  In any event I suspect that "7" can
be a variety of different constructs at the psychological level, such as an
array of dots or an array of dwarfs in a fairy tale.  However, some sort of
categorization mechanism seems capable of forming associations among all those
different constructs;  and "7" may be nothing more than the transitive closure
of those associations.

>  On the other hand, the mathematical notion '7' is a useful fiction
>that can be defined in a variety of ways, depending on the axiom
>set you're using:  in the scheme of Principia, as I recall, it is
>a set consisting of the union of the empty set and the set '6'.
>The fiction is useful because it has two nice properties:  1) the
>mathematical and psychological notions are related by an
>isomorphism; and 2) the mathematical notions can be manipulated
>by very powerful formal reasoning procedures.
>
Rather than calling it a "fiction," why not go along with Searle and call it a
"syntactic structure?"  Rather than viewing syntax in terms of characterizing
which strings of symbols are well-formed, we could just as easily view it as
characterizing which strings are valid according to the rules of PRINCIPIA
MATHEMATICA.  In other words there is really not any difference between those
"very powerful formal reasoning procedures" and what Searle has chosen to call
syntactic manipulation.

>  The isomorphism is what makes mathematical theorems relevant to
>reality, regardless of whether the mathematical entities appearing
>in them actually "exist". 
>
>  But the isomorphism is not perfect.  The axioms defining mathematical
>numbers are only approximately true of the cognitively-defined
>equivalents of those numbers -- in particular, the preverbal cognitive
>operations yielding the notion of "number" cannot be performed upon
>very large groups of things.  However, the formal operations that
>can be performed upon the mathematical notions are so useful and
>powerful that we find it advantageous to ignore the psychological
>intuitions, and treat the mathematical fictions as though they are
>the real thing.
>
Perhaps this is what Searle is really trying to get at.  Human psychology is
too sloppy for the isomorphism to be perfect.  For that matter, I suspect that
no human psychology has a concept for every integer (notwithstanding an
anecdote about Ramanujan).  So is that how Searle has chosen to view strong
AI:  as an isomorphism between symbol systems with their syntactic
manipulations and human psychology?  Would anyone else buy that as
a characterization of strong AI?  (I sometimes suspect Allen Newell
might.  McCarthy can speak for himself.  However, I am not sure how
many other proponents would take quite such an extreme position.)
-- 
Stephen W. Smoliar; Institute of Systems Science
National University of Singapore; Heng Mui Keng Terrace
Kent Ridge, SINGAPORE 0511
Internet:  smoliar@iss.nus.sg


