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Article 4921 of comp.ai.philosophy:
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>From: bill@NSMA.AriZonA.EdU (Bill Skaggs)
Newsgroups: comp.ai.philosophy
Subject: Re: ai
Keywords: ai,realism,Goedel,platonism
Message-ID: <1992Apr4.211235.20693@organpipe.uug.arizona.edu>
Date: 4 Apr 92 21:12:35 GMT
References: <atten.702298596@groucho.phil.ruu.nl> <ktp7d0INNgna@exodus.Eng.Sun.COM> <1992Apr3.200218.26197@guinness.idbsu.edu>
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Reply-To: bill@NSMA.AriZonA.EdU (Bill Skaggs)
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In article <1992Apr3.200218.26197@guinness.idbsu.edu> 
holmes@opal.idbsu.edu (Randall Holmes) writes:
>
>	The underlying problem is the notion that truths of logic and
>mathematics are "constraints on our minds".  I don't think that these
>subjects actually have anything to do with psychology at all.  (From
>my standpoint, Godel is making the same mistake in referring to
>"possibilities of _thought_[my italics]").  I maintain that truths of
>logic and mathematics govern the possible arrangements of objects,
>properties and relations in all domains.  The reason psychology comes
>into it is that to do mathematics _we_ have to think about it.  17 is
>prime independently of our thinking about it, and its primality is not
>a fact of psychology.  [ . . . ]


In some circumstances words like "reality" and "existence" can be more
trouble than they're worth.  Consider the question, "Does Sherlock
Holmes exist?"  Once I know that Sherlock Holmes is a fictional
character created by Arthur Conan Doyle and modeled after a teacher
Doyle knew in medical school, I know everything of interest about the
ontological status of Sherlock Holmes.  If we expend our energy arguing
about whether Sherlock Holmes exists, then our language has become more
of a burden than a help, and we had best move on to more significant
questions.

Similarly, in the case of mathematical objects, it is less important to
defend postions regarding the question of whether they are "real" and
"exist" than to understand their ontological status in the same way we
understand the ontological status of Sherlock Holmes and Abraham
Lincoln.

I would like to argue that mathematics arises from formalizing
certain very general properties of our cognitive representations of
space and of physical objects.  The concept of number, for example,
arises from the notion of "set" and the concept of one-to-one 
correspondence.  (Two sets have the same number of members if their
elements can be placed in one-to-one correspondence.)  The notion
of "set" is identical with the notion of "containment", which is
a crucial part of the way we represent spatial relationships.  The
concept of one-to-one correspondence derives from the notion of
object constancy (i.e. the fact that physical objects tend to
maintain roughly the same form and position) and from the operation
of moving things around.  (Two sets are in one-to-one correspondence
if their members can be moved into an arrangement where they are
perfectly paired up.)  Piaget, in fact, showed that young children
cannot grasp the concept of number until they are old enough to
be able to move objects around in their minds.

All of the basic notions of mathematics can be analyzed in the
same way.  The most basic notion of all, namely "truth",
can be derived, I believe, from the essentially spatial notion
of "containment".  (After all, logic, which is the mathematics
of "truth", can be mapped to set theory, which is
the mathematics of "containment".)

Now, the properties that mathematics is based on are very
general features of space and physics:  features like
continuity, object constancy, betweenness, etc.  These
features are likely to hold in any environment in the 
universe capable of giving rise to intelligent life, so
any intelligence we come in contact with is likely to share
our mathematics.  But if intelligence could arise in the
absence of one of these features (say, object constancy), such an
intelligence could not possibly share our mathematics.  Actually,
it isn't clear to me that we could ever communicate with such
an intelligence, or even recognize it as intelligent; in fact,
it isn't clear that intelligence is possible without the features
of the world that mathematics is based on.

	-- Bill


