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Article 1966 of comp.ai.philosophy:
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>From: bill@NSMA.AriZonA.EdU (Bill Skaggs)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: Existence
Message-ID: <1991Dec8.181146.2226@arizona.edu>
Date: 9 Dec 91 01:11:44 GMT
References: <1991Dec7.123551.2220@arizona.edu> <1991Dec7.173732.6277@husc3.harvard.edu> 
 <1991Dec7.205153.2222@arizona.edu> <1991Dec8.103340.6300@husc3.harvard.edu> <1991Dec8.175957.2225@arizona.edu>
Reply-To: bill@NSMA.AriZonA.EdU (Bill Skaggs)
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Once again my idiotic version of rn has posted a quoted copy of
the article I was replying to, rather than my reply -- and it
won't let my cancel it, either.  Here is what I meant to
write (sorry):


In article <1991Dec8.103340.6300@husc3.harvard.edu> 
zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>
>  (Several interesting things, for a change -- too many to
>   follow up on, so I'll just pick the most interesting.)
>
>The problem with treating mathematical entities as useful fictions is that
>we want them to be true in an ordinary context of the proof-theoretic
>turnstile (\vdash for the TeX-literate) : `|-- 7 + 5 = 12'; we don't want
>to say that only `Mathematicians claim that 7 + 5 = 12' is literally true.
>Of course, for a deconstructionist, the latter alternative would be
>preferable, simply because she wants to deny the truth of scientific
>discourse. 
>
  What is at issue here is the relationship between mathematics and
reality.  This is a difficult problem:  if it puzzled Einstein, we
can safely assume that it is puzzling.  Nevertheless I'd like to
take a shot at an answer.

  To make things concrete, let us try to understand the relationship
between the mathematical entity '7' and a pile of 7 pebbles.

  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.

  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.

  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.

  To sum up, the equation '7+5=12' is literally true if the terms
of it are the psychological concepts '7', '5', '12', '+', and '='.
The equation is also fictionally "true" in the fictional world of
Principia.  In practice, because of the isomorphism between the
world of Principia and the cognitive domain of number, there
is no need to distinguish between the two.

	-- Bill


