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Article 1926 of comp.ai.philosophy:
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>From: bill@NSMA.AriZonA.EdU (Bill Skaggs)
Newsgroups: rec.arts.books,sci.philosophy.tech,comp.ai.philosophy
Subject: Existence
Message-ID: <1991Dec6.204854.2218@arizona.edu>
Date: 7 Dec 91 03:48:52 GMT
References: <12563@pitt.UUCP> <1991Dec4.212727.6154@husc3.harvard.edu> 
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Reply-To: bill@NSMA.AriZonA.EdU (Bill Skaggs)
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	William Chesters, Wadham College, Oxford:
>Platonism isn't the only philosophy of mathematics you know!  There are 
>people who (horror) don't think real numbers exist.

	Mikhail Zeleny, Harvard, Mass:
>Their opinions are absolutely irrelevant to mathematics as it stands.


This is a fantastic exchange, because it wonderfully illustrates the
most common trap philosophers fall into, which I call the "In the
beginning was the word" fallacy:  the assumption that words have
well-defined meanings, independent of how they are learned and used,
and that by careful analysis we can uncover those meanings.

The word at issue here is "exist".

What does it mean to say "Real numbers exist"?  Both Chesters and
Zeleny believe that this sentence is either true or false.  In fact
it is merely meaningless.

The meanings of words are learned by watching how other people use
them.  Many words are only used when certain conditions are present,
and in the absence of those conditions are meaningless.  "Existence"
is one of those words.  When it is applied to a name or to a concrete
description, we all have some notion of what it means:  when I say
"Benjamin Disraeli existed but Sherlock Holmes did not", we all know
that one of the consequences of this is that an archaeological
study of 221B Baker Street is unlikely to turn up traces of the
presence of Sherlock Holmes, whereas a study of Disraeli's residence
(wherever it was) is likely to yield something.  When I say "The
present king of France does not exist", we all know that one of the 
consequences is that it makes no sense to ask "What is his name?"

When the attempt is made, though, to apply "existence" to abstract
descriptions, we find no consensus.  People ordinarily don't use
the word under those conditions, and when somebody tries to, we
don't understand what is meant.  We don't know what the consequences
are.  ***"Real numbers exist."***  So what?  What are the consequences?
What difference does it make whether the number pi "exists"?

It is of course possible to extend a word into a domain in which it
is not ordinarily used, by explicitly agreeing to use it under
specific conditions.  Mathematicians are doing this when they make
axioms such as "Through any two points there exists exactly one
straight line."  This axiom is an agreement on how to use the words
"point", "line", and "exist".   To think of it as asserting a fact
about points and lines is to be very confused. 

If we remember that words are only defined as precisely as they
are used -- including words such as "existence" and "truth", we
will avoid a lot of useless squabbling.

	-- Bill


