From: sasghm@theseus.unx.sas.com (Gary Merrill)
Subject: Re: The Mathematical Universe
Originator: sasghm@theseus.unx.sas.com
Date: Mon, 5 Oct 1992 13:17:59 GMT
Organization: SAS Institute Inc.
Keywords: inconsistent multiplicities


In article <1992Oct2.231700.5065@guinness.idbsu.edu>, holmes@opal.idbsu.edu (Randall Holmes) writes:

|> >
|> >What more specifically worries me is that mathematicians in recent
|> >decades have adopted a way of talking about "universes", as if this
|> >relieved them of the obligation of attending to the mathematical
|> >structure of *the* universe in which their "universes" exist.
|> 
|> It certainly is an appalling semantic breach to refer to several
|> universes!  It does seem that there should be at most one.

This puzzles me Randall.  Having confessed (to me at least) that you
are a possible worlds realist, you surprise me with this apparent
denial of multiple universes (which under at least one common interpretation
would mean the same as multiple possible worlds).  Could you clarify
this a bit?  Do you mean, for example, to countenance multiple possible
universes (all of which you concede the "reality" of), but assert the
existence of only one *actual* universe?

How is reference to several universes more appalling than reference
to several possible worlds?  Or is your view that there is only *one*
universe and this comprises all the possible worlds (only one of which
is actual)?

|> >
|> >Thus, if "universes" are things, and U is one of these things,
|> >and x is something which "exists in" U, then there *is* really
|> >something there, which could be denoted by, e.g., "U.x" - and
|> >which simply exists, period.
|> >
|> 
|> This is not a paradoxical situation.  It is perfectly consistent for
|> the universe (the set of all x such that x = x, i.e., everything) to
|> exist and be a member of itself.  (Just not in ZFC!)

This raises some interesting questions about just what *does* exist.
First, I suppose, the *actual* things exist (one is tempted to say,
"The things that really exist exist").  Then there are the mere possibilia
(which under most semantics sastisfy reflexivity of identity).  However,
it can be (has been) argued that *impossibilia* satisfy this principle as
well.  One *might* want to block existential generalization in this case,
or one (Meinong?) might not.  But if you are going to allow quantification
over possibilia, why not impossibilia as well?  Of course, this does
change the sense that we normally attach to "there is ...".  And then
there is the annoying problem of principles of individuation for
impossibilia.

|> >
|> >But it is for this very reason that religion is not, nor ever
|> >can be, part of science. But mathematics, whatever it is, gets
|> >whatever respect it has by being, or pretending to be, part of
|> >science.
|> 
|> Be careful.  Mathematics is _older_ than natural science (in the
|> modern sense) and it is ___NOT___ part of it.  Attempts to incorporate
|> mathematics into the natural sciences do neither math nor science any
|> good.

Be careful again.  Distinguish "is a tool used in science" with "is part
of science", "is a tool that may be applied to empirical domains" with
"has direct empirical content" or "derives its meaning from its empirical
applications."

|> 
|> >
|> >So the contents of mathematical "universes" ought to be open
|> >to the scrutiny even of those who do not (in whatever sense one
|> >can) "live" or "work" in those universes: otherwise they are simply 
|> >not scientific concepts, but more like religious or artistic ideas,

Certainly there are similarities between Platonism in mathematics and
various aspects of religious faith -- particularly if one wants to
countenance the "existence" of possibilia or impossibilia.  Was it
Augustine who said something like "I believe *because* it is absurd"?





-- 
Gary H. Merrill  [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC  27513 / (919) 677-8000
sasghm@theseus.unx.sas.com ... !mcnc!sas!sasghm