From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: The Mathematical Universe
Keywords: inconsistent multiplicities
Organization: Boise State University
Date: Fri, 2 Oct 1992 23:17:00 GMT

In article <1992Oct2.195218.26232@dcs.qmw.ac.uk> arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:
>zeleny@husc9.harvard.edu (Michael Zeleny) writes:
>
>>AHR:
>>>What I was actually saying, though, is that if there are several
>>>*different* things (or contexts) called "universes", then we do
>>>need a theory of such things (or contexts).
>
>>Theories of contexts would appear to fail in the provenance of formal
>>semantics; I don't know why you assimilate them to universes. My argument
>>has to do with the thesis, due in modern times to Spinoza and developed by
>>Cantor, that one cannot meaningfully talk about such things as universes.
>
>I was just trying to avoid the word "thing"! (In case it was carrying
>some meaning I didn't intend.) Unfortunately I seem to have fouled up
>in the same way with "context". (I'll try "sort-of-entityish-thingy"
>next. :-) )

As long as you use a noun, you have a problem...

[...]
>
>Since the discovery or invention (I'm not arguing about which!)
>of non-Euclidean geometry, and even more so since the discovery
>or invention of nonstandard models of first-order theories,
>there has been a (to me) alarmimg and depressing loss of faith
>in the scientific meaning of mathematics. Faith abhors a vacuum;
>and, like Kline, I'm worried by what has taken the place of the old
>(wrong!) ideas abou maths as science. -- Just as I'm worried by what
>has taken the place of the old (wrong, I think!) ideas about "God".

The "wrong" idea is that math is science at all.

[...]

>
>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.

[...]
>
>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!)

[...]

>
>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.
  

>
>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,
>and mathematicians then should not profess to have the authority
>of scientists.

I don't (at least). I profess to have the authority of a
mathematician, which is different in character.

>
>I suspect that the use, by mathematicians, of the phrase "a
>universe" is a way of deflecting this sort of scrutiny, by
>the manouevre of saying, "These things belong to a world you
>know nothing about."

No, I think it only means "domain of discourse" or "what we are
currently talking about".  It's just bad usage.  Since the real
universe isn't there in modern mathematics, the word is just lying
around, available for this purpose.

[...]

>--
>Angus H. Rodgers  (PhD student), | arodgers@dcs.qmw.ac.uk  
>Dept. of Computer Science, Queen | [ +44 | 0 ] 71 975 5241   
>Mary & Westfield College,  Mile  | This space has not been left blank,
>End Rd., London E1 4NS, England  | intentionally or otherwise.


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu