From: zeleny@husc9.harvard.edu (Michael Zeleny)
Newsgroups: sci.math,sci.philosophy.meta,sci.logic
Subject: The Mathematical Universe (was re: Topos theory for mere mortals (?))
Summary: does it exist?
Keywords: inconsistent multiplicities
Date: 29 Sep 92 01:13:14 GMT
Article-I.D.: husc3.1992Sep28.211315.15922
Followup-To: sci.logic
Organization: The Phallogocentric Cabal

In article <1992Sep28.151315.18034@dcs.qmw.ac.uk>
arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:

>zeleny@husc9.harvard.edu (Michael Zeleny) writes:

MZ:
>>[...]  However, I wish to go on record as stating that,
>>though it probably is a sociological fact that most realists who bother
>>thinking about it, think that there is exactly one universe of all sets,
>>and that, conversely, some people have gone on record announcing their
>>conversion to formalism on the grounds of their loss of belief in the
>>unique existence of such a universe, I do not myself believe that a realist
>>philosophy of mathematics cannot accomodate a multiplicity of set-theoretic
>>(or type-theoretic) universes possessed of different size and structural
>>properties. 

AHR:
>Even if we modestly bow to the apparent inevitability of having to
>face the existence of a plurality of "universes" of "sets", then,
>unless we give up the use of our rational faculties altogether,
>we are forced (aren't we?) to develop a "theory of universes", with
>pretensions to absolute truth.

No, we are not forced to do anything of the sort.  Though there may be but
a single mathematical universe, it does not follow that there ought to
exist a theory of the same.  As said Spinoza (whose ideas on the infinite
are known to have inspired Cantor), _definitio est negatio_; and if the
universe is conceived as encompassing everything, there would seem to be no
way of meaningfully defining it.

AHR:
>What sort of thing do people mean by the phrase "a universe"? How do
>we reason about such things? Are they mathematical objects?

Probably not; it is not even clear that they are objects of any sort.

AHR:
>Perhaps the common noun "universe" here can be replaced by "topos", and
>perhaps it can't; either way,I'm not technically qualified to say.
>Be that as it may, my (non-technical) question is whether there is any 
>way of understanding this word (in the sort of context in which you have
>just used it) as referring to anything other than a structured set. If
>there is, my whole argument collapses. (Of course, merely replacing the
>word "set" here by "class" or "collection" makes no semantic difference.)

There is also a way of interpreting the word in question as _a priori_
non-referring.

MZ:
>>There is, after all, no good reason to suppose that everything
>>in the mathematical universe is a set [...]

AHR:
>Naively, I just can't see how the human mind could possibly manage
>without some such concept of "*the* universe" as you have just
>appealed to (without irony, I take it).

Note that the same things have been claimed on behalf of the concept of a
personal God.  As for the question of my own usage, I use the term in
question in much the same way as I would talk about a spiffy haircut,
without thereby committing myself to an ontology of haircuts.  It is, after
all, an elementary Russellian point that not every singular term of a
natural language calls for a literal interpretation based on its surface
grammatical structure.  And in this case I would be very happy to admit the
meaninglessness of my above remark, by accepting a reasoned argument (some
are reported in Michael Hallett's _Cantorian Set Theory and Limitation of
Size_) to the effect that the term `the mathematical universe' is bereft of
any denotation, on the grounds that the putative totality which it is meant
to denote is far too large to be a consistent multiplicity in the Cantorian
sense.

>--
>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.

cordially,
mikhail zeleny@husc.harvard.edu
"Un de mes plus grands plaisirs est de jurer Dieu quand je bande."

P.S. The Phallogocentric Cabal will hold its first meeting for the
1992-3 school year at 7 p.m. on Thursday, October 1 in the Emerson 
Hall of Harvard University.  Free memberships are still available.