From: arodgers@dcs.qmw.ac.uk (Angus H Rodgers)
Newsgroups: sci.logic
Subject: Re: The Mathematical Universe (was re: Topos theory for mere mortals (?))
Keywords: inconsistent multiplicities
Date: 29 Sep 92 12:21:09 GMT
Organization: Computer Science Dept, QMW, University of London

Michael Zeleny writes:

>[...] Though there may be but
>a single mathematical universe, it does not follow that there ought to
>exist a theory of the same.

Indeed, if there is only a single mathematical universe, then,
assuming that it is accessible to all non-brain-damaged humans,
we can probably manage quite well without having a theory of it.

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).

If people can inhabit different mathematical universes, then in
view of the role of mathematics in science, a Popperian vision
of science loses all plausibility: which, in *my* universe, is
an intellectual catastrophe.

I came to think that Morris Kline, in "Mathematics: the Loss of
Certainty", was being a mite hysterical about the prospects for
rationality in mathematics, mainly because he overestimated the
significance of the non-categoricity of infinite models of
first-order theories, and never mentioned second-order logic
(I think - my recollection of the book is a litle hazy);
but the truth may be, rather, than in my complacent ignorance of
the true present state of set theory, I was being too sanguine
about the security of the "universe" I thought I was living in;
and the gates of the city are open to the troops of the
sociology of ignorance, and nothing is sacred. (Pardon the
apocalyptic rant; but I do feel quite strongly about this, and,
worse, I seem to have reason to do so.)

>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.

It sometimes seems as if people use the name "God" when they want 
to refer to the personal aspect of the universe. And it can be
necessary to refer to some such thing (pardon the word) in those
cases, all too frequent, where some aspect of human nature (whether
thought of as divine or not) is being described as if it were a mere
artefact.

And it seems to me that something very like this is happening to
our understanding of mathematics.

>[...] 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.

I don't think that there has to be a specifically *mathematical*
universe: meaning either a universe purely for mathematics alone,
or a mathematical object (such as a set) which is so damned big
that it is felt to deserve the honorific title of "universe".

Nor do I feel compelled to regard the universe as a totality. I
can't even imagine doing so.

I'd be happy with just one (non-totalized) Universe, plus perhaps
(as objects which, it goes without ssying, exist in the Universe)
a plurality of things which mathematicians, for some reason, want
to call "universes" -- just so long as I know what they mean.
Presently, I don't.
--
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.