From: zeleny@husc9.harvard.edu (Michael Zeleny)
Newsgroups: sci.logic
Subject: Re: The Mathematical Universe
Keywords: inconsistent multiplicities
Date: 2 Oct 92 15:50:04 GMT
Article-I.D.: husc3.1992Oct2.115007.16014
Organization: The Phallogocentric Cabal

In article <1992Sep29.122109.6758@dcs.qmw.ac.uk> 
arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:

>Michael Zeleny writes:

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

AHR:
>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.

My point was not of a pragmatic nature.

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.
(The prima facie self-contradiction implicit in this claim is easily
avoided by a charitable interpretation, or eradicated by circumlocution.)

AHR:
>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.

This admittedly catastrophic possibility was not what I meant to suggest.

AHR:
>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.)

I share your distrust of first-order theories, and recommend the Stewart
Shapiro book I cited elsewhere as an easy remedy.  You might also want to
look at the classic compendium _Model-Theoretic Logics_, edited by Barwise
and Shapiro.  Most set-theoretic research is not to be trusted for
foundational purposes precisely because of its dependence on the infamous
first-order thesis, whose full strength (the only sort of quantification
that makes sense is quantification over the individuals) makes a mockery of
the basic premisses of set theory (cf. Zermelo's views on this subject, as
recorded in Moore's book on the Axiom of Choice, and elsewhere in the
numerous recent discussions of the Zermelo-Skolem controversy).

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).

MZ:
>>Note that the same things have been claimed on behalf of the concept of a
>>personal God.

AHR:
>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.

I'll grant you the validity your point only if you reciprocate by allowing
Gaunilo to refer to the most perfect island.

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

Something like what?  I am quite baffled by this transition.

MZ:
>>[...] 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.

AHR:
>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".

Mutatis mutandis, the argument goes for any sort of universe.

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

How else do you propose to regard it?

AHR:
>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.

I don't understand your use of the term `non-totalized'.

AHR:
>Presently, I don't.

Nor do I.

>--
>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
"Nous donnons la mort, nous saurons la subir."

P.S.  The Phallogocentric Cabal will be meeting on Mondays at 6 p.m., in
the 3rd floor graduate student lounge of the Harvard University Emerson
Hall, proceeding promptly to the nearest unoccupied seminar room.  This
month's project is forcing and Boolean-valued models, as discussed in
Bell's book and elsewhere.  More model theory, categories, topoi, types,
admissible ordinals, and other nifty stuff will be coming up shortly.
Free memberships are still available; inquire within.
Newsgroups: sci.logic