From: arodgers@dcs.qmw.ac.uk (Angus H Rodgers)
Newsgroups: sci.logic
Subject: Re: The Mathematical Universe
Keywords: inconsistent multiplicities
Date: 2 Oct 92 19:52:18 GMT
Organization: Computer Science Dept, QMW, University of London

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. :-) )

>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.

I wasn't saying that people were right to use the word "God" in the
way that they do; only that it is difficult to find alternative ways
of speaking about the sacred. (Witness Heidegger?)

>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.

It was probably a bit schizophrenic! I was trying to say that the
human faculties necessary for doing mathematics (reason, intuition, 
etc.) are forever being treated as if they were either nonexistent, 
or reducible to formal systems.

This is probably still not clear; but at the moment I don't feel
capable of doing anything more than directing attention back to
your original point, which was that my talk about the Universe
was reminiscent of some other people's talk about a personal God.

I accept this point; and I was making a somewhat befuddled effort
to explain why there is that connection.

The more sophisticated religious believers often point out that
the trouble with atheists is not that they believe in nothing,
but that they'll believe in anything.

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".

(But that's a whole other thread, and one even less appropriate for
this newsgroup. - My posting here instead of to sci.math was a 
technical accident, BTW.)

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.

(I made an even less successful effort to bring this
topic up on sci.philosophy.tech recently.)

In particular, I cannot see how these mathematical "universes"
could be anything other than structured sets, where the term
"sets" refers to things in *the* Universe: and it is these
things which must be what mathematics is really about.

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.

That is, it exists in the same mode as my wife's idea of God 
exists, whether or not God exists. Ideas of God are potential
objects of study even for those who don't believe in them.

It can be objected, to that last point, that religious ideas
are mysteries, which it is arrogant and stupid to assume will
be equally accessible to all; and this is true.

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.

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 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." But this is just the "Emperor's New
Clothes" argument. ("One day two swindlers came. They told
everybody that they could weave the most marvellous cloth.
Not only were the colours and the patterns of their material
extraordinarily beautiful, but the cloth had the strange
quality of being invisible to anyone who was unfit for his
office or unforgivably stupid.") And we know what to make of
that.

(But I've flogged this point more than enough already; and
McClarty's replies have given me a lot to think about, before
I either venture on this topic again, or decide that I'd
better stop making a fool of myself in public - in the
opposite way to the foolish Emperor!)

>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.

I was trying to explain that when I use the phrase "the universe",
I do not mean to refer to anything other than that which we all
cannot help living in - be we mathematicians, maniacs, both, or
neither. So I should not be taken as asserting the existence
either of some impossibly big mathematical object, or of some
strange and special place where only mathematicians can go.

I think it was William James who pointed out that religion seems
mysterious because it refers not to things far off, but to those
things which are closest to us. Similarly, I just don't mean
anything special at all by the phrase "the universe". (But 
mathematicians do, clearly, mean something special by the
phrase "a universe", and that's what I keep trying to get at.)

>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?

I was just trying to avoid the sort of confusion seen at work in the
"First Cause" argument; but my philosophically uneducated language
may have resulted in more confusion, rather than less.

>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'.

Nothing technical; as I said a moment ago, I just mean the familiar,
mysterious Universe, not seen mistakenly as an object in something
larger.

>AHR:
>>Presently, I don't.

>Nor do I.

>mikhail zeleny@husc.harvard.edu

I'm glad we agree about something!

(Finally, I must apologize, firstly for the excessive length of this
posting - which I seem constitutionally unable to avoid - and secondly
for any new confusions I may have introduced, which may be worse than
usual, because I am tired and my concentration is poor today. I hope
that *some* clear point has emerged from the general fog, nevertheless.)
--
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.
Newsgroups: sci.logic,sci.math,rec.games.chess