IJ footnote

[Andrew Dinn]

Jeffrey Reid writes:

> > On Thu, 21 Mar 1996, Adam Lou Stephanides wrote:
> > > There is at least one other math error in the footnotes: Cantor's
> > > Diagonal Proof doesn't mean that between any two objects you can put
> > > an infinity of other objects.

> It does if those objects are numbers on the real line...

Sorry, but I have to disagree with both of you.

Cantor's proof concerns more than just real numbers (that's basically
decimals in layman's speak). The property mentioned in DFW's footnote
(as presented by Adam) also holds e.g. for the rational numbers
(fractions) - between any two fractions there are (speaking
classically, that is, although there are mathematicians who argue that
you *do* have to put them there) an infinite number of other
fractions.

However, the way I read the quote attributed DFW he claims Cantor's
proof shows that no infinite collection is complete, in the sense that
that no matter what infinite cardinality you care to identify there is
always some infinite cardinality which is infinitely larger. This
latter is the standard pop-speak way of bending people's brains with
what are actually quite simple and mundane proofs in transfinite set
theory. I actually much prefer DFW's phrasing in terms of the any pair
property which is essentially equivalent.

And note that Cantor's proof does rely on taking an *ordering* of some
collection with the first cardinality and showing that there are
ommissions in the ordering. The jump to assume that all the missing
elements are `between' the ones in the ordering (which Cantor does
make) is actually rather a large one (look at it real closely, because
it's the one that gets you from the rationals to the reals and that's
where the rot creeps in).

> Actually, I think this is a general problem when writing eloquently
> about math and the hard sciences.  Can any generalizations or
> interpretations of math and science ever be exactly correct?  I
> don't think so. Technical definitions of objects like the Cantor
> set or, yes, even TRP's fave Entropy, are economic. Anything extra
> is by definition imprecise or at worst, outright wrong. Does this
> make these subjects impossible to include in an artistic work while
> still staying true to the concepts? Maybe. One thing I am sure of,
> it is rarely rewarded.  Scientists turn up their noses at the
> imprecise specification of some pet concept, and the art community
> doesn't like it because it isn't part of their world.  However, this
> is what I like most about Pynchon (I laughed uncontrollably at Mr.
> Hilbert-Spaess in GR as well as countless other science 'in jokes'
> which lesser authors would avoid) he isn't afraid to tackle some
> tough but interesting science in his books.  DFW also has this (to a
> lesser degree) in what I've read of IJ.

I agree that maths and much of science are exceptionally precise
disciplines. But I don't think scientists are any more anal than most
people about popularisation of their discipline. Imagine how
intolerant lit types would be of a sit-com based on lit-crit - endless
wisecracks and arguments about what Derrida actually said (and what he
actually meant?).

I was reading today a review of a new book, `The Physics of Star Trek'
which shows how tolerant most scientists are of some utterly
ridiculous things which appear in the shows, things which clearly are
nonsense - e.g. faster than light travel. These are just accepted as
cliches of the genre. What this reveals to me is that most scientists,
like most people, leave their critical faculties at home when fed
comforting, familiar pap and engage them when fed stuff which, other
than the bits which touch their discipline, goes way over their
heads. How else do you account for the success of all those sword and
sorcery books?


Andrew Dinn
-----------
And though Earthliness forget you,
To the stilled Earth say:  I flow.
To the rushing water speak:  I am.