Re: 20th Century Mathematicssowa <firstname.lastname@example.org>
Date: Tue, 4 May 93 02:32:29 EDT
From: sowa <email@example.com>
To: firstname.lastname@example.org, email@example.com
Subject: Re: 20th Century Mathematics
Cc: firstname.lastname@example.org, interlingua@ISI.EDU
I just want to emphasize how totally disjoint Cantor's diagonal is from
anything in mathematics that has any useful applicability to anything.
NOTHING, absolutely NOTHING, in analysis depends in any way on Cantor's
diagonal proof. All the work on epsilon's and delta's, not to mention
infinitesimals, was completed long before by Cauchy, Riemann, Weirstrass,
etc. They never made any assumptions about orders of inifinity, and in
fact, they never even talked about infinity as a completed whole. All
of modern analysis follows directly in their footsteps and does not in
any way depend on anything that Cantor did or said about uncountable sets.
Some comments on your comments:
> ... You referred rather
> casually to 'Cantor's paradise', which is usually taken to refer to
> modern mathematics....
That was Hilbert's phrase, and he used it only to refer to Cantor's
work on transfinite numbers (cardinals and ordinals). Very little
of "modern mathematics" deals with uncountable sets.
> You cite the diagonalisation argument as though it were a mere
> bywater, but its consequences permeate contemporary mathematics.
> The definition of the real line depends on it: all of analysis
> would have to be reworked if we reject this. The ideas of limits
> of infinite series, used fundamentally throughout physics and
> chaos theory and topology would need to be revised. It's very
No. None of those subfields of mathematics depend in any way on
Cantor's diagonal "proof". As I pointed out above, all of analysis
was very thoroughly established long before Cantor's work. Limits,
infinite series, etc., never refer to notions of uncountability in
any way. The term "real line", by the way, is an interesting case
in point. A number of mathematicians, including Kurt Goedel, by
the way, have observed some dubious properties about that identification
of the real numbers with points on a line. Suppose, for example, that
you take a line segment of length 1 and break it in half. Intuitively,
you might imagine that you would get two identical segments of length
0.5 -- at least that is what everyone from Aristotle up to the late
nineteenth century would have assumed. But if you "identify" that
line segment with the reals from 0.0 to 1.0, you have to ask what
happens to the point at 0.5 -- which "half" does it belong to?
If it stays with only one side, then you don't have two identical
line segments. Instead, you have one segment that is closed at
both ends and another segment that is open at one end. This is a
very unpleasant consequence that Goedel was not at all pleased about.
My hero, C. S. Peirce, had a solution: he did not "identify" the
real numbers with points on a line. In fact, he did not believe that
a line could be identified with a collection of points; instead, he
believed that points were determined by intersecting lines -- they
were not the constituents that made up the material of which the line
was constructed. Anyway, that is getting off on another tangent,
but it does illustrate the point that there are a lot of serious
questions that remain unanswered, and even someone like Goedel who
worked long and hard in Cantor's paradise was still bothered by them.
> Your citing Whitehead as having worries about the construction
> is revealing. Indeed, when these results were new they were
> controversial, and many older philosophers and mathematicians
> had doubts and reservations. But no convincing arguments were ever
> produced to suggest that the argument is faulty: and after
> nearly a century of critical testing and analysis, 'Cantor's
> paradise' (Hilbert's phrase) seems to be pretty secure. Certainly
> no coherent alternative has ever been constructed (except the
> intuitionist's smaller place.) So again, your rejection of this
> established and well-understood piece of basic mathmatics,
> citing the opinions of somone considered even in his day a
> bit out in left field is, well, idiosyncratic.
No. Whitehead didn't start discussing his doubts until after he
finished his work on the Principia with Russell -- over 30 years
after Cantor's work. And I don't want to go into another litany
of "Big Names" or more "amateur scholarly" citations, but this whole
area has never been one that is anywhere near as solid as the rest
of mathematics. Frank Ramsey, for example, whom Russell considered
one of the most brilliant of his students, made a number of contributions
to the theory of large cardinals. But after thinking about the whole
subject in greater depth, he began to have doubts about it (he was a
friend of Wittgenstein's and I don't know who influenced whom on this
point). If Ramsey hadn't met his tragic death at age 28, he would
probably have debunked the whole theory long ago, and we wouldn't have
to be arguing about it now.
> But you don't give any argument against it, and you ignore all the
> thinking that has been done about it.
On the contrary, there hasn't been much thinking about it at all.
There are three kinds of mathematicians: people who work in fields
like analysis, algebra, chaos theory, etc., who don't depend in any
way on the existence of uncountable sets; students whose professors
gave them a classical theorem in mathematics and told them to generalize
it to a higher order of infinity for their PhD theses; and really
profound thinkers like Goedel, Wittgenstein, Ramsey, etc., who have
worried about it. Of the third group, Wittgenstein represents the
skeptic, Goedel represents the true believer who wrestled with doubts
but overcame them, and Ramsey represents an ardent disciple who later
lost his faith. I consider myself an agnostic who admires Goedel,
but sympathizes with Wittgenstein and Ramsey.
> That is the conclusion of a very pointed and convincing argument, not
> just a vague feeling that things aren't right (which is all that you
> offer here).(I share your gut feeling, and would be interested in finding
> out how you propose to get past Cantor's argument.)
Your reference is vague. Whose argument did you consider "very
pointed and convincing"? Mine or Cantor's? I am not claiming to
be a better mathematician than Cantor, but I am claiming to have the
benefit of the work by Turing and others on noncomputability. If
Turing's work had preceded Cantor's, I doubt that anyone would have
considered it to be anything more than another variation of a
In any case, I have been cleaning out my old office at IBM, and I
unearthed a collection of volumes of the Journal of Symbolic Logic,
which I used to subscribe to until I got fed up with some large
cardinal number of dissertations on large cardinals. If you know
of anyone who is fond of such things, I'll send them to anyone who
is willing to pay the freight.