Re: 20th Century Mathematicsphayes@cs.uiuc.edu
Date: Sun, 2 May 1993 23:57:06 +0000
To: sowa <email@example.com>
Subject: Re: 20th Century Mathematics
Cc: interlingua@ISI.EDU, firstname.lastname@example.org
OK, I admit that "reject 20th century mathematics" was a debating
ploy, and I did say that you seemed to do this. You referred rather
casually to 'Cantor's paradise', which is usually taken to refer to
modern mathematics. But let me comment on your note.
I agree that there seems to be something very peculiar, and
certainly unintuitive, about modern ideas of infinity. But its
important to appreciate just how all-pervasive these ideas are.
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
I don't mean to claim that this could not be done - indeed,
much of it has been done by followers of the intuitionists -
but it is definitely an unusual position to adopt, well outside
the mainstream of modern mathematical thinking. And the intuitionist
program has some unusual consequences itself.
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.
And it should be said that all the things that make you suspicious:
the Skolem-Lowenheim result, for example and its apparent contradiction -
have been thoroughly analysed and discussed in the literature, and
are now pretty well understood. (It is clear that 'uncountable' has
to be understood as relative to an ability to define counting functions,
for example.).You talk of your deep 'suspicion' about Cantor's result.
But you don't give any argument against it, and you ignore all the
thinking that has been done about it.
>If Cantor had simply said that the whole field of infinite sets is
>rife with noncomputable functions and nonconstructive definitions,
>then I would certainly agree. What I don't accept is the conclusion
>that THEREFORE there must exist uncountable sets.
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.)
>As far as the Knowledge Sharing Effort or the ANSI standards, I am
>perfectly willing to let people use either KIF or CGs to state any
>kind of nonsense they please, including axioms for "uncountable" sets.
>But I don't want to let anything in the foundations of either KIF or
>CGs depend on any assumptions of their existence.
I agree: in case anyone else is reading this, let me immediately point out
that TMT does not depend on this, of course. Lets keep the rhetoric
Let me turn to your worry about Cantor's argument. It can be given
in a simpler form, with fewer assumptions. We do not need the
complete table, only a process of generating it.
Here's another version
1. A set can be put into 1:1 correspondence with the integers
iff a sequence can be defined which will eventually mention any
member of the set.(Lemma: We can assume there are no repetitions)
2. Any sequence of digits defines a unique real number.
Thats all we need. Now, if the reals are denumerable, then by (1)
they can be put into a sequence. Define a sequence of digits
as follows: take the decimal expansion of the Nth real generated,
let the Nth digit in that expansion be M, and let the defined digit
in the sequence be (M+1 modulo 10). This sequence defines a real,
by (2), and evidently this real is not in the sequence generated
by the process.
The point is that we do not need to talk of a completed infinite
table (contrast for example such tricks as Richards paradox), so
the proof does not depend on any complexities of definitions.
Moreover, we do not need your assumption about one infinite set
being 'larger': the proof merely establishes that the reals are
Its harder now to see what could be wrong. Both assumptions seem
pretty reasonable to me. (If I had to part with one of them, I'd
abandon 2.. It amounts to the claim that any infinite sequence
of smaller and smaller 'shifts' must converge to a unique real
number. This is false in the actual world, because eventually
we get to quantum dimensions. So maybe the idea of the continuum
is just incoherent. Or
maybe after a while different decimal series really get
indistinguishable, so the construction works but the conclusion -
that the number so defined is not in the set - does not follow.
Its fun to try these ideas out, but they all have very strange
consequences, unfortunately. And they play hell with analysis,
calculus, and most everyday maths.)
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