Re: 20th Century Mathematicsphayes@cs.uiuc.edu
Date: Tue, 4 May 1993 14:56:03 +0000
To: sowa <email@example.com>
Subject: Re: 20th Century Mathematics
Cc: firstname.lastname@example.org, interlingua@ISI.EDU, email@example.com
John, I don't want to keep arguing with you about the history of mathematics,
but I can't let your last message go unanswered. You will have the last
word on this as I won't spend any more email time on it.
>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.
But you omit a vital aspect of the historical story. While these ideas were
being developed, serious worries were arising about whether any of it made
sense, in particular whether or not it was consistent. Arguments about
whether series converged or not were often felt to be only marginally
coherent, and intuitions often failed or disagreed. These worries largely
motivated the development of set theory and the famous attempts to reconstruct
mathematics in set-theoretical terms. And 'Cantor's paradise' is a conceptual
framework which provides a consistent foundation for all that machinery.
Cantor thought there was only one kind of infinity - countable infinity -
and spent years trying to prove it: he was as surprised as anyone by his
argument. I agree, analysis does not explicitly depend on Cantor's
construction, but it does rely on notions of infinity - infinite series,
for example - which seem to clearly allow it.
If one decides to expel oneself from paradise then one is obliged to
explain how these ideas make sense. Or perhaps I should say, that I
won't follow you until I hear such an explanation.
The real problem is that the basic idea of Cantor's construction is
very clear and can apparently be used with any other attempt to define
limits. For example, take Whitehead's 'extensive abstraction' concept,
which identifies a point with a suitably convergent infinite nested
collection of spatial patches. It is easy to give a Cantorian argument
for this kind of definition. The slender thread you talk about seems
to be remarkably tough, in fact.
But anyway, we can argue about this for ever. All I want to do here
is make it clear that ordinary working mathematics, these days, accepts
these ideas without fuss over foundational issues. While
we are playing citation games, try these, from the Encylopedia of
First, Mostowski's article on Tarski:
"It should be noted that in these papers, Tarski has not criticised
the assumptions of set theory. Like most mathematicians, he has simply
accepted them as true."
Second, James Thomson on Infinity:
"Much of Cantor's theory is now almost universally accepted. There
are philosphers who still ask whether there really are infinite numbers,
or whether (a very slight improvement) whether what are called infinte
numbers deserve to be called numbers,....Such questions do not seem
very profitable or interesting. As a mere fact of anthropology (but
nonetheless interesting for that), one may mention that it is now
virtually impossible to instil a general skepticism about infinite
numbers among freshmen who have had a good high-school education."
>Your reference is vague. Whose argument did you consider "very
>pointed and convincing"? Mine or Cantor's?
Cantor's. You didnt give an argument.
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