Subject: Re: [HM] Mathematics and Time
From: Gordon Fisher (gfisher@shentel.net)
Date: Fri Mar 10 2000 - 12:25:17 EST
I ask, only a little facetiously, have we come to understand completed
infinities in C20? I know we've come to manipulate various kinds of them, as
in the tradition of Cantor, or in non-standard analysis, etc., but there seem
at least to be some gaps in our understanding.
For example, I've never been able to be at peace with the idea that the real
numbers can be well-ordered. How are we supposed to go about this, anyway?
Or, as another example, as I once said long ago to my dissertation adviser (R
D Anderson, a student of R L Moore) when I was a graduate student, I feel
sometimes that a Cantor set (i.e., the totally disconnected thingamabobby) is
a measure of how far we have so far failed to *really* understand the notions
of continuity and discreteness.
Anyhow, while Kant did consider so-called completed infinities under the aegis
of "paradoxes of pure reason" (which isn't the same as saying he rejected
them), many mathematicians besides Brouwer were openly influenced by him in
the 20th century (Hilbert, for example), and many more have been influenced by
him without realizing it, or at least without admitting it.
In related news, Kant has often been accused of being an old stick-in-the-mud
for having proclaimed Euclidean geometry to be the only kind there could be,
due to the way human minds work. However, I concluded some years ago, after a
rather long excursion into the works of Kant, that (1) there are possible
interpretations of what he said which make him correct in this statement, and
(2) it is possible to adjust what he said to make his theories about the way
our minds work valid and to include non-euclidean geometies as valid (but
probably not both of these!).
Some of Kant's most salient ideas and critiques are still influential today,
and not only among philosophers. Some of his ideas perennially get
rediscovered by people who aren't aware they are fishing in waters already
stocked by Kant. I'm reminded of something I once read in a book by P
Sorotkin, a (once?) well known sociologist. After a passage in which he
described his views on the nature of causality, he made a remark in a footnote
along these lines: It is remarkable that the ancient philosoper Aristotle
formulated much the same ideas about causality .....
I'm not arguing that Kant (or Aristotle!) was right about everything, or even
that he (or they) settled anything in particular, but the way he (they) tried
to settle things seems to me to have perennial appeal and relevance -- so far,
anyway.
Gordon Fisher
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