17.655 history of 20C mathematics

From: Humanist Discussion Group (by way of Willard McCarty willard.mccarty@kcl.ac.uk)
Date: Wed Feb 18 2004 - 03:32:37 EST

               Humanist Discussion Group, Vol. 17, No. 655.
       Centre for Computing in the Humanities, King's College London
                     Submit to: humanist@princeton.edu

         Date: Wed, 18 Feb 2004 08:25:51 +0000
         From: Anne Mahoney <amahoney@perseus.tufts.edu>
         Subject: Re: 17.647 history of 20C mathematics?

Willard --

No one has yet mentioned "The Honors Class: Hilbert's Problems and
their Solvers," by Ben Yandell (A. K. Peters: 2002). It addresses
precisely your questions, about the influence of Hilbert's problems on
20th-c. mathematics. Another book on the subject came out at the same
time, Jeremy Gray's "The Hilbert Challenge" (Oxford: 2000), but I have
not read it. Both are reviewed together in Notices of the AMS for
September 2002; see http://www.ams.org/notices/200208/rev-blank.pdf for
the review.

Hilbert's address was re-printed (in English) on its centenary in Bull.
AMS 37 (2000), 407-436; it's available on line at http://www.ams.org.

I'm not sure I'd agree that "Hilbert's project ran aground" -- it's just
that the safe harbor wasn't where he thought it was. The goal was to
systematize all of mathematics. In a very strict sense this isn't
possible (that's Gödel's theorem), but this result in itself is already
a form of systematization. Model theory, then, is about what you can do
in the spaces opened up by the incompleteness theorem. Take a model of
Euclid's first 4 postulates, but with a different version of the 5th;
what do you get? Is it interesting? Does it correspond to anything in
the real world? Yes, in fact -- and there's 20th c. geometry for you.
(Well, roughly.)

--Anne Mahoney
Tufts University

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