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Humanist Discussion Group, Vol. 17, No. 655.

Centre for Computing in the Humanities, King's College London

www.kcl.ac.uk/humanities/cch/humanist/

www.princeton.edu/humanist/

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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