Subject: Re: [HM] Hilbert's Foundations of Geometry
From: James T. Smith (smith@math.sfsu.edu)
Date: Fri Jan 07 2000 - 13:01:04 EST
Here are some citations from the bibliography of my forthcoming Wiley book,
Methods of geometry. They do discuss Hilbert's intent.
Frege, Gottlob. 1980. Philosophical and mathematical correspondence.
Edited by Gottfried Gabriel, Hans Hermes, Friedrich Kambartel, Christian
Thiel, and Albert Veraart, abridged from the German edition by Brian
McGuinness, translated by Hans Kaal. Chicago: University of Chicago
Press.
Freudenthal, Hans. 1962. The main trends in the foundations of geometry in
the 19th century. In Logic, methodology and philosophy of science:
Proceedings of the 1960 International Congress, edited by Ernest Nagel,
Patrick Suppes, and Alfred Tarski, 613-621. Stanford: Stanford University
Press.
Peckhaus, Volker. 1994. Hilbert's axiomatic programme and philosophy.
In The history of modern mathematics, volume 3, Images, ideas, and com-
munities, edited by Eberhard Knobloch and David E. Rowe, Boston:
Academic Press, 91-111.
Steiner, Hans Georg. 1964. Frege und die Grundlagen der Geometrie.
Mathematisch-physikalische Semesterberichte 10:175-186, 11:35-47.
Toepell, Michael-Markus. 1986. Ueber die Entstehung von David Hilberts
Grundlagen der Geometrie. Goettingen: Vandenhoeck & Ruprecht.
Comments.
1. I'm away from my files, hence can't give further more precise
information. (My book may be out already--it's that close!)
2. I think one needs to be careful with the Hilbert-Frege correspondence.
I don't believe that Hilbert described there what he was really doing:
axiomatic mathematics as we understand it today. I don't know when
Hilbert's expository style caught up with his mathematics. This difficulty
is compounded by the numerous early editions of the Grundlagen, the fact
that Frege published a famously savage review of some edition several years
later, and that Bernays and Hilbert both edited later editions in some of
the critical places. In any case, the appropriate citation for the
Hilbert-Frege correspondence is the German edition, not the one cited here
for an English-language text. And the appropriate citation for the
Grundlagen is the first German edition.
3. The Freudenthal citation here points to a much shorter version of the
paper referred to in Bernd Buldt's earlier HM posting. I provide it for the
benefit of those who don't read German. Freudenthal wrote still another
German version, in a Gedenkband edited by Wilhelm Klingenberg. I can't
remember whether that's entitled Gauss-Gedenkband or Hilbert-Gedenkband.
4. In spite of Freudenthal's absolutely elegant presentation, one should
be careful there, too. For instance, he states in one of those papers that
the Italians took the 1900 International Congress of Philosophers by storm.
You get that impression from Russell's autobiography, too. But close
inspection of the list of attendees reveals that several of the Italians'
papers were delivered in absentia.
Greetings from Hornbrook on the Klamath, where we're hoping for snow.
James T. Smith
Professor of Mathematics
San Francisco State University
smith@math.sfsu.edu
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