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Hilbert and his twenty-four problems. (English) Zbl 1086.01028
Van Brummelen, Glen (ed.) et al., Mathematics and the historian’s craft. The Kenneth O. May Lectures. New York, NY: Springer (ISBN 0-387-25284-3/hbk). CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 21, 243-295 (2005).
First the biography and work of Hilbert, of a “man of problems” is shortly summarized, with several illustrating figures. Then follows the part: “How did Hilbert’s Paris talk come about?”, where some excerpts from the correspondence with Minkowski and Hurwitz are given. In the part “On the problems” all 24 problems are analyzed and divided into four groups: Actual Lecture (7, 8, 13, 16; 19, 21, 22). Published Version (1, 2, 6; 20, 23; 3, 4, 13, 18; 3, 9, 10, 11, 12, 14, 17), Cancelled (24). Some later compilations of problems in books and papers by other authors in 1949–1998 are listed, specially the “Millennium Prize Problems” by the Clay Institute (2000) are mentioned; also the important problems from the 17th and 18th centuries are reminded of. Some remarks are given on Hilbert’s philosophy of mathematics: It is stressed that Hilbert’s device was “Noscemus”, which means “We can and shall know”.
The final part is “The 24th problem”. As far as the author knows the cancelled 24th problem has remained unpublished until now. He came across the problem in the Hilbert Nachlass at the Niedersächsische Staats- und Universitätsbibliothek Göttingen, Handschriftenabteilung, while he was studying Hilbert’s notices on the 23rd problem. The entry in Hilbert’s notebook begins: “The 24th problem in my Paris lecture was to be: Criteria of simplicity, or proof of the greatest simplicity of certain proofs. Develop a theory of the method of proof in mathematics in general.” The author will not risk more than a few disjointed remarks, and refers for more details to his paper in [Am. Math. Mon. 110, 1–24 (2003; Zbl 1031.01011)]. As an example there is indicated a criterion for simplicity of geometric constructions proposed by Émile Lemoine (1840–1912). The well-known Four-Color Problem is considered as to the complexity of technical details for proofs. Hilbert’s notebook is concluded with: “Mathematics stalks on earthly ground and at the same time touches the divine firmament”.
For the entire collection see [Zbl 1078.01004].

01A60 History of mathematics in the 20th century
01A70 Biographies, obituaries, personalia, bibliographies