On the origins of David Hilbert's ?Grundlagen der Geometrie?

Abstract

Toward the end of the 19th century a remarkable change came about in the field of the foundations of geometry. Whereas geometry had hitherto been based on empirical facts, it was now seen as a purely formal deductive system. Hilbert (1862-1943) perfected this method in his Grundlagen der Geometrie, which first appeared in June 1899 in a Festschrift commemorating the unveiling of the GaussWeber monument in Gottingen. The ninth edition appeared in 1962 for the Hilbert centenary, and the twelfth for the bicentenary of Gauss' birth in 1977. Hilbert's method immediately gave a new direction to mathematical thought in the 20th century. Its impact on contemporaries has been studied and further developed in numerous publications, for instance in the works by A. Schmidt (S 1), Freudenthal (F 1) and van der Waerden (W 1) cited in the foreword of the newer editions of the Grundlagen, However, little is known about the origins of Hilbert's Grundlagen or about the development that led him to it. According to the biographies (Tl ;D2;B1;B2;B3;W3;R1), Hilbert appeared to have worked almost exclusively with algebra and questions concerning the theory of numbers in the years prior to 1899. His publications on the theory of invariants suggest as much. With the Grundlagen der Geometrie, however, he presented to the public a thoroughly mature work about an entirely different subject. Referring to the break with the number-theoretical works of 1897/1898 (GA 1), Weyl has remarked (W 3, 264):