Mathematics Made in Germany: On the Background to Hilbert’s Paris Lecture

Abstract

Submissions should be uploaded to http://tmin.edmgr.com or to be sent directly to David E. Rowe, rowe@mathematik.uni-mainz.de M uch has been written about the famous lecture on ‘‘Mathematical Problems’’ (Hilbert 1901) that David Hilbert delivered at the Second International Congress of Mathematicians, which took place in Paris during the summer of 1900 (Alexandrov 1979, Browder 1976). Not that the event itself evoked such great interest, nor have many writers paid particularly close attention to what Hilbert had to sayon thatoccasion.Whatmattered—both for the text and the larger context—came afterward. Mathematicians remember ICM II and Hilbert’s role in it for just one reason: this was the occasion when he unveiled a famous list of twenty-three problems, a challenge to thosewhowished tomakenames for themselves in the coming century (Gray 2000). These ‘‘Hilbert Problems’’ and ‘‘their solvers’’ have long served as a central theme around which numerous stories have been written (Yandell 2002, Rowe 2004a). They have also served as a convenient peg for describing important mathematical developments of the twentieth century (Struik 1987). Yet relatively little has been written about the events that led up to Hilbert’s lecture or the larger themes he set forth in themainbodyof his text. With this in mind, the present essay aims to address these less familiar parts of the story by sketching some of the relevant historical and mathematical background. In accounts of Hilbert’s life, his Paris lecture has rightly been seen asmarking the great turning point in his spectacular career (Blumenthal 1935, Reid 1970). That career began quietly enough in Hilbert’s native Konigsberg where he emerged as an expert on algebraic invariants; the famous finiteness theorems for arbitrary systems of invariants stem from the late 1880s and early 1890s (Rowe 2003, 2005). Then, beginning around 1893, he broadened his terrain by taking in the theory of algebraic numberfields,work that culminatedaround1899, two years after the publication of his Zahlbericht (Hilbert 1998). In the meantime, Felix Klein managed to arrange Hilbert’s appointment to a prestigious chair in Gottingen, where he taught from 1895 until his retirement in 1930. Gottingen’s subsequent success had everything to do with Hilbert’s intriguing role within the context of a remarkable research community (Rowe 2004b). Indeed, when Klein first brought him toGottingen, hedid so because he regarded Hilbert as the foremost pure mathematician of his generation and hence the ideal person to counter Berlin’s traditional strength. No one could have anticipated—with the possible exception of Hilbert’s good friend, Hermann Minkowski—how thoroughly Gottingen would come to dominate German mathematics after 1900 (for an overview of the Berlin-Gottingen rivalry, see Rowe 2000a). During the 1880s, when Hilbert began his studies in his nativeKonigsberg, it wasnoeasymatter formathematicians in Germany tomeetwithoneanother todiscuss theirworkor just to gossip about the latest news in the profession. The only forumat that time for regular formal gatheringswas the annual