From mechanical to electromagnetic reductionism
1910–1914
pp. 227-285
in: , David Hilbert and the axiomatization of physics (1898–1918), Berlin, Springer, 2004Abstract
Minkowski died on January 12, 1909, at the age of 44 of a sudden and violent attack of appendicitis. The loss of his good friend enormously affected Hilbert. He had just returned to full creative activity, solving a classical, long-standing open problem in number theory, the so-called Waring's problem. In content, this problem was not too distant from Minkowski's own early investigations, as it deals with the possibility of representing any given integer as a sum of a specified number of powers of a certain kind (e.g., as a sum of four squares, as a sum of nine cubes, as a sum of 19 fourth powers, etc.). Hilbert's solution, however, was thoroughly analytical in character and, curiously enough, it was triggered by an idea recently introduced, but not fully elaborated, by the third side of the old Königsberg triangle: Hurwitz. Very much like Hilbert's first solution of the Gordan problem, this one was not a constructive one. The extremely complicated proof, which lacked the kind of conceptual clarity typical of Hilbert's work in algebraic number fields, implied the existence, for each positive integer exponent n, of an integer G(n) such that any given integer is a sum of at most G(n) non-negative n-th powers. The proof used an identity in 25-fold multiple integrals, and did not provide a way to actually find the value of G(n). 3 This fact, of course, did by no means diminish the magnitude of the achievement, which was soon adopted by other, leading mathematicians working in this field who used it later as a starting point for important improvements over the next years.4