Hilbert, David

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Hilbert, David,

(1862–1943), German mathematician, professor at Königsberg (1886–95) and Göttingen (1895–1930), b. Königsberg, Germany. His proof of the theorum of invariants (1890) supplanted earlier computational work on the subject and paved the way for modern algebraic geometry. His report on algebraic number theory (1897) inspired many of the subsequent developments in that area, and he also made significant contributions to multivariable calculus, theoretical physics, mathematical logic, and Euclidean geometry. At the International Mathematical Congress in Paris (1900), he put forth 23 research problems, which launched much of the research agenda of 20th-century mathematics; some of the problems remain unsolved.

Hilbert, David


Born Jan. 23, 1862, in Velau, near Königsberg; died Feb. 14, 1943, in Göttingen. German mathematician.

Hilbert graduated from the University of Königsberg and was a professor there from 1893 to 1895. From 1895 to 1930 he was a professor at the University of Göttingen, although he continued to lecture there until 1933. After Hitler came to power (1933), Hilbert lived in Göttingen but was not involved in university affairs. His work strongly influenced the development of many branches of mathematics, and his activities at the University of Göttingen contributed considerably to Göttingen’s reputation in the first third of the 20th century as one of the world’s primary centers of mathematical thought. Hilbert supervised the dissertations of a large number of noted mathematicians, including H. Weyl and R. Courant.

Hilbert’s work falls into sharply delineated periods, each devoted to some aspect of mathematics: (1) the theory of invariants (1885-93), (2) the theory of algebraic numbers (1893-98), (3) the foundations of geometry (1898-1902), (4) Dirichlet’s principle and the related problems of the calculus of variations and differential equations (1900-06), (5) the theory of integral equations (1900-10), (6) the solution of Waring’s problem in number theory (1908-09), (7) the bases of mathematical physics (1910-22), and (8) the logical bases of mathematics (1922-39).

Hilbert’s studies in the theory of invariants culminated a stormy period of development in that field of mathematics in the second half of the 19th century. He proved the basic theorem of the existence of a finite basis for the system of invariants. His work in the theory of algebraic numbers transformed that field of mathematics and became the starting point for its future development. Hilbert’s solution of Dirichlet’s problem laid the basis for working out the so-called direct methods in the calculus of variations. The theory of integral equations with a symmetrical kernel that he constructed constituted one of the bases of modern functional analysis and especially of the spectral theory of linear operators. His Foundations of Geometry (1899) became the model for future work on the axiomatic structure of geometry.

By 1922 Hilbert had devised a much more extensive plan for formulating the basis of mathematics through its complete formalization with the subsequent “metamathematical” proof of the noncontradictibility of the formalized mathematics. The two-volume Foundations of Mathematics, written by Hilbert with P. Bernays, in which this concept is developed in detail, was published in 1934 and 1939. Hilbert’s initial hopes in this area were not borne out: the problem of the noncontradictibility of formalized mathematical theories proved to be deeper and more difficult than he anticipated. But all further work on the logical bases of mathematics has largely followed the path outlined by Hilbert and has used the concepts that he formulated. Although he considered the complete formalization of mathematics necessary from the logical point of view, he at the same time believed in the strength of creative mathematical intuition. He was a grand master of a highly visual exposition of mathematical theories. Visual Geometry, written with S. Cohn-Vossen, is an outstanding work on that subject. Hilbert’s works are characterized by confidence in the unlimited strength of human reason and a belief in the unity of the mathematical science and the unity of mathematics and the natural sciences. Hilbert’s complete works, published under his supervision (1932-35), ends with the article “Understanding Nature,” which concludes with the statement “We must know—we shall know.”


Gesammelte Abhandlungen, vols. 1-3. Berlin, 1932-35.
In Russian translation:
Osnovaniia geometrii. Moscow-Leningrad, 1948.
Osnovy teoreticheskoilogiki. Moscow, 1947. (With W. Ackermann.)
Nagliadnaia geometriia, 2nd ed. Moscow-Leningrad, 1951. (With S. Cohn-Vossen.)


Problemy Gil’berta. An anthology edited by P. S. Aleksandrov. Moscow, 1969.
Weyl, H. “David Hilbert and His Mathematical Work.” Bulletin of the American Mathematical Society, 1944, vol. 50, pp. 612-54.
Reid, C. Hilbert. Berlin, 1970.


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