Hilbert's Mathematical Problems

Hilberts Probleme (deutsch)
Photo In 1900, DAVID HILBERT outlined 23 mathematical problems to the International Congress of Mathematicians in Paris. His famous address influenced, and still today influence, mathematical research all over the world.

The original address Mathematische Probleme appeared in Göttinger Nachrichten, 1900, and in Archiv der Mathematik und Physik, 1901. The French translation by M. L. Laugel Sur les problèmes futurs des mathématiques appeared in Compte Rendu du Deuxième Congrès International des Mathématiciens, Gauthier-Villars, Paris, 1902.

Mary Winston Newson translated Hilbert's address into English for Bulletin of the American Mathematical Society, 1902. A reprint of which appeared in Mathematical Developments Arising from Hilbert Problems, edited by Felix E. Browder, American Mathematical Society, 1976. There is also a collection on Hilbert's Problems, edited by P. S. Alexandrov, 1969, in Russian, which has been translated into German.

Further Reading:
Ivor Grattan-Guinness: A Sideways Look at Hilbert's Twenty-three Problems of 1900 (pdf file), Notices of the AMS, 47, 2000.
Jeremy J.Gray: We must know, we shall know; a History of the Hilbert Problems, European Math. Soc.: Newsletter 36, and Oxford Univ. Press, 2000.

David Joyce, Clark University, produced a list of Hilbert's problems and a web version of Hilbert's 1900 address in March 1997.

Biographical notes on Hilbert are contained in the History of Mathematics Archives, School of Mathematics and Statistics, University of St Andrews, Scotland.

Problem 1 Cantor's problem of the cardinal number of the continuum. Cohen 1963/64: the continuum hypothesis is formally indecidable in the Zermelo-Fraenkel set theory.
Problem 2 The compatibility of the arithmetical axioms.
Problem 3 The equality of two volumes of two tetrahedra of equal bases and equal altitudes. Solved by Dehn in 1902.
Problem 4 Problem of the straight line as the shortest distance between two points.
Problem 5 Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
Problem 6 Mathematical treatment of the axioms of physics.
Problem 7 Irrationality and transcendence of certain numbers. Gelfond-Schneider 1934, Baker 1966.
Problem 8 Problems of prime numbers. The distribution of primes and the Riemann hypothesis.
Problem 9 Proof of the most general law of reciprocity in any number field. See class field theory developed by Hilbert, Takagi, Artin, and others; norm rest symbols computed by Shafarevich in 1950, and further developments as in algebraic K-theory.
Problem 10 Determination of the solvability of a diophantine equation. Solved negatively by Matiyasevich in 1970.
Problem 11 Quadratic forms with any algebraic numerical coefficients. The Hasse principle 1923/24; arithmetic and algebraic theory of quadratic forms.
Problem 12 Extension of Kroneker's theorem on abelian fields to any algebraic realm of rationality.
Problem 13 Impossibility of the solution of the general equation of the 7-th degree by means of functions of only two arguments.
Problem 14 Proof of the finiteness of certain complete systems of functions. First counter example by Nagata in 1958.
Problem 15 Rigorous foundation of Schubert's enumerative calculus.
Problem 16 Problem of the topology of algebraic curves and surfaces.
Problem 17 Expression of definite forms by squares. Solved by Artin in 1927.
Problem 18 Building up of space from congruent polyhedra. Crystallographic groups, fundamental domains, sphere packing problem.
Problem 19 Are the solutions of regular problems in the calculus of variations always necessarily analytic?
Problem 20 The general problem of boundary values.
Problem 21 Proof of the existence of linear differential equations having a prescribed monodromic group.
Problem 22 Uniformization of analytic relations by means of automorphic functions.
Problem 23 Further development of the methods of the calculus of variations.

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Copyright © 2000 by Ina Kersten
kersten@mathematik.uni-bielefeld.de