David Hilbert of the University of Göttingen speaks at the International Congress of Mathematicians at the Sorbonne in Paris. He lists 23 unproven problems, which he believes will shape the course of future mathematical research. Hilbert’s 23 problems are: 1) (Partly accepted) The continuum hypothesis; 2) (Partly accepted) Can the set of axioms of arithmetic be shown to be consistent?; 3) (Solved) Given two polyhedra of the same volume, can they both be cut into the same set of smaller polyhedra?; 4) (Too vague) Construct all metrics in which lines are geodesics; 5) (Partly accepted) Are all continuous groups automatically differential groups?; 6) (Too vague) Axiomatize all of physics; 7) (Partly Solved) Given a ≠ 0.1 algebraic and b irrational, is the number ab always transcendental?; 8) (Open) Prove the Riemann hypothesis; 9) (Partly accepted solution) Generalize the reciprocity law in any algebraic number field; 10) (Unsolvable) Determine the general solutions of a Diophantine equation; 11) (Solved) Extension of the results of quadratic forms in the case of an algebraic coefficient; 12) (Open) Extend Kronecker’s Theorem on Abelian fields to arbitrary algebraic fields; 13) (Solved) Solution of the general seventh degree equation using functions with only two arguments; 14) (Solved) Prove the finiteness of some complete systems of functions; 15) (Partly accepted solution) Rigorous foundation of Schubert’s enumerative calculus; 16) (Too vague) Topology of curves and algebraic surfaces; 17) (Solved) Expression of rational functions defined as quotients of sums of squares; 18) (Partly accepted solution) Is there a non-regular, space-filling polyhedron? What is the densest packing of spheres?; 19) (Solved) Are solutions of Lagrangians always analytic?; 20) (Solved) Do all variational problems with given boundary conditions have solutions?; 21) (Partly accepted solution) Proof of the existence of linear differential equations having a prescribed monodromic group; 22) (Partly accepted solution) Uniformization of analytic relations by means of automorphic functions; 23) (Too vague) Further development of the calculus of variations
