In 1900 Professor David Hilbert delivered a lecture before the International Congress of Mathematicians in Paris. In his lecture he outlined the following 23 problems that he believed should be tackled in the following century.
1. Cantor's problem of the cardinal number of the continuum
2. The compatibility of the arithmetical axioms
3. The equality of two volumes of two tetrahedra of equal bases and equal altitudes
4. Problem of the straight line as the shortest distance between two points
5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group
6. Mathematical treatment of the axioms of physics
7. Irrationality and transcendence of certain numbers
8. Problems of prime numbers
9. Proof of the most general law of reciprocity in any number field
10. Determination of the solvability of a diophantine equation
11. Quadratic forms with any algebraic numerical coefficients
12. Extension of Kroneker's theorem on abelian fields to any algebraic realm of rationality
13. Impossibility of the solution of the general equation of the 7th degree by means of functions of only two arguments
14. Proof of the finiteness of certain complete systems of functions
15. Rigorous foundation of Schubert's enumerative calculus
16. Problem of the topology of algebraic curves and surfaces
17. Expression of definite forms by squares
18. Building up of space from congruent polyhedra
19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
20. The general problem of boundary values
21. Proof of the existence of linear differential equations having a prescribed monodromic group
22. Uniformization of analytic relations by means of automorphic functions
23. Further development of the methods of the calculus of variations
Click here for further details of these 23 problems and for a full transcript of Hilbert's famous lecture.
