- Solving several important problems in the theory of invariants. Hilberts basis theorem solved the principal problem in the 1800s invariant theory by showing that any form of a given number of variables and of a given degree has a finite, yet complete system of independent rational integral invariants and covariants.

- Unifying the field of algebraic number theory with his 1897 treatise
*Zahlbericht*(literally "report on numbers").

- Providing the first correct and complete axiomatization of geometry with his 1899 book
*Grundlagen der Geometrie*("Foundations of Geometry").

- His suggestion in 1920 that mathematics be formulated on a solid logical foundation (by showing that all of mathematics follows from a system of axioms, and that that axiom system is consistent). Unfortunately, Gödel's Incompleteness Theorem showed that his grand plan was impossible.

- [Hilberts paradox]?, (also called the infinite hotel paradox), a musing about strange properties of the infinite.

- Laying the foundations of functional analysis by studying integral equations and Hilbert spaces.

- Putting forth a list of 23 unsolved problems in the Paris conference of the [International Congress of Mathematicians]? in 1900.

Hilbert's 23 problems are:

- Problem 1 The continuum hypothesis
- Problem 2 Are the axioms of arithmetic consistent?
- [Problem 3]? Can two tetrahedra be proved to have equal volume (under certain assumptions)?
- [Problem 4]? Construct all metrics where lines are geodesic?s
- Problem 5 Are continuous groups automatically differential groups?
- [Problem 6]? Axiomatize all of physics
- [Problem 7]? Is a
^{b}transcendental, for algebraic a and irrational b? - Problem 8 The Riemann hypothesis
- [Problem 9]? Find most general law of reciprocity in any algebraic number field
- Problem 10 Determination of the solvability of a [diophantine equation]?
- [Problem 11]? Quadratic forms with algebraic numerical coefficients
- [Problem 12]? Algebraic number field extensions
- [Problem 13]? Solve all 7-th degree equations using functions of two arguments
- [Problem 14]? Proof of the finiteness of certain complete systems of functions
- [Problem 15]? Rigorous foundation of Schubert's enumerative calculus
- [Problem 16]? Topology of algebraic curves and surfaces
- [Problem 17]? Expression of definite rational function as quotient of sums of squares
- [Problem 18]? Is there a non-regular, space-filling polyhedron?
- [Problem 19]? Are the solutions of Lagrangians always analytic?
- [Problem 20]? Do all variational problems with certain boundary conditions have solutions?
- [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 calculus of variations

Currently problems 1, 2, 3, 5, 10, 14, 21 are solved and 7, 8, 9 are not.(*what about problems 4,6,11,12,13,15,16,17,18,19,20,21,22,23 ? PME link doesn't say anything about them or is vague*)

Further information:

- Listing of the 23 problems, with descriptions of which have been solved [Prime Math Encyclopedia]
- A transcript of Hilbert's 1900 address, with a useful table of contents, and a listing of books addressing each question, is available at [http://aleph0.clarku.edu/~djoyce/hilbert/toc.html]
- [MacTutor Hilbert biography]

/Talk