David Hilbert
   "Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?"

With these words David Hilbert opened the International Congress of Mathematicians in Paris in the year 1900.  Hilbert outlined 23 major mathematical problems that he felt was essential to provide solutions for in the coming new century.

It was perhaps the most influential speech ever given to mathematicians, given by a mathematician, or given about mathematics.

The collection consisted of Cantor's problem of the cardinal number of the continuum (the continuum hypothesis); the problem of the straight line as the shortest distance between two points (alternative geometries); determination of the solvability of a diophantine equation (brought about because of everyone's inability to solve Fermat's Last Theorem); the distribution of primes; and so on.

However, Hilbert's address was more than just a collection of problems. It outlined his philosophy of mathematics and proposed problems important to his philosophy.

Although almost a century old, Hilbert's address is still important and should be read (at least in part) by anyone interested in pursuing research in mathematics.

In 1974 a symposium was held at Northern Illinois University on the Mathematical developments arising from Hilbert problems. A major mathematician discussed progress on each problem and how work on the problem has influenced mathematics. Also, 23 new problems of importance were described. The two-volume proceedings of the symposium was edited by Felix Browder and published by the American mathematical Society in 1976. See also Irving Kaplansky's Hilbert's problems, University of Chicago, Chicago, 1977.

Below is a Table of Contents from which you can view Hilbert's opening address and/or the 23 individual problems themselves.
 
 

Hilbert's 23 Mathematical Problems

Opening Address
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Problem 1 - Cantor's problem of the cardinal number of the continuum.
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Problem 2 - The compatibility of the arithmetic axioms.
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Problem 3 - The equality of two volumes of two tetrahedra of equal bases and equal altitudes.
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Problem 4 - Problem of the straight line as the shortest distance between two points.
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Problem 5 - Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group. (i.e., are continuous groups automatically differential groups?)
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Problem 6 - Mathematical treatment of the axioms of physics.
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Problem 7 - Irrationality and transcendence of certain numbers.
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Problem 8 - Problems (with the distribution) of prime numbers.
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Problem 9 - Proof of the most general law of reciprocity in any number field.
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Problem 10 - Determination of the solvability of a diophantine equation.
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Problem 11 - Quadratic forms with any algebraic numerical coefficients.
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Problem 12 -   Extension of Kronecker's theorem on abelian fields.
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Problem 13 - Impossibility of the solution of the general equation of the 7th degree.
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Problem 14 - Proof of the finiteness of certain complete systems of functions.
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Problem 15 - Rigorous foundation of Schubert's calculus.
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Problem 16 - Problem of the topology of algebraic curves and surfaces.
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Problem 17 - Expression of definite forms by squares.
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Problem 18 - Building space from congruent polyhedra.
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Problem 19 - Are the solutions of regular problems in the calculus of variations always necessarily analytic?
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Problem 20 - The general problem of boundary curves.
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Problem 21 - Proof of the existence of linear differential equations having a prescribed monodromic group.
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Problem 22 - Uniformization of analytic relations by means of automorphic functions.
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Problem 23 - Further development of the methods of the calculus of variations.