## David Hilbert's 24 Problems

David Hilbert gave a talk at the International Congress of Mathematicians in Paris on 8 August 1900 in which he described 10 from a list of 23 problems. The full list of 23 problems appeared in the paper published in the

*Proceedings*of the conference. The talk was delivered in German but the paper in the conference proceedings is in French. Hilbert began his talk with these words (translated into English):-

He ended his talk with these words (translated into English):-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 towards 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?

A fuller extract from his speech is at THIS LINKThe problems mentioned are merely samples of problems; yet they are sufficient to show how rich, how manifold and how extensive mathematical science is today, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches. ... I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole. ... with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science. The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted prophets and many zealous and enthusiastic disciples.

We have entitled this page "Hilbert's 24 Problems" but have only said that his paper contains 23 problems. The 24th Problem appears in a draft of Hilbert's paper, but he then decided to cancel it.

We do not state these problems in full, merely give the reader the context of each problem:

1. The cardinality of the continuum, including well-ordering.

2. The consistency of the axioms of arithmetic.

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

4. The straight line as shortest connection between two points.

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining a group.

6. The axioms of physics.

7. Irrationality and transcendence of certain numbers.

8. Prime number theorems (including the Riemann hypothesis).

9. The proof of the most general reciprocity law in arbitrary number fields.

10. Decision on the solvability of a Diophantine equation.

11. Quadratic forms with any algebraic coefficients.

12. The extension of Kronecker's theorem on Abelian fields to arbitrary algebraic fields.

13. Impossibility of solving the general seventh degree equation by means of functions of only two variables.

14. Finiteness of systems of relative integral functions.

15. A rigorous foundation of Schubert's enumerative calculus.

16. Topology of real algebraic curves and surfaces.

17. Representation of definite forms by squares.

18. The building up of space from congruent polyhedra.

19. The analytic character of solutions of variation problems.

20. General boundary value problems.

21. Linear differential equations with a given monodromy group.

22. Uniformization of analytic relations by means of automorphic functions.

23. The further development of the methods of the calculus of variations.

[24.] The simplicity of proofs (omitted).

2. The consistency of the axioms of arithmetic.

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

4. The straight line as shortest connection between two points.

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining a group.

6. The axioms of physics.

7. Irrationality and transcendence of certain numbers.

8. Prime number theorems (including the Riemann hypothesis).

9. The proof of the most general reciprocity law in arbitrary number fields.

10. Decision on the solvability of a Diophantine equation.

11. Quadratic forms with any algebraic coefficients.

12. The extension of Kronecker's theorem on Abelian fields to arbitrary algebraic fields.

13. Impossibility of solving the general seventh degree equation by means of functions of only two variables.

14. Finiteness of systems of relative integral functions.

15. A rigorous foundation of Schubert's enumerative calculus.

16. Topology of real algebraic curves and surfaces.

17. Representation of definite forms by squares.

18. The building up of space from congruent polyhedra.

19. The analytic character of solutions of variation problems.

20. General boundary value problems.

21. Linear differential equations with a given monodromy group.

22. Uniformization of analytic relations by means of automorphic functions.

23. The further development of the methods of the calculus of variations.

[24.] The simplicity of proofs (omitted).