On 8 August 1900, at the Second International Congress of Mathematicians in Paris, David Hilbert delivered a lecture titled “The Problems of Mathematics.” In it he presented a list of 23 unsolved problems, covering nearly every area of the field, and argued that the liveliness of a mathematical discipline could be measured by the problems it faced.
The list became the defining research agenda of the 20th century.
The context
Hilbert was 38, at the peak of his powers. His 1899 Grundlagen der Geometrie had just rebuilt Euclidean geometry on modern axiomatic foundations. His earlier work on invariant theory had essentially closed a major area of 19th-century algebra. He was the dominant figure at Göttingen, the world’s leading mathematical center.
He used his authority deliberately. “Who of us,” he asked in the lecture, “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?”
He then named 23 problems that, in his view, would shape the century.
The problems
The full lecture was published in English in 1902. The problems spanned:
Foundations (1–6)
- 1. The continuum hypothesis: is there a set of size strictly between and ?
- 2. Prove the consistency of arithmetic.
- 3. Prove that two tetrahedra of equal volume are necessarily scissors-congruent (decomposable into matching pieces).
- 4. Investigate the foundations of geometry.
- 5. Investigate the concept of continuous groups (Lie groups without differentiability assumptions).
- 6. Axiomatize the physical sciences, starting with probability and mechanics.
Number theory (7–11)
- 7. Show that is transcendental for algebraic and irrational algebraic .
- 8. The Riemann hypothesis and related problems about primes.
- 9. Generalize the reciprocity law.
- 10. Find an algorithm to determine whether a Diophantine equation has integer solutions.
- 11. Investigate quadratic forms over algebraic number fields.
Algebra and geometry (12–18)
- 12. Extend Kronecker’s theorem on abelian extensions of the rationals.
- 13. Is every continuous function of three variables expressible as a superposition of continuous functions of two?
- 14. Are finitely generated invariant rings always finite?
- 15. Rigorize Schubert’s enumerative calculus.
- 16. Investigate the topology of real algebraic curves and surfaces.
- 17. Can every positive semi-definite polynomial be written as a sum of squares of rational functions?
- 18. Investigate space-filling polyhedra.
Analysis and physics (19–23)
- 19. Are solutions of “regular” problems in the calculus of variations always analytic?
- 20. Do all variational problems with boundary conditions have solutions?
- 21. Prove existence of linear differential equations with a given monodromy group.
- 22. Uniformize analytic relations by automorphic functions.
- 23. Further develop the calculus of variations.
How did they fare?
A rough tally:
- Clearly solved: Problems 3, 7, 10, 14, 17, 19, 20, 22. Most were solved within a few decades.
- Answered in unexpected ways: Problem 1 (the continuum hypothesis) was shown by Gödel (1940) and Cohen (1963) to be independent of standard set theory — neither provable nor disprovable. Problem 2 was undercut by Gödel’s incompleteness theorems. Problem 10 was solved negatively: Matiyasevich (1970) showed no such algorithm exists.
- Partially solved: Problems 9, 15, 16, 18, 21.
- Still open: Problem 8 (the Riemann hypothesis) remains the biggest unsolved problem in mathematics. Problem 6 (axiomatizing physics) is arguably not a well-posed mathematical question but continues to inspire work.
Legacy
Hilbert’s list did what he hoped. It focused mathematical attention, structured careers, and provided a shared sense of what mathematics was for. Several of the problems produced entire fields in the course of being solved: the work on Problem 10 created modern recursion theory; the work on Problem 2 produced Gödel’s incompleteness theorems and all of modern mathematical logic.
Hilbert ended his 1900 lecture with a statement that became his epitaph: Wir müssen wissen. Wir werden wissen. “We must know. We will know.” It was a declaration of faith in the solvability of problems, a faith that Gödel and others would complicate but not destroy.
In 2000, the Clay Mathematics Institute announced seven Millennium Prize Problems, explicitly modeled on Hilbert’s list — with the addition of a one-million-dollar reward for each.
Frequently asked
Why is this list so famous?
Because Hilbert used his authority — he was the most influential mathematician of his day — to focus the discipline's attention. Many of the problems were genuinely open questions at the frontier, and solving one became a mark of first-rank achievement. Several Fields Medals were awarded for work on Hilbert's list.
How many have been solved?
It depends on how you count. Some are clearly solved, some have been answered with heavy caveats, some have been resolved in unexpected ways (Problem 1, the continuum hypothesis, turned out to be independent of standard set theory), and a handful remain open — most notably Problem 8, which contains the Riemann hypothesis.