David Hilbert (1862–1943) is the towering figure of early 20th-century mathematics. He worked in virtually every field of pure mathematics, led the influential Göttingen school, and set the agenda for the century with his famous list of 23 problems presented in 1900.
Life
Hilbert was born in Königsberg (then Prussia, now Kaliningrad, Russia), where he also received his doctorate in 1885. He taught at Königsberg until 1895, when he moved to Göttingen — at the time the world’s premier mathematical center — where he remained until his retirement.
Under Hilbert, Göttingen attracted the best mathematical talent in Europe: Hermann Weyl, Emmy Noether, John von Neumann, Richard Courant, Hans Lewy. The 1920s and early 1930s saw Göttingen at the peak of its influence — until the Nazi purges of 1933 scattered its Jewish and politically suspect faculty across Europe and America. When a Nazi official asked Hilbert at a banquet in 1934 whether the mathematics institute had suffered from the removal of Jewish influence, Hilbert replied: “It hasn’t suffered, Minister. It just doesn’t exist any more.”
Contributions
Foundations of geometry
Hilbert’s 1899 book Grundlagen der Geometrie (“Foundations of Geometry”) gave a modern axiomatic treatment of Euclidean geometry, improving on Euclid by filling in logical gaps and making all assumptions explicit. The book is a model of the axiomatic method and was immensely influential in 20th-century mathematics’ turn toward abstract foundations.
The 23 problems
On 8 August 1900, Hilbert gave a lecture at the International Congress of Mathematicians in Paris titled “The Problems of Mathematics.” He presented a list of 23 important unsolved problems, covering nearly every area of the field.
The problems ranged across:
- Foundations: the continuum hypothesis (Problem 1), the consistency of arithmetic (Problem 2)
- Number theory: including the distribution of primes and the Riemann hypothesis (Problem 8)
- Algebra: including the solvability of Diophantine equations (Problem 10)
- Analysis and physics: axiomatization of physics (Problem 6)
By setting this agenda, Hilbert effectively chose the direction of 20th-century mathematics. Solving a Hilbert problem, or making substantial progress on one, became one of the highest honors in the field.
Hilbert spaces and functional analysis
Hilbert’s work on integral equations introduced what are now called Hilbert spaces — infinite-dimensional vector spaces with an inner product. These became the mathematical framework for quantum mechanics and modern functional analysis.
Hilbert’s program
Starting around 1920, Hilbert launched an ambitious program to place all of mathematics on rigorous formal foundations. He wanted to prove, within a finite formal system, that:
- Mathematics is consistent (no contradictions)
- Mathematics is complete (every true statement is provable)
- Mathematics is decidable (there is an algorithm to determine truth)
In 1931, Kurt Gödel shattered the program. His incompleteness theorems showed that any consistent formal system rich enough to contain arithmetic must contain statements that are true but unprovable within the system. Hilbert’s program, in its strong form, was impossible.
Yet the program’s legacy is immense. It gave birth to modern mathematical logic, model theory, proof theory, and recursion theory — and ultimately to theoretical computer science, via Alan Turing’s 1936 work.
Legacy
Hilbert’s grave in Göttingen bears his own words: Wir müssen wissen — wir werden wissen. “We must know — we will know.” It was a statement of faith in the solvability of mathematical problems, a faith that Gödel complicated but did not destroy.
Hilbert shaped 20th-century mathematics as a whole: its problems, its methods, its aspirations. He is arguably the most influential mathematician of the period.
Known for
- Hilbert's 23 problems (1900)
- Hilbert space (functional analysis)
- Hilbert's program (formalism)
- Foundations of geometry
Frequently asked
What were Hilbert's 23 problems?
A list of unsolved problems Hilbert presented at the 1900 International Congress of Mathematicians in Paris. They set the research agenda for much of the 20th century. Several have been solved (like the first, about the continuum hypothesis — sort of), several remain open (like the eighth, which includes the Riemann hypothesis).
What is Hilbert's program?
An attempt, starting around 1920, to prove that all of mathematics could be derived from a finite set of axioms and that the system would be provably consistent. Kurt Gödel's incompleteness theorems (1931) showed this was impossible in the strong form Hilbert had hoped for — a historic moment in the foundations of mathematics.