Hilbert’s twenty-three problems still pace modern math
In 1900 David Hilbert published twenty-three problems meant to match Henri Poincaré’s stature and set an agenda. A 1902 English Bulletin translation spread the list. Some items now have community-accepted solutions; others remain contested, partial or open—including flavors of the continuum problem.
Only ten of the problems were actually spoken aloud, in Hilbert's address to the International Congress of Mathematicians at the Sorbonne on August 8; the full list appeared later, and Mary Frances Winston Newson produced the English version. There was nearly a 24th, on simplicity criteria in proof theory, which Hilbert cut and which the historian Rüdiger Thiele rediscovered in the manuscript notes in 2000.
The topics range widely and so does their precision. The third, about cutting polyhedra of equal volume into matching pieces, was the first solved; the eighth contains the Riemann hypothesis and is still open. The tenth asked for a procedure deciding whether any Diophantine equation has integer solutions, and in 1970 Yuri Matiyasevich, finishing work by Julia Robinson, Hilary Putnam and Martin Davis, proved no such procedure can exist. The first problem earned Paul Cohen a Fields Medal in 1966. The fourth, on foundations of geometry, and the sixth, on putting physics on axioms, may never be answerable at all, and the 23rd was meant only as a nudge toward the neglected calculus of variations.
The second problem, a finitistic proof that arithmetic is consistent, sat at the heart of Hilbert's program of formalizing mathematics after Frege and Russell. Gödel's second incompleteness theorem showed such a proof is impossible in a precise sense, yet Hilbert, who lived twelve more years, apparently never answered it in writing. He had insisted that mathematics contains no ignorabimus, no question whose answer must stay forever unknown, although he allowed that a solution could take the form of an impossibility proof.
Other famous lists have rarely matched this influence. André Weil's four conjectures of the late 1940s, finally settled by Dwork, Grothendieck and Deligne, were closer in scope to a single Hilbert problem. Paul Erdős attached cash prizes to his many problems, and 22 of William Thurston's 24 questions from 1982 fell within three decades.
Source: Hilbert's problems
