David Hilbert Listed 23 Mathematical Problems in 1900 — and Hid a 24th

On August 8, 1900, the German mathematician David Hilbert stood up in front of the International Congress of Mathematicians at the Sorbonne in Paris and laid out a program. He gave ten unsolved problems aloud — time was limited — and published the full list of twenty-three in the conference proceedings shortly afterward. The translation that brought the list into English in 1902 was by the American Mary Frances Winston Newson, the first American woman to receive a PhD in mathematics from a European university. Hilbert's twenty-three were a deliberate prospectus for the field. They covered foundations (Problem 1 was the continuum hypothesis), number theory (Problem 8, the Riemann hypothesis), geometry, and physics.

A century of mathematical labor followed. Roughly half the problems are now considered solved by consensus. Some, like Problem 1, were resolved unexpectedly: Paul Cohen showed in 1963 that the continuum hypothesis is independent of the standard axioms of set theory — neither provable nor disprovable from them. Several remain open, the Riemann hypothesis most famously. Hilbert reportedly told colleagues that, if awakened from a thousand-year sleep, his first question would be whether anyone had proved it.

The surprising final note arrived in 2000. The German mathematical historian Rüdiger Thiele, going through Hilbert's working notebooks at Göttingen for a centennial paper, found a draft entry for a twenty-fourth problem that Hilbert had crossed out before publication. The problem asked for a rigorous theory of "simplicity" of mathematical proofs — a way to mathematically rank one proof of a theorem as cleaner than another. Hilbert apparently decided the formulation wasn't sharp enough to be a proper problem and dropped it. Modern proof-complexity theory has come around to it on its own.