Hilbert's 23 Problems

In 1900, David Hilbert presented 23 unsolved problems at the International Congress of Mathematicians in Paris — a bold attempt to set the agenda for 20th-century mathematics. Many have been solved: the 4th (non-Euclidean geometry), 14th (algebraic invariants), 17th (positive polynomials as sums of squares). Some remain open: the 8th (Riemann Hypothesis) and 6th (axiomatise physics). Gödel's incompleteness theorems dealt a fatal blow to the 2nd problem — proving the consistency of arithmetic — in 1931. Hilbert's list shaped an entire century.