Hilbert & Poincaré
The two titans who defined the mathematical landscape of the 20th century
19.1 David Hilbert
David Hilbert (1862–1943) presented 23 problems at the 1900 International Congress that guided mathematics for a century. He made fundamental contributions to invariant theory, algebraic number theory, functional analysis (Hilbert spaces), and the foundations of mathematics.
19.2 Henri Poincaré
Henri Poincaré (1854–1912), the last universalist, founded algebraic topology, discovered chaos in deterministic systems, and came close to special relativity. His topological conjecture was not proved until 2003 by Perelman.
19.3 The Foundational Crisis
Three competing philosophies emerged: logicism (Russell), formalism (Hilbert), and intuitionism (Brouwer). Hilbert's program to prove mathematics consistent, complete, and decidable seemed within reach — until Gödel shattered it in 1931.