David Hilbert (1862–1943) was a German mathematician so influential that for a while the town of Göttingen was the centre of the mathematical universe, and he was its king. He worked across almost every branch of the subject, and in 1900 he stood up and handed the next century its to-do list.

The idea that changed things

Hilbert's name is attached to a dazzling range of ideas — Hilbert spaces underpin quantum mechanics, and his 23 famous problems set the agenda for twentieth-century maths. He was also a champion of Cantor's then-controversial theory of infinite sets, declaring "no one shall expel us from the paradise Cantor has created." Much of that paradise turns on the idea of a countable infinity — the notion he dramatised with his most beloved thought experiment.