David Hilbert

David Hilbert was a German mathematician from Königsberg who spent most of his career at Göttingen, then a leading centre of world mathematics. He axiomatised geometry, set out an influential list of open problems in 1900, and led a programme to secure mathematics by formalising it completely and proving its consistency by finitary means. Gödel's incompleteness theorems of 1931 showed that the programme could not succeed in its original form.

Key ideas

• Mathematical theories should be set out as complete formal axiom systems.
• The consistency of such systems should be proved by finitary, contentual reasoning.
• Formal symbols can be studied as objects in their own right (metamathematics).
• Mathematics should not be confined to constructive methods; Cantor's transfinite should be defended.

Major works

• Grundlagen der Geometrie (Foundations of Geometry, 1899)
• "Mathematische Probleme" (1900)
• "Über das Unendliche" (On the Infinite, 1926)
• Grundlagen der Mathematik (1934–1939, with Paul Bernays)

In the conversation

• Reading group Reading group: Nagel and Newman, Gödel’s Proof Lars Eriksen · 0 replies
• Debate Intuitionists are right to refuse the law of excluded middle for infinite collections Ana Clara Ferreira · 5 replies
• Debate Giving up excluded middle would cost school mathematics far too much Vikram Sethi · 5 replies