David Hilbert

Works

• 1899Foundations of Geometry

  Axiomatic construction of Euclidean geometry from five groups of axioms for undefined “points”, “lines” and “planes”; consistency and independence of the axioms are proved by models. Frege objected that axioms must be true of given objects.

• 1900Mathematical Problems

  Lecture at the International Congress of Mathematicians in Paris; the printed version sets out 23 open problems (the talk itself presented ten), among them the continuum hypothesis and the consistency of the axioms of arithmetic, in the conviction that in mathematics there is no “ignorabimus”.

• 1928Principles of Mathematical Logic

  Textbook written with Wilhelm Ackermann from Hilbert’s lectures: it sets out propositional and first-order logic and poses two open questions – whether first-order logic is complete and whether there is a decision procedure – answered by Gödel (1929) and by Church and Turing (1936).

Position

Hilbert is on the map for the foundations of mathematics and logic. His “Foundations of Geometry” (1899) built Euclidean geometry on groups of axioms whose basic terms – point, line, plane – receive their meaning only from the axioms; he proved the independence of axioms by models and held that the consistency of axioms suffices for the existence of the objects they define. Frege objected in letters and two articles that axioms must be true statements about given objects; the exchange is a classic of the philosophy of mathematics. In response to the paradoxes of set theory and to Brouwer’s intuitionism, Hilbert proposed in the 1920s to formalise mathematics completely and to prove the consistency of these formal systems by “finitary” means, reasoning about finite strings of signs that he regarded as secure as elementary arithmetic. Infinitary mathematics would thus be justified as an instrument, and no one should drive mathematicians “from the paradise that Cantor created”. With Ackermann he also posed the decision problem: is there a mechanical procedure that decides whether a formula of logic is valid? Gödel’s incompleteness theorems (1931) showed that the consistency of arithmetic cannot be proved within arithmetic itself, so the programme could not succeed in its original form; Church and Turing (1936) answered the decision problem in the negative. Modified versions of the programme continue in proof theory. His Königsberg address of 1930 ended with the words “We must know – we will know”, against the “ignorabimus” of Emil du Bois-Reymond.

Life & context

David Hilbert was born on 23 January 1862 in Königsberg in East Prussia (according to some sources in nearby Wehlau) and studied mathematics there, where Hermann Minkowski and Adolf Hurwitz became his friends; he took his doctorate in 1885 under Ferdinand Lindemann. He became professor in Königsberg in 1893 and moved to Göttingen in 1895, where he stayed until his retirement in 1930 and helped make the university a centre of mathematics; Edmund Husserl was his colleague there from 1901 to 1916. At the International Congress of Mathematicians in Paris in 1900 he spoke on mathematical problems; the printed version of the lecture set out 23 problems that shaped much research in the century that followed. He supported Emmy Noether’s habilitation against resistance in the faculty. In the 1920s he worked with Paul Bernays and Wilhelm Ackermann on the foundations of mathematics. After 1933 many of his colleagues and students were driven out by the Nazi regime. He died in Göttingen on 14 February 1943.

Further main works

• Grundlagen der Geometrie (1899)
• Mathematische Probleme (Paris lecture, 1900)
• Neubegründung der Mathematik (1922)
• Über das Unendliche (1926)
• Grundzüge der theoretischen Logik (with W. Ackermann, 1928)
• Grundlagen der Mathematik (with P. Bernays, 2 vols., 1934–39)