Brouwer–Hilbert controversy participant David Hilbert
Background
The background for the controversy was set with David Hilbert's axiomatization of geometry in the late 1890s.
In his biography of Kurt Gödel, John W. Dawson, Jr observed that "partisans of three principal philosophical positions took part in the debate" – the logicists (Gottlob Frege and Bertrand Russell), the formalists (David Hilbert and his school of collaborators), and the constructivists (Henri Poincaré and Hermann Weyl); within this constructivist school was the radical self-named "intuitionist" L. E. J. Brouwer.Brief history of Brouwer and intuitionism
Brouwer in effect founded the mathematical philosophy of intuitionism as a challenge to the then-prevailing formalism of David Hilbert and his collaborators Paul Bernays, Wilhelm Ackermann, John von Neumann and others. As a variety of constructive mathematics, intuitionism is a philosophy of the foundations of mathematics which rejects the law of excluded middle in mathematical reasoning.
After completing his dissertation, Brouwer decided not to share his philosophy until he had established his career. By 1910, he had published a number of important papers, in particular the fixed-point theorem. Hilbert admired Brouwer and helped him receive a regular academic appointment in 1912 at the University of Amsterdam. Then, Brouwer decided to return to intuitionism.In the later 1920s, Brouwer became involved in a public controversy with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.Origins of disagreement
The nature of Hilbert's proof of the Hilbert basis theorem from 1888 was controversial. Although Leopold Kronecker, a constructivist, had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea" — in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object".Brouwer was not convinced and in particular objected to the use of the law of excluded middle over infinite sets. Hilbert would respond: "Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists."