Much of the controversy took place while both were involved with Mathematische Annalen, the leading mathematical journal of the time, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board.
In his biography of Kurt Gödel, John W. Dawson, Jr, observed that "partisans of three principal philosophical positions took part in the debate"[1] – these three being the logicists (Gottlob Frege and Bertrand Russell), the formalists (David Hilbert and his colleagues), and the constructivists (Henri Poincaré and Hermann Weyl); within this constructivist school was the radical self-named "intuitionist" L. E. J. Brouwer.
Brouwer founded the mathematical philosophy of intuitionism as a challenge to the prevailing formalism of David Hilbert and his colleagues, Paul Bernays, Wilhelm Ackermann, John von Neumann, and others.
[3] 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.
"[6] In an address delivered in 1927, Hilbert attempted to defend his axiomatic system as having "important general philosophical significance.
"[1] Hilbert views his system as having no tacit assumptions admitted, stating, "After all, it is part of the task of science to liberate us from arbitrariness, sentiment and habit and to protect us from the subjectivism that... finds its culmination in intuitionism.
"[2] Finally, Hilbert singled out Brouwer, by implication rather than name, as the cause of his present tribulation: "I am astonished that a mathematician should doubt that the principle of excluded middle is strictly valid as a mode of inference.
I am most astonished by the fact that even in mathematical circles, the power of suggestion of a single man, however full of temperament and inventiveness, is capable of having the most improbable and eccentric effects.
The formalistic school should therefore accord some recognition to intuitionism instead of polemicizing against it in sneering tones while not even observing proper mention of authorship.
Weyl concluded "consistency is indeed a necessary but not sufficient condition" and stated "If Hilbert's view prevails over intuitionism, as appears to be the case, then I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics.
Brouwer viewed this loss of constructibility as bad, but worse when applied to a generalized "proof of consistency" for all of mathematics.
In his 1900 address Hilbert had specified, as the second of his 23 problems for the twentieth century, the quest for a generalized proof of (procedure for determining) the consistency of the axioms of arithmetic.
Hilbert, unlike Brouwer, believed that the formalized notion of mathematical induction could be applied in the search for the generalized consistency proof.
[17] If successful the quest would result in a remarkable outcome: Given such a generalized proof, all mathematics could be replaced by an automaton consisting of two parts: (i) a formula-generator to create formulas one after the other, followed by (ii) the generalized consistency proof, which would yield "Yes – valid (i.e. provable)" or "No – not valid (not provable)" for each formula submitted to it (and every possible assignment of numbers to its variables).
[20] The gang of three – Poincaré, Weyl, and Brouwer – claimed that Hilbert tacitly, and unjustifiably, adopted as one of his premises formal induction (the Kleen symbol string).
[21] Weyl's (1927) agreement and Brouwer's polemics ultimately forced Hilbert and his disciples Herbrand, Bernays, and Ackermann to reexamine their notion of "induction" – to eschew the assumption of a "totality of all the objects x of an infinite collection" and (intuitionistically) assume that the general argument proceeds one x after another, ad infinitum (van Heijenoort p. 481, footnote a).
(p. 217)Brouwer's insistence on "constructibility" in the search for a "consistency proof for arithmetic" resulted in sensitivity to the issue as reflected by the work of Finsler and Gödel.
These are not incompatible truths, as long as the law of the excluded middle over the completed infinite isn't invoked anywhere in the proofs.
A hard look at the premises of Turing's 1936–1937 work led Robin Gandy (1980) to propose his "principles for mechanisms" that throw in the speed of light as a constraint.
Secondly, Breger (2000) in his "Tacit Knowledge and Mathematical Progress" delves deeply into the matter of "semantics versus syntax" – in his paper Hilbert, Poincaré, Frege, and Weyl duly make their appearances.
Specifically, he claims a mind must come to the argument equipped with prior knowledge of the symbols and their use (the semantics behind the mindless syntax): "Mathematics as a purely formal system of symbols without a human being possessing the know-how for dealing with the symbols is impossible [according to the chemist Polanyi (1969, 195), the ideal of a form of knowledge that is strictly explicit is contradictory because without tacit knowledge all formulas, words, and illustrations would become meaningless]" (brackets in the original, Breger 2000: 229).
A serious study of this controversy can be found in Stephen Kleene's Introduction to Metamathematics, particularly in Chapter III: A critique of mathematical reasoning.