---
title: "Brouwer–Hilbert controversy"
chunk: 1/5
source: "https://en.wikipedia.org/wiki/Brouwer–Hilbert_controversy"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T04:17:31.712335+00:00"
instance: "kb-cron"
---

The Brouwer–Hilbert controversy (German: Grundlagenstreit, lit. 'foundational debate') was a debate in twentieth-century mathematics over fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. 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 1928, Hilbert had Brouwer removed from the editorial board of Mathematische Annalen.

== Background ==
The controversy started with 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" – 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.

=== History of Intuitionism ===
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. 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. After becoming established, 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 Hilbert's biographer Constance 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 responded: "Taking the Principle of the Excluded Middle from the mathematician... is the same as... prohibiting the boxer the use of his fists."

=== Validity of the law of excluded middle ===
In an address delivered in 1927, Hilbert attempted to defend his axiomatic system as having "important general philosophical significance." 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."
Later in the address, Hilbert deals with the rejection of the law of excluded middle: "Intuitionism's sharpest and most passionate challenge is the one it flings at the validity of the principle of excluded middle..." Rejecting the law of the excluded middle, as extended over Cantor's completed infinite, implied rejecting Hilbert's axiomatic system, in particular his "logical ε-axiom."
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 even more astonished that, as it seems, a whole community of mathematicians who do the same has so constituted itself. 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."
Brouwer responded to this, saying: "Formalism has received nothing but benefactions from intuitionism and may expect further benefactions. 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."

== Deeper philosophic differences ==

=== Truth of axioms ===
Until Hilbert proposed his formalism, axioms of mathematics were chosen on an intuitive basis in an attempt to use mathematics to find truth. Aristotelian logic is one such example – it seems "logical" that an object either has a stated property (e.g. "This truck is yellow") or it does not have that property ("This truck is not yellow") but not both simultaneously (the Aristotelian Law of Non-Contradiction). The primitive form of the induction axiom is another example: if a predicate P(n) is true for n = 0 and if for all natural numbers n, if P(n) being true implies that P(n+1) is true, then P(n) is true for all natural numbers n.
Hilbert's axiomatic system is different. At the outset it declares its axioms, and any (arbitrary, abstract) collection of axioms is free to be chosen. Weyl criticized Hilbert's formalization, saying it transformed mathematics "from a system of intuitive results into a game with formulas that proceeds according to fixed rules" and asking what might guide the choice of these rules. 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."