The Brouwer–Hilbert controversy () 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"Dawson 1997:48 – 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.Kleene (1952), pp. 46–59 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.Davis, p. 96 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".Reid 1996, p. 37
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."This quote appears in numerous sources. A translation of the original can be found in van Heijenoort: Hilbert (1927) p. 476 and reads as follows: "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether."
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,Hilbert's writing is clean and accessible: for a list of his axioms and his "construction" see the first pages of van Heijenoort: Hilbert (1927). 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."van Heijenoort p. 483
The law of excluded middle extended to the infinite Cantor (1897) extended the intuitive notion of "the infinite" – one foot placed after the other in a never-ending march toward the horizon – to the notion of "a completed infinite" – the arrival "all the way, way out there" in one fell swoop, and he symbolized this notion with a single sign ℵ0 (aleph-null). Hilbert's adoption of the notion wholesale was "thoughtless", Brouwer alleged. Brouwer in his (1927a) "Intuitionistic reflections on formalism" states: "SECOND INSIGHT The rejection of the thoughtless use of the logical principle of the excluded middle, as well as the recognition, first, of the fact that the investigation of the question why the principle mentioned is justified and to what extent it is valid constitutes an essential object of research in the foundations of mathematics, and, second, of the fact that in intuitive (contentual) mathematics this principle is valid only for finite systems. THIRD INSIGHT. The identification of the principle of excluded middle with the principle of the solvability of every mathematical problem."van Heijenoort, p. 491
This Third Insight is referring to Hilbert's second problem and Hilbert's ongoing attempt to axiomatize all of arithmetic, and with this system, to discover a "consistency proof" for all of mathematics. So into this fray (started by Poincaré) Brouwer plunged head-long, with Weyl as back-up.
Their first complaint (Brouwer's Second Insight, above) arose from Hilbert's extension of Aristotle's "Law of Excluded Middle" (and "double negation") – hitherto restricted to finite domains of Aristotelian discourse – to infinite domains of discourse.See the lead paragraphs of van Heijenoort: Brouwer (1923b) p. 335. In the late 1890s Hilbert axiomatized geometry.Breger states that "Modern mathematics starts with Hilbert's Grundlagen der Geometrie" (p. 226). Then he went on to use the Cantorian-inspired notion of the completed infinity to produce elegant, radically abbreviated proofs in analysis (1896 and afterwards).Brouwer itemizes the other places where he thinks Hilbert has gone wrong (van Heijenoort p. 491–492) In his own words of defense, Hilbert believed himself justified in what he had done (in the following he calls this type of proof an existence proof): "...I stated a general theorem (1896) on algebraic forms that is a pure existence statement and by its very nature cannot be transformed into a statement involving constructibility. Purely by use of this existence theorem I avoided the lengthy and unclear argumentation of Weierstrass and the highly complicated calculations of Dedekind, and in addition, I believe, only my proof uncovers the inner reason for the validity of the assertions adumbrated by GaussThis is a sly poke at the finitists: "Empiricist philosophers, such as Hobbes, Locke, and Hume, had convinced some mathematicians, such as Gauss, that there is no infinite in mathematics" (Anglin p. 213). and formulated by Weierstrass and Dedekind."Anglin, p. 474 "The value of pure existence proofs consists precisely in that the individual construction is eliminated by them and that many different constructions are subsumed under one fundamental idea, so that only what is essential to the proof stands out clearly; brevity and economy of thought are the raison d'être of existence proofs."Anglin, p. 475
What Hilbert had to give up was "constructibility." His proofs would not produce "objects" (except for the proofs themselves – i.e., symbol strings), but rather they would produce contradictions of the premises and have to proceed by reductio ad absurdum extended over the infinite.
Hilbert's quest for a generalized proof of consistency of the axioms of arithmetic 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.