---
title: "Brouwer–Hilbert controversy"
chunk: 2/5
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 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."
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. In the late 1890s Hilbert axiomatized geometry. 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). 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 Gauss and formulated by Weierstrass and Dedekind." "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."
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.
If this proof/procedure P was found, given any arbitrary mathematical theorem T (formula, procedure, proof) put to P (thus P(T)) including P itself (thus P(P)), P would determine conclusively whether or not the theorem T (and P) was provable – i.e. derivable from its premises, the axioms of arithmetic. Thus for all T, T would be provable by P or not provable by P and under all conditions (i.e. for any assignment of numerical values to T's variables). This requires the use of the Law of Excluded Middle extended over the infinite, in fact extended twice – first over all theorems (formulas, procedures, proofs) and secondly for a given theorem, for all assignment of its variables. This point, missed by Hilbert, was first pointed out to him by Poincaré and later by Weyl in his 1927 comments on Hilbert's lecture: "For after all Hilbert, too, is not merely concerned with, say 0' or 0' ', but with any 0' ... ', with an arbitrarily concretely given numeral. One may here stress the "concretely given"; on the other hand, it is just as essential that the contentual arguments in proof theory be carried out in hypothetical generality, on any proof, on any numeral. ... It seems to me that Hilbert's proof theory shows Poincaré to have been completely right on this point."
In his discussion preceding Weyl's 1927 comments, van Heijenoort explains that Hilbert insisted that he had addressed the issue of "whether a formula, taken as an axiom, leads to a contradiction, the question is whether a proof that leads to a contradiction can be presented to me".

"But [writes van Heijenoort] in a consistency proof the argument does not deal with one single specific formula; it has to be extended to all formulas. This is the point that Weyl has in mind ... ."
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). In other words: mathematics would cease as a creative enterprise and become a machine.