---
title: "Brouwer–Hilbert controversy"
chunk: 4/5
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"
---

He does not appear to use this in the formalist's sense, but there is some contention around this point. Gödel specifies this symbol string in his I.3., i.e., the formalized inductive axiom appears as shown above – yet even this string can be "numeralized" using Gödel's method. On the other hand, he doesn't appear to use this axiom. Rather, his recursion steps through integers assigned to variable k (cf his (2) on page 602). His skeleton-proof of Theorem V, however, "use(s) induction on the degree of φ," and uses "the induction hypothesis." Without a full proof of this, the "induction hypothesis" could be assumed to be the intuitive version, not the symbolic axiom. His recursion simply steps up the degree of the functions, an intuitive act, ad infinitum. Gödel's proofs being intuitionistically satisfactory and infinitary are not incompatible truths, as long as the law of the excluded middle over the completed infinite isn't invoked anywhere in the proofs.
Despite the last-half-twentieth century's continued abstraction of mathematics, the issue has not entirely gone away. A hard look at the premises of Turing's 1936–1937 work led Robin Gandy (1980) to propose his "principles for mechanisms" that have speed of light as a constraint. As another example, 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. Breger asserts that axiomatic proofs assume an experienced, thinking mind. 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).

== Kleene on Brouwer–Hilbert ==
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. He discusses §11. The paradoxes, §12. First inferences from the paradoxes [impredicative definitions, Logicism etc.], §13. Intuitionism, §14. Formalism, §15. Formalization of a theory. Kleene takes the debate seriously, and throughout his book he actually builds the two "formal systems" (e.g., on page 119 he discusses logical laws, such as double negation elimination, which are disallowed in the intuitionist system).

== Notes ==