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

== Bibliography ==
W.S. Anglin 1994, Mathematics: A Concise History and Philosophy, Springer–Verlag, New York. ISBN 0-387-94280-7.
Herbert Breger, 2000. "Tacit Knowledge and Mathematical Progress", appearing in E. Groshoz and H. Breger (eds.) 2000, The Growth of Mathematical Knowledge, Kluwer Academic Publishers, Dordrecht Netherlands, ISBN 0-7923-6151-2, pages 221–230.
Martin Davis, 1965. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems, and Computable Functions, Raven Press, New York, no ISBN. This includes:
Emil Post, 1936. "Finite Combinatory Process. Formulation I", with commentary (pages 288ff)
Emil Post, 1941 unpublished until 1965. "Absolutely Unsolvable Problems and Relatively Undecidable Propositions: Account of an Anticipation", with commentary, (pages 338ff)
van Dalen, Dirk (1990). "The war of the frogs and the mice, or the crisis of the Mathematische annalen". The Mathematical Intelligencer. 12 (4): 17–31. doi:10.1007/BF03024028. S2CID 123400249. On the battle for editorial control of the journal Mathematische Annalen between Hilbert and Brouwer, stemming in part from their foundational differences.
Martin Davis, 2000. The Engines of Logic, W. W. Norton, London, ISBN 0-393-32229-7 pbk. Cf. Chapter Five: "Hilbert to the Rescue" wherein Davis discusses Brouwer and his relationship with Hilbert and Weyl with brief biographical information of Brouwer.
John W. Dawson, Jr, 1997. Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters, Wellesley, MA, ISBN 1-56881-256-6.
Robin Gandy, 1980. "Church's Thesis and Principles for Mechanisms", appearing in J. Barwise, H. J. Keisler and K. Kunen, eds., 1980, The Kleene Symposium, North-Holland Publishing Company, pages 123–148.
Stephen Hawking, 2005. God Created the Integers: The Mathematical Breakthroughs that Changed History: edited, with commentary, by Stephen Hawking, Running Press, Philadelphia, ISBN 978-0-7624-1922-7. Hawking's commentary on, and an excerpt from Cantor's "Contributions to the Founding of the Theory of Transfinite Numbers" appears on pp. 971ff.
David Hilbert (1927), "The foundations of mathematics" appearing at http://www.marxists.org/reference/subject/philosophy/works/ge/hilbert.htm and apparently derived from Sohotra Sarkar (ed.) 1996, The Emergence of Logical Empiricism: From 1900 to the Vienna Circle, Garland Publishing Inc, [no publisher's location, no ISBN]. Hilbert's famous address wherein he presents and discusses in some depth his formalism axioms, with particular attention paid to double negation and the Law of Excluded Middle and his "e-axiom. [This on-line document contains typographical errors; a better version is van Heijenoort's Hilbert (1927).]
Stephen Kleene, 1952 with corrections 1971, 10th reprint 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam Netherlands, ISBN 0-7204-2103-9. Cf. in particular Chapter III: A Critique of Mathematical Reasoning, §13 "Intuitionism" and §14 "Formalism".
Jean van Heijenoort, 1976 (2nd printing with corrections), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, Massachusetts, ISBN 0-674-32449-8 (pbk.). The following papers and commentary are pertinent and offer a brief time-line of publication. (Important further addenda of Gödel's regarding his acceptance of Turing's machines as a formal logical system to replace his system (Peano Axioms + recursion) appear in Martin Davis, The Undecidable):
Hilbert (1904). On the foundations of logic and arithmetic, p. 129
Brouwer (1923, 1954, 1954a). On the significance of the principle of excluded middle in mathematics, especially in function theory, p. 334
Brouwer (1927) . On the domains of definition of functions p. 446
Hilbert (1927). The foundations of mathematics p. 464. (Hilbert's famous address).
Weyl (1927). Comments on Hilbert's second lecture on the foundations of mathematics p. 480.
Bernays (1927). Appendix to Hilbert's lecture "The foundations of mathematics" p. 485
Brouwer (1927a). Intuitionistic reflections on formalism p. 490
Gödel (1930a, 1931, 1931a). Some metamathematical results on completeness and consistency. On formally undecidable propositions of Principia mathematica and related systems I, and on compleness and consistency p. 592
Brouwer (1954, 1954a). Addenda and corrigenda, and Further addenda and corrigenda, p. 334ff
Ernest Nagel and James Newmann 1958, Gödel's Proof, New York University Press, no ISBN, Library of Congress card catalog number 58-5610.
Constance Reid 1996. Hilbert, Springer, ISBN 0-387-94674-8. The biography in English.
Bertrand Russell, originally published 1912, with commentary by John Perry 1997. The Problems of Philosophy, Oxford University Press, New York, New York, ISBN 0-19-511552-X.