Notes on van Dalen’s biography of L. E. J. Brouwer

Tuesday, 17th November, 2015

book details

L. E. J. Brouwer — Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life
Dirk van Dalen

the biography

This is a very good biography. Exasperatingly thorough and comprehensive (830 pages not including bibliography and index). It’s a very humane book, and a good example of what a humanistic project a biography is, or should be.

the maths

During Brouwer’s work on his PhD thesis — perhaps especially in correspondence with his PhD supervisor Korteweg, and in parts of the thesis that were later rejected — we see already many of the foundation stones of Brouwer’s intuitionism (p. 86-7): e.g., that “the points of departure of the theory” should be determined by “how mathematics roots in life” (p. 86), that “the primeval phenomenon is simply the intuition of time”, the “shift from goals to means”.

The “life” in which maths roots is not a social life, and the rooting is not an instrumentalism. It seems to be a kind of subjective idealist solipsism: “my mathematical thinking is non-sensory internal architecture”, “consciousness gains access to free creation — which is my mathematics — as soon as it knows itself autonomous and immortal, ignoring objective knowledge and common sense.” (p. 190-1, interview with Weissing ca 1913).

Brouwer was aware of and did tackle preceding work on mathematical foundations, including Kant, and especially the constructivist approaches of Kronecker, Poincare and Borel (p. 232f). Brouwer’s position differed from these earlier constructivists in important ways.

The constructivists’ main concern seems to have been that mathematical objects should be constructible in finitely many steps from the natural numbers — e.g., irrational numbers were admissable mathematical objects only if derivable from natural numbers by an algorithm with finitely many steps. Brouwer held to this rather weakly even in the beginning, for example including “free selection” as an algorithmic step. He does seem to have kept the “finitely many steps” idea, but in a less technical form.

Brouwer also differed from the earlier constructivists in his rejection of the principle of the excluded middle (p. 104f, 196). See his 1908 paper. In 1918 he went further and demonstrated that the law of double negation cannot be proved (p. 307). n.b. these constraints on logic are for infinite domains like mathematics.

However, Brouwer’s focus seems not to have been on the objects of mathematics, but on the subject of mathematics — the mathematician. Van Dalen refers to Brouwer’s “subjectivist approach to mathematics” (p. 306). The creating subject at the centre of Brouwer’s maths is an idealised mathematician (p. 738-9).

next steps

The two points most interesting to me are (a) the nature of the mathematical subject and (b) the constraints on logic.

Unlike Husserl’s “transcendental” subject, Brouwer’s subject exists in time. The “intuition” of time is a foundational experience for this subject, but what is meant by an “intuition” and why the subject’s appreciation of time can’t be ordinary learned experience is not explored here. Of course I am tempted to reach for some kind of psychologism, or maybe a kind of “abstract” psychologism a la Hegel.

Reading about Brouwer’s constraints on logic I kept thinking about Hegel’s logic (all those old slogans: “the unity of opposites”, “the negation of the negation”).


  • read more on the nature of the subject and the constraints on logic in Brouwer’s maths (see references below)
  • swot up on Hegel’s “Science of Logic”
  • it might be useful to read an introduction to the philosophy of maths, as it is a completely new field to me. Two likely-looking candidates are “Thinking About Mathematics” (Shapiro, 2001) and “Introducing Philosophy of Mathematics” (Friend, 2007).

annotated references

Brouwer, L. E. J. (1908). De onbetrouwbaarheid der logische principes (The unreliability of the logical principles). Tijdschr. Wijsb. 2, 152-58. (There are two English translations available: one in volume 1 of Brouwer’s Collected Works (1975); and a new translation by Mark van Atten & Göran Sundholm

Brouwer’s “revolutionary rejection of the general validity of the principle of the excluded third” (p. 104f, 106).

Brouwer, L. E. J. (1913). Intuitionism and formalism. Bull. Am. Math. Soc. 20, 81-96. (English translation of “Wiskunde, Waarheid, Werkelijkheid”).

This was Brouwer’s inaugural lecture, in which he differentiates his own intuitionism from the contructivism of Kroneker, Poincare, et al. (p. 218-20, 233f).

Brouwer, L. E. J. (1929). Mathematik, Wissenschaft und Sprache. Monatschefte Math. Phys. 36, 153-164. English translation in Mancosu (1998).

Presents “Brouwer’s views on the genesis of the basic entities of the subject’s inner and outer world”, motivated and developed more clearly here than in Brouwer’s PhD thesis (according to van Dalen. Plausible).

Brouwer, L. E. J. (1949). Consciousness, Philosophy and Mathematics. Proceedings of the 10th International Congress of Philosophy, Amsterdam, 1948, pp. 1235-1249.

This paper sounds like a summing up and a restatement of B’s fundamental intuitionist positions (p. 755-6): the phases of consciousness, the “move of time”, the “cunning act” (aka the jump from ends to means), the creating subject, the solipsism.

There also seems to be an innovation, in the “… phenomenon of play, occurring when conative activity or causal thinking or acting is performed playfully, i.e. without inducement of either desire or apprehension …”

Johan Huizinga is only mentioned once in the biography, when Brouwer mourns Huizinga’s death in a 1945 letter (p. 702), but I couldn’t help thinking of Homo Ludens when I read this passage. Huizinga published Homo Ludens in 1938.

Brouwer, L. E. J. (1955). The effect of intuitionism on classical algebra of logic. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences Vol. 57 (1954 – 1956), pp. 113-116.

Brouwer’s last paper, stresses “the basic differences between classical and intuitionist logic”, “worth reading … because of its reflections on the nature of logic” (p. 798-800).

Heyting, A. (1956). Intuitionism, an introduction. North-Holland.

“Its role should not be underestimated, … its readability has always been praised.” (p. 800)

Mancosu, P. (1998). From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s. Oxford UP.

Translated papers of Brouwer, Weyl and others. Contains Brouwer (1929) and Weyl (1921).

van Dalen, D. (2004). Kolmogorov and Brouwer on constructive implication and the Ex Falso rule. Russ. Math. Surv. 59, 247-257.

This paper traces” the construction-meaning of implication in Brouwer’s famous ‘jump from ends to means'” (p. 612).

Weyl, H. (1921). Uber die neue Grundlagenkrise der Mathematik. Math. Z. 10, 39-79. English translation in Mancosu (1998).

This sounds like quite an exciting read, “beautiful provocative paper” (p. 311).


One Response to “Notes on van Dalen’s biography of L. E. J. Brouwer”

  1. […] bring? Part of the light shining on this exploration will be the differences between Husserl and Brouwer (in which my preference is closer to […]

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