Notes on van Atten, van Dalen & Tieszen on Brouwer and Weyl

Monday, 14th September, 2015

Paper details

M. van Atten, D. van Dalen, and R. Tieszen
Brouwer and Weyl: the phenomenology and mathematics of the intuitive continuum
Philosophia mathematica 10(3), 203-226


I thought this was a nice accessible paper (hardly any maths!). As well as “doing what it says on the tin”, it provides an introductory overview and a discussion of choice sequences.

Next stop for me is van Dalen’s biography.


The “intuition” in Brouwer’s intuitionism seems to be to do with our appreciation of time. Weyl’s definitions seem to be explicitly Husserlian, but Brouwer doesn’t explicitly use Husserl’s intuitionism. ADT quote Brouwer describing (what they repeatedly call) “the primordial intuition of mathematics” as “the substratum … of any perception of change, a unity of continuity and discreteness”. Related “intuitions” are the role of memory, and the asymmetry between past and future.

My questions here would be:

  • What does Brouwer mean by (an) intuition? Does it follow a Kantian distinction between learned knowledge and direct appreciation?
  • Why choose these particular things to be the “primordial” intuitions of mathematics? Time is a good candidate I concede. An even more “primordial” foundation might be eye movement: saccadic movement for the discrete counting; smooth pursuit movement for the continuum.
  • (Why) is it important that these things be unlearned intuitions? From my reading in developmental psychology, I would say these things — appreciation of time, memory, appreciation of an asymmetry between past and future — are learned by the human infant.

Related is the question of how Brouwer’s intuitionism differs from other constructivist approaches to mathematics. How much of this talk of “intuition” is really necessary?

Brouwer vs Weyl; Brouwer vs Husserl

Weyl seems to be much closer to Husserl than Brouwer does. It wasn’t clear to me from reading this paper whether Brouwer was just unfamiliar with Husserl or actively disagreed with Husserl’s position. Where differences are outlined in this paper, Brouwer’s position seems preferable to me.

Bits I like

I like Brouwer’s emphasis on the “creative subject” (or “transcendental ego”). This subject seems to exist in time: e.g., it can carry out only “finitely many acts of intuition”.

In what sense is this subject “transcendental”? What does it “transcend”? (and, again, why?)

Why am I interested in all this anyway?

My interest in Brouwer — and more broadly in the Constructivist approach to mathematics — is a digression from my main interests. However, it’s a digression from three of my main interests at the same time, which makes it hard to ignore.

  • Work: I am a computer programmer. The importance and role of type systems is a hot topic atm, and consequently type theories and proof systems. The constructivist approach to maths is relevant to computer programming because, like Brouwer’s creative subject, the computer program exists in time.
  • Schrödinger: I want to understand Schrödinger’s wave mechanics! That is the maths I am really interested in. I think I first heard about Brouwer — especially the Brouwer-Weyl connection — when I was reading Schrödinger’s biography late last year. Weyl was Schrodinger’s best friend in Zurich, and helped S with some of the maths for his breakthrough wave mechanics paper (1926).
  • Marx & Hegel: As well as the reference to Hegel on nLab, lots of the Brouwer in this paper is very reminiscent to me of Hegel: Brouwer’s “unity of continuity and discreteness” recalls Hegel’s “unity of unity and disunity”; the “two-ity” described early in ADT recalls the fundamental place Hegel gave to contradiction.

One Response to “Notes on van Atten, van Dalen & Tieszen on Brouwer and Weyl”

  1. llaisdy Says:

    c.f. Hegel’s Phenomenology of Mind B.IV. The truth of self-certainty, esp. §166-177: consciousness of self (and time §169) is a development from earlier forms of consciousness.

    c.f. also Vygotsky passim (and my PhD thesis).

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