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).

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.

Notes on “The Young Hegel”

Friday, 28th August, 2015

Book details

The Young Hegel: studies in the relations between dialectics and economics
Georg Lukács
1966 (English translation 1975)
MIT Press


The point of “Young Hegel” is to show the development of Hegel’s thinking up to (and including) his Phenomenology of Mind. The special focus is to show how Hegel’s study of economics (primarily Adam Smith and David Ricardo) affected this development.

This is the most biographical treatment of Hegel I’ve read so far. It’s divided into sections with Hegel at Berne (1793-96), Frankfurt (1797-1800) and Jena (1801-03, 1803-07). The “very” young Hegel in Berne is depicted as idealising Classical civilisation, especially of Greece, which he would contrast absolutely with the “positivity” and dogma of the Christian world. In Frankfurt, Hegel became “reconciled” to the bourgeois/Christian reality, and this reconciliation brought with it (or was partly) a need to link the two worlds — for example to explain the historical changes (Hegel’s main source seems to have been Gibbon!).

I think “reconciliation” is Hegel’s word but, in Lukács’ telling, the process was more than only intellectual. Lukács talks of Hegel’s Frankfurt “crisis”, and how Hegel took much of the source material from his own personal experience of life.

Hegel’s debates with Kant, Fichte and Schelling are covered in detail. So Hegel is placed in historical context in terms of philosophy; also the political situation in Germany and France, and the political/economic situation in England are shown to influence (determine, Lukács would have it) Hegel’s thinking.

The exploration of Hegel’s study and use of economics is very good. Obviously at least part of the point of this exploration is to confirm and consolidate our understanding of the similarities between Hegel and Marx. Pinkard stressed how Hegel’s “idealism” (as in not materialism) is not quite as straightforwardly “idealist” as you might think (e.g., Hegel describes animals as idealists). What you might call Hegel’s “proto-materialism” is drawn out by Lukács.

Marx turns up from time to time in this book, but not really until the final chapter. After a synopsis of the Phenomenology, Lukács reviews Marx’s Economic and Philosophical Manuscripts of 1844 (EPM), which culminate in a critique by Marx of the Hegel’s Phenomenology.

Another frequent visitor in the book is Goethe. Lukács is always stressing the similarity of thinking between Goethe and Hegel.

For all the foibles of his style (see below), Lukács knows how to put a book together, and this book finishes in a kind of triple climax. The first is the one you expect and that the whole book has been building up to: the Phenomenology, dealt with in a huge crescendo of exposition. For the second, Marx steps out of the shadows and we plunge in to EPM. The third and final climax is announced on the very last page — “Historically, only one figure may be placed on a par with Hegel: …” — and it is Goethe who takes the final bow and receives the bouquet.

Lukács’ style

Lukács does have a tendency

  • to be a bit unmediated/deterministic with his explanations
  • not to shilly-shally about with minor issues
  • to make sure Stalin gets a mention every so often.

For the unsympathetic reader there is plenty not to like about Lukács. I like to think Lukács was writing assuming two particular audiences: (a) Marxists, so certain things don’t have to be explained, and certain things can be short-circuited, and (b) the censor and the KGB.

Next steps

Marx’s EPM is an obvious next stop.

For strengthening understanding of the development of the dialectic, with Hegel and Marx as two points on the timeline as it were, I should re-read Ilyenkov’s Dialectical Logic.

Goethe: the work most mentioned is Wilhelm Meister. Iphigenia on Tauris and Confessions of a beautiful soul also get interesting mentions (as does Faust obvs).

Smart Army Helmet

Thursday, 23rd July, 2015

Sometimes quite astounding where your work can end up:

Ivan Uemlianin, 2009, Developing speech recognition software for command-and-control applications.

J. Alejandro Betancur, Gilberto Osorio-Gómez, Alejandro Mejía, & Carlos A. Rodriguez, 2014, Smart army helmet: a glance in what soldier helmets can become in the near future by integrating present technologies.

book details

Schrödinger: life and thought
Walter Moore
Cambridge University Press


This was a great book. Schrödinger had such a crazy love life!

I’m interested in Schrödinger because, naively, I like the feel of wave mechanics much more than I do that of quantum mechanics. I don’t like what I think of as the particle fetishism. However, the maths is hard, so I’m easing myself around the subject, looking for ways in.

This book was a thorough and enjoyable portrait of the man and his work. There are three main points of interest for me:

Space-Time Structure (1950)

Described as a “lovely little book” (p. 426) on “the geometry of space-time and its affine and metric connections” (p. 450). Grew out of extended correspondence with Einstein.


Hermann Weyl was Schrödinger’s best friend in Zürich during the twenties, and helped S with some of the maths for his breakthrough wave mechanics paper (1926). (Weyl was also a longtime lover of Schrödinger’s wife Anny.)

Weyl’s 1918 book on general relativity, “Space-Time-Matter” is described as “a complete account of relativity theory including all the necessary mathematical background” (p. 146). Its table of contents is accessible on the book’s Amazon page. This might be more accessible than Space-Time Structure.

Weyl also “became the strongest supporter of the intuitionist theory of mathematics, founded by Luitzen Brouwer of Amsterdam”. I’ve been reading a lot about intuitionism and constructivism in maths recently (at least a lot of tweets LOL). I’ve even seen a reference linking it back to Hegel.


From 1918 to 1920 the focus of Schrödinger’s work was on colour theory (pp 120-129). Describing the colour space, S uses a three-dimensional affine geometry. This treatment develops and the affine geometry becomes Reimannian. His work on colour theory seems to use similar terminology and concepts as his later work on general relativity — manifolds, {affine,differential,Reimannian} geometry. His colour space is three-dimensional (e.g., red, green, blue), while relativistic space-time is four-dimensional.

I am thinking that colour theory along these lines might be an accessible introduction to the mathematics required for general relativity. I haven’t had much luck finding modern mathematical introductions to colour theory.

Schrodinger’s main paper on the subject was:

“Grundlinien einer Theorie der Farbenmetrik im Tagessehen”,
Annalen der Physik, (4), 63, (1920), 397–456; 481–520

available in English (according to Wikipedia) as “Outline of a Theory of Colour Measurement for Daylight Vision” in Sources of Colour Science, Ed. David L. MacAdam, The MIT Press (1970), 134–82.

Notes on “The Perfect Theory”

Saturday, 10th January, 2015

book details

The Perfect Theory: a century of geniuses and the battle over general relativity
Pedro G. Ferreira
Little, Brown


The best “popular science” book I’ve read in a very long time — possibly ever: calm, readable prose; clear explanations; no silly metaphors or analogies. Possibly string theory has had its day, and I was pleased to see that particle fetishism was not treated as the be-all-and-end-all of general relativity. What stood out to me most of all was the broad international perspective.

The Soviet Union

The contributions of Soviet scientists to general relativity was given a prominent place. Four names look especially interesting:

Gennady Gorelik, Scientific American 1997, The Top Secret Life of Lev Landau,

Gorobets, B., (2008), “Круг Ландау: жизнь гения” (“The Landau Circle: the life of a genius”), URSS. n.b., this is the first book in a trilogy. The second and third volumes are “Круг Ландау: Физика войны и мира” (“The Landau Circle: Physics in war and peace”) and “Круг Ландау и Лифшица” (“The Landau Circle and Lifshitz”).

Ioffe, B. L., (2002), “Landau’s Theoretical Minimum, Landau’s Seminar, ITEP in the Beginning of the 1950’s”,

Tropp, E., Frenkel, V. & Chernin, A., (1988; tr. 1993), “Alexander A Friedmann: The Man who Made the Universe Expand”, CUP.


Jacob Bekenstein, Mordehai Milgrom and “Modified Newtonian Dynamics” (MOND).

Bekenstein, J., (2004), “Relativistic gravitation theory for the MOND paradigm”, Phys.Rev. D70 (2004),

Bekenstein, J., (2007), “The modified Newtonian dynamics — MOND — and its implications for new physics”, Contemporary Physics 47, 387 (2006)


The Square Kilometer Array.

The core of the beast will be laid out in the Karoo desert, but a number of these dishes will be scattered throughout the continent in places like Namibia, Mozambique, Ghana, Kenya, Madagascar. It will be a truly continental, /African/ endeavor.

(p. 235)

book details

Aristotle: a very short introduction
Jonathan Barnes
Oxford University Press


I found this a very pleasant, readable overview of Aristotle’s philosophy. It is sensitive to his context and to his distance from our own culture and perspectives.

Before reading this I knew little about Aristotle. I’d read his Nicomachean Ethics and Jonathan Lear’s “Aristotle: The Desire to Understand”.

Three points stood out for me:


Aristotle’s Prior Analytics and Posterior Analytics are the foundation works of the study of formal logic. The roots of formal logic are in Aristotle the empirical scientist, the explorer, the man of the world — not e.g. in Plato the idealist.

I don’t think I need to read Aristotle’s logical works, but it might be interesting to explore how his more formal thought related to his more exploratory work (e.g. biology, politics).


For Aristotle, change seems to be fundamental — at least to things in the sublunary world. That’s interesting to me given my interest in Hegel and Marx.

Reading: Physics, which Barnes thinks is “one of the best places to start reading Aristotle.”


Bodies in the superlunary world are not subject to change, and Aristotle’s work on the heavens contain some of his speculations on divine creatures and on the source of change: the creator.

Reading: On the heavens; Metaphysics I & XII.


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