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Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer, Vol. 1: The Dawning Revolution

Dirk van Dalen
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Bonnie Shulman
, on

When he saw me carrying around my copy of The Life of L.E.J. Brouwer, Volume 1, a colleague remarked, "Brouwer, isn't he the one who wanted to reject half of mathematics?" This is the typical impression that most mathematicians have of Brouwer (and this colleague was a specialist in Nielsen fixed point theory!). Van Dalen's biography provides a welcome antidote to this oversimplified view of a complicated and complex man, a brilliant mathematician who wanted to make mathematics more honest.

The book is organized like a patchwork quilt, according to topics, rather than in chronological order. Thus, events taking place in various chapters actually occurred in Brouwer's life in parallel. In addition, there are some major themes in Brouwer's life that weave like threads throughout the different chapters: mysticism, topology, intuitionism, philosophy of language. Then there are his quarrels with Lebesgue, Denjoy, Menger, Hilbert--many of these conflicts persisted for years, and some even occurred simultaneously with others. As van Dalen makes clear, "Brouwer was a high-strung nervous person, who could easily exaggerate matters, when under stress. On top of that he had an extreme passion for justice; as Bieberbach put it: he was a justice fanatic..." (p. ix).

As the attentive reader will realize, at any given time, Brouwer was involved in an impressive and wide-ranging multitude of projects (as well as battles). But this "almost compulsive multiple activity" was not obvious to most observers during Brouwer's lifetime, even those who knew him fairly well. "As a rule, informants were greatly surprised to learn that at the time of the events reported by them, so many other things were going on simultaneously" (p. ix). Brouwer was a very private individual, and kept not only his thoughts and emotions, but even his involvement in public affairs, very much to himself.

Brouwer's stepdaughter, Louise Peijpers, was interviewed at length by van Dalen and his assistants, and, even though she was bedridden at the time, her memory was clear and she helped flesh out the details and facts of Brouwer's past. It is to her credit that even though she herself was a strong (and often difficult) character, and thus had her own tempestuous conflicts with her stepfather, these did not influence the reliability of her information. (Van Dalen, ever the careful researcher, tells us that most of her stories were independently verified by other sources.)

This is a dense and many-layered biography, like a rich dessert, and I found that I wanted to savor it in small bites. I came away understanding how Brouwer had earned his reputation as eccentric, revolutionary, reactionary, and prophet all at once. For instance, although his mathematical and philosophical works unquestionably broke ground for much 20th century research, he held some decidedly 19th century views on the role of women in the world. (See pp. 73 ff.) Women were temptresses, endangering male purity, and woe to him that succumbed to the distracting charm of the female. Ever the man of contradictions, it should also be noted that Brouwer was also one of the first mathematicians to employ a female assistant (Cor Jongejan) and admired and was on good terms with both Emmy Noether and Olga Taussky. And his contempt for the distractions posed by the female sex notwithstanding, he turned out to be quite a ladies' man in his private life. His expressed views on Jews were also quite clichéd and reactionary. From the summary of Brouwer's notes for his dissertation we read: "How the Jews dominate the farmers with the help of mathematics, and how the farmers do the same to the animals" (p. 84).

From these notes and the dissertation itself, van Dalen pieces together Brouwer's sources, and one is able to see by whom he was influenced. For mathematicians and historians of mathematics alike, this will be one of the more fascinating and useful aspects of the book. In fact, this biography is as much a sophisticated analysis of Brouwer's mathematical and philosophical work as it is a narrative of his life. To whet your appetite, here is a tidbit that particularly intrigued me: "[O]n page 5 [of the paper The force field of the non-Euclidian [sic] spaces with negative curvature] Brouwer introduced the parallel displacement (without giving it a name) years before the notion officially entered into the literature" (p. 87). Chapter 5, The New Topology, is a masterful account of the birth of 20th century topology, Brouwer's role in the process, and a plausible explanation as to why he did not develop algebraic topology, which was easily within his reach.

I was charmed by van Dalen's description of the context in which Brouwer's work took place: "The whole episode of the preparation of the dissertation is typical of a dignified and noble academic past, a time when letters could be exchanged the same day, when professors had time to read drafts of dissertations on short notice, when publishers produced hand-composed proofs within days" (p. 90). It was also quite startling to realize what it meant for Brouwer to be an editor at the Mathematische Annalen before the invention of copying machines. "[I]n those days the making of a simple copy was nothing less than the complete retyping or the copying by hand of the original" and Brouwer, with his "almost exaggerated sense of the scientific responsibilities of editors... found it his duty to keep a precise record of the various versions of manuscripts. To that end he had copies made of most papers, a job mostly performed by Cor Jongejan" (pp. 297, 298).

It should be clear by now that I thoroughly enjoyed reading this book, and am looking forward to Volume 2. I hope many mathematicians read it, and thereby gain a much more accurate picture of Brouwer the mystic, geometer and intuitionist, as well as some understanding of the enigma of Brouwer, the man. My only complaint has to do with the numerous typographical, grammatical, spelling and syntactical errors that plague the text. These include minor things like transposing footnotes 13 and 12 and then repeating number 13, footnote 43 missing altogether, transposition of letters, disagreement of tense, misspellings (as above, "Euclidian" instead of "Euclidean"), incorrect punctuation, run-on sentences as well as sentence fragments, wrong words ("intentions" instead of "intuitions," "incidents" instead of "accidents," "parallel displacement" instead of "parallel transport") to the more annoying typos in mathematics (p. 178, f(z_1) should be f_1(z), the same for f(z_2), and a phi that belongs in an exponent is instead part of the base). However, editorial lapses aside, I enthusiastically recommend this book for professors and lovers of mathematics and its history. Bright college students should also enjoy it, and especially appreciate the sections that convey the flavor of what it meant to do mathematics in those heady times. There will be technical parts they may want to skip, but because of the book's organization by topics, this should not significantly mar their enjoyment of the whole.

Bonnie Shulman ( is associate professor of mathematics at Bates College in Lewiston, ME. Her training is in mathematical physics, and her current interests include the history and philosophy of mathematics. She lives in a geodesic dome in the woods in Poland Spring, ME with her husband who is also a mathematician, and her cat Mozart. She practices t'ai chi, and yoga, and enjoys running on back roads and dirt trails.
1. Child and student
2. Mathematics and mysticism
3. The dissertation
4. Cantor-Schoenflies topology
5. The new topology
6. Making a career
7. The war years
8. Mathematics after the war
9. Politics and mathematics
10. The breakthrough
11. Bibliography of Brouwer's writingsReferences Index