Those of you who read my review of volume 1 know how much I have been looking forward to reading (and reviewing) volume 2 of Dirk van Dalen’s biography of L. E. J. Brouwer (1881-1966). So you will understand how thrilled I was in June to finally have finished the ritual clearing-of-the-desk-for-the-semester and be sitting on my deck opening up to the first page of *Mystic, Geometer, and Intuitionist*. Ah, summertime!

It generally comes as a surprise to my colleagues who know Brouwer primarily for his fixed point theorem, to learn that the father of modern topology had another, greater love. He was a key player in the foundational drama that played out in the early 20th century. In fact this undisputed mathematical genius devoted the majority of his considerable energies to developing and advocating intuitionism, a philosophy which views mathematics as a mental activity constructed by a “creating subject”. Van Dalen captures this incredulity nicely: “The working mathematician, more concerned with practicing mathematics than with reflecting on its soundness, could not stop wondering why Brouwer, the wizard of topology, had given up the riches of traditional mathematics for a life in the arid desert of the foundations” (p. 643 — note that there is continuous pagination between volumes 1 and 2, the second volume starting at page 441).

I myself have often wondered if Brouwer disavowed his earlier fixed point theorem which depended on what was clearly not an intuitionistic proof. In this volume van Dalen has unearthed an answer to this question that never made its way into the mathematical literature. In a postcard to Heinz Hopf (1894-1971), Brouwer explains that fixed point theorems, although false from a constructive viewpoint, do have an intuitionistic interpretation. The basic approach in the constructivization of classical theorems is to replace precise results by results “up to an ε”. As Brouwer admits, “[t]hese theorems are, however, no fixed point theorems, for one has no means to indicate, i.e. to approximate, a fixed point.” You can read the mathematical technicalities of his recasting of the result in intuitionistic form for yourself on pages 549–50. However, even though Brouwer’s topological work is firmly situated in traditional mathematics, it is not wholly devoid of an intuitionistic flavor. As Dale Johnson points out, “[h]is way of doing topology stressed geometrical *construction*, geometrical thought-construction that depended upon a primitive intuition.” (“L.E.J. Brouwer’s Coming of Age as a Topologist” in Studies in the History of Mathematics, ed. Esther R. Philips, 1987, Mathematical Association of America, p. 94.) Johnson is referring to Brouwer’s “ur-intuition” of time, from which he claimed “all mathematical systems, including spaces with their geometries, have been built up.” (L.E.J. Brouwer, *Collected Works, I*, ed. A. Heyting, Amsterdam, 1975, p. 116.) But Brouwer never explicitly connected this basic intuition to his topological work, nor did he use intuitionistic logic to derive his results. It remained for later generations to come up with constructive algorithms (in n-dimensions) to approximate the fixed points whose existence is asserted in Brouwer’s theorem. (See, for example, S. Karamardian, ed., *Fixed Points: Algorithms and Applications*, Academic Press, 1977.)

The first volume of van Dalen’s biography (subtitled “The Dawning Revolution”) does an excellent job of explaining the basic ideas and principles of intuitionism. The second volume (like the second half of Brouwer’s life) has less mathematical content, but is just as engaging as the author meticulously recounts the battles Brouwer waged (both ideological and personal) in his attempts to disseminate his program. The subtitle, “Hope and Disillusion”, aptly captures the story of this great man, a legend in his own lifetime, who began his campaign as an enthusiastic idealist, a conquering hero carrying the intuitionistic banner, hell-bent on reforming mathematics, and ended his life as a resigned, embittered and paranoid (but still quite intense) elderly gentleman.

This is a very scholarly work, make no mistake. The author combed through correspondence and archives; he interviewed colleagues, friends and family of the protagonist; and he does not shrink from detailed mathematical exegesis. (He is, by the way, superbly qualified for the latter as he wrote his Ph.D. with Arend Heyting, Brouwer’s intellectual heir and founder of intuitionistic logic.) Van Dalen’s craft is in making the characters come alive. He is a consummate story-teller and keen observer of human nature who paints portraits of complex, often self-contradictory personalities participating in the conflicts and controversies of an era of tumultuous intellectual, political and social upheaval. The book is as lively and compelling as a good novel – I admit that at several points I skipped to the Epilogue in order to see how things turned out in the end. I really cared what happened to these people!

At the crux of the foundational dispute between Brouwer and Hilbert (a.k.a.the *Grundlagenstreit*) is the warrant for mathematical truth. Hilbert maintained that if the introduction of a particular theory was consistent (i.e., did not lead to a contradiction) then this was enough to justify it. In short, if it works, it’s true! For Brouwer, justification of theories required an explicit mathematical construction. In an address to the Tenth International Congress of Philosophy at Amsterdam in 1948, titled *Consciousness, Philosophy and Mathematics*, Brouwer concluded that “there are no non-experienced truths” (p. 833). Van Dalen astutely describes this dispute as “a typical example of one of those scientific tragic-comedies, where both parties run around the stage, shouting but not listening” (p. 498).

Another bitter conflict between the two giants Brouwer and Hilbert was the battle over the makeup of the editorial board of the *Mathematische Annalen* (the premier mathematical journal at that time). This struggle had repercussions that sundered friendships and for a while threatened to split the German mathematical community in two. In hindsight we can see this, too, as a tragedy of errors, but for Brouwer personally, and for the recognition of intuitionism as a viable mathematical activity, the aftermath was devastating.

Van Dalen has clearly been conducting his research for decades. As early as 1990 the *Mathematical Intelligencer* (vol 12, no. 4, pp. 17-31) published a lengthy article by him (an excellent complement to the treatment in the book), on the war between Hilbert and Brouwer (which Einstein dubbed “The War of the Frogs and the Mice”) over the editorship of the *Annalen*. Interestingly, a picture and description of Paul Urysohn’s grave appears in the same issue of the *Intelligencer* (in Ian Stewart’s column “The Mathematical Tourist”). The premature death by drowning of this great Russian topologist is the first act (Chapter 12) in the volume under review, setting the stage for the Menger priority dispute, a protracted battle meticulously described by van Dalen in chapter 15. I will not dwell on these episodes in this review, but hopefully I have piqued your curiosity.

Another major fight in which Brouwer became embroiled was the boycott of German mathematicians instituted by the *Conseil International de Recherche* after World War I. Brouwer championed the reintegration of German mathematicians into the international community. He was not alone in this quest. In 1882, just after the Franco-Prussian War, the Swedish mathematician Gosta Mittag-Leffler (1846-1927) founded *Acta Mathematica*, an explicitly international journal of mathematics. By actively soliciting and publishing the work of the most brilliant mathematicians from both Germany and France, he set out to affect some reconciliation between the two countries. During and after World War I, he once again eagerly assumed the role of mediator, this time working through the pages of his journal to reunite both the English and the French with the Germans. The English mathematician G.H. Hardy (1877-1947) shared Mittag-Leffler’s desire to re-establish friendly relations with German mathematicians and scientists as soon as possible after the First World War. But their efforts, as well as Brouwer’s, to return to normalcy after the cataclysm met with stiff resistance.

One may wonder why Brouwer’s life is the story of a man involved in one conflict after another. In my review of volume one, I quoted van Dalen as saying that Brouwer… had an extreme passion for justice; as Bieberbach put it: he was a justice fanatic…”(p. ix, volume 1). It was in large part this commitment to political and social justice that drove him to become involved in so many controversies. And Brouwer was not an easy man to get along with: “…although he could be magnanimous and warm hearted, there were moments when he could not govern his sharp tongue” (p. 554). Coupled with his urge to defend the underdog, these qualities led to his being both admired and reviled by his contemporaries.

Brouwer is not the only complicated character illuminated by van Dalen’s piercing insight. On the e-mail discussion list *historia-matematica*, I was chastised by a reader of my earlier review for citing Bieberbach “the Nazi racist editor of *Deutsche Mathematik,* as the arbiter in matters of justice!” Indeed, in light of his subsequent turn to Nazism, it is at best ironic, and at worst (especially for victims of the *in*justices of the holocaust) quite offensive to quote Bieberbach in this context. However, as van Dalen points out, Ludwig Bieberbach (1886-1982) has “gone down in history as an evil man, [but] it should not be forgotten that he was an active member of the mathematical community, with a talent for organizing and excellent social graces” (p. 507). Bieberbach’s characterization of Brouwer was cited because it concisely encapsulated the general impression held by many. I think van Dalen’s point was that Brouwer could be extremely stubborn and unbending when fighting for what he thought was just. Although his principles were often in conflict with those of his peers, his integrity demanded that he not surrender and he kept fighting long after many others would have let a matter drop. He was indeed a “justice junkie”. The same qualities that make heroes out of those fighting for principles we share, make stubborn fools out of those fighting for causes we do not embrace.

Nevertheless, the “tendency to dismiss Bieberbach’s statements and opinions on the basis of his later political views and actions” (p. 617) is quite understandable. As the author said to me in a private e-mail, “The problem is that a person may have silly or even dangerous views on some matters (e.g. political ones), and still be completely rational in other matters.” Van Dalen’s portraits are nuanced; he does not tell a simple story of heroes and villains. I believe he has discharged his responsibility as an author and historian in scrupulously documenting his sources and sharing with readers the fruits of his labors — information we can use to contextualize and form our own judgments of people and their actions.

How does van Dalen make sense of Bieberbach? In the early 1920s Bieberbach was still rather ambiguous about politics. He was a “*deutschnational*” (a member of one of the chauvinistic German movements formed partly in reaction to the boycott and harsh treatment of German scientists) and at the same time somewhat left of center at least as compared to most of his Berlin colleagues. “His house was open to guests of all political persuasions and he had a reputation among the students for unorthodox open-mindedness” (p. 697). Drawing on the research of Herbert Mehrtens, who interviewed Bieberbach a year before he died, van Dalen assesses Bieberbach’s conversion to National Socialism in this way: “Bieberbach’s political extremism was not dictated by an iron fate, but rather the result of a mixture of personal preferences and ill-founded political views” (p. 698). (See H. Mehrtens, “Ludwig Bieberbach and Deutsch Mathematik” in *Studies in History of Mathematics*, ed. E.R. Philips, 1987, Mathematical Association of America, pp. 195-241). In a lecture in 1926, held for a society of friends and supporters of education in the exact sciences, titled *On the mathematician’s ideal of science* , Bieberbach reversed his earlier position as a strong pro-formalist, and sided with Brouwer’s intuitionism. But by this time he was using intuitionism opportunistically to support his own (highly politicized) ideal of a “German” mathematics. This “shows how intuitionism could be drawn into a political debate, where Brouwer would have been very reticent” (p. 700). Reviewing the obvious inconsistencies in Bieberbach’s philosophy, the discrepancies in his behavior, his insistence that his views were scientifically-based, while simultaneously drawing political conclusions from his theories — conclusions with material consequences he chose to ignore — it is not much of a stretch to see parallels with our own tumultuous times in this cautionary tale (lessons to be drawn left to reader).

Above all, this book is a good read, and a little easier going than the first volume for those not conversant with the fine points of intuitionistic mathematics and logic. Although the two volumes dovetail perfectly as one long narrative, anyone with some familiarity with the foundational debate and the personalities involved could probably read the second volume without having read the previous one. There are also many fewer typos in volume 2, their frequency increasing towards the end (I’m guessing there was a competent editor who ran out of time as the deadline approached). Finally, if you still haven’t ordered a copy from your library or bookstore, I’ll entice you with one more tidbit: the author suggests a couple of interesting open questions in the history of mathematics (see, for example, p. 496 and p. 502) for other scholars to pursue.

Bonnie Shulman (bshulman@abacus.bates.edu) is associate professor of mathematics at Bates College in Lewiston, ME. Trained as a mathematical physicist, her current research is in history and philosophy of mathematics. Recently she has been exploring the connections between mathematics and ethics, in particular the work of Karl Menger.