L(uitzen) E(gbertus, a.k.a. Bertus) J(an) Brouwer, one of the premier Dutch mathematicians in history, was a major force in topology from its early post-Poincaré days, and is in fact considered to be one of the very founders of this young but incomparably fecund subject. Indeed, almost every introductory course in topology these days includes coverage of the Brouwer fixed point theorem, to name but one of this fine scholar’s fundamental contributions. Another is invariance of dimension.
However, Brouwer’s greatest fame, or perhaps notoriety, consists in the fact that he was also the prime mover of the mathematico-philosophical movement of intuitionism, often mentioned in contrast to logicism or (Hilbert’s) formalism, and equally possessed of an aura of arcane anachronism, at least as far as today’s mathematical mainstream is concerned. But this is not to say that the corresponding foundations crisis of the early 20th century, involving not just Hilbert and Brouwer, but a number of other major players as well, Hermann Weyl most notable among them, was ever fully resolved. It ended in something of a détente coupled with a sort of tolerant neglect on the part of the aforementioned mainstream, and it cannot be denied that, as the unchallenged leader of the profession, Hilbert won: his authority was simply too great for it to have gone otherwise. Of course, we should qualify this victory with the observation that Kurt Gödel did succeed in reining in Hilbert’s hopes regarding the foundations of mathematics. But Hilbert certainly emerged unscathed, whereas the same cannot be said about Brouwer: their once close friendship (with the Hilberts even vacationing in the Netherlands) disintegrated and, in due course, escalated into what Einstein called “the frog and mouse battle between the mathematicians” (see below). Additionally Brouwer’s followers, again with Weyl most notable, began to fall away in considerable numbers.
Nonetheless two points should be made: first, no one in his right mind would claim that Hilbert’s posited formalism reflected in any real way his actual creative activity or his mathematical inner dialogue. Second, when it comes to the nuts and bolts of carrying out mathematical creative activity, i.e. actually doing mathematics, the intuitionists doubtless had, and have, something very interesting and relevant to contribute. For one thing, their position is that mathematics cannot be reduced to logic at all, and in a most profound sense it is a very different flavor of ratiocination. But “taking the use of his fists away from the boxer,” as Hilbert famously described Brouwer’s opposition to the (full) law of the excluded middle, proved to be too much to accept. Ditto for other tenets of Brouwer’s program: it didn’t succeed in doing anything ordinary mathematics couldn’t do, and, conversely, straightjacketed its adherents into bizarre logical contortions the mainstream ultimately couldn’t buy into.
Nonetheless, intuitionism is still alive, especially in the Netherlands. Consider, for instance, the influence exercised by the long-lived Arend Heyting, a direct student of Brouwer at the University of Amsterdam, who spent his entire (long) professional career at his alma mater. Furthermore, the editor of The Selected Correspondence of L.E.J. Brouwer, Dirk van Dalen, in turn wrote his 1963 Amsterdam thesis on intuitionistic plane projective geometry under Heyting, whence he is Brouwer’s “grandson.” Thus, Amsterdam emerges as the surviving nucleus of this intuitionistic cell, although Brouwer, himself also a long-time member of the faculty there, spent most of his time at his home in Laren. As related by B. L. van der Waerden, this was rather exceptional, given that Amsterdam Universiteit academics were expected to live in Amsterdam, but Brouwer rated an exception and appeared on campus only once every week. Laren, though only a half hour car-ride from Amsterdam, is really part of another world, as any Dutchman will confirm, and this cultural divide was even more pronounced in Brouwer’s time, when car-rides were still rare curiosities. (A long time ago, your reviewer spent six years in this tiny but in many ways dramatically heterogeneous country.)
In any case, both as a human being and as an academic Brouwer was indeed anything but ordinary, and this is certainly also reflected in his correspondence, only a selection of which populates the pages of the book under review. Happily, Brouwer was in touch with a large number of scholars, and involved with a variety of different themes, be they mathematical, academic, cultural, or personal. This makes the Selected Correspondence interesting to a wider audience, of course, and there is the additional feature of getting a peek behind the scenes of some major conflicts, given Brouwer’s dramatically polemical tendencies. And those he locked horns with, particularly Hilbert himself, were often by no means less inclined to give an inch either once the battle was joined.
Van Dalen delineates a dozen major themes by way of partitioning the correspondence coherently, so we encounter Brouwer and his correspondents holding forth on quite a spectrum of interesting topics. For example, there is the matter of Lebesgue’s role in the search for the invariance of dimension theorem, Brouwer vs. Menger regarding dimension theory (with several other luminaries joining in), the question of the participation of Germany at international mathematical congresses as a former enemy to victorious European nations in the post World War I era, and of course the aforementioned “frog and mouse” affair, which ultimately led to Hilbert’s complete restructuring of the editorial board of Mathematische Annalen. This latter event is covered, in her usual pithy manner, by the late Constance Reid in Hilbert, still a sine qua non for any one even remotely interested in modern mathematics and mathematicians.
I guess most of us have a lot of books like Reid’s on our shelves, attesting to our species’ constitutional predisposition to seek a vicarious participation in great events and adventures. Certainly when I was 18 or so and it came hot off the presses, I devoured Reid’s book at the speed of a CERN neutrino racing to Italy. And this is the rule and not the exception: we mathematicians all (or almost all) love to read tales of this type, involving our favorites and heroes, including the players we encounter in the book under review: Hilbert (of course), Blumenthal, Weyl, Alexandroff, Kneser, Hahn, Freudenthal, Hasse, and Coxeter, as well as a number of physicists such as Ehrenfest, Sommerfeld, and Planck.
And here we get it straight from the horse’s mouth.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.