Ernst Zermelo’s name, which, as we learned already in Constance Reid’s Hilbert, he joked was derived from Waltzermelodie , is famously attached to the best-known axiomatization of set theory, as well as to the first proof of the marvelous theorem that any set can be well-ordered. As regards the axiomatization of set theory, i.e., the now all-but-commonplace systems “ZF” or “ZFC” (if we include the Axiom of Choice), here Zermelo is of course linked with Abraham Fraenkel, who, as we learn in the fine book under review, provided Zermelo a close friendship which supported him throughout his later life.
Zermelo’s academic career, which started brilliantly, earning him the enthusiastic support of Hilbert, ended prematurely, in early retirement forced by failing health. His health notwithstanding, Zermelo went on to live into his eighties, supported by a state pension, but academically largely on the outside, looking in. Still, even with his retired status Zermelo kept very much alive mathematically (modulo health setbacks) and he maintained professional contact with a number of prominent mathematicians, Fraenkel among them, of course, but also Skolem, Gödel, and Scholz.
We also learn in Ebbinghaus’ well-researched book that Zermelo was something of a multiple mathematical threat, being also an expert on the calculus of variations, and a good mathematical physicist. Indeed, Zermelo took his doctorate in Berlin as an intellectual heir to Weierstrass, with his major professors being Lazarus Fuchs and Hermann Amandus Schwarz. His PhD thesis made quite an impact on the subject of the calculus of variations, and, partly based on its strength, Zermelo came to Göttingen and presently became Hilbert’s interlocutor in this branch of analysis.
Even before this, however, Zermelo had served as an assistant to none other than Max Planck (at Berlin), where he managed to become embroiled in a somewhat fiery polemical exchange with Ludwig Boltzmann. Zermelo’s interest in mathematical physics and in the calculus of variations stayed with him even after his primary focus had shifted (due to Hilbert’s tone-setting influence) to set theory and logic. Furthermore, the early battle with Boltzmann turned out to be something of a bellwether: in due course Zermelo would engage both Gödel and Skolem in rather ferocious mathematical combat. Ebbinghaus does a fine job in conveying the details of these polemical episodes, mathematical and otherwise.
In fact, for mathematical content in particular, Ernst Zermelo: An Approach to His Life and Work stands out as something of an exemplar. The author succeeds beautifully in bridging the pernicious gap between serious and precise mathematical exposition, on the one hand, and general, non-expert, readership, on the other. His treatment of Zermelo’s proof of the well-ordering principle is a particularly apt case in point.
Ebbinghaus also manages the difficult, indeed ineffable, task of painting an evocative picture of the mileu of early- and middle-twentieth century German academic life: it all certainly rings true in light of what we gleaned from Reid’s biographies of Hilbert and Courant. To boot, Ernst Zermelo: An Approach to His Life and Work is a beautifully produced book, replete with a large number of photographs and a few reproductions of Zermelo’s handwritten communications.
Ernst Zermelo was an important mathematician, a gunfighter as well as an architect, about whom any contemporary mathematician should learn a good deal more than what is presented in university courses on set theory and logic. Ebbinghaus’ book is perfect for this purpose.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.