Labyrinth of Thought, by José Ferreirrós, published this year in its second revised edition, is the English translation of the author’s 1999 contribution, volume 23, to the Science Networks – Historical Studies series. The original, El Nacimiento de la Teoría de Conjuntos, was very well received (see, for example, Roger Cooke’s comments in Modern Logic Review), and the present English version is well worth reading, not just by historians of mathematics specializing in set theory and logic, but by any mathematician with an interest in the origins of set theory. The history of this fundamental and critical branch of modern mathematics is intrinsically fascinating, both because the material is accessible (at least at a superficial level) and due to what is hiding in the shadows. Every mathematician should be familiar with the subject’s mainstays, including the paradoxes and antinomies, their resolutions or “fixes,” and the broad features, at least, of what Hilbert called Cantor’s paradise. The pages of Labyrinth of Thought(what a terrific title! See page xv: it’s from Jorge Luis Borges) richly feature these themes and the indicated players, from Frege and Russell to Cantor and Hilbert — and beyond: Zermelo, Fraenkel, Von Neumann, and Gödel appear in the book’s last chapter.
Revisiting these familiar figures and their works from Ferreirós’ fascinating historical perspective is reason enough to read the book, but there is much more to recommend it. Specifically, Ferrierós’ presents the thesis that it really didn’t all begin with Cantor (Hilbert’s propaganda notwithstanding): crypto-set theory (the unfortunate phrase is mine: I’m sure Ferreirós would never say something like this…) can already be discerned in the work of Dirichlet, Riemann, and Dedekind, well before Cantor’s appearance on stage. Indeed, on page xiii of the Introduction to Labyrinth of Thought the author states that whereas “[t]he traditional historiography of set theory has reinforced … misconceptions [like the claim] that set theory originated in the needs of analysis … [t]he present work will show, on the contrary, that [already] during the second half of the 19th century the notion of set was crucial for emerging new conceptions of algebra, the foundation of arithmetic, and even geometry … [antedating] Cantor’s earliest investigations in set theory…”
Ferreirós makes his case via a two-pronged analysis: Part One of Labyrinth of Thought concerns the way in which crypto-set theory answered the indicated calls from algebra, arithmetic, and geometry; Part Two then addresses Cantor’s work on “infinity and the continuum” with Dedekind also cast in a major role. Thereafter the book’s Part Three addresses the further evolution of set theory through the 1950’s, offering, “perhaps for the first time, a comprehensive overview.” Part Three concerns the work of Russell on the theory of types, the work of Zermelo, the famous “foundations crisis” (to which Einstein referred as a “frog and mouse battle,” with Hilbert and Brouwer pitted against each other and Weyl doing fifth-column work), metamathematics according to Gödel, and the emergence of mathematical logic in the contemporary sense. This is doubly interesting material in light of Ferreirós’ novel perspective.
Furthermore, for mainstream mathematicians, as opposed to historians in the aforementioned sense, Labyrinth of Thought, given the wealth of biographical material Ferreirós includes in the book, is also an exciting return to the lives and labors of some very familiar titans of the past. His slant on things only adds to the pleasure of it all. As already noted, the mathematics discussed is of course to some degree very familiar, but in this regard Ferreirós’ skill as a historian enhances things considerably in that a wealth of material usually vouchsafed only to the experts is woven into the narrative.
Thus, Labyrinth of Thought succeeds on two counts, as a scholarly work aimed at historians of set theory and as a wonderful excursion into the mathematical world of the second half of the 19th and first half of the 20th century: it should appeal to all of us.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.