Books

World Books Review: The Birth of Infinity

The contest between French and Russian mathematicians who sought new answers to one of the oldest puzzles in math, the nature of infinity, inspired this intriguing book. The French school chased rationalist solutions, while the Russian mathematicians were reportedly inspired by mystical insights attained through their religious practice, visions into the infinite that led to the founding of descriptive set theory.

Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity By Loren Graham, Jean-Michel Kantor. The Belknap Press of Harvard University Press, 239 pp., $25.95

Reviewed by Anna Razumnaya

Naming Infinity is a short, accessible book about mathematical imagination. It focuses on one—albeit prolonged—moment in the history of mathematics: the first three decades in the development of set theory. Loren Graham and Jean-Michel Kantor, who collaborated on Naming Infinity across the Atlantic, propose that these new developments in mathematics came about as the result of the Russian mathematicians’ exposure to the ideas of their colleagues in France and, crucially, to the mystical beliefs of the schismatic religious movement whose followers were known as Name Worshipers.

The central intuition of the Name Worshipers was a recognition of the intimate connection between a name and the existence of the object named. Name Worshipers consequently saw the name of God as a point of connection for human beings with divinity itself. The official Orthodox Church saw Name Worshiping (imyaslaviye) as a heresy and sought its eradication. Once confined primarily to St. Panteleimon Monastery on Mt. Athos in Greece, imyaslaviye was inadvertently disseminated across Russia when, in 1913, Nicholas II ordered to remove heretical monks from the monastery and transport them by ship to the Russian mainland.

Russian mathematicians familiar with Name Worshiping—principally, Dmitri Egorov, the founder of what became known as the Moscow School of Mathematics—used the metaphysical intuition of imyaslaviye as a step toward new development in set theory. This new branch of mathematics captivated Egorov during his travels in Germany and France, where he attended lectures by Poincaré, Darboux, Hadamard, Lebesgue, Hilbert, and Minkowski.

Founded by Georg Cantor, set theory enjoyed a phase of exuberant development in France, but it soon became clear that the strong French rationalist tradition closed off the imaginative possibilities necessary for its further growth. Conservative mathematicians saw the new theory as a species of speculative metaphysics rather than mathematics in the proper sense. It took a new sensibility and a new style of doing mathematics to overcome these imaginative constraints. Dmitri Egorov and his mathematical circle—or, rather, the triangle uniting Egorov himself, Nikolai Luzin, and Pavel Florensky—possessed just the right kind of intellectual temperament. Incidentally, all three of these mathematicians happened to sympathize with the mystical views of Name Worshipers.

The authors of Naming Infinity struggle to balance this interesting fact against the risks of it being misinterpreted as grounding mathematics in mysticism. Consequently, they never quite bring themselves to articulate the precise extent to which the ideas of the Name Worshipers affected the members of the Moscow School of Mathematics. Name Worshiping is mentioned—“named”—but then abandoned without due exploration.

The fruitfulness of the encounter between the religious and mathematical types of imagination is an intriguing topic. However, the authors never articulate fully which of the Name Worshippers’ beliefs spurred the problem-solving in a stagnant area of mathematics. It also remains unclear whether those beliefs became permanently incorporated into the Moscow style of doing mathematics, or whether they were discarded, like Wittgenstein’s ladder, after they provided the necessary imaginative impulse.

What remains of the plot is reducible to a series of formally notated theorems where commentary is, in a sense, superfluous. Here Naming Infinity faces the same challenge as any other popular book about mathematics: the challenge of divided readership. The mathematically savvy readers will already know the formal side of the story and so will have no need for explanations beyond a brief sketch. The non-specialists will be similarly more interested in the social and historical contexts of the ideas. In both cases, it is the flare and verve of the way the story is told that makes or breaks the book.

Luckily, mathematics owes the bulk of its progress to personalities with such a surplus of eccentricity that personal tics alone are often enough for an exciting book. Naming Infinity has some of that anecdotal quality. Its real strength, however, is in the authors’ skill at isolating the key actors in the story, their personalities, motives, and intellectual styles—and fitting these elements together in a way that bares the machinery of creativity and ambition. To the reader’s dismay, one may find that human destinies are routinely trapped by that machinery.

To tame the complexity of their plot, the authors have organized it in an series of elegant triangles: the three key French mathematicians—Émile Borel, Henri Lebesgue, René Bair—form “the French trio”; likewise, Egorov, Luzin, and Florensky, are framed as “the Russian trio”. These two triangles are in turn triangulated with set theory itself, showing how its growth came to be catalyzed by the shifting approaches of the French and the Russian schools.

Naming Infinity is a straightforward, kinetic, and seductive read. Cause gears into effect, fueled by the aspirations and propensities of the large cast of characters surrounding the “Russian trio”: mathematicians with varying proportions of talent and ambition, political manipulators—exemplified by the slanderous Ernst Kol’man—and the manipulated. There are deeply moving biographical sketches of figures like Nikolai Chebotaryov, who sacrificed his academic career in the name of principle, Pyotr Kapitsa, a physicist kidnapped by the Soviets from England, and Nina Bari, who edited Luzin’s posthumous papers before throwing herself under a subway train.

Dmitir Egorov, the founder of what became known as the Moscow School of Mathematics

At their best, Graham and Kantor expose, elegantly and economically, the motives of seemingly hard-to-explain actions. Movingly, they uncover the reasons behind the political acquiescence of Pavel Alexandrov and Andrei Kolmogorov, whose homosexual orientation gave the regime a convenient fulcrum for blackmail, and behind the suicide of Lev Shnirel’man, elected a corresponding member of the Academy of Sciences at the astonishing age of 28.

In describing the life trajectories of their subjects, the authors are unafraid to take sides, show their sympathies, even judge. There is something refreshingly honest in their striving to be fair to their real-life characters without feigned impartiality. This considered generosity and the passion that shows itself in the copious quantities of documentary and anecdotal evidence gathered by Loren Graham in Russia, make the book a fascinating read despite its shortcomings. Just as a stimulating conversation, even when left incomplete, opens the mind to new ideas, Naming Infinity suggests new ways of thinking about mathematical creativity and intellectual excellence.

Harvard University Press could have been more thorough in checking difficult spellings in the book. The Solovetsky Archipelago in the White Sea is repeatedly referred to as “Solovetsk” Islands. (The single-letter suffix “k” is common at the end of city names but does not work that way in names referring to landmasses.) The name of St. Panteleimon is spelled as “Pantaleimon” throughout. While the illustrations supplied by the authors are carefully attributed, the publisher has not bothered to credit Mikhail Nesterov’s 1917 painting, “Philosophers,” reproduced on the jacket. The painting is a double portrait of Pavel Florensky—a member of “the Russian trio” (wearing a white priest’s cassock)—in conversation with philosopher Sergey Bulgakov. It is part of the Tretyakov Gallery’s collection in Moscow.

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Anna Razumnaya is a freelance translator and a doctoral student at the Editorial Institute at Boston University. Her translations of two poems by Osip Mandelstam are forthcoming in Pusteblume.