Books and Arts

Nature 458, 971-972 (23 April 2009) | doi:10.1038/458971a; Published online 22 April 2009

Pursuing the infinite

Jascha Hoffman1

BOOK REVIEWEDNaming Infinity: A True Story of Religious Mysticism and Mathematical Creativity

by Loren Graham & Jean-Michel Kantor

Belknap/Harvard University Press: 2009. 256 pp. $25.95, £19.95

Pursuing the infinite

C. E. FORD MATH. INTELLIGENCER 13, 24–30; 1991/SPRINGER SCI. & BUS. MEDIA; N. MALYSHEV/ITAR-TASS

Secret-police archives document the execution of mathematician Pavel Florensky by the Soviet regime.

Religious mystics have a long history of borrowing from mathematics. It is less common for mathematicians to draw on religion. In Naming Infinity, historian of science Loren Graham and mathematician Jean-Michel Kantor argue that an esoteric Christian sect contributed to advances in set theory in Russia in the first decades of the twentieth century. In pursuing their claim, they reveal a much larger drama: the flourishing of mathematics under the repression of the early Soviet regime.

Graham and Kantor begin in 1913, when the Imperial Russian Navy stormed a monastery on a Greek peninsula where a sect of Russian Orthodox monks had fled to pursue a mystical practice known as name worshipping. Holding the heretical view that God comes into existence when named, these monks believed that repeating the name of Jesus while controlling their breath and heartbeat would bring them closer to the infinite. Their persecution at the hands of the Tsar in the ensuing years aroused the sympathy of a number of Russian intellectuals. Among them was a handful of mathematicians in Moscow who, working in the young field of set theory, also found themselves dealing with the infinite.

These Russian mathematicians had been racing their French colleagues to take the measure of infinite sets of real numbers. In 1891 the German mathematician Georg Cantor made a crucial advance when he proved that some infinite sets were larger than others. A group of French mathematicians at the turn of the century — including Emile Borel, Henri Lebesgue and René Baire — were searching for a systematic way to determine the size of these infinite sets. This aroused the scepticism of colleagues such as Henri Poincaré, who claimed that Cantor's hierarchy of infinities had "a whiff of form without matter, which is repugnant to the French spirit". According to the authors, the French researchers found themselves at the edge of an "intellectual abyss" where, "under the influence of their ultra-rationalistic traditions, they lost their nerve" and abandoned their work.

The situation was different in Moscow, where the Russian mathematicians took up the same problems with zeal and eventually resolved them, advancing the far-reaching field of measure theory and launching descriptive set theory. Graham and Kantor argue that the spiritual views of these mathematicians were crucial to their scholarly work. That there were ties between some of the mathematicians and the heretical sect is not in doubt. The geometer Dmitri Egorov believed in name worshipping. His student Pavel Florensky, a mathematician turned theologian, held that the 'set of all sets' might be God himself. The eminent mathematician Nikolai Luzin was privately sympathetic to the sect.

None of this illuminates a substantive connection between the ideas of the monks and the mathematicians. These Russian scholars did push forward where the French would not, so it is reasonable to ask whether their religion gave them an edge: did their belief that both God and sets could be named into existence help them deal more creatively with the infinite? The authors do not settle this question, and never fully explain why the work of the Russians should have required a belief in name worshipping as opposed to another spiritual belief. In the end, they backpedal to say they are "not claiming a unique or necessary relationship" between mysticism and mathematics but are merely saying that the heresy of name worshipping "played a role in their conceptions". They don't, however, say what that role was.

Whatever their ties, the mathematicians and the heretics suffered similar fates under the Soviet regime. For a time both escaped the worst treatment. The name-worshippers hid in the shadows as Vladimir Lenin went after the mainstream Orthodox church. Mathematicians survived longer than other academics because, unlike physicists or chemists, they did not need special equipment, and unlike historians or philosophers, their findings did not immediately fall foul of Soviet dogma.

Eventually the Stalinist state caught up with everyone. Egorov was detained in 1930 for "mixing mathematics and religion" and died in prison. Florensky confessed under torture and was sent to the Gulag, where he studied permafrost and seaweed before his execution in 1937. The case of Luzin is a miraculous exception. In 1936 he was accused of collaborating with foreigners by the Marxist mathematician Ernst Kolman, who proclaimed, "Soviet science will rip away your mask!" He was saved by a letter to Joseph Stalin from the physicist Peter Kapista, who argued that Luzin might yet be useful to the government. It is not clear why Stalin listened, but his whim ensured the future of a discipline.

It will be hard for the uninitiated to follow Naming Infinity, owing to the book's uneven exposition and narrow biographical focus. The connection between mathematics and mysticism is tenuous. The real drama appears around the edges, as the researchers survive famine, repression and war long enough to set the direction for a century of mathematics. It is a story of the persistence of intellectual life against the wrecking tide of history.

  1. Jascha Hoffman is a writer based in New York.
    Email: jascha@jaschahoffman.com