Book of the week: Naming Infinity

This absorbing book tells astonishing stories about some of the most important developments in mathematics of the past century. It begins with an episode that might seem to have little to do with the discipline: the attack in 1913 by a Russian gunboat on the Pantaleimon Monastery on Mount Athos. This was an attempt to suppress the controversial doctrine of "Name Worshipping", which centred on the practice of chanting the names of Christ and God to achieve an ecstatic trance. After the storming of the monastery, the Name Worshippers went underground in Russia, but adherents continued the practice even after the Revolution.

The authors' thesis is that Name Worshipping underlay the development of descriptive set theory, the study of subsets of the real number line and one of the major areas of 20th-century mathematics: this arose from the work of Georg Cantor on infinity and was developed by the French mathematicians Emile Borel, Henri Lebesgue and Rene Baire, before being taken up in Russia by Dmitri Egorov and others.

Naming Infinity argues that Egorov and his colleagues Pavel Florensky and Nikolai Luzin were influenced by Name Worshipping and by analogies between the naming of sets, which in a sense brings them into existence, and the mystical power of the name of God in the "Jesus Prayer" incanted by Name Worshippers.

First, we are told about Borel, Lebesgue and Baire. This story is not an entirely happy one: we hear of Baire's chronic depression and eventual suicide, and how Lebesgue and Borel fell out over mathematical and social issues. Developments in descriptive set theory then moved from France to Russia. There is an argument that the Name-Worshipping sympathies of Egorov, Florensky and Luzin helped them to deal with this abstract and demanding area of mathematics, while the French, lacking this religious inspiration, were unable to make progress.

The authors certainly make a plausible case for the importance of mystical belief in inspiring the Russian mathematicians: their provocative claim is that "Two different cultural contexts led to contrasting results: French skepticism and hesitation, Russian creativity and advancement ... a religious heresy was instrumental in helping the birth of a new field of modern mathematics".

Perhaps the most moving section of the book is that dealing with the famous Moscow School of Mathematics in Soviet times. Its origins are traced to the Lusitania seminar established by Egorov and Luzin (the source of the name "Lusitania" is obscure). The enthusiasm that these teachers inspired in their students is clearly conveyed, as is the atmosphere of intellectual excitement, despite the freezing lecture rooms (the rule that lectures could not take place if the room temperature fell below -5C was ignored).

The excitement was not just mathematical: the charismatic Luzin attracted female students, and there seems to have been a strong homosexual community at the heart of Lusitania, too. It is good to have these factors acknowledged: mathematics is placed in its human context.

The cold and the food shortages were not the only problems facing mathematics in Stalin's Moscow, and those with known religious beliefs were in an especially precarious position. Egorov, perhaps naively, argued that universities should tolerate diverse beliefs: he was imprisoned in the city of Kazan, where he died in tragic circumstances in 1931.

Florensky, who habitually wore his priest's robes at scientific congresses, was sent to a prison camp and, despite his significant scientific contributions to the Soviet Union, was shot in 1937. Luzin, much more discreet about his religious beliefs, nevertheless fell foul of the authorities and was tried as an enemy of the state. The book's description of Luzin's trial is riveting: he was saved by the secret intervention of Peter Kapitsa, the high-profile physicist, who interceded with Stalin.

Luzin's trial heard evidence from many junior members of the Moscow School of Mathematics, and this graphically illustrates the shattering dilemmas facing scientists at the time.

The brilliant young Lev Schnirel'man, who made a significant breakthrough in Goldbach's conjecture (still unproven today), committed suicide after being interrogated by the secret police and coerced to incriminate colleagues. Many others criticised Luzin at the time of his trial: they had little choice.

The book has its heroes, most notably Nikolai Chebotaryov, who sacrificed his career by resigning his post in Moscow when he discovered that his predecessor, Egorov, had been unfairly dismissed. In a remark-able coincidence, Chebotaryov and his wife cared for Egorov before his death in Kazan.

The book has one prominent villain, the careerist Marxist mathematician Ernst Kol'man, who led the campaigns against Egorov and Luzin. But all the characters are drawn with human virtues and demerits, as they did what was necessary to survive in impossible conditions, and also pursued more mundane objectives such as career progression and priority claims. In Graham and Kantor's telling, mathematics appears as a thoroughly human activity.

Two truly great figures in 20th-century mathematics, Pavel Alexandrov and Andrei Kolmogorov, who as homosexuals were particularly vulnerable, were forced publicly to support the Soviet biologist Trofim Lysenko and to criticise Alexander Solzhenitsyn. Kolmogorov spoke at the end of his life of his perpetual fear of the secret police. More happily, we hear of the importance of swimming to Alexandrov and Kolmogorov, and its connection with mathematical inspiration.

We are given sympathetic accounts of other members of the Moscow School of Mathematics, including the tragic figures of Pavel Uryson, who drowned in an accident in France at the age of 26, and Nina Bari, who committed suicide by throwing herself under a train after editing her lover Luzin's mathematical papers for publication after his death. Even Kol'man is perhaps partially redeemed by his later confession that he was "sincerely deluded, nourished by illusions which later deceived me".

At this point the book has wandered some way from the topic of the religious inspiration for descriptive set theory. But this reinforces the book's theme that mathematics is a human activity, influenced decisively by the beliefs and life choices of practitioners.

The book is generously illustrated, with many photographs and, on the cover, an evocative painting of Florensky in his robes. However, it betrays its double authorship in a number of minor ways. Names are not consistently transliterated - for example, is the poet Andrei Bely or Andrey? - and occasionally someone is mentioned before they are introduced by a later paragraph. Little attempt is made to explain the mathematics, although it could hardly be otherwise in a book such as this: specialists will know the details already, and the mathematics is too difficult for non-specialists.

But, for the reader who is prepared to take on trust the value of the mathematics, the stories told are fascinating. The conclusion draws parallels between the Name-Worshipping beliefs of Egorov, Florensky and perhaps Luzin, and the mystical philosophy of Alexander Grothendieck, one of the most enigmatic mathematicians of the late 20th century.

This is a remarkable book, illuminating the history of 20th-century mathematics and its practitioners. The stories it tells are important and too little known. It is clearly a labour of love and deserves a wide audience: it is an outstanding portrayal of mathematics as a fundamentally human activity and mathematicians as human beings.

THE AUTHORS

Jean-Michel Kantor is a mathematician and historian of mathematics at the Institut de Mathematiques de Jussieu in Paris, Jussieu, and writes on the philosophy of science for the literary journal La Quinzaine Litteraire. He was previously curator of the mathematics section of the National Museum of Science and Industry at Paris La Villette, and worked on programmes with Russian scientists.

Loren Graham is professor emeritus of the history of science at MIT and research associate at Harvard University. He is a fellow of the American Association for the Advancement of Science and the American Academy of Arts and Sciences, a member of the American Philosophical Society and a foreign member of the Russian Academy of Natural Science. His book, A Face in the Rock: The Tale of a Grand Island Chippewa (1998), is being made into a film.