Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity

In an age where science and religion have the potential to enlighten a holistic perspective of reality, the theme of this book is phenomenal. And, since both mathematics and religion address knowledge that transcends finite limits, "naming infinity" is a potential nexus, an opening point of agreement. Alas, in an effort to simplify the mathematics for laymen, the challenges faced by the math mavens of the late 19th / early 20th centuries are largely reduced to the notion of the infinite set of integers being larger than the infinite set of even integers. Perhaps that is enough... But, the idea that infinity divided by 2 presents a challenge that axiomatically requires God's assistance in terms of nomenclature is not compelling. And, in an effort to simplify the practices of a specific sect of the Russian Orthodox Church, we get an equally vague notion of the mystical process of repeatedly invoking specific names of God to access an alternate world of insight. Entering a transcendent state by ritual chanting has a broad and deep history within the domain of religion. To watch mathematicians explain sacred ecstasy is like watching George W. Bush explain ethics. What we do get is the social and political milieu of Europe as the scientific community struggled with mathematics, culture, language, and the impact of the Russian revolution. We also get fascinating insights into the personalities of some of the eccentric and brilliant minds of the era. Mathematicians, some of whom remained deeply religious, despite the onslaught of Communism in the Soviet Union, made great strides in understanding and teaching. The success of the Russian math professors may have resulted less from their Orthodox rapport with incantation (premise of authors), but more from their open and consultative pedagogy (implication of authors). A fascinating subject - worthy of continued research and discussion.

My abilities in mathematics are decidedly pre-Euclidean. A scientist friend of mine used the visual metaphor of an established tree to explain mathematics: arithmetic, geometry, algebra and trig are the roots, the many developments in modern math are the branches and leaves. The trunk connecting roots and branches and supporting the tree is the calculus. Given this metaphor, I'm still scrambling among the roots for acorns of understanding from the top of the tree, because I never climbed past calculus. This limited my capacity to understand the math concepts Graham and Kantor describe in "Naming Infinity". Other reviewers have commented on the book's lack of equations to demonstrate the math propositions discussed in it. I wish some simple clear definitions of the building blocks of set theory had been available in an appendix. Beyond the few figures which elucidate Cantor's discoveries in the second chapter, and a discussion of the conflict between Platonic and Aristotelian notions of mathematics and how these played out in both the French and Moscow Schools of math in the early XXth century, there are precious few tools to help the untutored reader develop a more profound comprehension of the subtleties of set theory and the mathematical continuum. It's also true that I sometimes wished for the authors to return to topics briefly discussed in earlier chapters: did the religious practices of the Name-Worshippers persist through the post-Stalin era, for example? What was Luzin's life like in his later years, after the discontinuity event of his pardon by Stalin? (Beyond his caustic insult to Kolmogorov, and his lover Bari's suicide after his death, there is precious little here about Luzin's twilight years.) These are minor cavils about a book which illumines an exciting time in European intellectual history. The history (indeed the existence) of the Name-Worshipper sect was unknown to me in Russian culture. The authors are to be thanked for their concise description of this movement's history and its leading exponents. I am very fond of the Silver Age in Russian cultural history--- and of the work of Symbolists like the remarkable Andrei Bely. Bely makes an appearance here, naturally, because he was the son of Nikolai Bugaev, the professor of mathematics at Moscow University who was the teacher of the trio of Russian mathematicians/name-worshippers considered in this book. The influence of the ideas propounded by the Name-Worshippers on Bely was another subject with which I was unfamiliar. Their obsession with "naming" does more to explain the numinous appeal of Bely's difficult works than that author's equally eccentric connection with the anthroposophy of Rudolf Steiner. The excerpts quoted from Bely's "First Encounter" bring to life the cruelly extirpated world of Russia's pre-revolutionary intelligentsia. Finally, this excellent small history vividly describes the lives of the protagonists of the Moscow School of Mathema

I read this book from cover to cover while flying on a plane from Dublin to St. Petersburg and back. That was so wonderful reading experience - I couldn't put the book down during those flights. I recall that I visited the Department of Mathematics a few times when I studied Chemistry in Moscow State University although at that time I knew next to nothing about Russian mathematicians. The book touched me so deeply that I bought the main work of Florensky: The Pillar and Ground of the Truth, the history of Russian philosophy and several books explaining Orthodox Church. This is the best mathematics history book I have ever read, my feelings perhaps comparable to those that I experienced when I finished reading Mathematics: The Loss of Certainty by Morris Kline but that was more than 20 years ago. Thanks, Dmitry Vostokov Founder of Literate Scientist Blog

I had the pleasant assignment of reviewing this book for The Mathematical Intelligencer (my review is to appear in a few months). In that review, I invoked the image of the Tower of Babel as a metaphor for the efforts (ultimately vain) of N.N. Luzin to find an effective way of enumerating the countable ordinal numbers. It is more than a coincidence that Luzin was an adherent of a splinter group in the Orthodox Church that called itself "onomatodoxy" (imeslavie, in Russian), meaning "name-glorification" or "name-worshipping". This group attached supreme importance to getting names exactly right in religious matters. Luzin carried this zeal into mathematics as well, trying desperately to break everything into clear, unambiguous definitions. The Polish mathematician Sierpinski had horrified him with a list of results in analysis that can't be proved without invoking the axiom of choice, and he actually had insomnia over that for many nights. It is true that Henri Lebesgue, who shared none of Luzin's religious mysticism, also tried to deal only with functions that are "analytic," but he didn't make a fetish of it. Luzin invented the contrasting terms "effective" and "auswahlistic" to emphasize the true epistemological status of results in analysis. If they used the Auswahlprinzip (axiom of choice), they weren't "effective." In order to make the book as widely accessible as possible, the authors do not go into any deep mathematical detail (there are no equations in the book), but they describe it in general terms well enough to give an adequate picture. In addition to the broad mathematical trends that are reflected in the book, there are gripping personal stories of individual mathematicians and their troubles. Alexandrov's tragic loss of his partner Uryson, who drowned in Brittany in 1924 is just one of them. (These two were known as "the PS's", because of their initials, Pavel Sergeevich Alexandrov and Pavel Samuilovich Urysohn. In Russian that is PSY, which means "the dogs", and Alexandrov once send a memo to Uryson headed "PSU ot PSA", that is, TO PSU FROM PSA. But in Russian the phrase can also be read "to a dog from a dog".) Alexandrov also figures prominently as one of Luzin's accusers in the tense hearing before a tribunal of the Academy of Sciences in 1936. At that trial he said, "I do not claim that Suslin named A-sets in my honor, because I never heard anything of the sort from him." Some 40 years later, that same Alexandrov wrote that "[Suslin] emphasized that he was naming the A-sets in my honor, in analogy with the B-sets [named for Borel]." Thus does memory play tricks on us all! Read this book. You'll enjoy it if philsophical questions, political intrigue, or mathematics can interest you at all.

This book is a wonderful and gripping account of a very important chapter in the history of 20th-century mathematics. Graham and Kantor challenge many "common wisdoms" and common myths about mathematics, religion, and mathematicians. It reminds us that the story behind the mathematics is often much more exciting than mathematics itself.