Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity
By Loren Graham and Jean-Michel Kantor

Math is the most abstract of subjects—a complex rendering of the world around us in symbols, expressions made from symbols, and in their manipulation. What minds first made sense of it all? Who are the men and women who first conceived of the world around us in these arcane ways? In their well-researched new book, Naming Infinity, Loren Graham, professor emeritus of the history of science at MIT, and Jean-Michel Kantor, a mathematician at the Institut de Mathématiques de Jussieu in Paris, try to remedy that situation. In this book they discuss the development of set theory, less by looking at theorems and definitions and more by looking at the people, politics, and religious ideas that helped shape this subject.

As the authors tell us, set theory began in December 1873, when Georg Cantor, “proved to his amazement, that the set of integers N (starting in 1895 Cantor would call its ‘number of elements’ , aleph-zero) and the set of real numbers R (the Continuum) had different kinds of infinite numbers of elements. The great German mathematician David Hilbert “gave full recognition to set theory and Cantor’s work since the 1870’s” in a speech he gave before the second International Congress of Mathematicians in Paris in 1900. Three young French mathematicians, Borel, Lebesgue, and Baire were aware of Hilbert’s speech and were strongly attracted to this new theory. However, it wasn’t long before problems with the new theory dampened the interest of the French.

“Already in 1895 Cantor realized that there were difficulties with what he called ‘sets that were too big to correspond to any cardinal’ . . . and he escaped from the resulting contradiction by introducing pluralities too big to be sets, corresponding to the theological notion, the ‘Absolute,’ which cannot be known, not even approximately.”

The underlying concept at the heart of the problem is infinity—what does it mean, how does one define it, is infinity a number, and, amazingly, are there different “size” infinities? These questions led to inquiries into the nature of what “number” itself is. The answers that were forthcoming all too often involved areas ‘outside the domain of mathematics’, areas that the French believed had no connection with mathematics. “Is mathematics a house built on sand, on the shaky foundations of psychology and philosophy?” Russian mathematicians of the time, in particular Egorov, Luzin, and Florensky, were eager to use both philosophy and religion to solve these vexing mathematical dilemmas. They were adherents to a tiny Christian sect, the Name Worshippers, branded heretics by the Russian Orthodox Church, who believed that saying the Jesus Prayer repeatedly brought one to a special “oneness” with God.

Both the French and the Russian mathematicians were wrestling with the problems of what a mathematical object is, what mathematicians are allowed to do. Lebesgue wrote to Borel in 1905, ‘Is it possible to convince oneself of the existence of a mathematical being without defining it?’ Florensky saw this question as the analogue of ‘Is it possible to convince oneself of the existence of God without defining him?’ The answer for Florensky—and, later, for Egorov and Luzin—was that the act of naming in itself gave the object existence. The Name Worshippers gave existence to God by worshipping his name; the mathematicians gave existence to sets by naming them.

The period prior to World War I until the mid-1920’s was a time of real advancement in set theory. These mathematicians, part of the Moscow circle known as “Lusitania,” created descriptive set theory to address the key problem in set theory – the Continuum Hypothesis. First posed by Cantor, CH states that there is no cardinal number between , the cardinal number of the natural numbers, and c, the cardinal of the R, the set of real numbers. It was a fruitful period of mathematical development but not destined to last long. The Revolution in 1917 brought Communism and atheism to Russia. The very religious beliefs that inspired Luzi, Egorov, and Florensky, and the others, brought persecution from the state that forced many of them out of their jobs and homes. Graham and Kantor paint a clear picture the petty intrigues as well as the serious harassment “the Russian Trio,” Egorov, Luzin, and Florensky, were forced to endure.

Most people, including more than a few mathematicians I suspect, believe that mathematics is created in a near vacuum by men and women sitting at a desk with (graph) paper and pencil, oblivious to and unaffected by the outside world. Kantor and Graham have succeeded admirably in portraying this episode in the history of mathematics in the most human terms, clearly showing how the development of set theory, as unworldly a collection of abstract concepts as one can imagine, was influenced by the psychological, philosophical, political, cultural, and religious beliefs of the day.

Reviewed by Stan Izen

Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity by Loren Graham and Jean-Michel Kantor
Harvard University Press, 2009
Cloth, 256 pp, $25.95
ISBN-10: 0674032934