What direct influence can, or should, a mathematician’s personal religious beliefs have on his or her technical mathematical work? The commonest answer to this question is also the simplest: None whatsoever. To be sure, we all understand that religious faith can influence our personal and professional conduct, as well as our career decisions, and in this way our beliefs may indirectly influence what we spend our time working on. But as for the actual practice of mathematics—will mathematicians with different religious beliefs prove different theorems, or prove them in different ways? If you read someone’s mathematical papers and books, will you be able to detect the influence of his or her religion? Surely the answer is no.
In their fascinating book Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity, the authors Loren Graham and Jean-Michel Kantor challenge this commonsensical point of view. Graham and Kantor give a historical account of several mathematicians who worked on the foundations of mathematics in the early twentieth century. At that time, the status of infinite sets was controversial. Cantor had famously pioneered the systematic study of so-called transfinite sets, and his ideas and methods were producing important new results, particularly in the realm of function theory and integration. Graham and Kantor emphasize the key roles of the French mathematicians Borel, Lebesgue, and Baire in these developments. Despite the success of Cantor’s theory of transfinite sets, however, many mathematicians harbored doubts about their legitimacy. Several now-famous paradoxes were discovered, casting doubt on the logical consistency of set theory. Around the same time, the German mathematician Ernst Zermelo proved his astonishing well-ordering theorem, and clarified that his theorem relied on a set-theoretic assumption called the Axiom of Choice. These developments seem to have caused Borel, Lebesgue, and Baire to back off from their former enthusiasm for transfinite sets; in particular, all three of them rejected the Axiom of Choice.
Part of what made the Axiom of Choice controversial was that it asserted the abstract existence of an infinite set of choices, without giving any hint of how those choices should be made. Graham and Kantor quote Lebesgue as asking, “Can we convince ourselves of the existence of a mathematical object without defining it? To define always means naming a characteristic property of what is being defined.” The Axiom of Choice did not meet Lebesgue’s threshold for defining a set.
Graham and Kantor then turn their attention to the main subject of the book, namely the Russian mathematicians who became interested in set theory, and who pioneered the field now known as descriptive set theory. Graham and Kantor’s main thesis is that a major reason that Dmitri Egorov, Nikolai Luzin, and others in the Moscow School of Mathematics were able to push ahead and find striking new results that went far beyond what their French predecessors had obtained was that they were aided by their mystical religious beliefs. Specifically, Egorov and Luzin were both involved in a religious movement (condemned by the Russian Orthodox Church) called “Imiaslavie” or “Name Worshipping.” Among other things, Name Worshippers believe that there is extraordinary divine power in the name of God. For example, many Name Worshippers bring themselves into ecstatic mystical trances by means of extended repetition of prayers such as the so-called Jesus Prayer (“Lord Jesus Christ, Son of God, have mercy on me, a sinner”). The constant repetition of Jesus’ name unlocks miraculous spiritual power.
As the term suggests, descriptive set theory is concerned with certain infinite sets of real numbers that can be described or “named” using specific logical expressions. Although a skeptic might see little connection between naming God and naming infinite sets, Graham and Kantor make a convincing case that Egorov, Luzin, Pavel Florensky (another mathematician who was also a priest), and others regarded the two activities as intimately related. Regarding Lebesgue’s question above, Graham and Kantor write:
The answer for Florensky—and, later, for Egorov and Luzin—was that the act of naming itself gave the object existence. Thus “naming” became the key to both religion and mathematics. The Name Worshippers gave existence to God by worshipping his name; the mathematicians gave existence to sets by naming them.
In other words, unlike Borel, Lebesgue, and Baire, the Name-Worshipping mathematicians were not turned off by the seeming intangibility of set-theoretic ideas like the Axiom of Choice. Quite the opposite: Emboldened by their beliefs about God, names, and infinity, they pursued transfinite set theory eagerly and invented an entire new subfield of mathematics.
It is an intriguing hypothesis. Graham and Kantor hasten to add that “when we emphasize the importance of Name Worshipping to men like Luzin, Egorov, and Florensky, we are not claiming a unique or necessary relationship. We are simply saying that in the cases of these thinkers, a religious heresy being talked about at the time when creative work was being done in set theory played a role in their conceptions. It could have happened another way; but it did not.” After reading this book, mathematicians with strong religious faith may want to think twice about whether that faith might have something to offer to their own mathematical investigations.
[Note: You may also be interested in the review by Alexey Glutsyuk that was published in the January 2014 issue of the Notices of the American Mathematical Society.]