Naming Infinity

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Last Updated on May 5, 2015, by eNotes Editorial. Word Count: 1842

For most people, exposure to mathematics begins with arithmetic in elementary school and may extend to the study of algebra, geometry, trigonometry, and calculus in high school. Math is usually viewed as a rational, scientific discipline with practical applications. The notion that mathematics may have a connection to the divine does not often enter into classroom discussions. After all the job of a mathematician is to define precisely and to prove theories and equations, while the purview of a theologian is to explore the transcendent. Throughout history and across cultures, however, mathematics and spirituality have intersected in humanity’s quest to know the absolute.

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The magic square number system formulated by the Chinese and the correspondence between Indian astronomy and sacred Hindu texts represent early examples of ways in which humans have linked mathematics and religion. During the classical Greek period, the Pythagoreans believed that creation was founded on the principles of mathematics. In the centuries that followed, Augustine, Galileo, Sir Isaac Newton, René Descartes, Blaise Pascal, Baruch Spinoza, and others explored the notion that the divine essence underlying the structure of the material universe could be unveiled through the study of numbers. With the advent of the Enlightenment in the seventeenth and eighteenth centuries, however, secular views predominated in the form of deism, agnosticism, and atheism, and mathematics was studied from a rationalist perspective. This approach continues to prevail, although some mathematicians continue to assert that metaphysics and mathematics are related.

The tug of war between the rationalist and metaphysical approaches to mathematics is at the center of the intellectual drama in Naming Infinity by Loren Graham and Jean-Michel Kantor. In the early twentieth century, mathematicians were wrestling with the problem of how to define the nature of infinity. They sought to determine what kind of thing infinity is, whether it should be considered a number, and whether it can be of different sizes, among other questions. Three French and three Russian mathematicians grappled with this problem, but their social and cultural contexts led them to tackle the situation from opposite sides of the spectrum. The French Émile Borel, Henri Lebesgue, and René Baire attempted to understand infinity from a rationalist viewpoint. Russians Dmitri Egorov, Pavel Florensky, and Nikolai Luzin drew on a spiritual practice called “name worshiping”a heretical practice according to the Russian Orthodox Churchto enhance their understanding of infinity.

The nature of infinity is central to set theory, which was introduced in an 1874 paper by German mathematician Georg Cantor titled “On a Characteristic Property of All Real Algebraic Numbers.” Even before Cantor began his study, the concept of infinity had occupied mathematicians for centuries. Cantor’s groundbreaking work on number theory, however, brought the nature of infinity into sharper focus. Cantor’s theory stirred controversy within the mathematical community but eventually garnered widespread interest if not acceptance. His notion of one-to-one correspondence among sets, his work with the real numbers and integers, and his idea of multiple infinities touched off a wave of excitement among mathematicians that over the years would transform mathematical inquiry.

German mathematician David Hilbert hailed Cantor’s ideas during his historic address to the International Congress of Mathematicians in Paris in 1900. Hilbert’s endorsement of Cantor’s work caught the attention of “the French trio” of Borel, Lebesgue, and Baire. However, the French mathematicians would eventually become less excited about set theory. Graham and Kantor note that one reason the three men lost their enthusiasm was that in 1895 Cantor himself had 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 a theological notion, the ’Absolute,’ which cannot be known, even approximately.

Troubled by the metaphysical aspects of Cantor’s ideas, the French believed that reason should be the sole basis for mathematical study and that anything that “cannot be known, even approximately” is beyond the realm of mathematics. The question became: “Is mathematics a house built on sand, on the shaky foundations of psychology and philosophy?”

Ernst Zemelo, a young German mathematician who proposed the “Axiom of Choice” in 1904, related this concern to a claim. His axiom raised some thorny questions such as “What does it mean to choose?” and “Is it possible to make an infinity of choices?” Lebesgue went further and asked “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.”

While the religious implication of set theory troubled the French, the Russians were intrigued by it. Egorov, Luzin, and Florensky, who was a priest as well as an accomplished mathematician, did not view the paradoxes inherent in set theory as insurmountable. Indeed, they believed that philosophy and religion could aid in mathematics. Specifically, they sought to prove the existence of sets by naming them. Followers of name worshiping, an obscure Christian sect, Egorov, Luzin, and Florensky believed that, by repeating the Jesus Prayer“Lord Jesus Christ, have mercy on me a sinner”they could eventually achieve oneness with God. The Jesus Prayer has been used for centuries by Christian mystics and contemplatives to achieve mystical union with God. Practitioners chant the prayer continuously until the rhythmic repetition coincides with the beating of their hearts and their breathing. Although repetition of the prayer is accepted practice in Russian Orthodox spirituality, the Name Worshipers went a step further. They claimed that the name of God is God himself. The Church, however, considered this idea heretical and tried to suppress Name Worshipers. The most notorious example of this purge was an attack by Russian imperial marines in 1913 on the Pantaleimon Orthodox monastery on Mt. Athos, Greece, where several hundred monks were expelled because they were adherents of the forbidden practice.

When early twentieth century mathematicians were grappling with the concept of infinity in relation to set theory, they were puzzled by Georg Cantor’s “infinity of infinities.” It seemed that Cantor, by naming setsin other words, by properly defining themgave them existence. Egorov, Luzin, and Florensky recognized the similarities between Cantor’s notions and their own religious beliefs. If they could make God real by worshiping God’s name, then they could also bring new infinities into existence by naming them. As Kantor and Graham point out, the notion of bringing an object into existence through naming is not as far-fetched as it sounds if one is familiar with the biblical account of creation, in which God said, “Let there be light” before light existed.

Florensky in particular saw Lebesgue’s inquiry to Borel in 1905“Is it possible to convince oneself of the existence of the question of a mathematical being without defining it?”as analogous to his query, “Is it possible to convince oneself of the existence of God without defining him?” The answer lies in the way Name Worshipers viewed the use of words. To them, words were not just products of the mind or intellect: They affect the very fabric of reality. This radical marriage of mathematics and religion went against the materialism and determinism that dominated the field at the time. It also proved to be the breakthrough that was needed to understand set theory as it related to infinity. The new insights on set theory derived from Name Worshiping had an enormous impact on the mathematical landscape for years to come and became a distinctive contribution of the Moscow School of Mathematics, of which Egorov, Luzin, and Florensky were members.

Graham and Kantor’s thesis that religious mysticism fueled creative mathematical insights is intriguing. Readers unschooled in the concepts of higher mathematics may find some ideas difficult to grasp, however. Terms such as “numerable and denumerable sets” and “continuous and discontinuous functions” are only summarily explained, and notions such as the Continuum Hypothesis or the Axiom of Choice could be better clarified. The authors also give the impression that the Jesus Prayer is mainly associated with name worshiping, when in fact the prayer has been practiced by non-Name Worshipers since the desert fathers first formulated it around the fifth century. Still, Graham and Kantor, both self-described secularists, present beliefs of the Russian Name Worshipers in a fair, unbiased way. It is evident that they respect the Russians not only for their mathematical genius but also for their dedication to their religious beliefs, which in turn led to important discoveries. The authors comment 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 case 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.

Graham and Kantor’s narrative is more than just a history of transformative events that had an impact on a particular field of mathematics. It is also a chronicle of social upheaval, political conflict, and personal sacrifice. In discussing the Russian trio, Graham and Kantor explore the political milieu in which the Russians lived and worked. The chaos following the Bolshevik Revolution and the Communist crackdown in the newly formed Soviet Union resulted in the persecution of people the government viewed as religious dissidents. Egorov, Luzin, and Florensky were among them.

Accused of participating in a counterrevolutionary organization (the Russian Orthodox Church), Egorov was arrested and sent to prison in Kazan. He refused to eat and subsequently died. Florensky was arrested three times and was tortured during his third arrest. Under pressure, he signed a forced confession, was exiled to a gulag in the Soviet far east, and was executed in 1937 for carrying out counterrevolutionary agitation in the labor camp. Luzin was the only one of the three who escaped prison. However, he was a victim of jealousy and was betrayed by Ernst Kol’man, one of his colleagues, who was a fervent Marxist. Luzin managed to escape imprisonment and death, however, through the efforts of well-placed friends. The authors also reveal details of the personal lives of other mathematicians who knew or were influenced by the Russians. Some of these stories include instances of sexual betrayal, homosexuality, and suicide.

Although some have argued that such anecdotes have no place in a history of mathematics, the personal profiles of the six men and their colleagues that Kantor and Graham provide contribute depth and drama to what could have been a dry account. One understands that these scholars were not simply sitting in an ivory tower with pen and pencil in hand, puzzling over esoteric concepts. They had families and friends, suffered losses and savored triumphs, and were subject to emotional, psychological, and spiritual trials. Graham and Kantor have succeeded in portraying not only the intellectual challenges that set theory presented to the French and Russians but also the human cost that may result from single-mindedly pursuing a goal in which one believes.


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Last Updated on May 5, 2015, by eNotes Editorial. Word Count: 28

Booklist 105, no. 13 (March 1, 2009): 12.

Christianity Today 53, no. 7 (July, 2009): 56.

London Review of Books 31, no. 16 (August 27, 2009): 28-30.

Nature 458, no. 7241 (April 2, 2009): 971-972.

Scientific American 300, no. 4 (April, 2009):83.

Times Higher Education, April 30, 2009, pp. 46-47.

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