Naming Infinity

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.

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...

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