Already around Galileo’s time mathematicians were giving names to the unthinkable. The word “absurd” was first used in English in 1557 to describe the product of 8-12 (or -4), since the word “surd,” meaning deaf or silent, had become known as the name for an irrational square root (expressing the coiner of the term’s attitude toward what it named). Symbols such as x and y, powers and roots, slowly rendered philosophical impossibilities into algebraic necessities: if x2 = 2 helped solve certain problems, then √2 no longer seemed all that crazy. If x2 = -1 somehow surfaced through manipulation, then √-1 was no longer unthinkable. In fact, the square root of negative one was soon termed an “imaginary” number: the name may have expressed a sense of incredulity, but it made the “imaginary” real.

Georg Cantor, the promising son of a successful St. Petersburg merchant who had moved his family to Germany, completed his studies in Berlin and Göttingen in 1869, and become a lecturer at the University of Halle, brought a new urgency to the problem of infinity. In fact, he soon learned, it was the problem not of infinity but of infinities, for (as Galileo had perceived) not all infinities are the same. Galileo, it turned out, had been mistaken: the series of infinite integers and the series of infinite even numbers were the same kind of infinity. Each term in one “set” (“every Many,” Cantor defined a set in 1883, “that can be thought of as One”) could be matched by a term in the other. What Cantor found by way of inventing set theory was that there were different powers, or cardinalities, to different sets of infinities.

Often kids try counting to infinity to resist falling asleep. They never make it. But Cantor counted to infinity and kept on going: there was the smallest countable infinity plus one, the smallest countable infinity plus two, the smallest countable infinity plus three, and so on. Then there was the smallest uncountable infinity, like points on a line from zero to one—called the “Continuum”-followed by the next-to-smallest uncountable infinity plus one, and so on. Cantor had created an entire universe of infinities by giving each different kind of set a name. Could he fit a set of infinity between the countable infinities and the uncountable ones? Were infinities like a smooth river or were they empty spaces, like the swaths of dark empyrean between sparkling white stars? This was the Continuum Hypothesis, and Cantor would spend the rest of his life trying unsuccessfully to prove it. Gradually he lost his mind, coming to believe that God, the set of all sets, had revealed set theory to him, and that all the sets he talked about existed preformed in God’s own mind. After the winter of 1902, he was in and out of the Nervenklinik, helplessly battling an infinity of madness.

Rocking in the belly of the Imperial Russian Navy ship as it sailed, in June 1913, through sparkling Aegean waters toward the Monastery of St. Pantaleimon on Mount Athos, the Archbishop Nikon of Vologda braced himself. He was determined. Even before hermits in the deserts of Palestine practiced the “Prayer of the Heart” in the fourth century, Christianity had known mystical sects. Later called hesychast monks from the Greekhesychia, or stillness, such mystics had believed in the power of glossalia, or “praying without ceasing,” with control of breathing and the heartbeat, to reach union with God. Already in the fourteenth century Gregory Palamas, a Constantine monk, had settled on Mount Athos preaching hesychasm as a true alternative to the staid rationalism of Byzantine Christianity. Now, in modern times, to the great consternation of leaders of the Russian Orthodox Church, a Russian monk named Ilarion had instituted the “Jesus Prayer” among his followers (“Lord Jesus Christ, Son of God, have mercy on me, a sinner”—sometimes shortened to “Lord Jesus Christ,” or just “Jesus”—repeated over and over again), a prayer considered heretical for harking back to mystical times. Ilarion admitted that when reciting the prayer worshippers needed to be careful. There were three “stages of immersion”—the oral, the mental, and finally the “Prayer of the Heart”: if one jumped between them prematurely, warm blood could descend to the lower parts of the body and lead to sexual arousal. Archbishop Nikon of Vologda clenched his fists.