Naming Infinity: A True Story of Religious Mysticism And Mathematical Creativity
By Loren Graham and Jean-Michel Kantor
(Belknap Press, 239 pp., $25.95)
A starry firmament, or sand cascading through one’s open fingers, or weeds springing up time after time: the first conception of infinity, of the uncountable and the unending, is not recorded, but it must have been stimulated by experiences such as these. It may have merged in the mind of an ancient progenitor with thoughts of a God, a possessor of unlimited might, an infinite being itself. But whether or not the idea of God was born with the first thoughts of what cannot be counted, this wonderful book by an American historian of science and a French mathematician teaches us that eons later, the divine and the infinite remain closely entangled. A mathematical understanding of infinity was a conundrum for rationalists, who believed it could be mastered by using only the methods of scientific logic, unsullied by eschatology or religion. But as Jean-Michel Kantor and Loren Graham show, they were wrong. Centuries after Bacon and Descartes, and the birth of the scientific method of the moderns, mysticism came to the rescue of one of the most intractable problems posed by abstract human thought. It was mysticism, not rationalism, that helped to crack infinity.
In the sixth century B.C.E. Anaximander of Miletus gave a name to the infinite, calling the indeterminate, or “something without bound, form, or quality,” apeiron. But limitlessness, and non-rationality, and ineffability were all descriptions of what infinity was not. The closest anyone came for centuries to a positive definition was “potentiality” as opposed to “actuality,” in the influential terms of Aristotle. But this formulation did little to help define the indefinable. Even Galileo, nearly two thousand years later, bowed his weighty head before the limitless. Contemplating the series of infinite integers (1,2,3,4...) and the series of infinite even numbers (2,4,6,8...), he gave up: clearly both could continue without limit, and yet wasn’t one precisely one half as large as the other?
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 Greek hesychia, 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.
The last thing Nicholas II wanted was for bickering monks to invite an invasion of the Greek army into the monastery; the czar didn’t care much about the theological dispute, but he was not about to lose a Russian protectorate in the Aegean. Later, after the gunboat Donets had lowered its anchor and Russian marines stormed the monastery with clubs, water hoses, and bayonets, each side would claim a different story. Whether monks were brutally murdered, soldiers were beaten, or only a small number of fanatics were rather quietly subdued didn’t in the end really matter: after all, nearly a thousand monks were hauled back on the ships to Russia, where their leadership was thrown in jail, and the rest were defrocked and banished to far-off provinces. The Name Worshippers of Mount Athos had been shut down. What mattered most were the defiant interruptions to the angry sermon of Archbishop Nikon of Vologda, who had marched into the monastery courtyard behind the troops. “You mistakenly believe that names are the same as God,” his voice trembled. “But I tell you that names, even of divine beings, are not God themselves.” Corralled, water-drenched, their arms twisted violently behind their backs, the monks would not be silenced. “Imia Bozhie est’ sam Bog!” some of them were clearly heard shouting, their eyes alight. “The Name of God is God!”
Few knew at the time that the events on Mount Athos would soon play a role in cracking the mysteries of infinity, least of all Henri Poincaré, France’s premier mathematician. Back on the mainland, Poincaré was aghast at Cantor’s sets. “The higher infinities,” Graham and Kantor quote him, “have a whiff of form without matter, which is repugnant to the French spirit.” Reared in the tradition of Descartes, Frenchmen knew better. Mathematics was a close cousin of physics. It dealt with things like triangles and populations and planets that could be counted and touched. It related to this world. Over their dead bodies would it be defiled by abstract philosophy.
And yet when Poincaré’s German rival, David Hilbert, gave the opening address at the Second International Congress of Mathematics in Paris back in 1900, he presented a brash challenge to his French hosts. At the top of a list of twenty-three fundamental mathematical problems that had yet to be solved, he placed the scourge that had driven Cantor to his wit’s end: the Continuum Hypothesis. Three young French mathematicians sitting in the crowd couldn’t help but get excited: the person who would solve the mathematical world’s greatest challenge would win eternal fame and instant celebrity.
Émile Borel, the eldest of the three, had taught the other two at the École Normale Supérieure. He was a Parisian socialite “consumed with all possible experience,” and would become director of the École, publisher of the Revue du mois (crucial to the birth of the Radical Left), mayor of his hometown, minister of the navy, an activist in the French Resistance, a prisoner of the Gestapo, and a winner of innumerable prizes. René Baire was the son of a tailor, whom a childhood acquaintance described as “a big fellow with an obviously weak bone structure, a wan complexion, and dark deep eyes that tended to stare in a disturbing manner.” And Henri Lebesgue, the son of a widowed seamstress, sported (according to Borel’s wife) a “malicious smile under a reddish moustache” and a pince-nez. All three were brilliant and ambitious. And each tackled the problem of infinity.
The challenge proved extreme. Already in 1901 Bertrand Russell pointed out a paradox in Cantor’s thinking: how could one countenance the set of all sets that do not belong to themselves? The paradox was maddening. So was the following definition of a number, published in 1905 by Jules Richard, a professor of mathematics from Dijon: “Consider the smallest number not definable in English in less than twenty words.” (Richard had just defined it in thirteen words.) Cantor himself had introduced pluralities too big to be sets so as to skirt such contradictions, calling them “Absolutes” and identifying them with God. At stake was the very reality of numbers, the ontology of mathematics: if giving names to sets of infinity led to unthinkable paradoxes, or to God, perhaps the sets of infinity were nothing but imaginary creations, impudent slaps in the face to the touchable and real. Weighed down by their Cartesian tradition, wary of losing themselves in a forest of abstraction, the three Frenchmen in the end could walk no further. Embittered, Lebesgue pulled back from his earlier intrepid sojourns into the abstract, breaking off a deep friendship with his former teacher. Wary of the mental consequences of such research, Borel simply gave up, turning to probability and statistics; “not much” compared to his earlier work, he wrote to his wife, but useful. And Baire’s lot was bleaker. Crippled by financial troubles, neurosis, and depression, he eventually committed suicide in 1932, alone in a hotel on Lake Geneva.
If the French sought to purge mathematics of philosophy, the Russians were just the opposite. Continuous functions are functions where small changes in the input result in small changes in the output; discontinuous functions are functions where this is not so. But to Nikolai Bugaev, professor of mathematics at Moscow University, they were much more than that. Speaking at the First International Congress of Mathematics in Zurich in 1897, he explained that discontinuous functions represent humanity’s escape from fatalism, manifestations of “independent individuality and autonomy,” protectors of “aesthetic” freedom. “Monsters,” one French mathematician called them, but Bugaev insisted. Discontinuity strengthened man’s moral fiber.
And so it wasn’t all that surprising that one of Bugaev’s brightest students should have developed a deep interest in the connection between mathematics and religion. Dmitri Egorov had spent 1902-1903 in Europe, where he had studied with some of the greatest mathematicians of the age, including Lebesgue. Back in Moscow, he married the daughter of one of Russia’s most famous violinists, Ivan Grzhimali, and settled into a comfortable life of lavish entertaining: Tchaikovsky, Rachmaninoff, Ilya Repin. Still, Egorov was a reserved and deeply religious man; friends saw him dutifully kiss the hands of priests lurking around his home.
At Moscow University, Egorov had two outstanding students. Nikolai Luzin, the grandson of a serf, nurtured a strong belief in the liberating power of science, a hope that philosophical materialism would make the world a better place some day. Pavel Florensky, on the other hand, the son of a railroad engineer from Yevlakh in present-day Azerbaijan, had just recently converted to religious faith following a revelation. Influenced by Bugaev, he became convinced that intellectually the nineteenth century had been a disaster. The culprit was continuous thinking, the notion that all transitions from one point to the next need to pass through all the intermediates. Mathematics, in fact, had been responsible for the atrocity, the “determinism” of differential functions having seeped into Lyell’s and Darwin’s gradualism in geology and evolution, into psychology and sociology, and worst of all, into the latest plague—Marxism. Thankfully Georg Cantor had at last challenged continuity, presenting his Continuum as a “mere set of points” rather than a dogmatic directed line. Now mutation in biology, electrons jumping orbitals in physics, and subliminal consciousness in psychology were all following suit. Humanity could finally graduate to higher places.
When the attempt at revolution was beaten down in 1905, Luzin suffered a mental breakdown; the trauma of the brutality had shaken his positivist aspirations to their core. Hoping that he might gain his balance abroad, Egorov sent his brilliant student to Paris to meet, among others, Lebesgue and Borel. “To see the misery of the people,” Luzin wrote home to his friend Florensky, who had since entered the Theological Academy, “to see the torment of life....—this is an unbearable sight.... I cannot live by science alone.... If I do not find a path to seek the truth ... I will not go on living.” Florensky replied that agnosticism and atheism were responsible for Russia’s chaos and confusion, and invited Luzin to stay near him in his monastery town. It was there that Luzin read his friend’s thesis “On Religious Truth,” and instantly abandoned all thoughts of suicide. He had caught a glimmer of a path to truth, an “intuitive-mystical understanding,” and Florensky had been his guide. “I owe my interest in life to you,” he wrote to him.
Throwing himself into set theory back in Moscow, Luzin maintained strong ties with Florensky, and here is where the escapades of the monks of the Aegean return to our story. It is not clear precisely when both men first learned of Name Worshipping, but already in 1906 they enjoyed calling each other by names other than their own. When news of the rebellion on Mount Athos reached Russia in 1913, Florensky spoke up publicly in its favor, and befriended monks who had endured firsthand the navy’s brutal attack on St. Pantaleimon. Soon two worlds were becoming entwined. Lebesgue had asked whether a mathematical object could exist without defining (meaning naming) it, and now the answer was becoming clear. Just as naming God via glossolalian repetition was a religious act that brought the deity into existence, so naming sets via increasingly recursive definitions was a mathematical act that conferred a reality in the world of numbers. Cantor and before him the ancient Neoplatonists had shown the way, but this was only the beginning. Infused with mysticism, Florensky believed, new forms of mathematics and religion were being born, ones that by rejecting determinism would rescue mankind from catastrophe. In both cases—God and infinity—the key to bringing abstractions into reality was bestowing upon them a name.
Before long, Egorov was onboard. He was blown away by his former student Florensky’s thesis, now published as a book, The Pillar and Foundation of Truth. Joining hands with Luzin, he formed an inner circle within the faculty of mathematics at Moscow University, and young brilliants, even very young ones, flocked from across the nation to study sets. Never mind the university rule that when classroom temperature dropped below freezing, teaching was canceled: the hungry brights bundled up, put pencil to paper, and paid no heed. With the revolution underway, massive purges hit the campus; idealism and religion had to be kept under the slide rule. But the outside danger fit the inner edge, unwittingly making things all the more exciting. The friends comprised a kind of secret order, and its name was “Lusitania.” In the inner circle no one called each other by their regular name. Each member was assigned an aleph of infinity, with breakthroughs and insights bestowing a higher cardinality. New recruits were א0, Luzin א17, and Egorov אω—the highest regular infinite before the Continuum. By this time Egorov was already a devout Name Worshipper: in his mind the mystery of numbers and the mystery of God had entirely merged through the power of the name.
On one occasion the group traveled by train to Petrograd for a scientific meeting, proceeding to invent appellations for nearly everything: “Commandant of the Carriage” for the student assigning compartments to the passengers; “Pegasus’ Stable” for the carriage carrying the bachelors; “the Lazaretto” for the host city’s Academy of Sciences. This was playful mischief, but behind it lurked a serious creed. To name was to individualize, to give existence, to create. Abstraction was a friend, not an enemy. Already, the approach was paying dividends: one way to tackle the Continuum Hypothesis was to try to imagine sub-sets of the continuum. Pavel Alexandrov, a student otherwise known as א5, had shown that a family of subsets known as B-sets, after Borel, satisfied the hypothesis—an advance Borel himself had failed to realize. But it was a discovery of another student, Mikhail Suslin, that really made headway. Lebesgue, it transpired, had made a mistake in a seminal paper. B-sets were not the only subset of the Continuum, not even by far. In fact it was possible to name a whole new hierarchy of subsets. “It was as if sets, of kinds not known before,” Graham and Kantor write, “were emerging from a secret cavern.” Infinity was slowly revealing its deepest catacombs.
Led by teachers who had become mystics, the Moscow School was making the greatest breakthroughs since Cantor. Realism had forced the Frenchmen to recoil, but a blend of philosophical Platonism and ancient glossolalian theology was now the Russians’ solution to the problem. The chants of the monks of Mount Athos had begun to do their magic.
The Continuum Hypothesis has never been proved, and it might never be. Still, the Russian School had its day of glory. Tragically, it was a short one. After the revolution, attacks on religion and non-Marxists grew. Soon Florensky was arrested. Courageously delivering scientific papers wearing a white priest’s cassock, he was found to be a pernicious contradiction in terms: a “priest-professor,” “an extreme right-wing monarchist,” a member of a “counterrevolutionary party.” Never mind that Florensky had never heard of The Party for the Rebirth of Russia, of which he was accused of being a leader. Ultimately he broke under torture, confessed to made-up crimes, and was shipped away to a hard labor camp on the Solovetsky Islands in the White Sea in the Arctic. He was shot on December 8, 1937, twenty miles south of Leningrad.
Luzin, too, soon felt the infernal breath of the Communist Party. Even the sanitarium could not save him. Lusitania had disbanded, and a number of jealous former students and careerist adversaries recognized the opportunity. “Unmasked” in Pravda, Luzin was accused of treason by a special investigative commission of the Soviet Academy of Sciences. Alongside the charge of “wrecking” (which included publishing his best articles unpatriotically in French) was the allegation of an “inability to understand the unity of continuous and discrete.” Only a merciful letter from the Nobel physicist Peter Kapitsa to Stalin seems to have saved him: even Newton, Kapitsa pleaded, had been a “religious maniac.” Miraculously, Luzin’s life was spared, but after 1936 he abandoned foreign publications and religion, and was never the same man again.
Luzin’s star student, Lev Schnirelmann, who had arrived years earlier at Lusitania when he was fifteen, wide-eyed and in love with sets, took his life in a Moscow apartment following his interrogation in 1938. Luzin’s student and lover, Nina Bari, jumped before an oncoming train years later. But it was the giant Egorov’s fate that was saddest of all. As the purges intensified, the revered father of the Moscow School of mathematics soon found himself betrayed by students and friends. Collecting all his dignity, he refused to back down, and was soon exiled to a prison near Kazan. Heartbroken, beaten, wasting away, אω died a torturous death. His last words were: “Save me, O God, by Thy name!”
Lusitania and its heroes are a salutary lesson in the ways religious and mystical thought can prove unexpectedly valuable in science. Graham and Kantor do a good job of balancing these lessons, cautioning against determinism and overplay. Still, the story of Baire and Lebesgue, of Luzin and Florensky, poses a challenge to the more pristine pretensions of reason. It reminds us that the roads to knowledge are many, and that no single way can claim to be the only way.
Contemplating his own life, begging his wife to take him home from the sanitarium in one of his last letters, Georg Cantor, the muse of the Name Worshipping mathematical mystics, wrote:
That was a winter cold and wild,
Like none one can recall;....
Cold would my winter have been;
To suffer gladly, pen a poem,
To escape the world I’m in.
He died alone, in the spring of 1917. Understanding what lies beyond the beyond is a problem that demands extraordinary talent, perhaps even genius. But as Naming Infinity so sensitively shows, escaping the world we live in, and the exacting parameters of reason, can sometimes lead to surprising results. As powerful as the gift of rationalism may be, there is still more in heaven and earth.
Oren Harman’s new book, The Price of Altruism: George Price and the Search for the Origins of Kindness, has just been published by Norton.