In Céline, Jünger, and Mishima — the doctor, the soldier, the samurai — the green-gloved man identifies three spirits of antiquity reborn in modernity. Where else did the classical man reappear? In this essay I identify another atavism: the mathematician. Like other ancient types, mathematicians are extinct — the last alive is Grigori Perelman, whose exploits will likely not be matched for centuries. Nevertheless, when studying a few mathematicians from the twilight of the West, we can recognize in their deeds and philosophy the rebirth of a savage spirit, and we can begin to grasp how antiquity, Nietzsche’s ancient fire, might reappear.

The true mathematician is an addict, a tyrannical “monster of will” who is cursed with a glimpse of pure power and beauty early in life, and who thereafter is compelled to spend the rest of his days seeking to recapture that infinitely brief moment of pure revelation — of pure valor — as a tortured and compulsive athlete. Schopenhauer justifies his mighty pessimism by comparing the joy of an animal devouring its prey with the suffering of the animal being eaten. The mathematician is the animal who experiences once or twice the infinitesimal and infinite pleasure of feasting — and who thereafter is burdened with the knowledge that he is the one being eaten, that all of his vain labor will hereafter amount to nothing more than food for a more rapacious successor. Like the Renaissance artist or German philosopher, the mathematician of the twilight of the West, “despite” being sensitive and spiritual, possessed a warlike intensity that is scarcely comprehensible today.

As a younger man I was drawn to mathematics for its purity and austerity; from a distance it seemed like a refuge of objectivity, without politics, hand-waving, and academic grift. Indeed, math does provide this for a little while, for those whose minds are ready to receive it — I can’t forget the night I came across a grainy PDF of G.H. Hardy’s A Mathematician’s Apology, huddled over the screen, reading it in one sitting. In this 1940 treatise, beneath a mask of acerbic schoolboy wit, I began to discern an unfamiliar, older spirit, uncanny and severe. In subject and composition, it’s a distinctly English essay, the musings of an unmarried Cambridge don looking back on a distinguished career — but it’s entirely devoid of English sentimentality or port-soaked romanticizing. Instead, the ruse of disinterested scientific objectivity serves as a vehicle for the presentation of a Spartan philosophy of life, animated by a fierce love of purity, and the motto: Wir müssen wissen, wir werden wissen (We must know, we shall know).

Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds…

If then I find myself writing, not mathematics, but ‘about’ mathematics, it is a confession of weakness, for which I may rightly be scorned or pitied by younger and more vigorous mathematicians. I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job.

This indicates self-knowledge of the highest order; a rare quality because it is painful; painful because it indicates (perhaps even an ascetic or erotic delight in) cruelty turned inward. Hardy throughout his life despised mirrors; he would instantly cover them upon entering a room. But what are the origins of his drives? Hardy soon offers an answer:

I shall assume that I am writing for readers who are full, or have in the past been full, of a proper spirit of ambition. A man’s first duty, a young man’s at any rate, is to be ambitious.

If we replace “mathematics” with “philosophy,” and “ambition” with “will to power,” Hardy’s view might look something like this —

But this is an ancient, eternal story: what formerly happened with the Stoics still happens today, too, as soon as any philosophy begins to believe in itself. It always creates the world in its own image; it cannot do otherwise. Philosophy is this tyrannical drive itself, the most spiritual will to power, to the “creation of the world,” to the causa prima.

After 1889, there have been no philosophers and no new philosophy. But it has been mere decades, not centuries, since the creation of new mathematics of any significance, and of all the premodern human types that reappeared in modern disguises in the late West, the mathematician died last. A mathematician’s psychology is interesting, but the real fruit of math lies in its practice, which sharpens the mind as lifting shapes the body. The cultivation of the psyche, the soul, must, following Nietzsche, begin with our physiology — and the physiological effects of mathematics can only be appreciated when the discipline is understood as the inborn pursuit of power and beauty in their most distilled and therefore most elusive form. Knowledge is recollection, and mathematics, alongside physical training and classical languages, provides a path back into nature for minds that have forgotten the primal aesthetic sense for beauty, a sense that is at heart anything but disinterested.

Hardy continues his Apology like Socrates, feigning innocence and a dispassionate approach. He draws a rigid distinction between “real” or “pure” mathematics and applied math, and asserts that pure math is both unfathomably more beautiful and also more useless than applied mathematics; pure math cannot be immediately used for war, and for this reason Hardy claims to be harmless. He also makes repeated reference to his friend Bertrand Russell, an ardent pacifist who focused on logic, and who gave us the finest description of mathematics: “Mathematics, rightly viewed, possesses not only truth but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” The mathematician devotes his life to this notion of beauty, a slave to his inborn aesthetic sense. I hope you can understand why I was drawn to it as a refuge of abstraction, objectivity and purity, and why I failed to perceive the inner meaning of Hardy’s essay. Hardy, like all true mathematicians, boasted of loathing the obligation to teach — “Whatever is profound loves masks; what is most profound even hates image and parable.”


Now I'd like to introduce you to a simple and profound mathematical idea that is central to the discussion to follow. It is the bizarre notion of two different kinds of infinity: "countable infinity" and "uncountable infinity."

Countable infinity describes the "size" of the set of positive integers, i.e. the positive whole numbers: 1, 2, 3, 4, … and so on.

The integers include all positive and negative whole numbers as well as zero: we can write them as {…, -3, -2, -1, 0, 1, 2, 3, …} and so on. The set of all integers is also — just like the set of strictly positive integers — "countably infinite." This is true because we can label every single one of the integers with a positive integer. For example:

We label the "1st" integer as 0,

the "2nd" is -1

the "3rd" is 1

the "4th" is -2

the "5th" is 2

the "6th" is -3

the "7th" is 3,

and so on, continuing infinitely until we’ve “counted” all of them, both negative and positive.

This should give an idea of the strange concept of "countable infinity." Georg Cantor, a titan of 19th-century mathematics, attempted to make such notions rigorous through the controversial Mengenlehre, or set theory. After helping to establish the theory of countable infinity, Cantor went even further in daring to define infinity: he introduced the notion of "uncountable infinity" to describe the real numbers. The real numbers constitute our reality — they include, in addition to the integers and the rational numbers (fractions and whole numbers), the irrational numbers such as π. Cantor launched an explosive dispute in mathematics by arguing that the real numbers are “uncountably infinite.” He proved that, unlike the set of integers, the set of real numbers is unfathomably more infinite than the “countably infinite” sets. Cantor did this by arguing by contradiction — he supposed that it was possible to “count” all real numbers in the way we “counted” all the integers above, and then proved that this attempt was logically impossible.

The implications of this project tore a deadly rift between Cantor and his teacher Leopold Kronecker. The battle between the sensitive, pious Cantor and the athletic Kronecker — a vehement, muscular manlet — is operatic in its tragedy; as one historian put it, “Kronecker’s attack broke the creator of the theory,” Cantor, who died in a mental asylum. The firestorm surrounding Cantor’s work — which has become absolutely dominant in modern analysis (the field that encompasses calculus) — hints at the true nature of “reason” versus “intuition” and also, perhaps, of “Apollo” versus “Dionysus.”

In his book Men of Mathematics, which follows Vasari and Plutarch as a masterpiece of “monumental” history, the mathematician E.T. Bell says of Cantor’s theory:

The controversial topic of Mengenlehre … typifies for mathematics the general collapse of those principles which the prescient seers of the nineteenth century, foreseeing everything but the grand débâcle, believed to be fundamentally sound in all things from physical science to democratic government.

In fact, Cantor was to mathematics what Darwin was to biology: the destroyer of comforting illusions and also, underneath the costume of extreme abstraction, a harbinger of Dionysus.

Cantor’s teacher Kronecker was himself a great mathematician and a brilliant student of philosophy and the classics. Kronecker was born into a Jewish family and converted to Lutheranism near death after a successful academic career. He was around five feet tall, fond of hiking and the outdoors, and vigorous and combative — “Possibly if Kronecker had been six or seven inches taller than he was,” writes Bell, “he would not have felt constrained to overemphasize his objections to analysis so vociferously.” One is reminded of BAP’s claim: “The manlet is the most revolutionary principle in world history,” and indeed, Kronecker was a revolutionary. Bell calls him “The Doubter”; Kronecker was said to have “no feeling for geometry,” and was more inclined to “make concise, expressive formulas tell the story and automatically reveal the action from one step to the next so that, when the climax was reached, it was possible to glance back over the whole development and see the apparent inevitability of the conclusion from the premises.” Kronecker’s revolution was an attack on the bedrock of modern analysis; he rejected totally the contemporary idea of “infinity” and of proofs that could not be completed in a finite number of steps. He is famous today for this aphorism: “God created the integers, all else is the work of man.”

Kronecker’s philosophy of mathematics was founded on this conviction, and he set out to demonstrate that, in his words, “all results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers.” Kronecker believed mathematics was corrupted by abstract and ill-defined notions of infinity, and he wanted to “arithmetize” mathematics, to return it to the foundation of the integers. There was also a mechanistic quality to his work — in the words of one writer, “Since [Kronecker] is of the view that numbers are merely systems of representations with which calculations are done, the essence of number theory is calculation.”

Georg Cantor was born into a pious Lutheran family. His career was that of a first-rate mathematician condemned to second-rate institutions. He was obsessive and spiritual, with grand ambition that corresponded to a strong neediness. Biographer Joseph Dauben writes:

With colleagues this ‘will to succeed’ was manifest in [Cantor’s] attempts to dominate conversation and in his delight at being the center of attention. He liked nothing better than to find himself surrounded by a group of listeners, and he delighted in leading discussion on a wide variety of subjects punctuated by outspoken pronouncements. But with his family he apparently made little effort to dominate at all. In fact, during meals he would often listen quietly, letting his children develop the subject at hand, and then, before leaving the table, he would make a point of turning to his wife, thanking her for the meal, and asking ‘Are you content with me and do you then also love me?’

Like Kronecker, it is difficult to immediately classify Cantor as guided primarily by “intuition” or by “logic” according to Poincaré’s famous typology. Cantor was fixated on medieval theology, and saw his Mengenlehre as a theory that might reunify Christianity and modern science. After his dramatic break with Kronecker and first psychic breakdown, Cantor spent the remainder of his life oscillating between prolific research and long stays at the asylum, where he would die. He claimed inspiration from God and the muses; he wrote to Catholic bishops asking them to accept his theory; moreover, “the amount of time [Cantor] devoted to various literary-historical problems increased steadily.” He wrote a heretical Biblical text called Ex Oriente Lux and developed obsessions with scripture, the Church fathers, freemasonry, Rosicrucianism, theosophy, Shakespeare, and English history. He tried to abandon mathematics entirely:

[Cantor] wrote to Ministry of Culture asking to leave math with a pension: “He emphasized his qualifications, his knowledge of history and literature, his publications on the Bacon-Shakespeare question, and even added the provocative news that he had come upon certain information in the course of his research concerning the first king of England, ‘which will not fail to terrify the English government as soon as the matter is published.’ Cantor tried to inject a note of urgency into the matter by asking the ministry to send him their reply within the next two days, for if they could offer him no alternative to teaching, then as a born Russian he would apply to the Russian diplomatic corps in hopes that he might be of service to Tzar Nicholas II.

But Cantor could not excise or escape his passions, his inborn will, and the divine madness returned to him, each time more vehement and sudden.

Dauben likens Cantor to Galileo, standing boldly against the authorities of his time, but this is a trite understanding that reeks of reddit scientism. The closest spiritual predecessor to Cantor is Plato — both were “tyrant’s lickspittles” who masked their world-reshaping tyrannical Dionysiac madness behind abstraction, and both successfully tricked a dying civilization into taking the bait. Cantor propounded a physics founded on the mathematical “point,” clothed in Leibniz’s language as “monads,” uncountably infinite and absolutely extensionless. Like some contemporaries, but also like a Gnostic dualist, Cantor divided the world into “matter” and “aether”; he believed that the “corporeal monads” that constituted matter were countably infinite, like the positive integers, and that the “aetherial monads” were uncountably infinite, as he had proven the real numbers to be. Thus, he considered the world beyond “matter” to be far more “infinite” than the perceivable physical world itself. He claimed that reconciling infinity with modern physics in this fashion would deliver man from the trap of Newtonian mechanics and materialism. Behind the arcane language of set theory, behind the protestations of innocence and objectivity, we can detect Cantor’s terrifying ambition from passages such as these, on “order types”:

Cantor frequently chose a simple example to demonstrate just how powerful a unifying concept order types could be. The most diverse things could be rendered mathematically comparable with great exactness. For the sake of illustration he liked to compare a painting with a symphonic score. Every point in the painting could be ordered according to a horizontal and a vertical axis, by the wave length of color and by color intensity at a given point. The symphony could also be quadruply ordered, every instant arranged in sequence by time elapsed from beginning to end, by the duration of a given tone, by its pitch and intensity. Though improbable, it might even turn out that such heterogenous objects as a Rembrandt portrait and a Beethoven symphony might have similar order types.

This is a naked display of the spiritual will to power, that seeks to remake the world in its image.

By now it should be clear that my youthful enthusiasm for “tight-assed reason” was mistaken. Despite the “austere beauty” there is something frightfully subjective and alive about mathematics; it is anything but an escape from life. Kronecker’s philosophy might seem like a noble attempt to lead math back towards the integers away from abstract folly; but at heart he promotes a base pleb’s mathematics that reduces all of life to calculation in order to avoid what Gauss termed “the horror of the infinite.” Cantor’s set theory might seem like an insane feat of abstraction; yet upon closer inspection the revelation of uncountable infinity is an acid that dissolves the illusions upon which calculus was constructed. Apollo and Dionysus appear in strange guises, and merely receiving their whispers is the work of generations — are they even distinct?

This is just a glimpse at the significance of mathematics for those who want old monsters to return. Beginning with sources like Nietzsche, the writings of Poincaré, or aphorism 22 of Bronze Age Mindset, we can approach math like one might approach Greek, as both a preview into the minds of greater men, but also as a vehicle for the reawakening of older spirits — out of the agon.

Things in the ambit of Dionysus became audible which had lain artificially hidden in the Apolline world: all the shimmering light of the Olympian gods paled before the wisdom of Silenus. A kind of art which spoke the truth in its ecstatic intoxication chased away the Muses of the arts of semblance; in the self-oblivion of the Dionysiac states the individual with all his limits and measures sank out of sight; a twilight of the gods was imminent.