A translation of a fresh Quanta Magazine piece popped up in my Habr feed — about a mathematician who calls infinity “complete nonsense” and compares belief in it to belief in God. The headline promised a philosophical curiosity, but beneath it a real war was unfolding — over the right of infinitely large numbers to be called numbers.
The movement is called ultrafinitism: the “ultra” here means radicalism. For centuries, mainstream mathematics has argued only over how to understand infinity. Ultrafinitism goes further and declares some ordinary numbers nonexistent — those that cannot be constructed and exhibited. The position is so extreme that, until recently, discussing it seriously meant risking your reputation.
And yet in April 2025, ultrafinitism was the subject of a three-day conference at Columbia University; in August that same year, New Scientist published a major article on the movement, and in April 2026 came Quanta’s analysis — the translation that showed up in my feed. That is the story: how the most despised heresy in mathematics began finding an audience — because it finally acquired working theories.
Aristotle regarded infinity as something one could approach without end, but could not declare to exist. “That the process of division never comes to an end,” he wrote, “ensures the potential existence of this activity, but not the separate existence of the infinite.” This “potential” infinity held sway for millennia.
At the end of the 19th century, Georg Cantor shattered it. He proposed treating a sequence of numbers, such as the integers, as a complete infinite set — not as a process receding into the distance, but as a finished object. Out of this grew Zermelo–Fraenkel set theory, the foundation of all modern mathematics. “Today, almost every mathematician is an actualist,” says set theorist Joel David Hamkins: infinity is accepted by default.
That faith comes with a measurable price. The assumption of actual infinity yields the Banach–Tarski theorem: a sphere can be cut into five pieces and reassembled into five spheres, each with the volume of the original. For some, that is beauty; for others, proof that something is rotten at the foundation.
Objections appeared at once. The intuitionists, led by Brouwer, insisted that a mathematical object could not simply be declared to exist — it had to be constructed. For them, π was not a number with an infinite decimal expansion, but an algorithm that generates digits. But intuitionism, too, prohibited only actual infinity while retaining potential infinity: construction had to be possible in principle. Ultrafinitism is the next step: a number exists only if it can actually be constructed.
The movement’s founder was Alexander Esenin-Volpin, and his biography deserves a book of its own. He was born on 12 May 1924 in Leningrad; his father was the poet Sergei Yesenin, whom he never met — Yesenin died in 1925. Alexander became a mathematician and graduated from Moscow State University in 1949 with a Candidate of Sciences degree.
That same year, he left for Chernivtsi — and less than a month after arriving, he was arrested for “systematic anti-Soviet agitation”: he wrote anti-Soviet poems and read them to acquaintances. Faced with a choice between a labor camp and a psychiatric hospital, he staged a suicide attempt in order to undergo psychiatric evaluation. The Serbsky Institute declared him insane, and he ended up in a Leningrad psychiatric prison, followed by five years of exile in Karaganda. An amnesty after Stalin’s death freed him — in December 1953, he returned to Moscow.
In 1959, he was invited to a mathematical symposium in Poland — and denied a passport on the grounds that mentally ill Soviet citizens were not permitted to travel abroad. He sent the text of his paper to Warsaw, where it was read aloud without its author. The paper was called “The Ultra-Intuitionist Program for the Foundations of Mathematics” — the manifesto that marked the beginning of ultrafinitism.
Then came the 1965 “glasnost rally” on Pushkin Square, demanding an open trial for Sinyavsky and Daniel; around 200 people attended, a substantial share of them KGB operatives. Esenin-Volpin practiced a “legalist” strategy: demanding that the state literally comply with its own laws. In 1968, he was sent back to a psychiatric hospital, prompting the “Letter of the Ninety-Nine” in his defense; after a Voice of America broadcast, he was released almost immediately. Vladimir Bukovsky described his diagnosis more accurately than any psychiatrist could: “pathological honesty.”
In 1972, he emigrated to the United States, taught at Buffalo and Boston University, then worked as a librarian. And this is no coincidence: his mathematics and his politics were one and the same gesture. In both, he demanded that a thing be precisely what could be constructed and exhibited, and refused to take anything on faith — whether infinity or the Soviet constitution.
The logician Harvey Friedman once decided to catch Esenin-Volpin in a contradiction. The objection “then name the boundary” sounds devastating: fine, numbers are not infinite, but exactly where do they end?
Friedman started with two: does 2¹ exist? Yes, immediately. 2²? Yes, but after a noticeable pause. 2³ — after a longer pause. This went on several times, until it became clear what was happening.
Esenin-Volpin was prepared to answer “yes” every time — but to 2¹⁰⁰ he intended to answer “yes” roughly 2¹⁰⁰ times more slowly than to 2¹. The objection fell apart: it demanded a sharp boundary, but there is none, nor should there be. For Esenin-Volpin, there is no single, once-and-for-all fixed sequence of natural numbers — there are different ones, and each has its own “too much.” A child grows until one day they cease to be a child; no one is obliged to name the exact day, only to acknowledge that there is an end somewhere.
Existence, by this logic, is determined by feasibility: by the time and memory required for a demonstration. A number that can neither be written down nor exhibited is an ellipsis with pretensions.
That Warsaw paper proposed the unthinkable: to prove the consistency of Zermelo–Fraenkel set theory — that is, of all mathematics — by ultrafinitist means. Esenin-Volpin himself developed the program no further than a sketch, but it gave rise to concrete results.
In 1971, Rohit Parikh gave the entire movement a formal foundation for the first time. He introduced the term “feasible numbers,” postulating that they were closed under addition and multiplication, and asked: where does the boundary lie?
The answer surprised the ultrafinitists themselves. You cannot modestly postulate a boundary such as “all feasible numbers are smaller than a tower of a thousand twos” — a tower of a thousand twos can itself be derived as feasible, so such an axiom leads straight to contradiction. The strongest claim about a boundary that can be postulated consistently is: “all feasible numbers are smaller than a tower of twos of height 2^1000.” Ultrafinitism had turned from philosophy into a theorem — and the theorem immediately pushed the boundary beyond all intuition.
In 1976, a similar revelation came over Edward Nelson of Princeton. “I felt the instantaneous overwhelming presence of One who reproved me for my arrogance in believing in the real existence of an infinite world of numbers,” he recalled, “leaving me, like a baby in a cradle, to count on my fingers.” Nelson tried to rebuild arithmetic without infinity. The result was very weak: in such a system, one cannot even prove that a + b always equals b + a; one can construct the number 100 or 1000, but not 100 to the 1000th power; and induction vanishes altogether.
Nelson hoped to use this to undermine ordinary Peano arithmetic — to show that it was inconsistent. In 2003, he announced that he had found a contradiction, and the result collapsed within weeks. But his weak systems, like Parikh’s constructions, did not disappear: they gave rise to bounded arithmetic, which complexity theorists use to understand what algorithms are capable of proving and what they are not.
The austere version had its price, too. In the survey volume Constructivism in Mathematics (1988), the logician Anne Troelstra dismissed it in a single sentence: “no satisfactory development exists at present.” In a rigorous work on mathematical logic, there was simply nothing to cite.
The most serious thing to happen to ultrafinitism in recent years was done by a man who believes in infinities of every kind — Hamkins himself. He built a theory he called “finite arithmetic”: numbers exist up to some greatest one, while addition and multiplication are partial functions, defined only so long as the result does not fly past the boundary.
An awkward question arises immediately: why do the numbers end abruptly, at precisely that point rather than a little further on? Hamkins gave an answer that the philosophy itself had lacked. It turned out that every model of such arithmetic describes within itself a strictly higher model, in which the former maximum can calmly be squared. Mathematics with a boundary has a built-in staircase leading outward: operations seem to reveal themselves gradually, step by step. What is more, every such model arises by cutting off a model of bounded induction — a result going back to Jeff Paris.
Thus, for the first time, ultrafinitism acquired what it had lacked for six decades: a coherent formal theory instead of a collection of opinions.
The movement acquired a face before it acquired a theory. Doron Zeilberger, a Rutgers University professor and celebrated combinatorialist, credits his own computer, Shalosh B. Ekhad, as coauthor of his papers. He declares mathematical analysis a degenerate case of discrete analysis: continuous curves are “fine sandpaper,” smooth only to the eye, and all of it can be replaced by a “discrete necklace” of numbers with tiny but non-infinite intervals. “I cannot be a tenured professor without doing this crazy nonsense,” he says of his 195 published opinions.
And here philosophy finds an unexpected ally — physics. We are accustomed to thinking of space and time as infinitely divisible, but physicists have the Planck scale, beyond which the very notion of distance loses its meaning: the “pixel of the universe.” Sean Carroll of Johns Hopkins University plays with finite models of quantum mechanics and says frankly that most cosmologists pretend the problem of an infinite universe does not exist.
Nicolas Gisin went still further. In his hypothesis, real numbers — those with infinite decimal expansions — are the hidden variables of classical mechanics. Our belief that a particle’s state can be specified with infinite precision is itself the source of the puzzle: determinism is an artifact of excessively precise data. Switch to intuitionistic mathematics, and the gulf between the predictable classical world and the random quantum one simply disappears. Gisin does retain potential infinity, however: information can accumulate and accumulate, it is just that this moment will never arrive.
In April 2025, Clarke-Doane brought together physicists, philosophers, logicians, and mathematicians at Columbia University — from Carroll to Parikh and Hamkins. The result was, in his own words, “a lesson in patience for everyone”: philosophers are used to arguing fiercely and then drinking beer together, mathematicians are not. Usually, if mathematicians disagree, someone has made a serious mistake.
There is still no unified theory of ultrafinitism, and that is the main criticism. Its supporters do not even share a common motivation: some are led by physics, others by logic, still others simply by distrust of ellipses. “I don’t think ultrafinitism is rejected because of strong arguments against it,” says Clarke-Doane. “People just think it is hopeless.” But that, he adds, is precisely what is changing: people have begun seriously considering how to place the view on solid foundations.
The finest duel here is the long-distance dispute between Zeilberger and Hugh Woodin, one of the leading researchers of higher infinities. Woodin said he pitied Zeilberger, unable to raise his eyes to the sky and see the beauty of infinite expanse. Zeilberger replied in his own style: he pitied Woodin, who needed the “opium of infinity” in order to live. “There is so much beauty in the trees and in the earth. You don’t have to look toward fiction.” And so they pity one another — each sees the other as a prisoner of his chosen faith.
Physicists, meanwhile, retain an ace up their sleeve: experiment. If it could be proved that the physical universe is finite, even the most convinced devotees of higher infinities would have to stop and think. If the opposite were proved, ultrafinitists would be in trouble. “It’s hard to be an ultrafinitist if there are infinities in the real physical world,” Carroll says.
There is also an amusing detail that says much about the movement’s maturity. It still has neither a journal nor a collected volume — only Zeilberger’s personal website, where he keeps his 195 opinions and links to the work of others. The movement is six decades old, and its catalog still lives in one enthusiast’s folder.
And yet the main thing to take from this story is that no one knows where a number ends. Neither supporters nor opponents. Defenders of infinity consider the question meaningless, while ultrafinitists consider it unanswered but real. For now, the answer can neither be proved nor calculated, and that is precisely what makes the dispute honest. Once, doubts like these ended in a psychiatric hospital or at the stake; now they have found a place on Columbia University’s program.
Zeilberger himself sums it up: had he lived in 1600, he would have been burned alive like Giordano Bruno, while today his punishment is simply to be ignored.