What Can We Gain by Losing Infinity? | Quanta Magazine

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond. Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” $latex e^{e^{e^{79}}}$. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe.

While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”That’s a problem that some ultrafinitists are still trying to address.Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”Dissident MathematicsAristotle saw infinity as something that you could move toward but never reach. “The fact that the process of dividing never comes to an end ensures that this activity exists potentially,” he wrote. “But not that the infinite exists separately.” For millennia, this “potential” version of infinity reigned supreme.

But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist. Cantor’s approach was to treat a series of numbers, such as the integers, as a complete infinite set. This approach would become essential in the creation of the foundational theory of mathematics, known as Zermelo-Fraenkel set theory, that mathematicians still use today. Infinity, he showed, is an actual object. Moreover, it can come in many different sizes; by manipulating and comparing these different infinities, mathematicians can prove surprising truths that on their face seem to have nothing to do with infinity at all. While few mathematicians spend much time in the realm of the higher infinite, “nowadays, almost every mathematician is an actualist,” Hamkins said. Infinity is assumed by default.But this foundation of modern math has inspired fierce arguments since it was first proposed. One reason is that accepting a core assumption about infinity allows you to construct strange paradoxes: It becomes possible, for instance, to carve a ball into five parts and use them to create five new balls, each with a volume equal to that of the first.Another objection is more philosophical. In the decades after Cantor’s revelations, some mathematicians argued that you cannot simply assert the existence of a mathematical structure — you must prove that it exists through a process of mental construction. In this “intuitionist” philosophy, for example, pi is less a number with an infinite non-repeating decimal expansion, and more a symbol that represents an algorithmic process for generating digits.But intuitionism only requires that a given mental construction be possible in theory: It prohibits actual infinity but permits potential infinity. Some mathematicians still weren’t satisfied with this. They remained troubled by Skewes’ number and other values so large they could never be written down. And so they sought to take intuitionist ideas to an extreme.“If you’re thinking, which numbers are going to exist in this view, those are going to have to be numbers that we can in practice construct,” not just theoretically construct, said Ofra Magidor, a philosopher at the University of Oxford.A new version of intuitionism — one that took these practical constraints to heart — crystallized in the 1960s and ’70s, with the work of Alexander Esenin-Volpin, a Soviet mathematician and poet.

Esenin-Volpin was known first and foremost as a political dissident. For leading protests and spreading anti-Soviet rhetoric and poetry, he was institutionalized. “He said, ‘I’m a human being. I have fundamental rights,’” said Rohit Parikh, a logician at the City University of New York who hosted Esenin-Volpin in his home after the Soviets forced him to emigrate in the 1970s. Esenin-Volpin was a strange houseguest, who would pace around Parikh’s attic all night and use his wife’s beloved ceramics as an ashtray while working on a strange theory that rejected not only potential infinity but even extremely large numbers — those that couldn’t be constructed in a person’s mind. Alexander Esenin-Volpin was a Soviet dissident, mathematician, and poet who was imprisoned several times for his human rights activism. Irene Caesar The logician Harvey Friedman once asked Esenin-Volpin to pinpoint a cutoff for what makes a number too large. Given an expression like 2n, at what value of n do numbers stop? Was 20 actually a number? What about 21, 22, and so on, up to 2100? Esenin-Volpin responded to each number in turn. Yes, 21 existed. Yes, 22 did. But each time, he waited longer to reply. The dialogue soon grew interminable.Esenin-Volpin had made his point. As Parikh and others would later put it, the limits of numbers were rooted in the limited resources needed to demonstrate their existence, like time. Or available computer memory, or the physical length of a proof. “Most ultrafinitists have the view that the distinction between the finite and infinite is inherently vague,” Clarke-Doane said.For Esenin-Volpin, a condition may be true for n, and for n + 1 — until it is not. A child grows and grows, until one day they’re no longer a child. One need not specify a specific end point. The important thing is that the end is in there, somewhere.Esenin-Volpin’s work was a call for a new kind of mathematics that could, in some sense, tolerate vagueness.