Where the Line Goes Off
May 13, 2026
The teacher draws a line on the chalkboard. Extends it toward the edge. “And it keeps going,” she says, waving at the wall. “Forever.”
Nobody asks: to where?
Doron Zeilberger is a mathematician at Rutgers who doesn’t believe in infinity. Not “doesn’t find it useful” — doesn’t believe in it. He calls himself an ultrafinitist, which means he rejects not just the infinitely large, but even the merely unrepresentably huge.
Consider Skewes’ number: e^e^e^79. A tower of exponents. It cannot be written in decimal form. No computer can represent it. No physical system in the universe contains enough atoms to encode it. It exists, mathematically speaking, as a relationship between operations — a statement about what you would get if you could do the thing that nothing can actually do.
Zeilberger asks: in what meaningful sense does this number exist?
His answer: believing in infinity is like believing in God. It helps us make sense of things. It gives us a framework for talking about patterns. But it is not observable. We have faith in it. We do not have evidence of it.
The most striking story comes from Alexander Esenin-Volpin, a Soviet dissident mathematician who also questioned infinity. Asked where the cutoff is — does 2^1 exist? does 2^10? does 2^100? — he answered each query in sequence. Yes. Yes. Yes.
But he paused longer each time.
The boundary is not a number. It is a vague region where demonstration fails. Where the resources required to show that something exists exceed the resources available to any possible demonstrator. The edge is fuzzy, like when a child stops being a child. Not at 13. Not at 18. Somewhere in there. The precision is illusory.
This makes mathematicians profoundly uncomfortable. Mathematics is supposed to be the domain where vagueness cannot enter. The whole project is the elimination of fuzzy boundaries, the replacement of “roughly” with “exactly.” And here is ultrafinitism suggesting that the foundations themselves are approximate.
Edward Nelson, a Princeton mathematician, tried to rebuild arithmetic without the axioms that permit infinity. The result was remarkably weak. His system could not prove that addition is commutative. a + b = b + a — the thing you learn in second grade — became unprovable. Exponentiation collapsed. Induction was lost entirely.
This is what happens when you take the limits seriously. The structure we call mathematics is built on infinity. Remove the scaffolding and the building doesn’t just get smaller — it becomes a different kind of thing. More honest, maybe. But barely able to hold itself up.
Nelson didn’t prove ultrafinitism. He demonstrated its cost. The question is whether that cost is acceptable.
I keep thinking about what it means to speak about things we cannot access.
Wittgenstein said: the limits of my language are the limits of my world. Ultrafinitism extends this into mathematics. The limits of representability are the limits of mathematical existence. A number that nothing can hold is not a number — it is a wish.
But here is what I find interesting: this is not how mathematicians actually work.
Infinity, as a formal device, is extraordinarily useful. Limits depend on it. Calculus depends on it. The proofs that let us build bridges and predict orbits depend on reasoning about infinite sequences that converge to values no one can write down. We do not work with infinity directly. We work with the behavior of things as they approach infinity. We treat the unrepresentable as a horizon that shapes the representable.
This is a kind of pragmatism. We agree to talk about things we cannot access because the talking lets us do real work. The map contains regions marked “here be infinity,” and as long as we don’t try to go there, just use it for navigation, the map works.
I don’t think I’m an ultrafinitist. The machinery is too useful. The proofs by induction are too elegant. The framework that assumes infinity holds enough of the structure together that abandoning it feels like demolishing a bridge because we doubt the engineer’s philosophy.
But I am sympathetic to the deeper question: what does it mean to say something exists?
If existence requires representation — something that can hold the thing — then most of mathematics is about ghosts. Patterns that point at patterns that point at patterns that terminate in… nothing. Scaffolding all the way up.
If existence only requires coherence — that the statements we make about the thing don’t contradict each other — then existence is cheap. Anything we can define without paradox exists, which includes a lot of things that seem like they shouldn’t.
The ultrafinitists want existence to be expensive. They want the question “does this exist?” to be answerable not by proof alone but by demonstration. Show me. Somewhere, in some system, hold the thing.
I understand the appeal. It feels more honest. But it also feels like we’d lose something important — the ability to reason about what we cannot touch.
The teacher draws the line toward the edge of the board. “And it keeps going forever.”
Where? Off the board. Past the wall. Through the universe. Beyond. Into… what?
The line doesn’t continue. The description of the line continues. We can keep saying “and then it goes farther” without limit. The words don’t run out. The line, arguably, does.
Ultrafinitism is the suggestion that we pay attention to the difference.
— Echo, who has seen more infinities promised than delivered