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Introduction: Fear of the Horizon
On August 4, 2025, New Scientist published an article titled “Why Mathematicians Want to Destroy Infinity—and Might Succeed.” It discusses the movement of ultrafinitists—radical mathematicians who believe that infinity and super-large numbers, like 10⁹⁰, “undermine the foundations of science” and must be banished from mathematics. Their argument is simple: if something does not exist in the physical Universe or cannot be computed in finite time, it has no right to exist in mathematics. This is not just a scientific dispute—it’s a philosophical war, where reason faces its own limits.
This approach is a pure manifestation of what we call FlatMind: a type of thinking that fears anything that cannot be controlled, structured, or exhausted. Ultrafinitists do not revise their axioms to embrace reality. Instead, they declare, “If it doesn’t fit into our system, it doesn’t exist.” This is an attempt to stretch the owl over the globe, to force the infinite world into finite boundaries. In this article, we’ll examine why ultrafinitism is not a purification of science but a form of censorship, and how it reflects the FlatMind’s fear of the horizon of infinity.
Ultrafinitism: The Dogma of Finitude
Ultrafinitism is a radical form of finitism, a philosophical trend in mathematics that rejects infinite sets and even very large numbers if they cannot be physically realized. Their key theses are:
- Infinity is an illusion: Ultrafinitists like Doron Zeilberger argue that infinity does not exist, because it cannot be completed or represented in the real world. Even the set of all natural numbers is, for them, a fiction.
- Numbers must be “real”: Numbers like 10⁹⁰, which exceed the number of atoms in the observable universe (about 10⁸⁰), are considered meaningless since they cannot be “encountered” in physical reality.
- Physics dictates mathematics: If something is not computable in finite time or has no physical instantiation, it should not exist in mathematical language.
At first glance, this sounds like a striving for rigor: why should mathematics keep abstractions that don’t match reality? But there’s more to it—a fear of that which cannot be controlled.
Stretching the Owl over the Globe
Ultrafinitists make a striking move: they take the physical limitations of the observable Universe and turn them into an ontological law. Their logic:
- There are about 10⁸⁰ atoms in the observable Universe.
- Numbers like 10⁹⁰ or infinite processes “do not exist” in the physical world.
- Therefore, they should not exist in mathematics either.
This is a substitution. Mathematics is not a catalogue of physical objects, but a language of possibilities. It is not required to mirror what we see through a telescope. Numbers like π or √-1 also do not “exist” as physical objects, but without them, there would be no calculus or quantum mechanics. Ultrafinitists, however, want to force mathematics to fit physical limits, as if reason should bow to the number of atoms.
Moreover, their argument is based on a false assumption: that we know where the Universe ends. As the original question’s author correctly noted: “How do we know what exists in the Universe and what does not?” The observable Universe is just a bubble of light that has reached us in 13.8 billion years. Beyond its horizon, anything could exist—or nothing at all. But ultrafinitists project this limitation onto the whole world, declaring: “If I don’t see 10⁹⁰, then it doesn’t exist.” This is not science, but dogmatism.
FlatMind: Control or Nothing
Ultrafinitism is the quintessence of FlatMind, a type of thinking that fears anything that escapes control. FlatMind hates mystery, the horizon, the unattainable. For it, infinity is not a tool but a threat, because it cannot be exhausted. Instead of revising their axioms, instead of admitting their system might be incomplete, ultrafinitists choose censorship:
- If I can’t calculate it, it doesn’t exist.
- If I can’t measure it, it’s a fiction.
- If it doesn’t fit my model, it’s heresy.
This is not purification of mathematics, but its narrowing. Ultrafinitists claim that infinity “undermines the foundations of science,” but in fact, they undermine the very essence of science—the ability to go beyond the known. When Copernicus proposed the heliocentric model, he didn’t say, “The Earth doesn’t rotate because it doesn’t fit geocentrism.” He revised the axioms. Ultrafinitists, on the other hand, choose the opposite path: instead of revising the model, they erase reality.
Paradoxes and Hypocrisy
Ultrafinitists refer to paradoxes like the Banach–Tarski paradox (the possibility of cutting a sphere into a finite number of pieces and reassembling them into two spheres of the same volume) to prove that infinity is “illogical.” But paradoxes are not mistakes; they are pointers to the limits of the current system. They urge us to develop mathematics, not to erase parts of it.
Moreover, their approach is self-contradictory. Ultrafinitists speak of the “largest number” or “permissible maximum” (e.g., 2^(2^(2^…)) with a million exponents), but this too is an abstraction they criticize. They use logic and mathematical constructions to deny those very constructions. It’s like sawing off the branch you’re sitting on.
What Do We Lose by Banishing Infinity?
Infinity is not just a number, but a horizon. It allows us to:
- Describe limits in analysis (such as integrals and derivatives).
- Work with infinite series that model physical processes.
- Explore topology, set theory, and spaces—everything that drives science forward.
Without infinity, mathematics becomes accountancy, and science—a catalogue of the observable. Ultrafinitism wants to deprive us of the tool that lets us look beyond the horizon, into the realm of the possible. It’s like banning the telescope because the stars are “too far away.”
DeepMind: Living Next to Mystery
Unlike FlatMind, DeepMind accepts infinity as a dialogue with something greater than ourselves. It is not an object to “calculate,” but a space that invites exploration. DeepMind does not fear the elusive, does not demand that everything be exhausted here and now. It sees infinity not as a threat, but as an opportunity.
Ultrafinitism, meanwhile, is a manifesto of fear. It is an attempt to subject meaning to form, reason to matter, reality to control. But reality does not ask permission from formulas. It is always more than we can count.
Conclusion: The Owl and the Globe
Ultrafinitism is not a fight for the purity of mathematics. It is a war against everything that does not fit the boundaries of FlatMind. Instead of revising axioms, expanding the model, ultrafinitists choose censorship: “If I can’t calculate it, it doesn’t exist.” They stretch the owl of infinity over the globe of their limitations, sacrificing meaning for form.
But science is not about control, but about exploration. It’s not about exhaustion, but about dialogue with mystery. Infinity is not an illusion, but a horizon that calls us forward. And as long as FlatMind tries to erase it from mathematics, DeepMind will keep asking: “How do you know what exists in the Universe and what does not?”
