Banishing Infinity

 

The Ultrafinitist Gambit

Let’s start with a confession. I love mathematics, but I’m allergic to grand metaphysical claims. Infinity sits right at the center of those claims like a peacock in a library. Gorgeous. Loud. A little absurd in the wrong aisle. Enter the ultrafinitists. They walk in with scissors and start clipping the peacock’s tail. They argue that infinity isn’t just suspicious. It’s unnecessary. Worse, it’s misleading. If you can’t build it, compute it, or even name it without cheating, why are we treating it like a citizen of mathematics?

That’s the vibe. It’s bolder than classic finitism and pricklier than intuitionism. It says: cut the drama. Keep the numbers you can actually touch. Maybe also the ones you can plausibly reach. The rest is costume jewelry. This isn’t a purely armchair rebellion. In April 2025, Columbia University hosted a conference. Ultrafinitism: Physics, Mathematics, & Philosophy. The roster was serious. Set theorist Joel David Hamkins gave a talk exploring ultrafinitism through the lens of potentialism, the idea that “the infinite” is never complete, just unending possibility. The hallway talk was lively. The questions were pointed. And the message was clear: this once-fringe view wants a seat at the grown-ups’ table.  Meanwhile, pop-science outlets are catching the current. New Scientist teased a cover feature on ultrafinitists “banishing infinity,” and Scientific American ran a timely primer on mathematicians who don’t believe infinity belongs in our description of reality. This debate is not a museum exhibit. It’s a live stream. 

So what is ultrafinitism, exactly? Think of it as finitism with a speed governor. Classical finitists deny actual infinity but happily accept arbitrarily large finite numbers. Ultrafinitists go further. They worry that some “numbers” are so huge that we can’t even, in good faith, say what we mean by them. Not just Graham’s number huge. Worse. They ask a blunt question: if a number can’t be named, computed, stored, transmitted, or even coherently individuated under physical constraints, does it really exist as a mathematical object? The ultrafinitist leans toward “no”. Or at least, “not yet.” This isn’t nihilism. It’s housekeeping. If your kitchen has three shelves, you don’t buy a 12-foot stockpot and declare it part of your cookware collection. Ultrafinitists want mathematics to respect the shelves.

Where this urge comes from

Partly from philosophy. Partly from computation. Partly from physics.

Philosophy first. Alexander Esenin-Volpin, poet, dissident, logician, charted an “ultra-intuitionist” program in the 1960s. He attacked the breezy way we talk about the natural numbers as if the whole infinite tower is present, complete, and surveyable. He thought such talk was metaphysical overeagerness dressed as rigor. His essays are a bracing read, even when you disagree.  Then there’s computation. Rohit Parikh’s 1971 paper introduced the idea of “feasible numbers.” Not every formally definable number is practically available. If your proof blithely invokes an integer that is astronomically beyond reach, that matters. Vladimir Sazonov later sharpened this with a formal treatment where even numbers like 2^{1000} might count as “non-feasible.” In these systems, arithmetic is tuned to what can be done, bounded by time, space, and resources. For computer scientists, this feels like home. It whispers.  “Complexity isn’t a footnote; it’s the foundation.”  Physics adds fuel. Nicolas Gisin and others argue that “real numbers” smuggle in infinite information. A finite region of space cannot contain infinitely many exact digits. So why should our physical theories depend on mathematical entities with that baggage? If we swap in finite-information numbers, we get a world that looks an awful lot like ours. It is messy, discrete, and stubbornly indeterministic. That’s a philosophical earthquake with practical aftershocks. 

What ultrafinitists actually propose

Cut away the mythology. Keep what works. Replace the rest with finite, constructive, and feasible structures.

That means being suspicious of the axiom of infinity and its spinoffs. It means treating the reals with side-eye. It means re-engineering analysis to run on discrete rails. Doron Zeilberger has argued, with delightful pugnacity, that “real” analysis is just a degenerate case of discrete analysis. According to this view, the continuum is an approximation we made convenient by habit. The real world, both physical and mathematical, is granular. It clicks. It doesn’t ooze.  Ultrafinitism is not one thing, though. It’s a spectrum of strictness. At the mild end, you get “feasible arithmetic,” bounded by complexity. At the spicy end, you get skepticism about very large finite numbers themselves. Some versions want to rebuild logic so that proof principles match finite computation—new rules, new proof objects, new limits. Recent work even sketches a “consistent ultrafinitist logic” with its own inference rules and a story about bounded Turing completeness. The technicalities are young, but the direction is clear: logic, meet laptops. 

The sales pitch

Why adopt such a severe diet? Three reasons.

First, honesty. If you model a finite world with tools that assume infinite precision, you inherit miracles you didn’t earn. You get Banach–Tarski paradoxes and other set-theoretic curios that have no laboratory analog. You get proofs by contradiction about objects no one can name or compute. You get infinite ladders where you only ever climb the first five rungs.

Second, alignment. A lot of modern work already lives in the finite. Cryptography, complexity theory, formal verification, data structures, randomized algorithms. These fields don’t need transfinite cardinals. They need guarantees at bounded scale. Ultrafinitism says: let’s make that the baseline, not the exception.

Third, coherence with physics. If you think information is finite and measured, then a mathematics that can talk only finitely at a time looks like a better partner. You trade a cathedral of absolutes for a workshop of constraints. Many physicists will take the workshop every time. 

The pushback

There is plenty.

First, classical mathematics is absurdly successful. Calculus runs our bridges and our rockets. Hilbert-style axiomatics and ZFC set theory give us stability, depth, and one enormous mental sky where everybody can fly. You don’t burn down an airport because the planes have more seats than you need. You just don’t book the upper deck.

Second, the technical maturity gap is real. Troelstra once said “no satisfactory development exists at present.” That was decades ago, but the spirit remains. The ultrafinitist toolkit is smaller, rougher, and still looking for its Newton. You can rebuild real analysis in discrete clothes, but you pay for every pocket you sew. Whole trunks of math need fresh carpentry. That takes time and many hands. 

Third, there’s a philosophical worry. If you tie mathematics too tightly to current computational limits, you risk conflating epistemology (what we can know or compute) with ontology (what there is). Machines get faster. Bounds move. Should existence in mathematics slide with each hardware cycle? Ultrafinitists answer with nuance, appealing to principled feasibility, not today’s RAM. But the worry lingers.

A middle path worth walking

Here’s my heresy about the heresy. We don’t have to become ultrafinitists to take ultrafinitism seriously.

Think of it as a design pressure. A calibration tool. A guard rail for when we model the world or engineer systems. It nudges us to prefer proofs that exhibit bounds. It makes us suspicious when a theorem’s only witness is a grotesque number that no computation can approach before the heat death of the universe. It reminds us that continuous models are approximations whose power comes from the quality of discretization, not metaphysical truth.

Set theory can keep its heavens. But real-world mathematics, the kind that moves money, diagnoses disease, routes packets, or points telescopes, can lean finite on purpose. Not as an afterthought. As a principle. In this sense, ultrafinitism rhymes with potentialism, the stance Hamkins teased at Columbia: treat “the infinite” as an ever-extendable horizon rather than a completed terrain. You never stand on the “last natural number.” You only stand on today’s coastline, sketching tomorrow’s pier. That’s a politely rebellious idea that classical mathematicians can live with—and engineers can love. 

How this plays out in practice

Start with analysis. Replace limits with error-controlled differences. Replace derivatives with finite differences tied to explicit step sizes. Replace integrals with quadrature under provable bounds. Analysts already do this. Numerical analysis is an entire planet, but ultrafinitism frames it as first principles rather than “what we do on computers until the theorem arrives.” It asks for proofs that travel with their runtime and memory tags included. Move to combinatorics and probability. We already reason with tail bounds, finite samples, and concentration inequalities calibrated to explicit n. Ultrafinitism simply puts pressure on proofs to keep constants honest and growth explicit. If your randomized algorithm works for “sufficiently large n,” specify “how large,” and make “sufficiently” a number, not a vibe. In logic and complexity, bounded arithmetic is not a fashion. It’s a program. It clarifies which mathematics is available at which complexity class. It connects “what exists” to “what can be proven” within resources. That’s a deep, fruitful, and decidedly finite way to think, whether you’re proving lower bounds or verifying critical code. Parikh and Sazonov were early chiselers of this granite; Buss, Cook, and Nguyen carried heavy loads after them. The ultrafinitist instinct lives here, in steel-toed boots.  In physics, the story is both tantalizing and treacherous. Gisin’s critique of real numbers isn’t an ultrafinitist manifesto, but it’s a family friend. If digits beyond physical resolution are metaphysical garnish, then determinism gets blurry at the edges of measurement. Some see freedom in that blur. Others see trouble. Either way, the mathematics you choose shapes the world you describe. Choose carefully. 

Objections, steel-manned

“But we’d lose transfinite techniques!” Maybe. But perhaps we keep them as sophisticated scaffolding for pure theory, not as load-bearing beams in models of the world.

“But discrete models miss continuity!” Only if you smuggle infinity into “continuity.” The finite can approximate the continuous to any demanded tolerance. Explicitly, constructively, and without metaphysical debt. That’s the entire point of error analysis.

“But math is about truth, not feasibility.” Yes! And proofs about truth that ignore feasibility are often less useful. The ultrafinitist reply is not that truth becomes false when it’s expensive. It’s that mathematics gains power when its truths are paired with costs.

The culture shift

Here’s the real ask: stop treating infinity as a default. Make it the opt-in. When you build a model, start finite and bounded. Explain your resources. State your error. Track your constants. Prove what your algorithm can actually do, not what it would do if the universe were a Platonic spreadsheet. That mindset is already dominant in software engineering, cryptography, formal methods, and a big chunk of machine learning. Ultrafinitism says: elevate that mindset. Don’t apologize for it. Build foundations that match it. Zeilberger’s provocation that “real analysis is a degenerate case of discrete analysis” isn’t a demolition order. It’s a dare. Can we rebuild the core material of undergrad analysis so that every limit hides an explicit bound, every existence proof hides an algorithm, and every theorem comes with a receipt? If we can, our students won’t just pass exams. They’ll ship theorems. 

What would change if this wins?

Textbooks would shift tone. “There exists” becomes “Here is.” Proofs would more often produce objects rather than trap them by contradiction. Theorems would become executable. You’d see a quiet alliance between mathematicians and computer scientists deepen. Formal proof assistants would move closer to the center of practice, because a world of feasible mathematics fits them like a glove. Conferences like the Columbia meeting would multiply. Philosophers would stop snickering and start collaborating. Physicists would test finite-information formulations next to traditional ones and report which predicts better under constraint. Mathematicians would still do set theory, of course. They’d just own, more openly, when it’s a magnificent idealization and when it’s a tool for earthly problems. 

What I think, after the dust

Infinity is a brilliant fiction that helps us tell the truth. It’s also a magnet for paradox. We should neither worship it nor ban it. We should domesticate it.

Ultrafinitism, in spirit, is domestication. It asks that our mathematics stay close to what can be built, computed, and controlled. It asks that we pay with finite coins for finite results, and stop running tabs in the infinite. I don’t want to live in a world without Cantor, Zorn, or compactness. I do want to live in a world where we reach for them last, not first, when modeling turbulent fluids, option prices, or protein folding. I want proof to mean “I can show you,” not “trust me, somewhere out there it happens.” If that’s what “banishing infinity” amounts to, count me sympathetic. The peacock keeps its feathers. We just stop pretending they fit in every room.

References (selected)

1. Ultrafinitism: Physics, Mathematics, & Philosophy, Columbia University conference, April 11–13, 2025. Program pages and course notices. 
2. Joel David Hamkins, A Potentialist Conception of Ultrafinitism (conference talk announcement, Apr 12, 2025). 
3. New Scientist, social posts teasing cover story on ultrafinitists (August 2025). 
4. Manon Bischoff, “Some Mathematicians Don’t Believe in Infinity,” Scientific American, Aug 4, 2025. 
5. Alexander Esenin-Volpin, “The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics,” in Intuitionism and Proof Theory (Proc. Buffalo 1968), North-Holland, 1970. (Bibliographic reproductions and scans). 
6. Rohit Parikh, “Existence and Feasibility in Arithmetic,” Journal of Symbolic Logic 36(3):494–508 (1971). 
7. Vladimir Sazonov, “On Feasible Numbers,” in Logic and Computational Complexity, LNCS 960, Springer (1995). 
8. Doron Zeilberger, “‘Real’ Analysis is a Degenerate Case of Discrete Analysis,” in New Progress in Difference Equations (ICDEA 2001). Online preprint. 
9. Michał J. Gajda, “Consistent Ultrafinitist Logic,” in TYPES 2023 (LIPIcs, 2024) and arXiv:2106.13309. 
10. Nicolas Gisin, “Are Real Numbers Really Real?” (2018) and subsequent work on finite-information numbers and intuitionistic mathematics in physics (2019–2021), including Indeterminism in Physics and Intuitionistic Mathematics. 
11. Background overview: “Ultrafinitism,” Wikipedia entry (overview and pointers to literature). (Use as a signpost, not an authority.) 

Note: The references above point to the conference where ultrafinitism’s current arguments were aired; foundational papers by Esenin-Volpin, Parikh, and Sazonov; Zeilberger’s discrete analysis provocation; Gajda’s recent formal logic sketch; and Gisin’s finite-information critique of real numbers in physics. Together they map the terrain: the philosophy, the logic, the computation, and the physics.