The observable universe contains an estimated 1080 atoms, a number represented as a 1 followed by 80 zeroes. Even if one counts every subatomic particle within these atoms, reaching a number like 1090 would far exceed the physical reality we can observe or measure. Such vast quantities, particularly the concept of infinity, have come under scrutiny from a dedicated group of thinkers.
For some, infinity signifies an endless process, like perpetual counting. For others, it represents an unimaginably large number. In either case, its connection to human experience becomes tenuous, even when considering the vastness of the cosmos. While cosmology often depicts an infinite universe, we also understand it as a “bubble” – the observable universe, demarcated by light’s travel time since the Big Bang. Anything beyond this extent remains effectively unknown.
The Ultrafinitist Stance
Since the 1960s, a persistent segment of mathematicians, philosophers, computer scientists, and physicists has argued that these abstract numerical concepts matter. Identifying as ultrafinitists, they caution against unquestioning acceptance of numbers such as 1090, which lack tangible real-world parallels. The concept of infinity, in particular, draws their strong skepticism.
Historically, the ultrafinitist movement has faced dismissal as both extreme and illogical. However, its proponents contend that the proliferation of extremely large numbers and the notion of infinity are destabilizing the foundations of science, impacting fields from logic to cosmology. The growing number of ultrafinitist sympathizers, though themselves finite in number, has reached a point where their arguments can no longer be easily ignored.
“There’s a critical mass now of people who have thought enough about these issues,” states Justin Clarke-Doane of Columbia University. “As it stands, there’s never been a collection or canonical text written on ultrafinitism, because the problem has been seen as too hard or too radical. There’s now potential for progress.”
In April, Clarke-Doane organized an ultrafinitist conference at Columbia, drawing researchers from diverse disciplines. While not all attendees were strict adherents to ultrafinitism, many shared concerns about the pervasive role of infinity in mathematics. “My hope is that this conference will mark a turning point in research on ultrafinitism,” Clarke-Doane remarked in his opening address. “Commentators should no longer be able to dismiss the view.”
Infinity’s Role in Modern Mathematics
Strong viewpoints were certainly expressed. “Infinity may or may not exist, God may or may not exist, but there is no need for either in mathematics,” declared Doron Zeilberger during the conference. This sentiment contrasts sharply with the prevailing mathematical consensus.
Most mathematicians operate within a framework known as Zermelo-Fraenkel set theory with the axiom of choice (ZFC). This axiomatic system, essentially a set of assumed truths, governs mathematical practice. Crucially, ZFC explicitly postulates the existence of infinity.
While ZFC proves effective for most applications, its consistency has been questioned for nearly a century. In 1931, Kurt Gödel demonstrated that ZFC’s axioms cannot be proven consistent within the system itself. “Nobody’s showed it’s inconsistent, but there’s no deep sense in which we’ll ever convince ourselves that it’s consistent,” Clarke-Doane explains.
Despite Gödel’s findings, this inconsistency often remains a background concern for researchers, as ZFC serves as a widely accepted foundation for contemporary mathematics. “Nowadays, mathematicians use the ZFC theory as foundation, without necessarily embracing it explicitly,” notes Zuzana Haniková of the Czech Academy of Sciences.
Rethinking Mathematical Foundations
Three decades after Gödel’s impactful theorem, Alexander Esenin-Volpin, a Russian mathematician, poet, and dissident, proposed a program to prove the consistency of ZF theory, a subset of ZFC, by abandoning infinity. His ambitious approach aimed to fortify contemporary mathematics through a radical departure.
The full implications of Esenin-Volpin’s ideas remain somewhat obscure, as they did not fully penetrate the mainstream of mathematical research. “These were not well-understood at the time and the details remain murky,” says Walter Dean of the University of Warwick. An early review, which coined the term “ultrafinitist,” deemed Esenin-Volpin’s 1961 paper “not particularly convincing.”
However, other mathematicians continued to explore ultrafinitist concepts. In 1971, Rohit Parikh published a paper that clarified some aspects, demonstrating that the idea of a “small number,” while difficult to define precisely, could be integrated into a functional theory. He developed a mathematical framework where all numbers were constrained to be less than a defined maximum, such as 2 tetrated to 1000. While vastly exceeding the estimated number of atoms in the universe, such numbers could be considered “feasible” within Parikh’s system. By stipulating that proofs also adhere to a feasible length, Parikh established the internal consistency of his theory. Though unable to fully replace standard mathematics, this marked a significant early success for the ultrafinitist proof methodology.
What constitutes a “feasible” number or proof lies at the core of the ultrafinitist endeavor. While this question touches upon ancient paradoxes, such as determining the exact number of grains needed to form a pile, Parikh’s primary concern was preserving mathematics’ connection to human activity. “You have to draw a line somewhere. Things have to be related to human activity,” he asserts. He believes this ultrafinitist perspective directs researchers toward lived experience, suggesting that an incomplete approach is still valuable.
Others find inspiration elsewhere. For computer scientist Doron Zeilberger, the fact that computers can only approximate infinity fuels his argument for its elimination. His interest in ultrafinitism began during his studies of calculus, a field he found problematic due to its reliance on infinitely large or small numbers. While the 17th-century introduction of calculus solidified infinity’s place in mathematics, Zeilberger views this as a historical contingency, influenced by the absence of early computing technology, and wishes to teach calculus without it.
Connections to Computation and Philosophy
Even those not identifying as ultrafinitists are concerned with the boundaries of computation. This is the subject of computational complexity theory. Walter Dean views ultrafinitism and computational complexity as two facets of the same concept: one more philosophical, the other more practical.
A prominent example within computational complexity is the P versus NP problem, widely considered the most significant problem in theoretical computer science. It addresses the computational resources required to solve certain mathematical problems and the ease with which their solutions can be verified.
In the 1980s, building on the work of pioneers like Parikh, Sam Buss developed “bounded arithmetic.” This set of tools bridges mathematical and computational limits when assessing problem-solvability. Using these tools, Buss identified problems that are both easy to solve and have easily verifiable solutions. Characterizing such relationships broadly is central to resolving the P versus NP conundrum. “This continues to be a fairly big deal and a central aim of complexity theory,” notes Dean. Buss emphasizes that this work has gained importance with advancements in fields like artificial intelligence and quantum computing, which introduce new questions about computational limitations.
From this perspective, computational complexity acts as a bridge translating abstract mathematical concepts into tangible reality, given that computers are physical objects. While mathematics is traditionally seen as a language for describing physical phenomena, some ultrafinitists propose a more integrated view. For instance, Pavel Pudlák argued in 2013 that any finite mathematical structure can be represented by a physical object, implying that a theorem about finite mathematical structures is also a physical law.
While this perspective is unconventional, Clarke-Doane finds it plausible. “No one has ever given an intelligible story of how to draw a sharp boundary between math and physics,” he observes. “If the physical world is partly mathematical, then you have to take the math seriously in a way that you don’t when you dismiss it as a language.”
A Finite Universe Hypothesis
If the ultrafinitist movement successfully removes infinity from mathematical frameworks, it may necessitate reconsidering the possibility of a finite universe, extending beyond the observable bounds. Physicist Sean Carroll explored this concept, outlining a physical model for an ultrafinite universe.
Within this quantum mechanics-based model, the universe remains spatially infinite but possesses a finite number of permissible quantum states. This results in a universe that is temporally periodic—it evolves but eventually returns to its initial condition. This starkly contrasts with the prevailing cosmological view of a universe originating in the Big Bang and continuously expanding according to physical laws, including those of thermodynamics.
Carroll demonstrated that by carefully calibrating his model, for example, by restricting entropy fluctuations, the laws of physics would not be violated. This provided a foundational blueprint for how a finite universe might operate and how the complexity of reality, such as spacetime geometry, could emerge. He did not assert that we inhabit such a universe but deemed it “perfectly conceivable.”
Carroll and colleagues have previously estimated the total number of possible quantum states in the observable cosmos to be approximately 1010122. “None of this is definite, but it gives us licence to think finite,” Carroll stated at the conference. Intriguingly, this estimate stems from efforts to unify gravity and quantum mechanics, suggesting a potential link between ultrafinitism and the quest for a theory of quantum gravity.
Infinity’s Persistence in Physics
Most physicists might hesitate to accept Carroll’s finite universe model as definitive. However, the notion of discarding or problematizing infinity in physics is not unprecedented. Quantum field theory, essential for understanding particles and forces, frequently generates infinite results.
As Clarke-Doane notes, even when physicists discuss the universe as if it were finite, “the math they’re using is up to its ears in infinity.” Physicists employ a technique called renormalization to manage these infinities, often by limiting analyses to particles with specific energy ranges or velocities, rather than considering all possibilities. This approach effectively bypasses the infinite outcomes.
Tim Maudlin from New York University counters that this does not indicate an inherent finitist bias in quantum physics. If a calculation for particle collision probability yields an infinite value, Maudlin argues, it simply signifies a failed calculation. Renormalization, in this view, is not about eliminating infinity but about using a mathematical tool to extract a meaningful answer. “To say you wanted a finite result isn’t because you say, ‘I reject infinite results’, but what you’re calling an infinite result just isn’t a result,” he contends. For Maudlin, these encounters with infinity do not inform whether space itself is finite or if there are infinite objects.
He posits that embracing an ultrafinite universe would need to stem from a novel physical theory, rather than serving as its foundational assumption, akin to how Einstein’s theory of special relativity revealed a finite speed of light. Without such a driving motivation, Maudlin likens the ultrafinitist project to an author attempting to write a novel without using the letter “e”—a significant technical challenge driven more by aesthetic preference than universal necessity.
The Future of Ultrafinitism
Given its potential applications across various scientific disciplines, is it time to seriously consider ultrafinitism as an alternative to standard mathematical foundations, even if only for comparative purposes? Zuzana Haniková, who is not an ultrafinitist, believes it could play a role. She points to the work of Czech mathematician Petr Vopěnka, who developed a similar alternative mathematical theory in the 1970s.
Vopěnka questioned why mathematical infinity could accurately model our finite real-world experience. While not a strict ultrafinitist aiming to eliminate infinity entirely, he viewed actual infinity as lying beyond our “perceptual horizon.” He introduced the concept of “natural infinity” and devised a theory that negates the axiom of infinity found in standard mathematics. This theory accommodates two types of mathematical objects: those that are precisely defined, and others that are more “blurred,” representing a path toward an infinite horizon. This allowed Vopěnka to conceptualize infinity not merely as something “beyond” any large collection of objects but also as an integral part of them, Haniková explains.
Could such reasoning shape the future of ultrafinitism? Does infinity truly need to be discarded, or can mathematics be redefined to acknowledge it differently? “This approach continues to be inspiring for mathematicians and philosophers alike,” says Haniková. It remains relevant for studies of vagueness that arise in linguistics, ethics, and mathematical logic.
Clarke-Doane is also receptive to the idea of incorporating the vagueness between the finite and the infinite into future foundational theories of mathematics. However, he acknowledges the substantial work still required by ultrafinitists. Developing a coherent ultrafinitist theory remains largely uncharted territory.
“It’s very often the case that you don’t need to care for practical purposes. But sometimes you do, and if there’s no one who has a coherent philosophical story about the foundations, that’s a serious problem when things go wrong,” Clarke-Doane states. He uses the analogy of a broken pipe in the “cellar of science,” where an expert is needed to prevent a flood.
Ultrafinitists are preparing themselves, building their theoretical tools to address potential philosophical crises in scientific reasoning.
Alexander Esenin-Volpin: The Rebel Mathematician
Born in the Soviet Union in 1924, Alexander Esenin-Volpin, an advocate of ultrafinitism, was the son of two poets and a multifaceted rebel. He earned a doctorate in mathematics and was an accomplished translator of European mathematical works due to his fluency in French. However, for decades, his mathematical career, including his developing ideas that would lead to the rejection of infinity, was sidelined by his activism against the Soviet regime. His anti-Soviet poetry and public demonstrations led to arrests and confinement in psychiatric institutions.
In 1972, he emigrated to the United States and began working at Boston University. Rohit Parikh, who later expanded upon Esenin-Volpin’s ultrafinitist concepts, met him at a conference in 1975. Esenin-Volpin stayed with Parikh for a month. “Unfortunately, even though I thought Volpin was a genius, I also came to realise that he didn’t quite understand his own work,” Parikh recalls.
Despite facing resistance, Esenin-Volpin consistently maintained his views on infinity, often with humor. Logician Harvey Friedman recounted an anecdote from a 2002 lecture where he asked Esenin-Volpin about the reality of a series of increasingly large numbers (21, 22, 23… up to 2100). Esenin-Volpin readily affirmed the reality of the first number. However, as Friedman posed questions about larger numbers, Esenin-Volpin’s responses became progressively delayed. “Sure, he was prepared to always answer yes, but he was going to take 2100 times as long to answer yes to 2100 [than] he would to answering 21. There is no way that I could get very far with this,” Friedman remembered.
Mathematician Roy Lisker described him in 1993 as “a frazzled dumpling of a man… his face and body marked by the vivid traces of what the Soviet system had done to him.” Esenin-Volpin passed away in 2016. His obituaries more frequently highlighted his political activism than his lifelong skepticism towards infinity.