Do infinities exist in nature? How mathematical abstraction brings us closer to reality
Impossible models underpin physics, while mathematics gives form to infinity, revealing the hidden structure within seemingly chaotic divergences
The Fibonacci sequence appears in the Mandelbrot set. The Mandelbrot set is the best-known visual representation of fractals—infinitely complex geometric structures in which patterns repeat at different scales. Fractals are widespread in nature: they appear in the patterns of plants, in the bronchial tree of the lungs, in the branching of lightning during storms, in snowflakes, and in the shapes of rivers, clouds, and mountain ranges, among many other examples – Photo: Sepitropova / Wikimedia Commons
Drawing on distribution theory, Pedro de Castro Diniz, a researcher at the IFSC, presents a mathematical framework for dealing with singular functions—objects that cease to be well defined at certain points. A classic example involves gravity: as the distance between two bodies decreases, the gravitational force increases dramatically. In idealized models, when the separation between the bodies approaches zero, the magnitude of the force diverges to infinity.
Distribution theory makes it possible to manipulate certain singularities that arise in mathematical physics. For example, the Dirac delta distribution used in the published study is suitable for describing several concepts in theoretical physics, such as a point mass or an electric charge – Photo: Oleg Alexandrov / Wikimedia Commons
For physics, however, this result does not necessarily represent a “real” infinity, but rather a limit associated with the way the system has been modeled. “The tools provided by distribution theory help us deal with these mathematically ‘ill-behaved’ functions. The mathematical description of a problem sometimes creates new problems of its own. In the case of singularities, on the one hand we gain descriptive power; on the other, we create the problem of infinity at that point. The study we published is about overcoming these problems that typically arise when we describe our models in this way”, Emanuel Alves de Lima Henn, Pedro Diniz’s advisor and co-author of the article Estrutura distribucional de derivadas singulares: aplicações em eletrostática e elasticidade, told Jornal da USP.
Consider a spherical cow
When we represent an electric charge, the mass of a planet, or the application of a force as being concentrated at a single point in space, we are making an idealization: we reduce extended objects to mathematical entities with no volume.
Idealizations of this kind lie at the heart of scientific modeling. Among physicists, a famous joke tells of a farmer, desperate because of his cows’ low milk production, who seeks help from a team of scientists. After weeks of study, the theoretical physicist steps onto the stage and announces: “Ladies and gentlemen, we have the solution! But it only works for spherical cows in a vacuum.”
Despite its humorous tone, the joke reveals something profound about science. Much of physics’ success lies precisely in reducing complex phenomena to simplified models capable of making the world more understandable, quantifiable, and predictable. To do so, scientists must decide which characteristics of a system are relevant for describing a given behavior—and, in that process, many aspects of reality are deliberately ignored.
Infinity emerges as an almost inevitable consequence of the idealizations that make science possible. “This is deeply characteristic of physics as a science”, said Emanuel Henn.
“We are trying to understand nature, which is extraordinarily complex; if every factor were taken into account, no solution would be possible. When we think about an ideal situation, we simplify it enough to solve a specific problem. We then move gradually toward greater complexity, enabling our models to describe reality more and more accurately”, he explained.
For astronauts orbiting the Moon, for example, treating the Earth as a point mass works perfectly well for calculating the gravitational effects acting on their spacecraft. For geography, however, it is necessary to take mountains, oceans, and irregularities in the Earth’s crust into account. Thus, the same object can be represented at different levels of abstraction in science depending on the phenomenon under study—and, even without corresponding exactly to the real world, these models remain extremely effective.
The Treachery of Images, a Surrealist work by the Belgian artist René Magritte. In a provocative tone, the painting bears the inscription Ceci n’est pas une pipe (“This is not a pipe”), prompting, among many other interpretations, reflection on the distance between human representations of the world and reality – Photo: Queen of Hearts / Wikimedia Commons
“Physics is not nature; it is a description of nature”, Wagner Lannes, professor and researcher at the Federal University of the Jequitinhonha and Mucuri Valleys (UFVJM), told Jornal da USP. According to him, mathematics is also an approximation that “works not because it faithfully reproduces the world, but because it creates organized ways of describing it.”
“Mathematicians often develop concepts to solve problems within mathematics itself, without imagining any practical application”, said Lannes.
Just as happened with complex numbers—with an “imaginary” component added to make it possible to solve specific problems—and with non-Euclidean geometries, such as elliptic and hyperbolic geometry, which exist outside a flat space, infinity also underwent a long process of legitimization before becoming a central tool in mathematics. According to the researcher, these examples show that abstractions seemingly far removed from reality can reveal profound structures underlying the natural world.
Does infinity, after all, have a shape?
As a historian of mathematics, Wagner Lannes argued that the way this abstraction was transformed into a tool is the result of a historical process. “It was not mathematics itself that came to deal with infinity. Rather, it was communities of mathematicians that developed languages capable of transforming it into something that could be defined and communicated.”
From Zeno’s paradoxes in Ancient Greece to the infinities of different sizes proposed by Georg Cantor in the nineteenth century, the concept has often been surrounded by philosophical and mathematical controversies. According to the mathematician, it was language that made it possible to deal consistently with something that seemed to elude everyday experience.
Zeno’s paradox, also known as Achilles and the Tortoise, describes a race in which the tortoise is given a head start. To catch up, Achilles must first reach the point where the tortoise was; but by then, it has already moved a little farther ahead. This process can continue indefinitely, since Achilles would have to pass through infinitely many intermediate points. For that very reason, Achilles could never catch up with—or even overtake—the tortoise. Although the argument appears to make sense on paper, it directly contradicts what we observe in reality. The flaw in the paradox lies in assuming that the time required to traverse an infinite sum of distances must also be infinite – Photo: Martin Grandjean / Wikimedia Commons
“The history of infinity is also a history of language”, Lannes summarized.
If mathematics has historically learned to tame infinity through language, physics continues to face situations in which it reappears as a concrete problem in the models being used. Pedro Diniz, author of the study that addresses infinity in physics, argued: “There should not be infinities in nature. Infinity indicates that something is probably being left out.”
With this statement, the researcher emphasizes that, in physics, the appearance of infinities usually signals the limits of the model being used. Despite these limitations, mathematics remains the primary language of physics. Confidence in mathematics as the language of nature has accompanied modern physics since Galileo, who argued that the world should be interpreted through mathematical language: “The book of the Universe cannot be understood unless one first learns its language and becomes familiar with the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures”, he wrote.
Nature by Numbers is a film by Cristóbal Vila/Etérea Estudios that illustrates the presence in nature of relationships derived from the Fibonacci sequence. This sequence is an infinite mathematical progression in which each number (beginning with the third) is the sum of the two preceding numbers. It begins with 0 and 1, generating the following sequence: 0, 1, 1, 2, 3, 5, 8, 13, … and so on.
The problem arises when the possible representation of reality challenges human intuition about continuity, infinity, and space. In such cases, formalisms such as distribution theory make it possible to deal with situations in which traditional models appear to lose their meaning. What once seemed to be a mathematical “monstrosity” haunting physical descriptions comes to reveal a necessity: the need to expand the tools used to describe reality.
According to Pedro Diniz, literal infinities do not exist in nature (such as infinite density), but they frequently appear when scientists extend an idealized model beyond the scale at which it remains interpretable. “The interesting thing is that this does not make the model useless. Quite often, even when the description diverges (that is, when an ‘infinity’ appears), the theory still allows us to extract a finite and well-defined physical quantity. That is precisely what distribution theory shows us. We are able to realize that there is, in fact, no physical infinity there”, he said.
The researcher also highlights the educational value of the study. “One of my professors used to say that mathematics should not be an obstacle to understanding physics, and that statement has stayed with me. The publication’s contribution lies in teaching, because it develops a complicated idea in detail and brings together several aspects of mathematical physics in more accessible language. Singular distributions arise naturally, and that is why I believe the work may be especially valuable for students”, stated the author.
Everything at a single point
“It is understood that we were all there, […] and where else could we have been? […] in fact, there was not even enough space to be crowded together. Every point of each one of us coincided with every point of every other one of us in a single point: the one where we all were.”
This is how Italo Calvino describes, in Cosmicomics, the primordial singularity preceding the Big Bang that gave rise to the Universe as we know it.
In physics, certain structures are also referred to as singularities, perhaps the best-known example being the black hole. “The singularities we deal with in the study, such as considering an electric charge, the entire mass of a body, or a force concentrated at a single point, are idealizations. More than that, these examples clearly look like idealizations. The type of singularity associated with a black hole, on the other hand, remains undecipherable”, said Emanuel Henn.
Although they share the same name, mathematical and physical singularities belong to different contexts. In mathematics, a singularity is an abstract concept related to functions and limits. In physics, it points to infinite curvature, infinite density, or infinite forces, actually representing a limitation of current theories—a sign that they cease to work under extreme conditions, indicating the need for more comprehensive descriptions of nature.
According to Wagner Lannes, in Ancient Greece infinity was, above all, an idea. Today, concepts such as limits, convergence, and infinitesimals help transform what was once a philosophical intuition into a manipulable mathematical object, representing a decisive development in modern mathematics.
A Möbius strip, a Surrealist work by the Dutch artist M. C. Escher. The Möbius strip is a non-orientable mathematical object (a topological surface). It appears to have two sides and two edges, but in fact it has only one side and a single edge. As a result, when we travel along its surface, we return to the starting point without ever encountering a boundary between “inside” and “outside”—a classic metaphor for continuity and infinity – Photo: Courtesy of Dror Bar-Natan
With infinity serving as the link, singularities reveal a longstanding tension between mathematics and physics. According to Vinicius Carvalho, a researcher in the Philosophy of Physics at the Federal University of Mato Grosso do Sul (UFMS), while mathematics is free to explore abstract entities, physics maintains a longstanding commitment to observation and experience. “Mathematicians can be bold”, he explained. “Physicists can also imagine freely, but, at some point, their theories must return to the observable world.”
It is precisely at this point that singularities become problematic. “A singularity represents the end of physics”, he said. This is because, at that point, the very quantities physics seeks to describe—space, time, mass, and energy—disappear. The result is a purely mathematical object: a dimensionless point with no observable physical properties. For this reason, singularities occupy a boundary region between two different ways of understanding the world: on the one hand, mathematical abstraction; on the other, the physical requirement that theories maintain some connection with experience.
Even so, the researcher emphasizes that infinities play an important role in the construction of theories. They make it possible to explore possibilities, formulate hypotheses, and develop new models, even helping to solve real-world problems. This does not mean, however, that every element introduced by these theories has an observable physical counterpart.
“All physics is mathematics, but not all mathematics is physics”, Carvalho stated.
Amid idealizations, singularities, and abstractions, infinity may reveal less about the existence of something endless in nature than about the process of constructing scientific knowledge, in a continuous effort to describe a world that is always more complex than the theories we build to explain it. In Einstein’s words: “One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike—and yet it is the most precious thing we have.”
The article Estrutura distribucional de derivadas singulares: aplicações em eletrostática e elasticidade was published in the Brazilian Journal of Physics Education and is available at this link.
For further information: Pedro de Castro Diniz, dinizcpedro@gmail.com
*Intern under the supervision of Luiza Caires
**Intern under the supervision of Simone Gomes
English version: Nexus Traduções, edited by Denis Pacheco
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