The great mystery of mathematics
Why is maths the language of the universe?
by Francisco Rodrigues, University of São Paulo
When we study mathematics at school, we often don’t realise its relevance in explaining the world and the philosophical questions behind its applicability. During these early years of learning, we approach mathematics as if we were assembling a set of tools, but we often don’t fully understand the importance of each tool and the purpose it serves. It’s like knowing what a screwdriver, a saw and a hammer are, but having no idea how to use them to build a wooden box. We don’t know why we study matrices, equations, functions or polynomials. It all seems so far away from our real world.
However, we will get an idea of the use of mathematics later when we use it to describe the laws of physics. We realise that a fairly simple algebraic equation can predict the position of an object in free fall, or the pressure of a gas from its temperature and volume. But when we learned these concepts, hardly any of our teachers asked us: “Why do these laws work?” That’s the great mystery. Why is mathematics so effective in explaining the world? Although this question seems rather abstract, it allows us to philosophise about our own reality. If mathematics works so well, was the universe built by mathematics? If so, who built it? If not, why is it able to describe the world so accurately? Do we live in a computer simulation? Are there parallel universes? In this article we’ll discuss this great mystery. Why does mathematics work?
“Nature is written in mathematical language.” — Galileo Galilei (1564–1642).
In 1960, Eugene Wigner (1902–1995), who won the Nobel Prize in 1962 for his contributions to the theory of the atomic nucleus and elementary particles, published the article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, in which he wrote: “The miracle of the adequacy of mathematical language for the formulation of the laws of physics is a wonderful gift that we neither understand nor deserve.” Wigner pointed to a great mystery that has haunted science since the ancient world: “The immense usefulness of mathematics in the natural sciences borders on the mysterious, and there is no rational explanation for it.” Mathematics grew out of the need to solve practical problems. We learned to calculate areas, perimeters and fractions. How is it possible to explain the universe with this tool? To try to explain the power of mathematics, let’s look at its origins.
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” — Bertrand Russell, A History of Western Philosophy.
The beginning
It all started with shepherds, perhaps in the Middle East, India or China, thousands of years ago, who were interested in adding up sheep without knowing how to count. They associated an object, such as a stone or stick, with each sheep that came out of the fencing. For example, a shepherd might have a group of five stones to represent five sheep. When they went out to graze, they would make a pile of stones to represent the number of sheep. When they returned with the flock, they would check that the quantities were the same to see if any sheep had been lost. In other words, they were solving a purely practical problem. Little did these shepherds realise that they were catching a glimpse of infinity.
A few thousand years later, mathematical activity began to be recorded. The first signs of mathematical notations date from around 30,000 BC, with records of marks on bones and stones suggesting counting practices and the notion of quantity. By 3,000 BC, in the region of the Fertile Crescent, where Mesopotamian civilisations such as Assyria, Babylon, Sumeria and Egypt emerged, there was a need to quantify production, collect taxes and distribute land. All very practical activities. These civilisations developed numbering systems, techniques for measuring land, calculations for building pyramids and temples, and mathematical knowledge applied to trade and engineering. In other words, in the beginning, mathematics was basically a product of the need to manage what was being produced in the first civilisations, as well as being used in engineering works. Mathematics was just a tool to help human life, like a hoe or a wheat grinder. Nothing more than that.
Between the 6th and 4th centuries BC, however, this practical approach began to take a different path in ancient Greece. There, the first mathematicians were interested in formulating mathematics in a more theoretical way, based on geometry. In other words, mathematics began to be used not just as a practical tool, but as a mental language to explain the world. Perhaps the first mathematician to observe the universality of mathematics was Pythagoras.
“All is number.” — Pythagoras.
Pythagoras was born on the Greek island of Samos, off the Ionian coast, in 570 BC. We don’t know for sure whether this mathematician really existed at that time, as records were not made until centuries after his death. These historical records say that Pythagoras created a kind of cult that worshipped mathematics. The members of this sect were called mathematikoi. Pythagoras claimed that numbers and mathematics were the origin of the whole universe. He said that “the beginning of all things is number” and “all things are numbers”. In other words, Pythagoras was so fascinated by the idea of using mathematics to explain the world that he even created a kind of religion. Aristotle himself wrote that “the Pythagoreans (…) thought that things were numbers (…) and that the whole cosmos was a ruler and a number”. For Pythagoras, the universe was not only described by mathematics, but was mathematics itself.
As Greek culture developed, various theorems helped to formulate the first mathematical ideas. Thales of Miletus, Euclid of Alexandria, Archimedes of Syracuse, Apollonius of Perga and Erastosthenes are just some of the names that contributed significantly to the birth of mathematics. This mathematics was admired and idolised, where mathematical concepts and forms existed in an ideal, Platonic and perfect world. For Plato, who suggested the idea we see in the Matrix film in his myth of the cave, mathematical concepts live in an ideal world, and our real universe is only an imperfect approximation of that world. In other words, for him, we live in a universe that is a crude approximation of a pure, ideal world. So a triangle that we draw on a piece of paper is just a representation of an ideal triangle that we don’t have access to. According to this idea, we live in a cave where we see only the shadows projected from this ideal world. In this way, we are explorers discovering the mathematics that already exists in this divine world. Just as America existed before it was discovered, theorems already exist in this ideal world. Mathematicians are explorers of this universe of ideas. In other words, for Plato, mathematics is discovered. It wasn’t invented.
A great mystery
So now we have a great mystery. In the Fertile Crescent, mathematics was invented to solve practical problems. In ancient Greece it was seen as a discovery, a concept that lived in an ideal world. So was mathematics invented or discovered? If it was invented by us humans who evolved on the plains of Ethiopia, why does it work so well, allowing us to explain everything from the motion of galaxies to the interactions between atoms and molecules? When it was discovered, who invented it? How can such precise rules and patterns emerge from the chaos and randomness that was the universe at the beginning of time? Where do the laws of physics come from? These are questions to which we have no answers.
“Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.” — Neil deGrasse Tyson
Mathematics meets the physical world
Pythagoras viewed the world through a mathematical lens, but he did not demonstrate this in practice. The ancient Greeks, in general, had little interest in experimentation. For example, Aristotle claimed that heavier objects fall faster than lighter ones but did not test this claim empirically. His observations, such as a feather falling more slowly than a stone, seemed sufficient to confirm his belief. He never explored whether tying a feather to a stone would alter the stone’s fall. Over a thousand years later, Galileo Galilei became the first Western scientist to rigorously demonstrate how mathematics could be applied to explain the physical world.
Galileo Galilei, born in 1564 in Pisa, Italy, was one of history’s most influential scientists and mathematicians. His father, a friend of Leonardo da Vinci, encouraged him to study medicine at the University of Pisa. Initially following this path, Galileo soon discovered his true passion for mathematics and the natural sciences, leading him to abandon medicine. He famously remarked that “nature is written in the language of mathematics,” expressing his belief that the universe operates according to mathematical principles.
To understand this idea, consider how virtual worlds, like those in computer games such as Minecraft or Roblox, are governed by programmed rules. In Minecraft, for instance, when you cut down a tree, its top remains suspended in midair, defying the laws of gravity as we know them. This phenomenon is not a flaw but a feature of the game’s programmed “laws of nature,” created by Markus Persson (Notch), the Swedish programmer behind Minecraft, released in 2009.
For a Minecraft character, this floating tree is as natural as gravity is in our reality. Similarly, our universe follows its own set of rules, governed by the laws of physics. These laws, meticulously described through mathematics, underpin everything we observe — from falling apples to planetary orbits. Just as Minecraft is built upon a framework of programmed logic, Galileo’s insights revealed that our reality is shaped and explained by the universal language of mathematics.
Galileo used an experimental approach to formulate the first laws of motion. Using mathematics, he constructed kinematics, the branch of physics that studies motion, in the form of equations. These equations are called mathematical models and they are used to explain the world. So instead of saying that the position of a body in free fall at time t (S(t)) is given by the initial position (S(0)), plus the initial velocity (V(0)) multiplied by time t, plus half the acceleration (a) due to gravity multiplied by time squared, you can simply write in a compact form:
- S(t) = S(0) + V(0)t + gt²/2.
This is a second-degree equation, and we solve it using the Bhaskara formula we learnt in primary school. An extremely simple equation, but one that describes a wide variety of movements. It’s the equation we use to calculate the position of a missile launched from a ship, or even the target to hit in the game Angry Birds. Something very simple that works here or on a moon in the Andromeda galaxy. Again, in the words of Eugene Wigner, “it’s a wonderful gift that we neither understand nor deserve”.
But the power of mathematics doesn’t stop there. Isaac Newton and Johannes Kepler discovered the laws governing the movement of the planets around the sun. Newton proposed the laws of mechanics, which explain how bodies move under the influence of gravity. To do this, Newton invented (or discovered) a new area of mathematics called differential and integral calculus, dedicated to the study of rates of change of quantities (such as the slope of a line) and the accumulation of quantities. The same tool had been independently discovered by the philosopher and mathematician Gottfried Wilhelm Leibniz. Newton had fallen out with Leibniz over the invention of calculus, claiming that Leibniz had copied his ideas, even though Leibniz’s approach was quite different and simpler than Newton’s.
“God is a pure mathematician!” — Sir James Jeans.
This simultaneous discovery of mathematical concepts is something that fascinates us and has happened many times in history. How do mathematicians working independently discover (or invent) the same tools? Moreover, the description of nature is often not unique. Newton’s laws were later reformulated using purely mathematical reasoning by Pierre-Simon, Marquis de Laplace, in his monumental masterpiece Mecanique Céleste. Laplace translated the geometric study of classical mechanics used by Isaac Newton into a calculus-based study, describing the laws of mechanics in a simpler and more elegant way. Laplace didn’t need drawings or geometry to describe the world. He formulated equations based on the concept of energy (proposed by Leibniz) and thus reformulated the whole of mechanics, creating analytical mechanics.
Napoleon asked Laplace where God fit into his mathematical work, and Laplace replied “Sir, I have no need of that hypothesis.”
Laplace, who believed in the efficacy of mathematics, proposed the existence of a demon who could predict the entire past and future of the universe based on the laws of mechanics. He said: “we may consider the present state of the universe as the result of its past and the cause of its future. If, at a given moment, an intellect had knowledge of all the forces that set nature in motion, and of the position of all the elements of which nature is composed, and if that intellect were great enough to submit such data to analysis, it would include in a single formula the movements of the largest bodies in the universe, and also those of the smallest atoms; for such an intellect nothing would be uncertain, and the future as well as the past would be within reach”. In other words, with mathematics everything is predictable, both the past and the future. All we need is a powerful computer and data. However, nature is not so simple and quantum mechanics, together with chaos theory, has shown that long-term determinism is impossible. We’ll look at this in another article in the future. For now, let’s get on with this great mystery.
From math to physics
An important point about the discoveries of Kepler, Galileo and Newton is that physics is not based on mathematics but on experimentation. In other words, physics uses mathematics to explain the world, but the laws of physics are formulated from experiments. In the words of the physicist Richard Feynman: “It doesn’t matter how beautiful your theory is, it doesn’t matter how intelligent you are. If it doesn’t agree with the experiment, it’s wrong.” So physics is not interested in making laws that are beautiful, but laws that are useful. Even so, the laws that do emerge are quite simple, and that impresses us. How can the movement of galaxies and planets be described by an equation that we can write down on a piece of paper? Or in electromagnetism, how can all of electromagnetic theory be described by four equations written by James Clerk Maxwell? It’s a great mystery…
Since the laws of physics are based on experiments, Newton’s laws were updated by Albert Einstein in 1905 with his theory of relativity. Einstein used a mathematical formulation to show that time and space are relative, not absolute, as Newton had proposed. Einstein used a tool developed by a mathematician, Hendrik Lorentz, to explain how measurements of time and space are related when examined by different observers in relative motion to each other. According to Einstein’s theory, gravity is the result of the curvature created in space by the presence of a very massive body. There is therefore no gravitational force as Newton proposed.
The most interesting thing about the theory of relativity is that it was developed mentally and only proved years later, when an eclipse was observed in May 1919 in Sobral, Ceará. Researchers found that matter and energy distort the fabric of space-time and can redirect the path of light travelling through it. In other words, using mathematical concepts developed by Lorentz, who was not interested in describing the universe, Einstein proposed a theory based not on observations but on mathematics, which was only proven years later.
Although Einstein used mathematics to propose his theories, he didn’t feel comfortable accepting certain results, as in the case of quantum mechanics. Although he helped develop it when he explained the photoelectric effect, which occurs when energy is transferred from photons to electrons, he never accepted the probabilistic interpretation of quantum mechanics. Einstein claimed that God “does not play dice with the universe” and thought it absurd that the world should be probabilistic rather than deterministic, as previously seen in mechanics and electromagnetism. In classical mechanics, the world is a predictable place. If you know the position and velocity of a particle, you can predict with certainty where it will be in the future, as Laplace suggested. In the quantum world, however, nature is a probabilistic place. If you know the position and velocity of a particle, you can only predict the probability that it will be in a particular place in the future. If we throw a dice, we don’t know what the next value will be, but we can calculate the probability of each side coming out. This indeterminacy haunted Einstein throughout his life, leading him to propose various interpretations that were refuted. Erwin Schrödinger himself, who introduced the equation that describes the behaviour of quantum physical systems, did not accept the probabilistic interpretation.
Schrödinger proposed a thought experiment called “Schrödinger’s cat” to illustrate his concerns about the probabilistic interpretation of quantum mechanics. In the thought experiment, a cat is placed in a closed box with a vial of poison. The vial of poison is connected to a device that can be triggered by a radioactive atom. When the radioactive atom decays, the poison bottle is broken and the cat dies. According to the probabilistic interpretation of quantum mechanics, the radioactive atom is in a superposition of states. In one state, the radioactive atom does not disintegrate and the cat lives. In another state, the radioactive atom decays and the cat dies. Until the box is opened, we can’t be sure whether the cat is alive or dead. The cat is in a state of superposition of states, alive and dead at the same time. Schrödinger argued that this was absurd. He didn’t think it was possible for a cat to be alive and dead at the same time. He believed that quantum mechanics must be incomplete and that ultimately there must be a more fundamental theory that explains the probabilistic nature of quantum mechanics.
Although Einstein and Schrödinger were against the probabilistic interpretation, it is a very accurate theory and agrees with data from experiments. In other words, although the wave function is a mathematical object, it is the best interpretation we have for the real world. It’s another demonstration of the incredible power of mathematics. Even if we can’t imagine some of the phenomena that occur in the quantum world, mathematics shows that the results are correct because they agree with the experiments in an extremely precise way.
In addition to the mathematical ideas inspired by physical experiments, there are instances where physics draws upon tools originally developed as purely mathematical constructs. In other words, mathematicians often create abstract theories that, while seemingly detached from reality, later prove to be fundamental in explaining natural phenomena. A prominent example of this is the concept of symmetry.
Symmetry is the property of an object that does not change when it is transformed in a particular way. Our brains have a special aesthetic appreciation for objects that are symmetrical. In art, the concept of symmetry is very important from an aesthetic point of view. In Leonardo da Vinci’s ‘Portrait of the Mona Lisa’, the painting is symmetrical in relation to the vertical axis, with Mona Lisa’s head and torso divided into two equal halves. This symmetry helps to create a sense of balance and harmony in the painting. In mathematics, symmetry is studied in geometry, algebra and group theory. For example, a square is symmetrical about four axes of rotation. This means that if you rotate the square 90 degrees, 180 degrees or 270 degrees, it still looks the same. A sphere is symmetrical about any axis of rotation.
The German mathematician Emmy Noether revolutionised the concept of symmetry in physics. She was interested in the study of algebra, far from any real applications. In 1918, Noether proposed one of the most important theorems in modern physics, stating that there is a connection between symmetry and the laws of conservation in physics. If a physical system does not change under a transformation, then a physical quantity related to that transformation is conserved. For example, the law of conservation of energy states that the total energy of the universe is always conserved. This law can be derived from Noether’s theorem, which tells us that the laws of physics are invariant under temporal transformations. In other words, the laws of physics don’t change over time. If you do an experiment today and repeat it in a year’s time, you’ll get the same results. The same applies to other quantities, such as linear or angular momentum or electric charge.
Therefore, the concept of symmetry, rooted in pure mathematics, has profoundly influenced our understanding of the physical world. It has enabled the discovery of new physical laws and the extension of existing ones. For example, gauge symmetry played a key role in the development of quantum field theory and the prediction of the Higgs boson — a particle responsible for giving mass to fundamental particles. Remarkably, this particle was theorised in 1964 on the basis of mathematical considerations alone. Symmetry has thus revolutionised physics, combining the abstract elegance of mathematics with experimental validation. As Werner Heisenberg noted, symmetry in physics embodies order and structure, revealing itself as a fundamental property of nature.
This effectiveness of mathematics has become a dogma in physics. Many researchers believe that mathematics is a fundamental part of nature, arguing that mathematics is inherent in the universe. As Paul Dirac stated in 1963, “it is more important to have beauty in your equations than to have them adequate to experience”. The laws of physics would be mathematical in nature, and mathematics would be necessary to understand nature.
The success of this idea is a great mystery. Mathematics has been used to predict the existence of the planet Neptune, radio waves, antimatter, neutrinos, black holes, gravitational waves and the Higgs boson, to name but a few. However, there are still those who disagree with this idea because there are natural phenomena that are not yet fully understood by mathematics, such as the nature of gravity or even turbulence in fluids.
Invented or discovered?
Those who believe that mathematics was invented argue that the human mind continues to invent new mathematical concepts to this day. In this case, there may be a limit to how far we can go in using mathematics to explain our world. One example is the Standard Model, whose equation is considered the “ugliest” ever produced in physics. Perhaps this beauty we have found so far is the limit, where we will have to choose between an effective theory or a “beautiful” equation. This path that physics must follow will help us to have more ideas about this great mystery of mathematics: invented or discovered?
“Geometry, algebra and calculus were created by humans, tailored to the needs and problems of their time.” — Florian Cajori
In addition to physics, it is important to note that the universality of mathematics occurs in economics, where we have the black hole equation; in complexity science, where we have power law distributions (see my previous post); in sociology, where we have the Pareto equation; in biology, where ecological models are highly accurate in describing the behaviour of ecosystems. If maths can be applied in so many areas, so universally, would it have been discovered or invented? We don't know…
Although we have not arrived at an answer to this great mystery, it is important to know it. In fact, everyone can try to answer it in their own way, following one line of reasoning or another — it is a philosophical problem. It helps us to see the beauty of mathematics, and gives some meaning to our existence by placing us as actors in a world governed by non-trivial rules. We are creators of a language that explains the cosmos, and at the same time creatures who can read this language, which was used to build nature. Who knew that the calculations we learnt in primary school were actually the language used to describe the universe? A few decades ago, no one could have imagined that we would have computers so fast that we could create virtual worlds like those in the games Minecraft or No Man’s Sky. It was all possible thanks to mathematics. If mathematics describes our universe, this leads to the hypothesis that we can develop simulated universes inside the computer. This has been done with games. Thinking philosophically, would our reality be a computer simulation where the creators of this universe used mathematical concepts just like we do? It’s an interesting idea, but we’ll leave it for another article.
If you’re curious about my research, visit this link: https://sites.icmc.usp.br/francisco.
See you next time!
To find out more:
- Book: Is God a Mathematician? Mario Livio.
- Paper: The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Eugene Wigner.
- Video: The Great Math Mystery, Nova.
