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Beyond the limits of the theoretical physics we currently understand, many of the most popular ideas have one thing in common: at their core, they’re mathematical extensions of what we presently know and understand. By adopting a mathematical framework that not only encompasses what we presently understand it, but that seeks to explain more things than our currently prevailing theories do, we hope to extend our understanding of the Universe past the current frontiers. We know that our current frameworks for General Relativity and Quantum Field Theory are great for what they do, but they don’t — and can’t — do everything.
Our theory of gravity and our quantum theories of particles and fields are fundamentally incompatible with one another: they cannot explain, for example, how the gravitational field of a quantum system behaves. Moreover, our current framework cannot sufficiently explain:
dark matter,
dark energy,
the ultimate origin of our Universe,
or the reason why our Universe is filled with matter and not antimatter,
among other puzzles. It’s true that mathematics enables us to quantitatively describe the Universe; it’s probably the most useful tool we have when it’s applied properly. But the Universe is a physical, not mathematical entity, and there’s a big difference between the two. Here’s why mathematics alone will always be insufficient to reach a fundamental theory of everything.
One of the great puzzles of the 1500s was how planets moved in an apparently retrograde fashion. This could either be explained through Ptolemy’s geocentric model (left), or Copernicus’ heliocentric one (right). However, getting the details right to arbitrary precision was something neither one could do. Both models have little predictive power; they could not detail the orbital properties of a hypothetical additional planet the way a more concrete physical theory, like Newtonian or Einsteinian gravity, would later do.
Credit : E. Siegel/Beyond the Galaxy
About 400 years ago, a great existential battle was unfolding about the nature of the Universe. For millennia, astronomers had accurately described the orbits of the planets using a geocentric model, where the Earth was stationary and all the other objects orbited around it. Armed with the mathematics of geometry and precise astronomical observations — including tools like circles, equants, deferents, and epicycles — the precise mathematical description of the heavenly bodies’ orbits matched what we saw spectacularly. For over a thousand years, this Ptolemaic model was largely undisputed for an underappreciated reason: because it was so much more successful, accurate, and had greater predictive power than any alternative.
The match between theory and observation wasn’t perfect, however, and attempts to improve upon it were frequently made. In most cases, this led to the addition of more epicycles, and then in the 16th century, the idea of Copernicus’ heliocentrism came about. By placing the Sun at the center, explanations of retrograde motion — as illustrated above — became simpler, but the fits to the data were worse than in the geocentric model. When Johannes Kepler first came along in the 1590s, he had a brilliant idea that sought to solve everything.
Kepler’s original model of the Solar System, the Mysterium Cosmographicum, consisted of the 5 Platonic solids defining the relative radii of 6 spheres, with the planets orbiting around the circumferences of those spheres. As beautiful as this is, it couldn’t describe the Solar System as well as ellipses could, or even as well as Ptolemy’s model could.
( Credit : Johannes Kepler, 1597)
He noticed that, on an astronomical front, there were six planets total, if you included Earth as a planet, but not Earth’s Moon. He also noticed that, mathematically, there were only five Platonic solids: five mathematical objects whose faces are all equal-sided polygons. By drawing a sphere inscribed inside and circumscribed outside each one, he could “nest” them in a way that fit the planetary orbits extremely well: better, honestly, than anything Copernicus had ever done. It was a brilliant, beautiful mathematical model, and arguably the first attempt at constructing what we might call “an elegant Universe” today.
But observationally, it was still a failure. It failed to even be as good as the ancient Ptolemaic model with its epicycles, equants, and deferents. The idea was one of brilliance, elegance, and simplicity, and marks our first attempt to argue — from pure mathematics alone — how the Universe ought to behave. But when we confronted the idea with reality, and the actual observations of the planets, it simply didn’t work. The prevailing model at the time, with all of its relative ugliness, still matched reality better than Kepler’s beautiful model did.
So what did Kepler do next? In many ways, he engaged in a stroke of genius that would ultimately define his legacy.
Even before we understood how the law of gravity worked, we were able to establish that any object in orbit around another obeyed Kepler’s second law: it traced out equal areas in equal amounts of time, indicating that it must move more slowly when it’s farther away and more quickly when it’s closer. At every point in a planet’s orbit, Kepler’s laws dictate at what speed that planet must move.
Credit : Gonfer/Wikimedia Commons, using Mathematica
He took his beautiful, elegant, compelling model that he had spend so long developing — a model that that he clearly loved — and, because it disagreed with observations, he did the most remarkable thing of all: he threw it away. Instead of working to refine and massage his model to better match what we saw, he dove into the data to find what types of orbits would match how the planets were actually seen to move. In the aftermath of his analysis, he came away with a set of three profound scientific (not purely mathematical) conclusions.
All six of the planets, despite assumptions, didn’t actually move in circles around the centrally located Sun, Instead, those planets moved in ellipses with the Sun at one focus, with a different set of parameters (like orbital speed and orbital eccentricity) describing the ellipse of each planet.
Because their orbits were non-circular, planets didn’t move at a constant speed in their orbits around the Sun. Instead, each planet moved at a speed that varied with the planet’s distance from the Sun — with faster speeds corresponding to closer approaches to the Sun and slower speeds corresponding to greater distances from the Sun — in such a way that each individual planet sweeps out an equal area over the same interval of time.
And finally, planets exhibited orbital periods that were directly proportional to the long axis (the major axis) of each planet’s orbital ellipse, their distance from the Sun, raised to a specific power (quantitatively determined to be 3/2).
This animation shows the four super-Jupiter planets directly imaged in orbit around the star HR 8799, whose light is blocked by a coronagraph. The four exoplanets shown here are among the easiest to directly image owing to their large size and brightness, as well as their huge separation from their parent star. Our ability to directly image exoplanets is constrained to giant exoplanets at great distances from bright stars, but improvements in coronagraph technology will dramatically change that story.
Credit : Jason Wang (Northwestern)/William Thompson (UVic)/Christian Marois (NRC Herzberg)/Quinn Konopacky (UCSD)
These three laws represented a pivotal moment in the history of science. It wasn’t philosophy, theology, or even mathematics that was at the root of the physical laws that governed nature. We couldn’t divine the “most perfect” mathematics that described nature, but instead could only use mathematics as a tool that describes how the physical laws of nature manifest. The key advance that happened was choosing to root our scientific models for reality in what was observable and measurable, and choosing the model that best fit our collected data.
This was the start of our since-unbroken tradition, for science, that any theory must confront itself with data from the Universe itself. Without this idea, further progress becomes impossible.
This idea has come up again and again throughout history, as new mathematical inventions and discoveries continuously empower us with new tools to attempt to describe physical systems. But each time, it wasn’t that the new mathematics told us how the Universe worked. Instead, new observations told us that something beyond our currently understood physics was required. Pure mathematics alone is insufficient to lead us to the next advance, but if we’re lucky, some of our already-discovered mathematical ideas will indeed wind up successfully describing the Universe we observe.
We often visualize space as a 3D grid, even though this is a frame-dependent oversimplification when we consider the concept of spacetime. In reality, spacetime is curved by the presence of matter-and-energy, and distances are not fixed but rather can evolve as the Universe expands or contracts. Prior to Einstein, space and time were thought to be fixed and absolute for everyone; today we know this cannot be true. If you place a particle on this grid and allow the Universe to expand, the grid will expand, too, and hence the particle will appear to recede away from you.
Credit : Reunmedia/Storyblocks
We can fast-forward to the early 1900s: when it was clear that Newtonian mechanics was in trouble. It could not explain how objects moved near the speed of light, which was better described by Einstein’s special theory of relativity. Newton’s theory of universal gravitation was in similarly hot water, as it could not explain the observed motion of the planet Mercury around the Sun. Concepts like spacetime were just being formulated, but the idea of non-Euclidean geometry (where space itself could be curved, rather than flat like the 3D grid shown above) had been floating around for decades among mathematicians.
Unfortunately, developing a mathematical framework to describe the possibility of curved spacetime (and gravitation) required more than pure mathematics, but the application of mathematics in a particular, tweaked way that would agree with observations of the Universe. There are many possible mathematical ways to write down equations that govern a curved manifold, but only one of those mathematical possibilities — at most — can correspond to the spacetime that describes our Universe. This, quite possibly, is the main reason why nearly everyone on Earth know the name “Albert Einstein,” but hardly anyone knows the name “David Hilbert.”
Instead of an empty, blank, three-dimensional grid, putting a mass down causes what would have been ‘straight’ lines to instead become curved by a specific amount. In general relativity, we treat space and time as continuous, but all forms of energy, including but not limited to mass, contribute to spacetime curvature. In addition, the distances between unbound objects evolve with time, owing to the expansion of the universe.
Credit : Christopher Vitale of Networkologies and the Pratt Institute
Both men — Einstein and Hilbert — had done substantial work on theories that linked spacetime curvature to gravity and the presence of matter and energy . They overlapped in location, spoke to each other, shared research ideas, and were quite fond of one another. Both of them had worked with similar mathematical formalisms; today one of the most important equations to come out of General Relativity is known as the Einstein-Hilbert action . But Hilbert, who had come up with his own, independent theory of gravity from Einstein, pursued bigger ambitions than Einstein: his theory attempted to include explanations for matter and electromagnetism as well as gravity, all in the same framework.
The problem was this: Hilbert’s ambitious theory simply didn’t agree with nature. Hilbert was constructing a mathematical theory as he thought it ought to apply to our physical reality, rather than seeking to let reality be the guide that led him to the equations that best described it. Because of Hilbert’s approach, now generally recognized as being a fundamentally backwards way to develop a theory, Hilbert never got successful equations out of his framework that accurately predicted the quantitative effects of gravity.
Einstein’s General Relativity, on the other hand, quite famously did, and that’s why the field equations are known as the Einstein field equations, with no mention of Hilbert at all. Unless your idea can survive a confrontation with reality, you aren’t doing physics at all.
The wave pattern for electrons passing through a double slit, one-at-a-time. If you measure “which slit” the electron goes through, you destroy the quantum interference pattern shown here. Regardless of the interpretation, quantum experiments appear to care whether we make certain observations and measurements (or force certain interactions) or not.
Credit : Dr. Tonomura; Belsazar/Wikimedia Commons
An almost identical situation would come up again just a few years later in physics history: this time, in the context of quantum physics. No matter how meticulously you set up your experiment, if you were to fire an electron through a double slit, it was impossible to know, based on the setup and its initial conditions, where it would wind up. You could only know what the odds were of it landing along a particular distribution of possible locations. A new type of mathematics — one rooted in wave mechanics and a set of probabilistic outcomes — was required. Today, we use the mathematics of vector spaces and operators, described by a term that might ring a bell to physics students: Hilbert space .
The same mathematician who worked on the physics of spacetime alongside Einstein, David Hilbert, had also discovered a set of mathematical vector spaces that was enormously promising for quantum physics. And interestingly enough, just as was the case for Hilbert’s attempt at a theory of gravity, the predictions arising from his Hilbert space didn’t quite make sense when confronted with physical reality.
To get a match with physical reality, some tweaks needed to be made to the math, creating what some call a rigged Hilbert space or a physical Hilbert space. (Where the “inner product” of that Hilbert space has additional physical constraints placed on it, but not for any mathematically motivated reasons.) The mathematical rules needed to be applied with certain specific caveats, or the results of our physical Universe would never be recoverable.
The pattern of weak isospin, T3, and weak hypercharge, Y_W, and color charge of all known elementary particles, rotated by the weak mixing angle to show electric charge, Q, roughly along the vertical. The neutral Higgs field (gray square) breaks the electroweak symmetry and interacts with other particles to give them mass. This diagram shows the structure of particles, but is rooted in both mathematics and physics.
Credit : Cjean42/Wikimedia Commons
Today, it’s grown very fashionable in theoretical physics to appeal to mathematics as a potential way forward to an even more fundamental theory of reality . A number of mathematical-based approaches have been tried over the years and decades, including to embed our currently understood reality into a grander framework by:
imposing additional symmetries,
adding extra dimensions,
adding new fields into General Relativity,
adding new fields into quantum theory,
or using larger groups (from mathematical group theory) to extend the Standard Model ,
along with many other scenarios.
Here’s the thing: mathematics is a powerful tool, but it is not a unique tool. When you have a mathematical framework, it generally admits many possible solutions. However, at the end of the day, there’s only one physical reality, and so you must impose something from physics — rules, constraints, selection effects, etc. — that allows you to pick out the one mathematical solution that corresponds to that observed, measured reality.
From a theorist’s perspective, the type of mathematical exploration that we can conduct are interesting and potentially relevant for physics: they may hold clues as to what secrets the Universe might have in store beyond what’s presently known. But mathematics alone cannot teach us how the Universe works. We will obtain no definitive answers without confronting its predictions with the physical Universe itself.
Visualizing the multiplication of the unit octonions, of which there are 8, requires thinking in higher-dimensional spaces (left). The multiplication table for any two unit octonions is also shown (right). Octonions are a fascinating mathematical structure, but offer non-unique solutions to a myriad of possible physical applications.
( Credit : Yannick Herfray (L); English Wikipedia (R))
This is not some lesson that only comes along at the frontiers or at the highest levels of theoretical physics. In one way, in fact, it’s a lesson that every physics student learns the first time they deal with projectile motion, and attempt to calculate the trajectory of an object thrown into the air. Questions you might want to answer include:
How far does the object go?
Where does the object land relative to your initial location?
How long does that object spend in the air before hitting the ground?
To be sure, there are mathematical equations, the equations of motion , that provide the answers. But when you solve those equations governing these objects, you don’t get “the answer.” You get several answers; that’s what you can arrive at from mathematics alone.
But in reality, there’s only one object. It follows one only trajectory, it lands in one and only location, and it does so at one specific, quantifiable time. Which of the possible answers that the equations give you corresponds to that reality? Mathematics not only won’t tell you, but it fundamentally can’t tell you. To get the answer that corresponds to reality, you need to understand the particulars of the physics problem in question, as that information is essential to determining which solution has a physical meaning behind it. Mathematics will get you very far in this world, but it won’t get you everything. Without a confrontation with reality, you won’t be able to tell the real world from worlds that are pure mathematical fantasy. If your goal is to understand the physical Universe, mathematics just isn’t enough on its own.
This article was first published in February of 2023. It was updated in July of 2026.
This article The physical Universe isn’t made of pure mathematics alone is featured on Big Think .
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