Invented, Then Unavoidable
Imaginary numbers were dismissed as useless for two centuries before they became load-bearing in physics. That history is the sharpest test we have of whether mathematics is found or made — and of why physicists trust beauty.
In 1545, Gerolamo Cardano set his readers a puzzle: divide ten into two parts whose product is forty. There is no such pair of ordinary numbers. Push the algebra through anyway and you get 5 + √−15 and 5 −√−15, two quantities whose square roots are of a kind nobody could point to. Cardano checked that they worked, called the whole manoeuvre “as subtle as it is useless,” told his reader to set aside the mental torture it caused, and moved on.
Three hundred and eighty-one years later, Erwin Schrödinger wrote down the equation that governs how every quantum system in the universe evolves in time, and that same √−1 was sitting in it — not as a shortcut, not as bookkeeping, but structurally, where it could not be removed. Schrödinger himself found this unpleasant and said so in a letter to Lorentz: surely, he wrote, the wave function ought fundamentally to be a real function.
Something has to be explained here. Either Cardano stumbled onto a piece of the furniture of reality and failed to recognise it — in which case mathematics is a landscape and mathematicians are explorers — or three centuries of mathematicians built a convenient fiction that later turned out to fit the world, which raises the harder question of why a fiction should fit anything at all. This is the oldest live argument in the philosophy of mathematics, and it is not merely academic: the two answers license different research heuristics, and physics has spent the last forty years betting heavily on one of them.
The landscape and the game
The realist position — usually called Platonism, after the theory of Forms it descends from — holds that mathematical objects exist independently of us. Numbers, sets and groups are not located anywhere, they did not begin at any moment, and they would be exactly as they are had no one ever thought about them. Proving a theorem is closer to charting a coastline than to writing a poem. G. H. Hardy put it without hedging: mathematical reality lies outside us, our function is to observe it, and the theorems we grandly call our creations are simply the notes of our observations. Kurt Gödel held a stronger version still, claiming that we have something like a perception of the objects of set theory, and that this perception is why certain axioms force themselves on us as true.
The opposing family of views is broader and less unified, which is part of why it wins fewer converts. John Stuart Mill treated arithmetic as very well-confirmed empirical generalisation about collections of pebbles. David Hilbert's formalism treated mathematics as the manipulation of symbols under rules, with consistency the only thing that needed guaranteeing. L. E. J. Brouwer's intuitionism made mathematics a mental construction, so that a statement is true only when we have built the object it asserts. Hartry Field's fictionalism grants that mathematical statements are, taken literally, false — there are no numbers — and argues that science does not actually need them to be true, only useful. Cognitive scientists such as George Lakoff and Rafael Núñez go further down: arithmetic grows out of embodied metaphors, object collection and motion along a path, elaborated far past their origins.
Notice that both positions have to account for the same phenomenology. Mathematicians overwhelmingly report the experience of finding rather than making — of being told no by the subject matter, repeatedly, in ways that feel nothing like a novelist deciding a plot. The realist explains this by saying the report is accurate. The constructivist has to explain it away, and the good version of that explanation is not dismissive: once you have fixed a set of rules, you have fixed an infinite space of consequences, and you have no more authority over which of them hold than a chess player has over whether a given endgame is winnable.
What actually forced √−1 into mathematics
The popular story about imaginary numbers is wrong in an instructive way. It says mathematicians wanted a solution to x² + 1 = 0 and invented one. But nobody needs a solution to that equation. You can simply say it has none, exactly as one says there is no largest prime, and nothing in the rest of mathematics complains.
What forced the issue was cubics. In 1572, Rafael Bombelli was working on equations of the form solved by Cardano's formula, and hit a case that Cardano had flagged and abandoned as irreducible. Take an equation with an obvious, honest, whole-number answer:
Apply Cardano's formula and it hands you back something that looks like nonsense: the cube root of 2 + 11i added to the cube root of 2 − 11i. The formula has routed a real question through territory that was not supposed to exist. Bombelli's move — which he called, with some embarrassment, a wild thought — was to carry on regardless and treat those objects as numbers with arithmetic of their own. Cube 2 + i and you get 2 + 11i. Cube 2 − i and you get 2 − 11i. Add the two cube roots and the imaginary parts annihilate:
This is the crux. The imaginary numbers did not enter as a wish. They entered as a detour that could not be avoided: a real problem, a real answer, and a route to it that passes through territory the mathematics of the day had declared empty. Whatever you think about invention and discovery, Bombelli did not get to choose that the route went that way.
It took another two centuries to make the detour respectable. Descartes coined the word “imaginary” in 1637, and meant it dismissively. Euler gave the objects their explosive utility in 1748, showing that eix = cos x + i sin x, which converted every question about rotation and oscillation into a question about exponentials. Caspar Wessel in 1799, Jean-Robert Argand in 1806, and Carl Friedrich Gauss in 1831 each realised that the “impossible” numbers were just points on a plane — Gauss complaining that the true metaphysics of √−1 had been needlessly obscured. In 1837 William Rowan Hamilton finished the demolition of the mystery by defining a complex number as nothing more than an ordered pair of ordinary real numbers with a particular multiplication rule. No new stuff, just a new structure over old stuff.
An invention with almost no room to invent
If complex numbers were freely invented, you would expect alternatives. Hamilton spent thirteen years looking for one. Having reduced complex numbers to pairs of reals, he tried for years to build a comparable arithmetic on triples — three-dimensional numbers he could use for rotations in space. He could add them. He could never make division work. His children reportedly asked him at breakfast whether he could multiply triples yet, and for years the answer was no.
The answer was no because it is impossible. On 16 October 1843, walking along the Royal Canal in Dublin, Hamilton realised he had to give up not a dimension but a law: with four components instead of three, and multiplication that is not commutative, everything works. He carved i² = j² = k² = ijk = −1 into the stone of Broome Bridge. In 1877 Ferdinand Georg Frobenius proved the general result: over the real numbers there are exactly three finite-dimensional associative division algebras — the reals, the complex numbers, and Hamilton's quaternions. That is the complete list. There is no fourth option and no three-dimensional option, and no amount of ingenuity will produce one. Give up associativity as well and you buy exactly one more — the eight-dimensional octonions, which John Graves found within two months of Hamilton's walk — and then the road ends: Adolf Hurwitz proved in 1898 that normed division algebras exist in dimensions 1, 2, 4 and 8, and nowhere else. Each law you surrender purchases a single further dimension, and the purse is empty after two.
This is the most interesting fact in the whole debate, and it cuts across both positions. Mathematicians did invent complex numbers, in the ordinary sense that they wrote down a definition that had not existed before and chose the notation. But the space of available inventions turned out to have almost no room in it. Once you insist on keeping arithmetic that behaves like arithmetic, the options are enumerable and few. An invention that constrained is hard to distinguish, in practice, from a discovery.
Then the physics arrived
Complex numbers earned their first industrial keep in the 1890s, when Charles Proteus Steinmetz introduced phasors and turned the differential equations of alternating-current circuits into ordinary algebra. That use is real but philosophically mild: it is a change of coordinates, and one could grind through with sines and cosines instead, badly and slowly.
Quantum mechanics is a different matter. In the Schrödinger equation,
the i is not a coordinate choice. It is what makes the equation generate unitary time evolution rather than diffusion; it is what makes probability amplitudes interfere; it is why a two-slit pattern exists. For decades one could still argue this was a formal convenience — complex numbers can always be simulated by pairs of reals with extra constraints, as Hamilton showed. Then in 2021, Renou and colleagues showed that the question has an experimental answer. Under a standard assumption about how independent quantum sources combine, a real-valued formulation of quantum theory makes different predictions from the complex one in a Bell-like network experiment. Two groups tested it in 2022, on photons and on superconducting qubits, and the real-valued version lost. The caveat matters — the result rules out real formulations that keep the usual tensor-product account of independence, not every conceivable one — but the headline stands: Cardano's useless subtlety is now something you can measure.
This is not an isolated case. Riemann's 1854 geometry of curved spaces, developed with no physical application in mind, was waiting when Einstein needed it in 1915. Group theory, invented to study the solvability of polynomial equations, turned out to classify elementary particles — Gell-Mann, and independently Yuval Ne'eman, used it in 1962 to predict a particle nobody had seen, with a specific mass, and the Ω− showed up at Brookhaven in 1964. Hardy boasted in 1940 that number theory had no warlike or practical use whatsoever; RSA encryption arrived in 1977 and now secures most of the internet. Eugene Wigner named the pattern in 1960: the unreasonable effectiveness of mathematics in the natural sciences, a gift we neither understand nor deserve.
The deflationary reply is worth taking seriously, though. We notice the hits. The overwhelming majority of mathematics has never applied to anything and probably never will, and physicists shop actively among existing structures until something fits — which makes eventual fit much less surprising. Richard Hamming, revisiting Wigner in 1980, added that we select the mathematics that works, that we select the physics we can do mathematics about, and that our evolved pattern-finding is doing quiet work in the background. That does not dissolve the puzzle. It does shrink it from a miracle to a strong regularity in need of explanation.
The aesthetic criterion, and what it is really tracking
Running underneath all of this is a working practice that both camps have to explain: mathematicians and physicists routinely use beauty as evidence. Not as decoration afterwards — as a reason to believe one line of enquiry over another, before the verdict is in.
Hardy was equally direct: there is no permanent place in the world for ugly mathematics. He was also unusually precise about what he meant. A beautiful proof, for Hardy, has unexpectedness, inevitability and economy — you did not see it coming, once seen it could not have gone otherwise, and there is nothing in it you could remove. Poincaré described aesthetic sensibility as a filter, the thing that decides which of the countless combinations thrown up by unconscious work are worth hauling into daylight. Steven Weinberg, writing about the theories he had helped build, described beauty in physics as rigidity: a beautiful theory is one you cannot adjust, where changing any piece breaks everything, so that it feels less designed than found.
There is even a measurement. In 2014, Semir Zeki and colleagues put mathematicians in a scanner while they rated equations for beauty. The equations they found beautiful activated the medial orbitofrontal cortex — the same field that lights up for music and painting. Euler's identity,
was the most consistently beautiful thing on the list. Whatever mathematical beauty is, it is not a metaphor; it runs through the brain's ordinary reward machinery for art.
For the Platonist this is easy and flattering. Beauty is what contact with the real structure feels like from the inside — a compass needle swinging towards something that was already there. For the constructivist it needs work, and the work is more interesting. Suppose beauty is a learned proxy for structural properties that have historically predicted success: compression (a beautiful theory says a great deal in a short description), generality (it applies far beyond the case it was built for), and rigidity (it has few free parameters, so it makes sharp, falsifiable claims). On this reading, beauty is not a window onto reality but a well-trained summary of what has worked before — a reliable guide inside explored territory, and much less reliable outside it.
The historical record is friendlier to the second reading than physicists like to admit. Kepler's model of the solar system as nested Platonic solids, from 1596, is genuinely exquisite and completely false. Parity conservation was so obviously beautiful that almost nobody thought to check it, until Wu's experiment in the winter of 1956–57 showed the weak interaction violates it flagrantly. The simplest grand unified theory, SU(5), was so elegant it was widely believed for years; it predicted proton decay that has never been seen, with the lifetime limit now pushed beyond 1034 years. Sabine Hossenfelder's argument in Lost in Math is that this failure mode is currently active: naturalness and elegance drove decades of particle physics towards supersymmetry, enormous experimental effort went into looking for superpartners at the Large Hadron Collider, and none arrived. Beauty as a heuristic has an excellent record when it is extrapolating slightly past well-tested ground, and a poor one when it is the only thing you have.
Choosing the rules, not the consequences
The most defensible position is not a compromise so much as a division of labour. What we invent are the definitions, the axioms, the notation and the questions — these are human artefacts, historically contingent, and they could have been otherwise. What we discover is what follows from them, which is not up to us at all. Bombelli chose to extend the number system. He did not choose that the extension is unique, that division fails in three dimensions, or that the cube root of 2 + 11i is 2 + i.
Structuralism sharpens this. On the structuralist account, associated with Paul Benacerraf, Stewart Shapiro and Michael Resnik, mathematics is not about objects at all but about patterns, and a number is simply a position in one. This dissolves a puzzle that had embarrassed realism — there are many equally good ways to build the number 2 out of sets, and no fact of the matter which one it “really” is, which suggests 2 was never a particular object to begin with. It also softens the applicability problem, because if mathematics describes structures rather than things, then the reason it fits the world is that the world instantiates structures, and structure is exactly what physics measures. You do not need a separate realm; you need reality to have shape.
What remains genuinely open is why the constraints run as tight as they do — why the list of division algebras has three entries rather than none or infinitely many, why the rules we chose for local reasons keep proving to have global consequences that match measurements taken centuries later. Neither camp has a satisfying answer, and both should admit it.
The practical upshot is about the compass. If you think beauty is perception of a pre-existing landscape, you should follow it even when the evidence is silent, because it is data of a kind. If you think it is a compressed record of what has worked, you should follow it too — it is the best prior available — but you should treat a long silence from experiment as a warning that you have walked off the map your intuitions were trained on. The last four decades in fundamental physics have been a large, expensive, still-unresolved experiment in which of those two attitudes is right. Cardano's useless subtlety took 381 years to reach the laboratory. It would be unwise to conclude either that everything beautiful eventually pays off, or that anything that has not paid off yet was never beautiful.
- Cardano, Ars Magna, 1545.
- Bombelli, L'Algebra, 1572.
- Gauss, Theoria residuorum biquadraticorum II, 1831.
- Hamilton, “Theory of Conjugate Functions, or Algebraic Couples”, 1837; quaternions, 1843.
- Frobenius, real division algebras, 1877.
- Hurwitz, normed division algebras, 1898.
- Schrödinger, letter to H. A. Lorentz, 6 June 1926, in Letters on Wave Mechanics, ed. Przibram, 1967.
- Hardy, A Mathematician's Apology, 1940.
- Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, 1960.
- Dirac, “The Evolution of the Physicist's Picture of Nature”, Scientific American, 1963.
- Benacerraf, “What Numbers Could Not Be”, 1965; “Mathematical Truth”, 1973.
- Hamming, “The Unreasonable Effectiveness of Mathematics”, 1980.
- Field, Science Without Numbers, 1980.
- Weinberg, Dreams of a Final Theory, 1992.
- Shapiro, Philosophy of Mathematics: Structure and Ontology, 1997.
- Lakoff & Núñez, Where Mathematics Comes From, 2000.
- Zeki et al., “The experience of mathematical beauty and its neural correlates”, Frontiers in Human Neuroscience, 2014.
- Hossenfelder, Lost in Math, 2018.
- Renou et al., “Quantum theory based on real numbers can be experimentally falsified”, Nature, 2021; tested by Li et al., Phys. Rev. Lett. 128, 040403, and Chen et al., Phys. Rev. Lett. 128, 040402, 2022.
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