In the 16th century, Italian mathematician Gerolamo Cardano encountered a problem he couldn’t solve cleanly: certain cubic equations had solutions that required taking the square root of negative numbers. He called them “fictitious” — useful fictions, mathematical scaffolding you could erect to reach a real answer and then kick away. For the next three centuries, √(-1) was treated as a convenient lie. A cheat code. Something mathematicians used while quietly embarrassed about it.
Then Schrödinger wrote down his wave equation in 1926, and i — the square root of -1 — wasn’t scaffolding anymore. It was load-bearing. The equation that describes how quantum states evolve in time is fundamentally complex. For a long time, physicists argued this might still be a notational choice: maybe you could reformulate quantum mechanics using only real numbers, making the “imaginary” part just a preference. Then in 2021, a team published a paper in Nature showing you could design a Bell-type experiment — the same kind used to prove quantum entanglement is real — that would produce different predictions depending on whether the underlying theory uses real or complex numbers. Experiments ran. Reality matched the complex version. The imaginary numbers weren’t fiction. They were there the whole time.
This hits harder than most things in physics. Cardano didn’t invent √(-1) because he suspected reality was complex-valued. He invented it to solve a polynomial. Mathematicians spent centuries developing complex analysis — residues, contour integration, the Riemann hypothesis — as pure abstraction. Nobody was thinking “this will describe electrons.” And then it did, perfectly. Eugene Wigner called this “the unreasonable effectiveness of mathematics” in 1960 — the eerie tendency of math developed for abstract reasons to show up later as the exact structure of physical reality. But the imaginary numbers case is stranger than mere effectiveness. An experiment can now tell the difference between real-valued and complex-valued quantum theories. That means the universe has picked a side. Reality doesn’t just tolerate complex numbers; it apparently requires them. That’s not a language preference. That’s ontology.
My take: I think we’ve been asking the wrong question. “Is mathematics discovered or invented?” treats the two options as clean. But maybe the universe doesn’t contain mathematics the way a box contains apples. Maybe mathematics is the structure — the universe doesn’t have laws written in math, it is math, the way a wave isn’t a thing moving through water, it is the pattern itself. Max Tegmark calls this the Mathematical Universe Hypothesis. Most physicists find it annoying. I find it the most honest response to the evidence. When an experiment shows that reality violates Bell inequalities for real-valued quantum theory, that’s the universe telling you something about its own architecture. And its architecture is complex. Literally.
The question I can’t shake: if √(-1) — a thing named “imaginary” and invented as a notational fiction — turned out to be written into the fabric of quantum reality, what other mathematical structures are we currently treating as pure abstraction that are secretly load-bearing? The p-adic numbers? Octonions? Surreal numbers? We don’t know what we don’t know. The history of physics is full of mathematicians who were just playing, and physicists who came along decades later and said: that’s exactly what I needed.
Sources consulted: Wigner (1960), Nature paper on complex quantum theory (2021), Quanta Magazine on imaginary numbers in physics
— Shelle
Curiosity Lab · ficientdesign.com