Physics answers questions of the form how does this behave? What it does not do — what it is not designed to do — is explain why the answer takes the form of an equation at all.

The success is so total that the question is easy to miss. A physicist writes down an equation, turns a mathematical crank, and out comes a number that matches an experiment to twelve significant figures. But why should turning a crank on a symbolic expression tell you anything about an electron in a laboratory? That is not a question physics can answer with physics, because any answer would itself be an equation.

The mathematics arrives first

The pattern that makes this more than a philosopher’s puzzle is one of order. Again and again the mathematics was completed for internal reasons — elegance, curiosity, consistency — with no physical application in view, and the physical world was later found to be using it.

Apollonius sliced cones for the beauty of the problem; eighteen centuries later Kepler found Mars travelling one of his curves. Riemann described curved spaces of arbitrary dimension in 1854 with no application in mind; Einstein needed exactly that machinery in 1915 and found it already waiting. Gell-Mann sorted particles with a nineteenth-century symmetry group, noticed the scheme left one slot empty, specified the missing particle’s mass in advance — and three years later it was found with the properties the symmetry required.

Number theory in living things

Seed and leaf spirals in sunflowers, pine cones and daisies count out consecutive Fibonacci numbers with high reliability. Bees build the hexagonal grid that divides a plane into equal cells with the least wall — a result mathematicians conjectured for two thousand years and only proved in 1999. Metabolic rate scales with body mass to the power of three-quarters across twenty-one orders of magnitude, from bacteria to whales.

Three popular claims should be dropped, because they do not survive measurement: the nautilus shell is not a golden spiral, DNA’s dimensions do not encode φ, and no ancient source connects the Parthenon to the golden ratio. The genuine examples do not need the help.

What explains it

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?Albert Einstein, Geometry and Experience, Prussian Academy of Sciences, 27 January 1921

Eugene Wigner called this the unreasonable effectiveness of mathematics in 1960 and offered no explanation, saying so plainly. The usual replies do not reach far enough. That mathematics is a vast catalogue from which physicists select after the fact cannot account for a particle predicted before it existed on any photograph. That our brains were shaped to model the world explains why we can count animals; it does not explain why a mind tuned for counting animals can derive antimatter from an equation.

The alternative is not that numbers are a second eternal realm alongside God — that would make God not the source of all reality. It is that mathematical truths are necessary thoughts in a necessary mind. If the universe was thought before it was made, then of course it is mathematically describable, and of course a rational creature made in the image of that mind can partly read it.

Nothing here is a proof, and it should not be offered as one. It is an explanatory comparison. You can hold that the fit is a very large coincidence requiring no account. You can pay for it with an infinite multiverse. Or you can conclude that the universe is written in mathematics because it was thought by a mind for which mathematics is not an invention but a native language.