Prime factorisation waited over 2,000 years to secure internet banking
Ancient mathematicians studied prime factors for their own sake. More than two thousand years later the same idea became the backbone of RSA encryption. Ellipses, likewise, sat in Greek geometry for millennia before Kepler found planets travelling along them. Why pure mathematics keeps describing the world is a genuine philosophical puzzle.
The physicist Eugene Wigner named it the unreasonable effectiveness of mathematics. Nineteenth-century geometers invented curved non-Euclidean spaces and extra dimensions with no practical aim, and early in the next century Einstein built relativity on them: special relativity uses a flat four-dimensional spacetime and general relativity a curved one. Sometimes the equations run ahead of the lab. The theories behind the positron and certain baryons produced unexplained solutions, physicists went looking for matching particles, and experiments found them a few years later.
Beneath that success lies an old question: is mathematics invented by human minds or real in its own right? Late in the nineteenth century the foundations themselves wobbled. Consistent geometries in which the parallel postulate fails, a function continuous everywhere yet smooth nowhere, and Georg Cantor's discovery of different sizes of infinity all defied intuition. Russell's paradox showed that the very notion of a set of all sets contradicts itself.
Responses split into schools. Formalism, intuitionism and logicism emerged early in the twentieth century; intuitionistic logic, for instance, rejects the law of excluded middle, and constructive mathematics demands an explicit example for every existence claim. The eventual settlement was mathematical logic, with theories built from a formal language, axioms and inference rules. Zermelo-Fraenkel set theory with the axiom of choice, known as ZFC, now quietly underpins almost every textbook. Category theory, created by Samuel Eilenberg and Saunders Mac Lane, later offered a rival language.
Whether mathematics counts as a science is still debated. Like physics it can be refuted, here by a single counterexample, and discovery often starts with experiment; Gauss said he found theorems through systematic trial. Yet it needs no empirical evidence. When predictions fail, the model is blamed rather than the mathematics, as when Mercury's orbit defied Newton until general relativity supplied a better model.
Source: Philosophy of mathematics
