Discovered or invented is the wrong question

Ask whether mathematics was discovered or invented and you will start a pleasant argument that cannot end. Both answers survive every observation anyone could ever make — which is a polite way of saying that neither one is doing any work. The productive question is standing just behind them.

The case for discovery — #platonism, in its classical form — is the stubbornness of the subject. Nobody voted on the distribution of the primes; pi was not designed; theorems push back against their authors' wishes with the resistance of granite. The case for invention is the visible human fingerprints. Axioms get chosen, definitions get revised, and non-Euclidean geometry sat unused in the toolbox for decades before anyone found a lock it opened. Each side treats its favourite evidence as decisive. Neither side can say what observation would prove it wrong.

That is the tell. A world where mathematical objects exist in some causally inert realm and a world where mathematics is an elaborate human construction are observationally identical: same theorems, same proofs, same physics built on top. When two metaphysical pictures predict exactly the same everything, the dispute between them is not a research programme. It is a matter of temperament — which is why it has run for twenty-five centuries without producing a casualty on either side.¹

The question that has consequences

In 1960 @Wigner put a finger on the fact the old dichotomy never touches: whichever answer you prefer, mathematics works in physics far better than it has any right to [1]. Concepts built with no empirical intent keep turning out to be the exact language of later physical theories, and there is no obvious reason why. He called this the #unreasonable-effectiveness of mathematics, and he did not mean the phrase as a compliment. He meant it as an open problem.²

The record really is strange. Riemann worked out the geometry of curved higher-dimensional spaces in 1854 as pure structure, with no application in sight; sixty years later it was exactly what @Einstein needed for general relativity, available off the shelf. Group representation theory was an algebraist's private garden until it turned out to classify elementary particles. "Invented" makes these anticipations look like miracles of coincidence. "Discovered" merely relocates the miracle — why should a realm of objects that cannot touch anything legislate for electrons?

The third path

There is a reading that dissolves the dichotomy rather than picking a side. #structuralism holds that mathematics is not about objects at all but about structures — patterns of relations that any collection of things whatsoever can instantiate. @Benacerraf exposed the crack in 1965: the number three cannot be any particular set-theoretic object, because there are several equally good candidates and no fact of the matter that chooses among them [2]. What is fixed is the position — third in the sequence. Numbers are places in a pattern, not things.

This splits the old question along a natural seam. The patterns are discovered, in the plain sense that anything instantiating them must obey them: take any collection you like, and there is a fact about its symmetries whether or not anyone looks. The presentations are invented: the notation, the axiom systems, the choice of which structures to name and study.³ And the effectiveness stops looking supernatural, because physics is in exactly the same business. A physical theory is a claim that the world instantiates a particular structure. Mathematics is the systematic inventory of the structures there are to instantiate. That the inventory keeps containing what physics needs is not a miracle. It is what an inventory is for.