CAT 2023 Slot 3 VARC: Discovered, invented, or both?

Ask a mathematician whether mathematics is discovered or invented, and you will typically get an answer delivered with more conviction than the question's actual difficulty warrants. The "discovery" camp points to a striking fact: mathematical structures, once formulated, display properties their inventors never anticipated and could not have simply stipulated into existence. Non-Euclidean geometry, worked out in the nineteenth century as an abstract curiosity with no evident physical referent, turned out decades later to be exactly the mathematics general relativity required to describe curved spacetime. Prime numbers behave in ways number theorists did not design and are still, centuries on, unable to fully predict. If mathematics were simply a human invention, the argument runs, it would be strange that inventions keep surprising their inventors — a novelist's characters do not usually reveal unplanned truths about the number of chromosomes in an unrelated species, but mathematical objects seem to do the analogous thing routinely.

The "invention" camp is not without its own strong cards. Mathematical notation, definitions and even which structures get studied at all are plainly human choices, shaped by culture and historical accident as much as by logical necessity. The number system with a zero and place value, so useful that its absence retarded Roman arithmetic for centuries, was a specific technological choice, not a metaphysical inevitability, first fully realized in India and transmitted onward. Whole branches of mathematics have been developed and then abandoned as fashions of interest shifted; whole other branches remain uninvented, not because they are false but because no one has yet found a reason to formulate them. Set theory's axioms were selected, after considerable debate, to avoid paradoxes discovered within earlier, differently chosen axioms — a process that looks a great deal like engineering a tool to specification rather than uncovering a fact that pre-existed the engineering.

A useful way through this apparent impasse is to notice that the dichotomy conflates two separate questions that need not have the same answer: are mathematical objects discovered, and is mathematical truth discovered? One can hold, consistently, that the definitions and notations are invented — a genuine human artifact, contingent on history — while insisting that once a structure is defined, its logical consequences are discovered, in the ordinary sense that they were true (or provably followable from the axioms) before anyone proved them, and remain so whether or not anyone ever does. On this view, a mathematician resembles less a novelist inventing a plot from nothing and more a cartographer: the decision of which territory to map, and in what projection, is a human and somewhat arbitrary choice; but once the territory and the map's rules are fixed, the coastline the cartographer finds is not up to her. She did not invent the coastline by choosing to survey it.

This hybrid picture does not resolve every philosophical difficulty attached to the question — it leaves open, for instance, why a universe should be describable at all by structures our species happens to find describable, and why certain "invented" formal systems keep turning out, unbidden, to model physical reality with an accuracy their inventors never sought. But it does explain, more economically than either pure position alone, why mathematicians can feel, in the same afternoon, the freedom of an author choosing where to begin and the discipline of an explorer who cannot simply decide the terrain will be otherwise.