Are mathematical objects discovered or invented?

Mathematicians themselves are split, and the split has a name. Platonists say mathematical objects are discovered: mathematical Platonism holds that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. It is often claimed to be the view most people have of numbers. The name comes from Plato's Theory of Forms: the everyday world only imperfectly approximates another, unchanging and ultimate reality, accessible to the intellect alone.

Formalists say mathematics is closer to something we make. Formalism holds that statements of mathematics and logic can be considered statements about the consequences of manipulating strings of symbols using established rules. A central idea is that mathematics is much more akin to a game, with no more commitment to an ontology of objects than ludo or chess. It emerged in the late nineteenth and early twentieth centuries as a response to paradoxes in set theory and worries about consistency, and David Hilbert was its most influential advocate.

Each side faces an awkward question. For Platonists: how do we come to know abstract objects? Benacerraf and Field pointed out that knowledge of physical objects rests on perceiving them, but there is no parallel account for knowledge of abstract ones. Gödel's own answer was a special kind of mathematical intuition. For formalists, the main critique is that the mathematical ideas that occupy mathematicians are far removed from string-manipulation games.

Many working mathematicians sit in between. Davis and Hersh suggested that most act as though they are Platonists but, if pressed to defend the position carefully, may retreat to formalism. So there is no agreed answer: it depends on which philosophy you hold, and sometimes on how hard you are pushed.