Mathematical Object

A mathematical object is any abstract entity that is treated as a subject of mathematical study or reasoning, such as a number, a set, a function, a space, or a structure such as a group, without necessarily corresponding to anything in the physical world. Whether mathematical objects genuinely exist independently of human thought, in the sense argued by the position known as mathematical Platonism, or are instead human constructions or useful fictions, as rival positions such as formalism and fictionalism hold, is a long-standing question in the philosophy of mathematics rather than a question mathematics itself can settle. Structuralism, an influential twentieth century position, argues that what matters about a mathematical object is not any intrinsic nature it might have but only the place it occupies within a larger structure of relations, so that, for instance, the number two is fully characterized by its relations to the other natural numbers rather than by any independent identity. The term is used informally across every branch of mathematics as a general way of referring to whatever kind of thing a given theory studies, from the simplest objects such as individual numbers to highly elaborate ones such as entire categories or infinite-dimensional spaces.