Structuralism (philosophy of mathematics)

Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the number 1 is exhaustively defined by being the successor of 0 in the structure of the theory of natural numbers. By generalization of this example, any natural number is defined by its respective place in that theory. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.

Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what kind of entity a mathematical object is, not to what kind of existence mathematical objects or structures have (not, in other words, to their ontology). The kind of existence that mathematical objects have would be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard.^[1]

Structuralism in the philosophy of mathematics is particularly associated with Paul Benacerraf, Geoffrey Hellman, Michael Resnik, Stewart Shapiro and James Franklin.