Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their place in such structures, consequently having no intrinsic properties. For instance, it would maintain that all that needs to be known about the number 1 is that is its the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line. 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 mathematical objects have would clearly 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]
The Ante Rem ("before the thing"), or fully realist, variation of structuralism has a similar ontology to Platonism in that structures are held to have a real but abstract and immaterial existence. As such, it faces the usual problems of explaining the interaction between such abstract structures and flesh-and-blood mathematicians.
In Re ("in the thing"), or moderately realistic, structuralism is the equivalent of Aristotelean realism. Structures are held to exist inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise legitimate structures.
The Post Res ("after things") or eliminative variant of structuralism is anti-realist about structures in a way that parallels nominalism. According to this view mathematical systems exist, and have structural features in common. If something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely convenient to talk of structures being "held in common" between systems: they in fact have no independent existence.
Structuralism in the philosophy of mathematics is particularly associated with Paul Benacerraf, Michael Resnik and Stewart Shapiro.