Editing Philosophy of mathematics (section)

===Structuralism===
{{Main|Mathematical structuralism}}
[[Mathematical structuralism|Structuralism]] is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their ''places'' in such structures, consequently having no [[intrinsic and extrinsic properties (philosophy)|intrinsic properties]]. For instance, it would maintain that all that needs to be known about the number 1 is that it is 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 [[line (geometry)|lines]] and [[plane (geometry)|planes]] in geometry, or elements and operations in [[abstract algebra]].

Structuralism is an [[epistemologically]] [[realism (philosophy)|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.<ref>{{cite book |last=Brown |first=James |title=Philosophy of Mathematics |publisher=Routledge |location=New York |year=2008 |isbn=978-0-415-96047-2}}</ref>

The ''ante rem'' structuralism ("before the thing") has a similar ontology to [[Mathematical Platonism|Platonism]].  Structures are held to have a real but abstract and immaterial existence. As such, it faces the standard epistemological problem of explaining the interaction between such abstract structures and flesh-and-blood mathematicians {{Crossreference|(see [[Benacerraf's identification problem]])}}.

The ''in re'' structuralism ("in the thing") is the equivalent of [[#Aristotelian realism|Aristotelian 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 rem'' structuralism ("after the thing") is [[anti-realism|anti-realist]] about structures in a way that parallels [[nominalism]]. Like nominalism, the ''post rem'' approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure.  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 instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence.