Structuralism in the philosophy of mathematics
Authors: Shapiro Stewart 
Published: September 2018

Define a system to be a collection of objects with certain relations. A basketball defence is a system of people under various positioning and defence-role relations; an extended family is a system of people under blood and marital relationships; and a chess configuration is a system of pieces under certain spatial and ‘possible move’ relations. A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations.
The slogan of structuralism is that mathematics is the science of structure. The idea is that the subject matter of a given branch of mathematics is a structure, or a kind of structure. Typically, the structures studied in mathematics are free-standing, in the sense that the various positions or places in the structure are characterized only with respect to each other, and anything at all can play the various roles, and stand in the various relations. Define a ‘natural number system’ to be a countably infinite collection of objects with a designated initial object, a one-to-one successor relation that satisfies the principle of mathematical induction and the other axioms of arithmetic. Examples of natural number systems are the Arabic numerals in their natural order, an infinite sequence of distinct moments of time, and the even natural numbers. According to structuralism, arithmetic is about the form or structure common to natural number systems. So a natural number is something like an office in an organization or a place in a pattern. Similarly, real analysis is about the real number structure, the form common to complete ordered fields.
Structuralism has certain affinities with functionalist views in, say, philosophy of mind. A functional definition is, in effect, a structural one. The difference, of course, is that mathematical structures are more abstract, and free-standing.
It is not possible to articulate the structuralist view much further without encountering issues that are contentious, even among those who call themselves structuralists. In some ways, a structure is like a traditional Form or universal. The main difference is that a universal typically applies to or holds of individual objects, or ‘particulars’, while a structure applies to, holds of or is exemplified by, systems – collections of objects under certain relations. Despite this important difference, the usual range of philosophical views concerning universals are available for structures. One can be a Platonist, or ante rem realist, holding that structures exist objectively, independently of, and metaphysically prior to, any systems that exemplify them. Or one can be an Aristotelian, in re realist holding that structures exist only in the systems that exemplify them. Or one can be a nominalist, holding that all talk of structure is to be paraphrased away, in a manner that does not commit one to the existence of structures. This is sometimes called ‘eliminative structuralism’, a structuralism without structures. Some eliminative structuralists think of mathematical assertions as talking about all systems of a certain type; others take such assertions to be talking about all possible systems of a certain type, a sort of modal view. On most of these nominalistic views, mathematical assertions end up objectively true or false, with their usual truth-values.
A structuralist’s views on other philosophical issues, concerning epistemology, semantics, methodology, applicability and the like, depend on the version of structuralism in question. The ante rem realist, for example, has a straightforward account of reference and semantics: the variables of a branch of mathematics range over the places in an ante rem structure; each singular term denotes one such place, etc. But the ante rem realist must account for how one obtains knowledge of structures, so construed, and for how statements about ante rem structures play a role in scientific theories of the physical world. The eliminative structuralist must account for how the reconstrued statements are known, how they figure in science, etc.; and the modal structuralist must articulate the nature of the invoked modality, and how it is known.