Structuralism, Mathematical

Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general conditions—typically spelled out in axioms defining the structure or structures of interest—characteristic of the branch of mathematics in question. Thus, in the basic case of arithmetic, the famous " axioms " of Richard Dedekind (taken over by Giuseppe Peano, as he acknowledged) were conditions in a definition of a " simply infinite system " , with an initial item, each item having a unique next one, no two with the same next one, and all items finitely many steps from the initial one. (The latter condition is guaranteed by the axiom of mathematical induction.) All such systems are structurally identical, and, in a sense to be made more precise, the shared structure is what mathematics investigates. (In other cases, multiple structures are allowed, as in abstract algebra with its many groups, rings, fields, and so forth.) This structuralist view of arithmetic thus contrasts with the absolutist view, associated with Gottlob Frege and Bertrand Russell, that natural numbers must in fact be certain definite objects, namely classes of equinumerous concepts or classes. Historically, structuralism can be traced to nineteenth century developments, including the rise of the axiomatic method and of non-Euclidean geometries leading to the recognition of multiple abstract spaces independent of physical space and of spatial intuition. (David Hilbert, whose work in the foundations of geometry (1899) was