Ante rem structuralism

Abstract

This paper is about structuralism, more precisely about the ante rem structuralism, version of which can be treated as a sort of (non-traditional) platonism and which tries to solve problems that platonism is concerned with, i.e. it allegedly solves both a) “the plight of the mathematical Platonist arising from the existence of multiple reductions of the major mathematical theories”, and b) the epistemological problem for platonism due to abstract mathematical entities being causally inert. Ante rem structuralism, more precisely, the version endorsed by Shapiro, is the doctrine according to which mathematics is concerned with abstract structures and the elements of the structures have no properties beside the structural ones; that is, they have no non-structural properties. Mathematical objects (numbers, sets, ...) are just places within structures; e.g. real analysis is about the real number structure and everything we can say about real numbers consists in their “structural” properties. According to Shapiro, there are three ways of grasping structure: abstraction or pattern recognition, linguistic abstraction and implicit definition. Author points out several difficulties with Shapiro's theory, concerned both with ontology and epistemology. These include the relativity of the objects of a theory to the theory itself, and problems concerned with grasping a structure.