Individuation of objects – a problem for structuralism?

Jessica Carter

pp. 291-307

in: Synthese 143 (3), 2005.

Abstract

This paper identifies two aspects of the structuralist position of S. Shapiro which are in conflict with the actual practice of mathematics. The first problem follows from Shapiro’s identification of isomorphic structures. Here I consider the so called K-group, as defined by A. Grothendieck in algebraic geometry, and a group which is isomorphic to the K-group, and I argue that these are not equal. The second problem concerns Shapiro’s claim that it is not possible to identify objects in a structure except through the relations and functions that are defined on the structure in which the object has a place. I argue that, in the case of the definition of the so called direct image of a function, it is possible to individuate objects in structures.