The author presents an alternative foundational project related to category and explain its advantages. He also explains how this categorical foundational project relates to structuralism and why it does not qualify as a variety of structuralism.
The paper is organized in ten sections and treats subjects like mathematical structuralism, isomorphisms and invariant forms, structures versus abstract object, collections versus transformations, homomorphisms, structuralist motivations behind category theory, category versus structures, embodiment of mathematical concepts, the category of categories, functorial semantics, sketch theory and internal language and a categorical perspective in and on mathematics.
First, the author briefly discusses mathematical structuralism, its historical origins and its relation to the set theory and category theory. The author explain reasons why Mac Lane, Awodey and some other people believe that category theory provides a support for mathematical structuralism. Then, he provides his critical arguments against this latter view, arguing that the notion of category should be viewed as a generalization of that of structure rather than as a specific kind of structure. Further the author analyzes Lowvere’s paper [F. W. Lawvere, Proc. Conf. Categor. Algebra, La Jolla 1965, 1–20 (1966; Zbl 0192.09702)] on categorical foundations and shows that he begins this paper with a version of structuralist foundations but then proceeds in a different direction. The author concludes with an attempt to outline the new categorical view of mathematics explicitly.