Categories as sets: problems and solutions

Abstract

The possibilities and problems attendant on the construction of a set-theoretical foundation for CT and the relevance of such foundations have been subject to extensive debates for many years. In this chapter, I will consider the historical development of these debates. So far, a detailed discussion of this subject matter is absent from the historical writing on CT; I do not know whether this lack of interest is but one more expression of the profound indifference exhibited by most mainstream mathematicians towards set-theoretical foundations of mathematics in general and of category theory in particular, or whether it indicates merely that the problem is an open one and hence in a trivial sense does not yet admit a conclusive historical treatment. Anyway, in a historical and philosophical analysis of a theory, one is not supposed to parrot uncritically the prejudices of the workers in the field. To the contrary, such prejudices are to be analyzed with priority; questions like: What are the motives underlying them? What basic convictions of the people active in the field do they reveal? What have been their consequences for the development of the theory and of the debates concerning it? The answers to these questions are most important both for an understanding of the theory’s history and for its philosophical interpretation.