Towards a Categorical Foundation of Mathematics

This paper is an expanded version of a talk with the same title given at the Logic Colloquium in Haifa in August, 1995. A year later, at the time of writing these lines, one of the papers, [M2], on which the talk was based has appeared in print. The present paper is, primarily, a somewhat philosophical introduction to the research monograph [M3], the other source of the talk. For understanding the story, it is necessary to keep the contents of [M2] in view; some of [M2] will be recapitulated here. The paper is written for readers with at least a slight familiarity with Category Theory. At certain points, we make remarks referring to more advanced notions; however, the understanding of these is not essential for the main part of the paper.