Foundations: structures, sets, and categories

Recent years have seen a wealth of discussion on the topic of the foundations of mathematics, and the extent to which category theory, set theory, or some other framework serves, or can serve, as a foundation, or the foundation of some, most, or all of mathematics. Of course, adjudications of these matters depend on what, exactly, a foundation is, and what it is for, and it depends on what mathematics is. It is like a game of Jeopardy. We are given some answers: set theory, category theory, abstraction principles, etc., and we have to figure out what the questions are. Most of the participants in this debate are at least fairly clear about what their questions are, but it seems that the participants do not have the same questions in mind. And some of the questions have disputable presuppositions concerning the nature of mathematics. My purpose here is to survey some of the terrain. The goal is to clarify the discussion, and perhaps to advance parts of it, without plumping for one or the other view.