Work on the Gentle Intro to Category Theory has been going slowly for a couple of months. I’d got to the point where I needed to backtrack and think through what I wanted to say, in a couple of preliminary chapters, about *sets* and then about *structures,* before going on to talk about category theory as (inter alia) a framework for talking about how structures-of-structures hang together.

The chapter on sets, in particular, has been causing me grief. This isn’t going to be one of those merely routine chapters about basic set theoretic notions and notations — I’m going to take it that anyone tackling even an intro on category theory is unlikely to need yet another run through those basics. Nor is this going to be a discussion of the fancy issues about big universes of sets that might or might not be needed for really big categories (after all, it’s only after we know quite a bit of category theory that it will become clear what the genuine issues are and how much of applicable category theory they affect).

No, what I want to write about are basic things that turn out to be relevant quite soon when we start talking about categories, like how much common-or-garden maths-talk seemingly about sets is really plural talk in thin disguise (and why plural talk should be taken at face value), and relatedly about how much set talk is about virtual classes in Quine’s sense. And then say something about what we get from having a more substantial set theory, and just how much substantial set theory is needed for ‘ordinary’ (non-set-theoretic) maths, and why there is no unique theory which delivers the goods. Fundamental but not-so-simple things like that, familiar to logicians but not necessary to mathematicians (who need to be convinced why, here at least, the should care)! I think I’m beginning to see how to shape such a chapter to say what needs to be said without getting too tendentious, after some fun reading around for more inspiration. And since I couldn’t see that a few weeks ago, there must have been progress, even if at a snail’s pace.

I’ll put the draft chapter online for comments when it’s done. But some very nice forthcoming events here mean I have to put further work on hold for a bit. So don’t hold your breath.

Why is it necessary to talk down set theory’s role in mathematics (“really” mathematicians are talking about plurals or Quinean virtual classes) in order to introduce category theory? You say it’s “relevant quite soon when we start talking about categories”, but Harold Simmons manages to avoid doing it in his

An Introduction to Category Theory. Awodey’sCategory Theoryseems to avoid it too. Even Leinster’sBasic Category Theoryleaves his version of it until Chapter 3.There are fascinating philosophical and foundational questions about the ontology of ordinary maths-talk, and about how much set theory is needed, but it could make more sense to treat them as a separate, though related, project. In any case, I don’t think it’s all that clear what mathematics are really talking about. For instance, I’ve also seen it claimed that they’re using “some sort of complicated (and informal) type theory.” (There seems to be a tendency for people to think that something that appeals to them is what mathematicians are really using.)

It might also be interesting to consider why some category theorists are so inclined to declare that certain questions are “meaningless”, and whether they have been influenced by whatever cultural and philosophical currents led to positivism, behaviourism, and other forms of verificationism in the past. It’s even quite common to see advocates of category theory say that one of its advantages over set theory is that it’s about how things “behave”.