Friday, August 11, 2006

Categories: episode three

Kai von Fintel has just e-mailed, to send a link to the Good Math, Bad Math blog on category theory which is excellent -- a series of mini-essays on concepts of category theory with some very helpful introductory explanations of some of the Big Ideas. Worth checking out, so thanks Kai!

I've now read quite a lot of Steve Awodey's new book. Disappointing: or at least, it doesn't do quite what I was hoping it would do. Awodey's two papers in Philosophia Mathematica were among the pieces that got me interested in category theory in the first place (see his 'Structure in mathematics and logic: a categorical perspective', 1996, and his reply to Hellman, 2004). So I suppose I was hoping for a book that had more of the discursive, explanatory, commentary that Awodey is good at. But there's very little of that. And although Awodey says in the preface that, if Mac Lane's book is for mathematicians, his is for 'everyone else', in fact Category Theory is still pretty well orientated to maths students (for example, there is a telegraphic proof sketch of Cayley's Theorem that every group is isomorphic to a permutation group by pp. 11-12!). Of course, there are good things: for just one example it helped me understand better the general idea of limits and colimits. But I wouldn't really recommend this book to the non-mathematician.

So over the next week or two, I think it's Goldblatt's lucid Topoi, and Lawvere and Rosebrugh's Sets for Mathematics for me.

1 comment:

Andy Fugard said...

Thanks for sharing the Structure in mathematics and logic: a categorical perspective reference. Looks interesting.

One of the things I'm struggling with at the moment is how make sense of the various notions of model/structure/interpretation and how they relate to inference systems. I feel I don't have the right language to talk about these things. For instance how best can the relationship between evaluating truth of a proposition, P, in a (finite first-order) model M, and what can be proved in say natural deduction, be described. Presumably it's easy to represent the finite model as a set of premises and then to turn the handle. There is some structure in common between these two representations of the solution to the problem: "is this sentence true in this possible world?" The structures diverge when in natural deduction you prove a general theorem and then instantiate the universally quantified variables with individuals.

For me, talking about syntax and semantics blurs the common structure. There is clearly syntax in the models. There is clearly semantics in dealing with the inferences.

Can category theory deal with these issues?