Friday, August 04, 2006

Categories: episode one

I said I'd post occasional progress reports on getting to grips with Category Theory, to pass on recommendations about what I've found helpful, etc. (Perhaps I should explain that -- when trying to get into a new chunk of maths or math. logic -- I find it works best to dive in and read a number of books quite quickly, skipping and skimming when the mood takes me, rather than plod very carefully through one key text. Also, though I know a few bits and pieces, I thought I should start over again from the very beginning.)

Two frequent recommendations for entry points are Lawvere and Schanuel's Conceptual Mathematics (CUP, 1997), and Lawvere and Rosebrugh's Sets for Mathematicians (CUP, 2003). But I can only say, give the first book a miss. It's not very good: the authors seem to have no clear idea about their intended audience, so they veer between the irritatingly condescending/extremely laboured on the one hand, and sudden jumps in difficulty/sheer opaqueness on the other. For just one example, it is very difficult to believe that someone who needs a noddy explanation of why proving if p, q establishes if not-q, not-p is going to make anything at all of the sudden excursus on Gödel and Tarski at p. 307. But worse, you can get to the end of the book with no clear idea about what it is supposed to have achieved, or why it might matter. The second book is so far proving also a bit uneven but much better.

I'm dipping into various expository/philosophical articles as I go along. One I've just come across which I found prokoving and illuminating is Barry Mazur's 'When is one thing equal to some other thing?'. Recommended.

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