Thursday, June 30, 2005

Day 57

[reading: Saunders Mac Lane, "Categories for the Working Mathematician"]

Still waiting for any signs of payoff in my explorations of category theory. On the surface, it sounded like a very interesting area to explore—a generalization of all of the concepts of morphism involved in mathematical structures. I like that kind of approach to mathematics: start with some interesting phenomenon (say, permutations of sets), but consider it in a more general setting so that the original problem is merely a special case (say: group theory). Then vary things further, sometimes by generalizing (say: semigroups), sometimes by specializing (say: rings, fields, topological groups), sometimes by changing the focus slightly (say: group actions, representations, morphisms).

I'm also old-fashioned enough to like a reasonably axiomatic approach to these things, so you know that if you're having a blonde moment you can always fall back to the basics to understand what's going on, step by step. Regardless of what Gödel might say, this is still a powerful and reliable way of going about things.

Combining the two, I found model theory to be one of my favourite bits of maths—exploring the whole class of axiomatized algebraic structures in one swell foop, to see what can be discovered in general about the things that satisfy those sets of axioms. Unlike category theory (at least so far), there are some genuinely surprising things that turn up quite early on in model theory—the Löwenheim-Skolem theorems, which show that you can't distinguish between different sizes of infinity in first-order logic (the downward version says any infinite model has a countably infinite submodel; the upward version says that any infinite model can be extended to a model of arbitrary size). This is unexpected (to me at least, although expectations involving transfinite numbers are often a little slippery), and leads to an almost-paradox. It also provides a rationale for why nonstandard analysis is valid.

Anyway, I'm still waiting to see anything equivalently startling show up in category theory, and to be honest my patience is running out.

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