On Tue, 2 Apr 2002, Tim May wrote: >I've been having a lot of fun reading up on "category theory," a >relatively new branch of math that offers a unified language for talking >about (and proving theorems about) the transformations between objects.
Baez convinced you, no? He seems to be a category freak. >I'll say a few words on why this is more than just the "generalized >abstract nonsense" that some wags have dubbed category theory as. It seemed like that at first, of course. However, some fairly deep observations have been made in the area, concerning the basic assumptions underlying math. Namely, the prevalence of sets, functions, first order logic and the like. There might just be something to categories, after all. >I won't try to explain what categories and toposes are here in this >e-mail message. Thank god. But isn't it "topoi"? >* http://math.ucr.edu/home/baez/ John Baez (a cousin of Joan) is >a mathematical physicist with wideranging interests...his site has all >sorts of good stuff I think Week's Finds are the ones best suited for nonlinear-agoric-geodesic, cypherpunkly consumption. >"Category theory is about how mathematicians draw diagrams on the >blackboard." A waggish, snide comment, but true. It becomes non-trivial once one starts to do true 2D algebra. In the absence of categories, one would never think there is anything "wrong" or "limiting" about the way algebra is now done. >Relativity was exciting--I took James Hartle's class using a preprint >edition of Misner, Thorne, and Wheeler's massive tome, "Gravitation." The Big Black Book. Tried it, didn't like it much. Somehow they manage to make the subject totally inaccessible to anyone used to the standard concept of tensor spaces. I mean, if they have a basis, why not simply talk about multilinear mappings? (They do, when talking about tangent spaces. I'm just wondering why tensors are needed at all.) >But the recent work on supergravity, loop spaces, knot theory, strings, >M-branes, etc., has re-ignited some interest...at the "amateur" level, >needless to say. It's well beyond a pipe dream, so why not? >The fact that we use "Alice and Bob" diagrams, with "Eve" and "Vinnie >the Verifier" and so on, with arrows showing the flow of signatures, or >digital money, or receipts....well, this is a hint that the >category-theoretic point of view may be extremely useful. (At other >levels, it's number theory...the stuff about Euler's totient function >and primes and all that. But at another level it's about commutative and >transitive mappings, and about _diagrams_.) I don't see the connection. Category theory mostly seems to be about questioning the way we represent and visualize mathematics. There, it is beginning to have some real influence. However, what you're describing above is well below that, in the realm of ordinary sets and functions. I seem to think categories have very little to do with such things. >* "game theory." We all know that most human and complex system >interactions have strong game-theoretic aspects. Cooperation, defection, >Prisoner's Dilemma, Axelrod, etc. But thinking that "all crypto is >basically game theory" has not been fruitful, so far. Axelrod? I just started reading up on basic game theory and the theory of oligopoly (Cournot, Nash, price vs. quantity selection, the works), but haven't bumped into that name, yet. What gives? >* the whole ball of wax that is complexity, fractals, chaos, >self-organized criticality, artificial life, etc. Tres trendy since >around 1985. But not terribly useful, so far. No? I seem to recall a couple of articles on how actual markets behave chaotically, based on time-series data. Such a conclusion is quite a feat, I'd say, and there's bound to be more out there. Besides, I'm not quite sure chaotics hasn't had an impact on e.g. cipher design -- current cipher design seems to concentrate a lot on diffusion, for instance. What is diffusion but a discretized version of a Lyapunov exponent-like characterization of chaotic blow-up? >* AI. 'Nuff said. We all know intelligence is real, and important, but >the results have not yet lived up to expectations. Maybe someday. The connectionist stuff seems interesting, here. So does silicon learning via genetic programming. >* object-oriented systems. In my view, this one _has_ basically lived up >to its billing, largely because it works for building more complex >systems (and is arguably how Mankind has usually built complex systems >like bridges and skyscrapers and chips). But some of the bolder claims >about "reusable software" and "software ICs" have yet to be realized. Of course. But how is this interesting? I view objects mainly as a logical extension of the analytic method: to-undestand-break-it-down. Not nearly as interesting as blind learning algos or the like. >* agoric, market-oriented approaches. I'm still very hopeful about this >one. It seems people never completely understand the power of emergent systems. If even the market is too much, how are people ever to get a hold of anything else? >A topos is a category imbued with additional properties. As always, in math. The trouble is, it's still too general to get us any truly applicable results. The curse of abstraction, you might call that. I view categories and topoi as something to challenge our notions about the basis on which we build mathematics, not an actual, useful tool. But what survives the challenge... Sampo Syreeni, aka decoy - mailto:[EMAIL PROTECTED], tel:+358-50-5756111 student/math+cs/helsinki university, http://www.iki.fi/~decoy/front openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
