Tim May
Tue, 02 Apr 2002 19:41:43 -0800
On Tuesday, April 2, 2002, at 02:58 PM, Sampo Syreeni wrote: > 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.
Yes, I believe there's a lot more. By the way, even though category theory may be about as foundational as set theory (a la Zermelo-Frankel axiomatization), it looks to be a _lot_ more useful in other areas. We all know what sets are, and use them every day, and use things like Venn diagrams more than almost any other tool (at least I do), but the axiomatic foundations are seldom used. The Axiom of Choice? > >> I won't try to explain what categories and toposes are here in this >> e-mail message. > > Thank god. But isn't it "topoi"? I was drinking coffee out of one of my thermoi and realized you were...of that camp. As I said, I'm also using Goldblatt's "Topoi." (But it's out of print, and unpurchasable, so far, so I use UCSC's copy.) Note that McClarty's book says "Toposes." One of these authors, maybe McClarty, maybe Johnstone, points out that plurals of words which were never Latin to begin with, like Thermos bottle, may be "thermoses," not "thermoi." I find "toposes" sounds better than "topoi." It's only topoi-logical, after all. >> >> 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 they were able to at least eliminate the "index gymnastics" of manipulating indices in, for example, the Riemann tensor. R-sub-ijk and all that rot. My copy of Sokolnikoff and Redheffer could be safely put away. > >> 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. Look at some of the "computer science" references, as opposed ot the "theory of math" references. Barr and Wells, or Pierce, for example. They point out that people are successfully using category theory terminology as a means of clarifying the unclear, not as a means of pushing the frontiers of math. The value of looking at functors (natural transformations between categories) as opposed to ordinary sets and functions is the ability to draw conclusions from other areas of math, it seems to me. > >> * "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? Axelrod, "The Evolution of Cooperation." >> * 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. I'm not saying chaos isn't real, just that it's not turning out to be very surprising or useful. >> * 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. I thought so in 1986-87, when I was involved in their neural net project. Got the expensive "PDP" 2-volume set. And Terry Sejnowski was a TA/instructor in my Relativity class many years back. Ho-hum. I'm glad _someone's_ still interested in this stuff, cuz I sure ain't. >> 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... > Personally, I don't think the "foundational" stuff is really even interesting. Chaitin explained Godel's Theorem in readily-understandable information-theoretic terms. As to whether categories and toposes are interesting and useful, aside from foundational issues, my belief is that they are. --Tim May "A complex system that works is invariably found to have evolved from a simple system that worked ...A complex system designed from scratch never works and cannot be patched up to make it work. You have to start over, beginning with a working simple system." -- Grady Booch