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 receiptswell, 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
