On Thursday, July 18, 2002, at 10:19 AM, Wei Dai wrote:
> I've been reading _Conceptual Mathematics_ but so far have not seen many
> connections with topics I'm most interested in learning right now
> (logic,
> recursion theory, decision theory). Perhaps category theory is more
> relevant in physics, or I should move on to topos theory.
Several points:
1. Perhaps you need to give it more time, as reading Lawvere and
Schanuel is not light reading. (It's at about the same level as Raymond
Smullyan's recent books on logic are. Or perhaps to Hofstadter's "Godel,
Escher, Bach." Not dense texts, but also not light reading.)
2. Or perhaps you can do what I do, which is to read several related
books at the same time, the various books illuminating the topic in
various ways.
3. As for fundamental relevance, I see it as important for mathematics
and computer science, as well as for certain (mostly frontier) issues in
physics. Not everyone does, even in math. You mention logic and
recursion theory, so you might want to do some Web searches on some of
these names: Martin Hyland (the effective topos, Eff), Joyal, Bell (not
the QM Bell), in connection with topos theory. The Paul Taylor book I
mentioned a while back has much material on logic, computability, with
lots of connections shown with category and topos theory.
(But, "moving on" to topos theory without knowing some category theory
is no more advisable than "moving on" to quantum theory without knowing
a fair amount of classical physics would be. Nothing wrong with dabbling
in several areas--QM, GR, classical--at the same time, using the "MBI"
(Many Books Interpretation).)
4. I find category and topos theory to be refreshing, stimulating, and
foundational. The Lawvere and Schanuel is a leisurely,
lots-of-exposition-and-motivational treatment which requires almost no
prerequisites, which is why both Bruno and myself recommend it. It is
definitely _not_ a "Why category theory is really important!" sort of
book.
5. Your mileage may vary. It may never be as interesting to you as it is
to me. That's a feature, not a bug. Or you may come back to it after a
while.
(Looking through my copy of Pearl's "Causality," simple graph theory
would be more immediately useful as background prep.)
> Topos theory seems to be motivated by intuitionistic logic, which is
> considered the logical basis of constructive mathematics (according to
> http://plato.stanford.edu/entries/logic-intuitionistic/). Does that
> mean I
> should learn something about intuitionistic logic and constructivism
> first
> before trying to tackle topos theory?
Yes, I think so. Doesn't mean you need to read an entire book on it,
though.
Smullyan's "Forever Undecided," 1987, is a good survey of parts of
modern logic. Not much on intuitionistic logc, but still good
preparation. A good chapter on possible worlds and Kripke's
contributions. (Which has strong connections to topos theory, as I've
mentioned.)
--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks
Received on Fri Jul 19 2002 - 10:14:37 PDT