On Sunday, December 1, 2002, at 10:00 AM, Osher Doctorow wrote:
>> From Osher Doctorow osher.domain.name.hidden Sunday Dec. 1, 2002 0958
>
> I agree again with Tim May.
>
> I also think that category theory and topos theory at least in its
> definition as a branch of category theory are too restrictive, largely
> because they are more abstract than concrete-oriented in their
> underlying
> formulations.
As I hope I had made clear in some of my earlier posts on this, mostly
this past summer, I'm not making any grandiose claims for category
theory and topos theory as being the sine qua non for understanding the
nature of reality. Rather, they are things I heard about a decade or so
ago and didn't look into at the time; now that I have, I am finding
them fascinating. Some engineering/programming efforts already make
good use of the notions [see next paragraph] and some quantum
cosmologists believe topos theory is the best framework for "partial
truths."
The lambda calculus is identical in form to cartesian closed
categories, program refinement forms a Heyting lattice and algebra,
much work on the fundamentals of computation by Dana Scott, Solovay,
Martin Hyland, and others is centered around this area, etc.
As is so often the case, the mathematical physicist John Baez has done
a fine job of introducing the subject to physicists and providing some
motivation. Here's one of his articles:
http://math.ucr.edu/home/baez/topos.html
As for the mix of concrete and abstract, I studied plenty of abstract
stuff on set theory, topology, analysis, and of course in physics. But
I also did a lot of applied physics and engineering during my career at
Intel. Believe me, I would have been in deep trouble had I proposed
that we look into applications of Tychonoff's Theorem when we having
problems with our dynamic RAMs and CCDs losing occasional stored bits
in what were called "soft errors."
But knowing a lot of abstractions helped me in countless ways. And now
that I am free to pursue what I wish (have been since 1986), studying
math that has some points of contact with ontology, physics, even AI,
is what I am enjoying. I should be receiving Peter Johnstone's massive
2-volume set, "Sketches of an Elephant: A Topos Theory Compendium," in
the next few days.
And ya gotta crawl before ya can walk. I'm only recently gaining a good
appreciation of S4, the logic system closely related to time and
causality. Had I not learned S4 vs. S5, more computability theory than
I used to know, a lot of stuff about lattices, quantum logic, and
category theory, I surely would not be able to make sense of _any_ of
what Bruno talks about!
>
> In fact, perhaps this is a key problem with computers. Most human
> beings
> whom I know have enormous difficulty in finding a Golden Mean between
> abstraction and concreteness insofar as the concrete reality and
> abstract
> reality are concerned if you get my meanings. The problem is only
> slightly
> less prevalent in academia. Computers seem to be nowhere near
> solving this
> problem - in fact, the more similar to human beings they get, the more
> difficult it may be for them to solve the problem. I am not even
> sure that
> most human beings in or out of academia think that there should be a
> Golden
> Mean between abstraction and concreteness [exclamation mark - several
> of my
> keys are out including that one].
I have experience in both of the areas you talk about. Now I'm not
saying this is why you should believe what I write, but at least my
background spans both the *applied* (in college, working in a Josephson
junction lab on superconductivity, and at Intel, working on microchips,
and with some startup companies I've been working with for the past
decade or so) and the *theoretical* (math, physics, computer science,
logic, topos theory, etc.).
Few things thrill me more than taking something which seems to be as
abstract as unworldly as anything imaginable and applying it in the
real world.
(P.S. Could I encourage you to not include the full text of the
messages you are replying to?)
--Tim May
Received on Sun Dec 01 2002 - 13:50:10 PST