On Sunday, December 1, 2002, at 10:00 AM, Osher Doctorow wrote:
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."I agree again with Tim May.From Osher Doctorow [EMAIL PROTECTED] Sunday Dec. 1, 2002 0958
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
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:
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!
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.).
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].
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?)