At 9:37 -0700 3/07/2002, Tim May wrote: >It'll be interesting and useful or it won't be (the future is >already set? Boolean logic really _does_ rule?). > >What I mean is that it's the particular area that's interesting to >me now. In fact, it's the most interesting thing I've seen in many >years. > >I mentioned it in some long articles on another list I've been >active in (Cypherpunks, a crypto list), but their issues and >concerns are pretty far-removed from logic, category theory, and >math (other than number theory). This list seems to have more points >of intersection. So, here I am.
This list is based on the idea that -more is simpler-. We are open to all many-things idea (many worlds, many computations, etc.). Category theory is interesting in that respect, but beware mathematical mermaids! :) >[Aside: I'm very intrigued by the possibility of building >category-based class systems which go way beyond object-oriented >class systems such as seen in Smalltalk, Java, ML, CLOS. Since >categories are generalized objects, with more of a focus on >_relations_ than objects, and since toposes are in some sense still >more general (logical worlds instead of just worlds), I look for the >computer languages of 20 years from now to look even more >categorical, or even more toposophical, than today's crude cuts.] Mmmh ... Years ago I found a reference on "Categoros" (?) a programming language based on category theory. I loose the reference (I remember only that it was a japanese work). I guess you know that typed lambda calculus have nice natural semantics in term of cartesian closed category. I prefer UNtyped lambda calculus, like pure lisp or like (axiomatic) recursion theory. I have study the Hyland topos which manage to have models both linked to recursion theory and sort of lambda calculus, but eventually I leave it because those models are not well fitting the problems I am working on. My current "hobby" is Knot Theory. Curiously some "quantum categories" seem to appear in knot theory ... Louis Kauffman wrote quite readable papers on that. My interest in knots stems from my reading of Kitaev Papers on anyonic quantum computing (see also Freedman about his "modular functor"). > >In the "new" much shorter version of my thesis I have suppressed >>all references to categories tough. > >I'm curious why. I know a handful of people still think of category >theory as "generalized abstract nonsense," but this is clearly not >the case. At least no more so than algebra and topology are >"generalized abstract nonsense." I have suppressed ref to categories because the technical part of my thesis was considered sufficiently difficult for not adding extra algebraical difficulties. Those cat were not useful for getting the results, only for linking those results with other works by others. >I mean in the sense that the history of modern science seems to me >to be a succession of "throwing out the "centered" object," throwing >out a world centered around the Sun, or centered around God, or >centered around even Newtonian physics. "throwing out the "centered" object" (or "subject" perhaps?) is quite in the spirit of this list. Have you read Everett? Quite important. He just embeds the physicist in the physical world. My own work is a (radical) generalisation of that idea in the sense that I embed the "arithmetician" in the arithmetical world, making it a first order citizen. >Euclidiean geometry, Riemannian geometry, Newtonian physics, >Lorentzian transforms in a Minkowski space (aka Einsteinian >physics), etc. are just special kinds of universes. Are they "real"? >(This gets into Tegmark territory, about "actual" (whatever that >means!) physical universes with different mathematics and then, >obviously, different physical laws. Egan, in "Distress," calls this >the "all topology model.") If we are machines I bet the physical laws emerge from all consistent possible Universes/Histories) and are necessarily unique! (more in my URL). I will search for "Distress". I have read and appreciate "Permutation City" by Egan. "All topology model"? nice! Especially because I begin to suspect a "many world interpretation" of knots and links! >Personally, I'm a partial Platonist (these things in mathematics >have some kind of existence) and a partial formalist (we are pushing >symbols around on paper and in our minds). I increasingly view the >constructivist (Intuitionist/Brouwerian) point of view as being >consistent with what we see in the world and what we can model or >simulate on computers. I am partial Platonist myself. I believe arithmetical truth is independent of me, all the rest are constructions by ... numbers. I must confess I am a sort of (neo)Pythagorician ... The "neo" comes from Church Turing Thesis. >[snip] There is much, much more to say on all of these topics. That is the least we can say ;) -Bruno -- http://iridia.ulb.ac.be/~marchal/
