Hi Tim, just some quick comments. On Tue, Aug 13, 2002 at 10:08:50AM -0700, Tim May wrote: > > * Because toposes are essentially mathematical universes in which >> various bits and pieces of mathematics can be assumed. A topos in which >> Euclid's Fifth Postulate is true, and many in which it is not. A topos >> where all functions are differentiable. A topos in which the Axiom of >> Choice is assumed--and ones where it is not assumed. In other words, as >> all of the major thinkers have realized over the past 30 years, topos >> theory is the natural theory of possible worlds.
Frankly I think you exaggerate :) I could be very long on this even without mentioning my thesis. But I want to be short, and in my thesis, toposes could only be used for first person semantics (and even this is still an exaggeration). Toposes are just (enlarged) S4 model, or (classical) model for intuitionist logic. At 11:15 -0700 13/08/2002, Tim May wrote: >Worlds _are_ propositions. This can be misleading. In modal context we have a duality: we can define world by set of propositions (the proposition true in the world), and we can define dually proposition by set of worlds (the world in which p is true). At 15:51 -0700 13/08/2002, Tim May wrote: >(You might also want to take a look at the paper by Guts, a Russian, >on a "Topos-Theoretic Model of the Deutsch Multiverse." Available at >the usual xxx.lanl.gov site.) Thank you very much for this interesting reference (and the reference therein, including a Russian website on Everett!). At 15:51 -0700 13/08/2002, Tim May wrote: >As far as the math of nonstandard logic goes, I think the most >interesting application within our lifetimes will come with AI. I agree. Perhaps Wei Dai should look at the non monotonic logics and to the logics of relevance. Especially if he want escape the problem of omniscience. >>At 21:29 -0700 13/08/2002, Tim May wrote: >>>Nor do I take Schmidhuber's "all running programs" notion very >seriously. Interesting ideas to play with, and to use some tools on. [...] At 21:29 -0700 13/08/2002, Tim May wrote: >Lack of even the slightest piece of evidence for "all possible >mathematical universes actually exist" and/or "the all runnable >computer programs.' > >I also don't believe there are gods or other supernatural beings, >for the same reason. > >If and when I see an experiment that points to there being other >universes which have tangible existence, then I'll start to believe. Then I urge you to read my thesis (which results, btw, has been published about ten years before Tegmark and Schmidhuber and which results goes far away beyond, ... :) Why. Because even *without experiment*, but with just a small amount of platonism in arithmetic and computationalism in the cognitive science, you will understand that the many computations are unavoidable, and that the physical laws necessarily emerges from simple elementary relation between integers ... I am more skeptical than you, I don't believe in a *physical* universe. Actually I show that with comp physics cannot be fundamental, but must emerge from numbers and "numbers as seen by numbers" ... Physicalism and materialism is *just incompatible* with mechanism. Perhaps read just my "Computation, Consciousness and the Quantum" loadable from my URL below. I will say more in a post which I am writing to you and where I make a comment on Yetter's "Functorial Knot theory". Bruno -- http://iridia.ulb.ac.be/~marchal/