On Tuesday, August 13, 2002, at 02:34 PM, Wei Dai wrote:
> Tim, I think I'm starting to understand what you're saying. However, it > still seems that anything you can do with intuitionistic logic, toposes, > etc., can also be done with classical logic and set theory. (I'm not > confident about this, but see my previous post in reponse to Bruno.) > Maybe > it's not as convenient or "natural" in some cases (similar to how modal > logic can be more convenient than explicitly quantifying over possible > worlds even when they are equivalent), but if one is not already > familiar > with intuitionistic logic and category theory, is it really worth the > trouble to learn them? I don't know. One learns a field for various reasons. Clearly a lot of people think "classical logic with the right exceptions and terminology" serves them well. A lot of others, though not as many, think program semantics and possible worlds semantics are best understood in the "natural" logic of time-varying sets and topos theory. I'm in the latter category (no pun intended). I also don't know what your goals are, despite reading many of your posts. If, for example, you are looking for tools to understand a possible multiverse, or how multiverses in general might be constructed, I'm not at all sure any such tools have ever existed or _will_ ever exist, except insofar as tools for understanding toposes, lattices, etc. exist. This is quite different from understanding, say, general relativity, where the tools of differential geometry and exterior calculus are immediately useful for understanding and for calculations. The MWI/Tegmark/Egan stuff is very far out on the fringes, as we know, and there is unlikely to be anything one can do calculations of. Still, it seems likely that a _lot_ of mathematics is needed...a lot more math than physics, almost certainly. Modal logic seems to me to be _exactly_ the right logic for talking about possible states of existence, for talking about possible worlds, for talking about branching universes. So the issue is not "But can't I find a way to do everything in ordinary logic?" but is, rather, to think in terms of modal logic offering a more efficient "basis" (in the conceptual vector space), a basis with a smaller semantic gap between the formalism and the hypothesized world. (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.) > > For example, posets can certainly be studied and understood using > classical logic. How much does intuitionistic logic buy you here? Posets can certainly be studied with classical logic. However, posets fail the law of trichotomy, that two things when compared by some ordering result in one of three outcomes: A is less than B, A is greater than B, or A is equal to B. This is the "common sense" comparison of objects, one with a "linear" or "total" order. However, posets are partially-ordered precisely because they don't follow the law of trichotomy. Is one more natural than the other? More common? More useful? I have my own beliefs at this point. A book I strongly recommend, though it is difficult, is Paul Taylor's "Practical Foundations of Mathematics." 1999. (I buy many books not to read straight through, but to consult, to draw insights and inspirations from, and to let me know what I need to learn more of. This is one of those books. The first 175 pages, which I've been reading from, is making more and more sense....the terms become familiar, I see connections with other areas, and by a process akin to "analytic continuation" the ensemble of ideas becomes more and more natural. Beyond these pages, though, it's mostly incomprehensible.) I recommend this book for the broad insights I am gaining from it, but not as any kind of manual for tinkering with multiverses! (insert silly smiley as one sees fit) My conclusion from Tegmark's paper, which dovetailed with Egan's treatment of "all topologies models" in "Distress," was that to make progress a lot of math needs to be learned. Which is my current approach. These are not my only inspirations. Indeed, I came to join this list after becoming fascinated (again, after a nearly 28-year absence) in topology, algebra, and the physics of time and cosmology. Seen this way, category and topos theory are worth studying for their own sake. I don't think it is likely that "every conceivable universe with consistent laws of mathematics has actual existence" (to nutshell my understanding of Tegmark's theory) is actually true (whatever that means). Nor do I take Schmidhuber's "all running programs" notion very seriously. Interesting ideas to play with, and to use some tools on. Strangely, then, I view these notions as places to apply the math I'm learning to. And I'm thinking small, in terms of simple systems. A paper I have mentioned a couple of times is directly in line with this approach: Fotini Markopoulou's "The internal description of a causal set: What the universe looks like from the inside." Here's the paper number and abstract: gr-qc/9811053 From: Fotini Markopoulou <[EMAIL PROTECTED]> Date (v1): Tue, 17 Nov 1998 19:28:10 GMT (41kb) Date (revised v2): Thu, 18 Nov 1999 17:32:41 GMT (42kb) The internal description of a causal set: What the universe looks like from the inside Authors: Fotini Markopoulou Comments: Version to appear in Comm.Math.Phys. (minor modifications). 37 pages, several eps figures Journal-ref: Commun.Math.Phys. 211 (2000) 559-583 We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss. --end of excerpt-- Take a look at these papers (hers, the Guts paper, the various Baez, Smolin, Crane, etc. papers). All free. Some are introductions. All have a fair amount to say about the nature of reality. The stuff on causal sets (lattices and posets) is of direct relevance to several areas of modern physics. The relevance to MWI and Tegmark-style meta-branchings seems clear to me. As far as the math of nonstandard logic goes, I think the most interesting application within our lifetimes will come with AI. --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
