On Tuesday, July 2, 2002, at 02:52 PM, Wei Dai wrote:

> On Thu, Jun 27, 2002 at 03:59:49PM +0200, Bruno Marchal wrote: >> Now, and we have discussed this before, I have no understanding of the >> expression "being inside a universe". > > Being inside a universe to me means having a causal relationship with > the > universe, in other words being able to affect it through decisions and > actions. That leads to the question of what causal relationships are and > how do you formalize them. > > Fortunately I've now read most of _The Foundations of Causal Decision > Theory_, by James M. Joyce, and can recommend it for a discussion of > causality. This is also a great book for learning about decision theory > in > general, and I highly recommend it to everyone here. > I haven't read Joyce's book, but it sounds similar to Judea Pearl's excellent "Causality" book, which I have read much of. Pearl's focus is on Bayesian-type models of contributing factors, with some interesting excursions into Kripke's "possible worlds" (in the "counterfactual" sense of talking about things that did not actually happen, but which might or could happen...of obvious interest in AI. linguistics, etc.). Pearl, by the way, the UCLA professor who is the father of the murdered journalist Danny Pearl (in Pakistan). On the subject of "being inside a universe," there are some exciting papers by Fotini Markopoulou and others on a "category theory" (more precisely, "topos theory") outlook on this. One of her papers is "The internal description of a causal set; What the universe looks like from the inside," 1999. Available at the xxx.lanl.gov arXiv site as paper gr-qc/9811053. (Henceforth, to cut down on giving URLs or arXiv numbers, I'll stick to giving author names and either exact paper names or at least enough keywords to allow recovery via Google (which is better than giving transient URLs in many case) or from persistent archive sites like arXiv. I'm not ready, yet, to write up my "first posting to the everything list" about category and topos theory, which are my current main interests. Well, I guess this obviously _is_ going to be my first post, by way of Markopoulou's paper dovetailing so directly with Wei Dai's "being inside a universe" point. So I'll at least say what category theory is about. (The book I recommend is Lawvere and Schanuel's "Conceptual Mathematics: A first introduction to categories.") In a nutshell, too small a nutshell to really educate you if you don't already know about it, categories are collections of objects and arrows going from one to another. For example, in the category of SET, the objects are elements of sets and the arrows (also called morphisms) are the functions mapping one element into another element of another set. In other words, all that "function box" and "bijection" and "injection" stuff of New Math. However, the use of categories unifies a lot of mathematics and the field has expanded dramatically since Eilenberg and Mac Lane developed the ideas for use in algebraic topology. The idea is that theorems developed in category-theoretic language in one domain can be "carried over" (with those arrows, between categories, and even between other sets of arrows, in ascending levels of abstraction). And in the 1960s the work of Grothendieck and Lawvere led to a category imbued with certain "notions of truth." This was dubbed a "topos." What's fascinating is that a topos is a kind of "micro universe.' Not in a physical sense, a la Egan or Tegmark, but in the sense of generating a consistent reality. More on this later. A popular treatment of the "what it means to be inside a universe" point of view is in the cosmologist Lee Smolin's book, "Three Roads to Quantum Gravity," less than a couple of years old. Smolin collaborates with Markopoulou, Chris Isham, C. Rovelli, and others, and he's associated with the "loop gravity" and "spin foam" schools of quantum gravity/TOE. By the way, Greg Egan is doing some work with some of these folks, including John Baez. (The John Baez site (he's the younger cousin of Joan) is a wonderful resource for pointers. His papers are relentlessly clear. Find it with Google. Or, here it is: http://math.ucr.edu/home/baez/README.html) The Isham and Markopoulou work is oriented toward replacing what I'll call "the manifold with a Boolean algebra" with a more general view which I'll call "a lattice with a Heyting algebra." The smooth spacetime of conventional relativity goes away, perhaps, at Planck-scale distances and energies (10^-33 cm, or near/inside event horizons, perhaps). Perhaps more strangely, the conventional Boolean algebra and logic get superceded by time-varying sets where the law of the excluded middle (A or not-A, not-not-A is A) is replaced by a richer logical system: Heyting algebra and logic. I'll get into this stuff more in future posts. In particular, Isham has a topos perspective on "consistent histories" (MWI) which is quite interesting. A streaming video lecture on "Quantum theory and reality" is available at http://www.newton.cam.ac.uk/webseminars/hartle60/1-isham/ This is not easy going, but watching it a couple of times may get across some of the ideas. And he and his main collaborator, Butterfield, have written several papers. My last comment will be that I am not really a Tegmarkian. Frankly, I thought Greg Egan treated the same ideas better than Tegmark did. In "Distress" we find the "all topologies model," yet another overloading of the acronym ATM. (AOL, acronym overload.) "Distress" was published in hardback in June 1997. Tegmark's TOE preprint appears in April 1997. So roughly simultaneous publication. Anyway, Tegmark is a professional physicist, and has done much good work on conventional cosmology, so I'm not dissing him. More on this later. --Tim May -- Timothy C. May [EMAIL PROTECTED] Corralitos, California Political: Co-founder Cypherpunks/crypto anarchy/Cyphernomicon Technical: physics/soft errors/Smalltalk/Squeak/ML/agents/games/Go Personal: b.1951/UCSB/Intel '74-'86/retired/investor/motorcycles/guns Recent interests: category theory, toposes, algebraic topology