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 

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

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