I've been reading Lee Smolin's 'Three Roads to Quantum Gravity' recently. For those of you who may not know about his work, Lee Smolin is a physicist working on efforts to unify quantum theory with general relativity. This book describes three paths to doing so, and is part of a new attempt to popularize some of the more radical theories on the cutting edge of physics (Brian Greene's 'Elegant Universe' is another). The March issue of Scientific American has a brief writeup on Smolin's work as well.
There is much that I don't understand in this book, and there are some fascinating and counter-intuitive ideas there. However, what intrigued me the most was this (from Chapter 8):
With matter there is a limit to how small we can divide something, for at some point we are left with individual atoms. Is the same true of space ? If we continue dividing, do we eventually come to a smallest unit of space, some smallest possible volume ? Or can we go on forever, dividing space into smaller and smaller bits, without ever having to stop ? All three of the roads I described in the Prologue favour the same answer to this question: that there is indeed a smallest unit of space. It is much smaller than an atom of matter, but nevertheless, [...snip...], there are good reasons to believe that the continuous appearance of space is as much an illusion as the smooth appearance of matter.
As he goes on to explain, it is not that good ol' Euclidean space is extra chunky; the notion of space itself breaks down at a certain level, giving rise to discrete graph-like structures. Now to a discrete geometer or a combinatorial person, the idea that our universe itself is discrete is a delicious idea. The idea of entropy (in the physical sense) has been related to entropy (in the information-theoretic sense) using such ideas.
Further, it begins to make me wonder about the Church-Turing thesis (well almost anything will, but that's a different story). One of the objections that has been raised regarding the universality of the Church-Turing thesis is the fact that Turing machines only operate on discrete entities, and have no way of dealing with real numbers. Now I am going way out on a limb here, but could a discrete universe make this a moot point ?
Update: I just picked up 'The Elegant Universe'. The key difference is in the approach each emphasizes: Greene stresses the importance of string theory, and Smolin, while discussing three different paths to quantum gravity, favors loop quantum graivity (both of these reflecting the authors' own expertise). As works of writing though, I have to say that I find 'The Elegant Universe' a bit more of a pleasant read. It feels like Smolin is struggling to hit the right metaphors to explain the theories he discusses (no fault of his - this is one of the deepest areas of science, after all)