Any reason to suppose that the state space of the universe *that we can manipulate and control* is something other than a Kaeler Manifold?

Exactly. Everybody who knows Math/Computer Science beyond the shallow end of the pool knows properties of infinite trees.

Heck, I took “Theory of Algorithms” from John Todd, still attending seminars at age 93. He was the man who got John von Neumann interested in computers.

Another well-known property of infinite trees of universes:

If the ‘tree’ of universes simulating other universes, is infinite, then at every level there exists a universe which at ALL future times will have a descendant.

In human geneology terms, this is to say that, if the human race lasts forever, there is at any time in the history of the human race, at least one person who has, at all future times, a living descendant.

Next, consider that the Harary graph reconstruction hypothesis, though true for finite trees, is untrue for infinite trees.

If one cannot see the original graph, but only the set of all subgraphs (created by removing a single vertex and all edges adjacent to that vertex), the problem is to reconstruct the original graph.

But we cannot tell, given an infinite set of isomorphic infinite trees, if the original was an isomorphic infinite tree or an infinite forest of those isomorphic infinite trees.

Now, what is the actual topology of the multiverse, if subtrees can be isomorphic to the entire tree? One need not see it as a tree structure any more.

]]>The problem is not so much to give a definition of intelligence. The problem is to prove that your definition really corresponds to your intuitive notion of intelligence, that it’s not too narrow, not too wide, but a precise fit.

I don’t think anybody knows how this can be done.

If you look on things from this viewpoint, Godel’s theorem will not seem very deep, after all. It just says that some form of definition of number (i.e., using the language of axiomatic theory) will not be a precise fit. I assume same will be the case for any attempt to define any real thing, too: e.g., any definition of electron will either be incomplete, or contradictory. Apparently, Greeks knew this. ]]>

Scott’s statement is fun to think about, IMHO.

The principle “no phase angle increase in open quantum systems” is, surely, a unifying principle of mathematics, physics, and (21st Century!) information theory, right?

So, as with any unifying principle, we ought to be able to express it in algebraic, geometric, differential, and (21st Century!) information-theoretic terms, right?

Scott’s post has (implicitly) linked the differential and information-theoretic aspects of this principle. Good.

The geometric interpretation is obvious: the state space of quantum computation must have a Kähler geometry (so that parallel transport of two states does not change their relative phase). Very good.

But what is the *algebraic* venue for Scott’s principle? Presumably, the Kähler manifolds appropriate to the state-spaces of quantum information theory must have a natural algebraic structure … which is … which might be … ???

Among other things, this implies that if the conclusion of the cosmological argument *fails*, then we get the conclusion of the simulation argument for free! In other words, if the chain of universes simulating universes extends infinitely far backwards, then *certainly* there must be another universe simulating ours, wherever we happen to be in the chain. And that’s all the simulation argument really cares about.

Re: the asymptotics of simulating a human, I wholly agree that this is an ill-defined question, but it may be definable in some sane sort of way: they study $n \times n$ chess, after all. I mean, I’m a physically bigger system than, say, Scott, but does that make me harder to simulate in a useful manner? Scott’s more brilliant than I am, does that make *him* harder to simulate than myself? Also, does this matter?

YK: Scott’s response to your question is a fairly accurate depiction of how he speaks. It’s not all bad, though: the class wouldn’t otherwise have time to keep up with him. Everyone remembers Feynman’s *Lectures on Physics*, but the people that were supposed to be in the class were, for the most part, lost or not attending by the end of it. I believe Scott’s given at least one talk at Perimeter Institute, however, and they put all their seminars online if you just want to see him speaking on a topic of some description.

Also: I don’t check the Shtetl for three days and Scott goes and posts something that gets 130-odd comments. Sheesh.

]]>Michael: OK, I’ve calmed down enough to understand you better :-), but my perspective is still very different. Here’s an analogy: as I’ve said before on this blog, if P≠NP but there were a fast algorithm to solve SAT on instances of size at most 10^{30}, then I would cheerfully admit that the P versus NP question had turned out to be irrelevant. In computer science, I see every asymptotic statement as just an idealized version of suitable finite statements. The difference is that with P versus NP,

(1) we know what the asymptotic statement *is*, and

(2) we have lots of historical evidence that thinking in terms of asymptotic statements is a good way to approach the finite questions we really care about.

In Penrose’s case, I claim that both (1) and (2) fail. Since “simulating a human being” is not mathematically well-defined to begin with, it’s not clear what it even *means* to define an asymptotic version of the problem. And even supposing we *could* somehow do this, we’d have no historical experience to indicate that such a thought-exercise would tell us anything about the finite question. So here it seems much more sensible to me just to ask the finite question directly — in which case we get a question about circuit complexity rather than computability.