A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).
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ETA: It would seem that rationality quotes are no longer desired. After several days this thread stands voted into the negatives. Wolud whoever chose to to downvote this below 0 would care to express their disapproval of the regular quotes tradition more explicitly? Or perhaps they may like to browse around for some alternative posts that they could downvote instead of this one? Or, since we're in the business of quotation, they could "come on if they think they're hard enough!"
Replying out of order:
2) A quick search of Google Scholar didn't net me a Chaitin definition of K-complexity for a structure. This doesn't surprise me much, as his uses of AIT in logic are much more oriented toward proof theory than model theory. Over here you can see some of the basic definitions. If you read page 7-10 and then my explanation to Silas here you can figure out what the K-complexity of a structure means. There's also a definition of algorithmic complexity of a theory in section 3 of the Chaitin.
According to these definitions, the complexity of N is about a few hundred bits for reasonable choices of machine, and the complexity of T(N) is &infty;.
1) It actually is pretty hard to characterize N extrinsically/intensionally; to characterize it with first-order statements takes infinite information (as above). The second-order characterization. by contrast, is a little hard to interpret. It takes a finite amount of information to pin down the model[*][PA2], but the second-order theory PA2 still has infinite K-complexity because of its lack of complete rules of inference.
Intrinsic/extensional characterizations, on the other hand, are simple to do, as referenced above. Really, Gödel Incompleteness wouldn't be all that shocking in the first place if we couldn't specify N any other way than its first-order theory! Interesting, yes, shocking, no. The real scandal of incompleteness is that you can so simply come up with a procedure for listing all the ground (quantifier-free) truths of arithmetic and yet passing either to or from the kind of generalizations that mathematicians would like to make is fraught with literally infinite peril.
3&4) Actually I don't think that Dawkins is talking about K-complexity, exactly. If that's all you're talking about, after all, an equal-weight puddle of boiling water has more K-complexity than a squirrel does. I think there's a more involved, composite notion at work that builds on K-complexity and which has so far resisted full formalization. Something like this, I'd venture.
The complexity of the natural numbers as a subject of mathematical study, while certainly well-attested, seems to be of a different sense than either K-complexity or the above. Further, it's unclear whether we should really be placing the onus of this complexity on N, on the semantics of quantification in infinite models (which N just happens to bring out), or on the properties of computation in general. In the latter case, some would say the root of the complexity lies in physics.
Also, I very much doubt that he had in mind mathematical structures as things that "exist". Whether it turns out that the difference in the way we experience abstractions like the natural numbers and concrete physical objects like squirrels is fundamental, as many would have it, or merely a matter of our perspective from within our singular mathematical context, as you among others suspect, it's clear that there is some perceptible difference involved. It doesn't seem entirely fair to press the point this much without acknowledging the unresolved difference in ontology as the main point of conflict.
Trying to quantify which thing is more complex is really kind of a sideshow, although an interesting one. If one forces both senses of complexity into the K-complexity box then Dawkins "wins", at the expense of both of your being turned into straw men. If one goes by what you both really mean, though, I think the complexity is probably incommensurable (no common definition or scale) and the comparison is off-point.
5) Thank you. I hope the discussion here continues to grow more constructive and helpful for all involved.
Relevant link: http://lesswrong.com/lw/vh/complexity_and_intelligence/