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I see that SIlas is back to insisting that you can't simulate a squirrel with a simple list of axioms, after having been told forty eight bajillion times (here and elsewhere) that nobody's asserting any such thing;
I didn't say that. Read it again. I said that there is some finite axiom list that can describe squirrels, but it's not just the axioms that suffice to let you use arithmetic. It's those, plus biological information about squirrels. But this arithmetic is not the infinitely complex arithmetic you talk about in other contexts!
my claim is that you can simulate a squirrel in the structure N, not in any particular axiomatic system.
You can't -- you need axioms beyond those that specify N. The fact that the biological model involving those axioms uses math, doesn't mean you've described it once you've described the structure N. So whether or not you call that "simulating it in the structure N", it's certainly more complex than just N.
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Splat:
1)
This depends on what you mean by "specify". To distinguish N from other mathematical structures requires either an infinite (indeed non-recursive) amount of information or a second order specification including some phrase like "all predicates". Are you referring to the latter? Or to something else I don't know about?
2) I do not know Chaitin's definition of the K-complexity of a structure. I'll try tracking it down, though if it's easy for you to post a quick definition, I'll be grateful. (I do think I know how to define the K-complexity of a theory.) I presume that if I knew this, I'd know your answer to question 1).
3) Whatever the definition, the question remains whether K-complexity is the right concept here. Dawkins's argument does not define complexity; he treats it as "we know it when we see it". My assertion has been that Dawkins's argument applies in a context where it leads to an incorrect conclusion, and therefore can't be right. To make this argument, I need to use Dawkins's intended notion of complexity, which might not be the same as Chaitin's or Kolmogorov's. And for this, the best I can do is to infer from context what Dawkins does and does not see as complex. (It is, clear from context that he sees complexity as a general phenomenon, not just a biological one.)
4) The natural numbers are certainly an extremely complex structure in the everyday sense of the word; after thousands of years of study, people are learning new and surprising things about them every day, and there is no expectation that we've even scratched the surface. This is, of course, a manifestation of the "wildly nonrecursive" nature of T(N), all of which is reflected in N itself. And this, again, seems pretty close to the way Dawkins uses the word.
5) I continue to be most grateful for your input. I see that SIlas is back to insisting that you can't simulate a squirrel with a simple list of axioms, after having been told forty eight bajillion times (here and elsewhere) that nobody's asserting any such thing; my claim is that you can simulate a squirrel in the structure N, not in any particular axiomatic system. Whether or not you agree, it's a pleasure to engage with someone who's not obsessed with pummelling straw men.
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 PA<sub>2</sub> 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.