Your biggest problem here, and in your blog posts, is that you equivocate between the structure of the standard natural numbers (N) and the theory of that structure (T(N), also known as True Arithmetic). The former is recursive and (a reasonable encoding of) it has pretty low Kolmogorov complexity. The latter is wildly nonrecursive and has infinite K-complexity. (See almost any of Chaitin's work on algorithmic information theory, especially the Omega papers, for definitions of the K-complexity of a formal system.)

The difference between these two structures comes from the process of translating between them. Once explained properly, it's almost intuitive to a recursion theorist, or a computer scientist versed in logic, that there's a computable reduction from any language in the Arithmetic Hierarchy to the language of true statements of True Arithmetic. This implies that going from a description of N to a truth-enumerator or decision procedure for T(N) requires a hypercomputer with an infinite tower of halting, meta-halting, ... meta^n-halting ... oracles.

However, it so happens that simulating the physical world (or rather, our best physical 'theories', which in a mathematical sense are structures, not theories) on a Turing machine does not actually require T(N), only N. We only use theories, as opposed to models, of arithmetic, when we go to actually reason from our description of physics to consequences. And any such reasoning we actually do, just like any pure mathematical reasoning we do, depends only on a finite-complexity fragment of T(N).

Now, how does this make biology more complex than arithmetic? Well, to simulate any biological creature, you need N plus a bunch of biological information, which together has more K-complexity than just N. To REASON about the biological creature, at any particular level of enlightenment, requires some finite fragment of T(N), plus that extra biological information. To enumerate all true statements about the creature (including deeply-alternating quantified statements about its counterfactual behaviour in every possible circumstance), you require the infinite information in T(N), plus, again, that extra biological information. (In the last case it's of course rather problematic to say there's more complexity there, but there's certainly at least as much.)

Note that I didn't know all this this morning until I read your blog argument with Silas and Snorri; I thank all three of you for a discussion that greatly clarified my grasp on the levels of abstraction in play here.

(This morning I would have argued strongly against your Platonism as well; tonight I'm not so sure...)

b) we have the subjective reports of mathematicians about the complexity of their subject matter, which I think should be given at least as much weight as the subjective reports of ecologists

Again, this word complexity is used in many ways. Complexity in the sense of humans find this complicated is a different concept from complexity in the sense of Kolmogorov complexity.

no set of axioms suffices to specify the standard model of arithmetic (i.e. to distinguish it from other models).

Then what do you mean when you say "integers"^H^H "natural numbers", if no set of premises suffices to talk about it as opposed to something else?

Anyway, no countable set of first-order axioms works. But a finite set of second-order axioms work. So to talk about the natural numbers, it suffices merely to think that when you say "Any predicate that is true of zero, and is true of the successor of every number it is true of, is true of all natural numbers" you made sense when you said "any predicate".

It is this sort of minor-seeming yet important technical inaccuracy that separates "The Big Questions" from "Good and Real", I'm afraid.

mattnewport: This would seem to put you in the opposite corner from Silas, who thinks (if I read him correctly) that all of physical reality is computably describable

No, it wouldn't -- he's saying basically the same thing I did. The laws of physics are computable. In describing observations, we use concepts from math. The reason we do so is that it allows simpler descriptions of the universe.

Map and territory

Less confusing than saying "belief and reality", "map and territory" reminds us that a map of Texas is not the same thing as Texas itself. Saying "map" also dispenses with possible meanings of "belief" apart from "representations of some part of reality".

Since our predictions don't always come true, we need different words to describe the thingy that generates our predictions and the thingy that generates our experimental results. The first thingy is called "belief", the second thingy "reality".

More: Map and Territory (sequence)

I agree that if arithmetic is a human invention, then my counterexample goes away.

Then you agree that your "counterexample" amounts to an assumption. If a Platonic realm exists (in some appropriate sense), and if Dawkins was haphazardly including that sense in the universe he is talking about when he describes complexity arising, then he wrong that complexity always comes from simplicity.

If you assume Dawkins is wrong, he's wrong. Was that supposed to be insightful?

On that reading, a key locus of our disagreement is whether arithmetic is a human invention. I think the answer is clearly no, for reasons I've written about so extensively that I'd rather not rehash them here.

It's a false dispute, though. When you clarify the substance of what these terms mean, there are meanings for which we agree, and meanings for which we don't. The only error is to refuse to "cash out" the meaning of "arithmetic" into well-defined predictions, but instead keep it boxed up into one ambiguous term, which you do here, and which you did for complexity. (And it's kind of strange to speak for hundreds of pages about complexity, and then claim insights on it, without stating your definition anywhere.)

One way we'd agree, for example, is if we take your statements about the Platonic realm to be counterfactual claims about phenomena isomorphic to certain mathematic formalisms (as I said at the beginning of the thread).

[Incidentally, it occurs to me that perhaps you are misreading my use of the word "model". I am using this word in the technical sense that it's used by logicians, not in any of its everyday senses.]

The definitions aren't incredibly different, which is why we have the same term for both of them. If you spell out that definition more explicitly, the same problems arise, or different ones will pop up.

(By the way, this doesn't surprise me. This is the fourth time you've had to define a term within a definition you gave in order to avoid being wrong. It doesn't mean you changed that "subdefinition". But genuine insights about the world don't look this contorted, where you have to keep saying, "No, I really meant this when I was saying what I meant by that.")