Thanks again for bringing insight and sanity to this discussion. A few points:
1) Your description of the structure N presupposes some knowledge of the structure N; the program that prints out the structure needs a first statement, a second statement, etc. This is, of course, unavoidable, and it’s therefore not a complaint; I doubt that there’s any way to give a formal description of the natural numbers without presupposing some informal understanding of the natural numbers. But what it does mean, I think, is that K-complexity (in the sense that you’re using it) is surely the wrong measure of complexity here—because when you say that N has low K-complexity, what you’re really saying is that “N is easy to describe provided you already know something about N”. What we really want to know is how much complexity is imbedded in that prior knowledge.
1A) On the other hand, I’m not clear on how much of the structure of N is necessarily assumed in any formal description, so my point 1) might be weaker than I’ve made it out to be.
2) It has been my position all along that K-complexity is largely a red herring here in the sense that it need not capture Dawkins’s meaning. Your observation that a pot of boiling water is more K-complex than a squirrel speaks directly to this point, and I will probably steal it for use in future discussions.
3) When you talk about T(N), I presume you mean the language of Peano arithmetic, together with the set of all true statements in that language. (Correct me if I’m wrong.) I would hesitate to call this a theory, because it’s not recursively axiomatizable, but that’s a quibble. In any event, we do know what we mean by T(N), but we don’t know what we mean by T(squirrel) until we specify a language for talking about squirrels—a set of constant symbols corresponding to tail, head, etc., or one for each atom, or....., and various relations, etc. So T(N) is well defined, while T(squirrel) is not. But whatever language you settle on, a squirrel is still going to be a finite structure, so T(squirrel) is not going to share the “wild nonrecursiveness” of T(N) (which is closely related to the difficulty of giving an extrinsic characterization). That seems to me to capture a large part of the intuition that the natural numbers are more complex than a squirrel,
4) You are probably right that Dawkins wasn’t thinking about mathematical structures when he made his argument. But because he does claim that his argument applies to complexity in general, not just to specific instances, he’s stuck (I think) either accepting applications he hadn’t thought about or backing off the generality of his claim. It’s of course hard to know exactly what he meant by complexity, but it’s hard for me to imagine any possible meaning consistent with Dawkins’s usage that doesn’t make arithmetic (literally) infinitely more complex than a squirrel.
5) Thanks for trying to explain to Silas that he doesn’t understand the difference between a structure and an axiomatic system. I’ve tried explaining it to him in many ways, at many times, in many forums, but have failed to make any headway. Maybe you’ll have better luck.
6) If any of this seems wrong to you, I’ll be glad to be set straight.
Splat:
Thanks again for bringing insight and sanity to this discussion. A few points:
1) Your description of the structure N presupposes some knowledge of the structure N; the program that prints out the structure needs a first statement, a second statement, etc. This is, of course, unavoidable, and it’s therefore not a complaint; I doubt that there’s any way to give a formal description of the natural numbers without presupposing some informal understanding of the natural numbers. But what it does mean, I think, is that K-complexity (in the sense that you’re using it) is surely the wrong measure of complexity here—because when you say that N has low K-complexity, what you’re really saying is that “N is easy to describe provided you already know something about N”. What we really want to know is how much complexity is imbedded in that prior knowledge.
1A) On the other hand, I’m not clear on how much of the structure of N is necessarily assumed in any formal description, so my point 1) might be weaker than I’ve made it out to be.
2) It has been my position all along that K-complexity is largely a red herring here in the sense that it need not capture Dawkins’s meaning. Your observation that a pot of boiling water is more K-complex than a squirrel speaks directly to this point, and I will probably steal it for use in future discussions.
3) When you talk about T(N), I presume you mean the language of Peano arithmetic, together with the set of all true statements in that language. (Correct me if I’m wrong.) I would hesitate to call this a theory, because it’s not recursively axiomatizable, but that’s a quibble. In any event, we do know what we mean by T(N), but we don’t know what we mean by T(squirrel) until we specify a language for talking about squirrels—a set of constant symbols corresponding to tail, head, etc., or one for each atom, or....., and various relations, etc. So T(N) is well defined, while T(squirrel) is not. But whatever language you settle on, a squirrel is still going to be a finite structure, so T(squirrel) is not going to share the “wild nonrecursiveness” of T(N) (which is closely related to the difficulty of giving an extrinsic characterization). That seems to me to capture a large part of the intuition that the natural numbers are more complex than a squirrel,
4) You are probably right that Dawkins wasn’t thinking about mathematical structures when he made his argument. But because he does claim that his argument applies to complexity in general, not just to specific instances, he’s stuck (I think) either accepting applications he hadn’t thought about or backing off the generality of his claim. It’s of course hard to know exactly what he meant by complexity, but it’s hard for me to imagine any possible meaning consistent with Dawkins’s usage that doesn’t make arithmetic (literally) infinitely more complex than a squirrel.
5) Thanks for trying to explain to Silas that he doesn’t understand the difference between a structure and an axiomatic system. I’ve tried explaining it to him in many ways, at many times, in many forums, but have failed to make any headway. Maybe you’ll have better luck.
6) If any of this seems wrong to you, I’ll be glad to be set straight.