I’m responding here to your invitation in the parent, since this post provides some good examples of what you’re not getting.
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.
Simulating squirrels and using arithmetic require information, but that information is not supplied in the form of axioms. The best way to imagine this in the case of arithmetic is in terms of a structure.
Starting from the definition in that wikipedia page, you can imagine giving the graphs of the universe and functions and relations as Datalog terms. (Using terms instead of tuples keeps the graphs disjoint, which will be important later.) Like so:
Universe:
is_number(0),is_number(1), …0:
zero(0)S:
next(0,1),next(1,2), …+:
add_up_to(0,0,0),add_up_to(0,1,1),add_up_to(1,0,1)…and so on.
Then you use some simple recursive coding of datalog terms as binary. What you’re left with is just a big (infinite) set of binary strings. The Kolmogorov complexity of the structure N, then (the thing you need to use arithmetic) is the size of the shortest program that enumerates the set, which is actually very small.
Note that this is actually the same arithmetic that Steve is talking about! It is just a different level of description, one that is much simpler but entirely sufficient to conduct simulations with. It is only in understanding the long-term behavior of simulations without running them that one requires any of the extra complexity embodied in T(N) (the theory). To actually run them you just need N (the structure).
The fact that you don’t seem to understand this point yet leads me to believe you were being a little unfair when you said:
By the way, I really hope your remark about Splat’s comment being “enlightening” was just politeness, and that you didn’t actually mean it. Because if you did, that would mean you’re just now learning the distinction between N and T(N), the equivocation between which undermines your claims about arithmetic’s relation to the universe.
Now, if you want to complete the comparison, imagine you’re creating a structure that includes a universe with squirrel-states and times, and a function from time to squirrel state. This would look something like:
is_time(1:00:00),is_time(1:00:01), …is_squirrel_state(<eating nut>),is_squirrel_state(<rippling tail>),is_squirrel_state(<road pizza>)squirrel_does(1:00:00, <rippling tail>), …
The squirrel states, though, will not be described by a couple of words like that, but by incredibly detailed descriptions of the squirrel’s internal state—what shape all its cells are, where all the mRNAs are on their way to the ribosomes, etc. The structure you come up with will take a much bigger program to enumerate than N will. (And I know you already agree with the conclusion here, but making the correct parallel matters.)
(Edit: fixed markup.)
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.