Haven’t had much board time recently, so my replies are slow.
Well, I agree that we need a better definition of complexity. From the link you posted, it appears that model theorists think so, too. Now if only one of them would stumble by this thread and give us one . . .
I appreciate the compliment (though I can’t remember the last time I was a pure anything, much less a mathematician). I am not sure I agree with the distinction you are drawing, though. Yes, I gave an example of derived “complexity” [sub]wish I knew what that really meant[/sub] and you are talking about building a model to predict the behavior of an existing system. But the result that arl argued for, and which you seemed to agree with as far as systems of “infinite complexity” go, implied not only a one-to-one mapping between model and system, it declared identity between them.
Now, when I think of the relationship between models and systems I generally consider it to e a many-to-one mapping. We can model plane figures in cartesian equations, polar equations, or with methods of Euclidean construction. We can model the positron with dirac’s equations or Feynman diagrams. I don’t see any compelling reason why modeling an infinite system will necessarily restrict us to a single construction.
Furthermore, even if that reduction holds, and only a single model of arbitrary accuracy is possible for an infinitely complex system, I cannot see any means of making the leap from that single model to identity with the system itself.
Actually, I take that back. I can see one means – if the reduction is not to a single case but to 0. If it is not possible to construct a model of arbitrary precision without replicating in exact detail all facets of the original and it is not possible to build that model as a separate but identical “copy”.
For instance, if we had to build a “miniature universe” to model teh Universe, and if we had to build it in the Universe. Then, of course, we would have to build a miniature model of the part of the Universe which held our model, which would have to contain a miniature model of the miniature model . . . ad infinitum. BANG! Then we hit the quantum wall and cannot miniaturize any further. Our model thus is not arbitrarily precise. The only “model” for the Universe is the original.
That seems a far cry, though, from the types of mathematical models which intiated this discussion. I wish I had a friend who was a model theorist, so I could go beat on him until he gave me a straight answer to the implications of complexity. [sub]Okay – I just wish I had a friend. We all have our wounds. Thanks so much for seasoning mine.[/sub]
We need an expert. Maybe I should ask Stephanie Seymour what she thinks.
No argument there, but I still don’t see how this takes us to identity of model and system. If we take our dynamic equations and set them in the middle of a hurricane, I doubt we will learn much of interest (though the idea has a certain visceral appeal). Likewise, the output of tose equations, no matter how copious the data we pass into them as parameters, will not drop water on farmer Brown’s fields.
I write that and fear I have engaged in reductio ad absurdum, but I really think not. I think, rather, for any finite model the identification is absurd. And I don’t see whatever line of reasoning would convince you that the absurdity necessarily vanishes for a system of infinite complexity.
This sounds like you are arguing for the reductive case I argued above: only a physical replication is fit to model the Universe and that replication cannot be achieved. QED the Universe can be modeled only by itself. (OR: the Universe has no model, which says the same thing.)
I am not sure how you would generalize this argument to any system of infinite complexity, though.