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Isn't it interesting that numerical relationships that are accumulations (recursive) were under-rated or under-recognized even though an important aspect of quantum theory since the 1920's?
Mitchell Feigenbaum discovered "Universality" in the 1970's after working to solve nonlinear equations (within physics) generally known to not be very solvable. He looked at the recursive-ness of the behavior of solutions to equations including bifurcation related oscillations in results and an eventual "chaos" with more highly non-linear circumstances. " . . . he believed that his theory expressed a natural law about systems at the point of transition between orderly and turbulent" (Gleick, p. 180). James Gleick explains that Feigenbaum's concepts were resisted but quotes a scientist's eventual opinion: "It was a very happy and shocking discovery that there were structures in nonlinear systems that are always the same if you looked at them the right way" (p. 183).
Feigenbaum later discussed with James Gleick some thoughts on his theory (which was proved in mathematicians' terms in 1979 by Oscar E. Lanford III) (Gleick, p. 183). The following quotes are from p. 185-7.
About quantum theory of the 1920's: "Some part of the imagery is missing. If you ask what the equations really mean and what is the description of the world according to this theory, it's not a description that entails your intuition of the world. You can't think of a particle moving as though it has a trajectory. You're not allowed to visualize it that way. If you start asking more and more subtle questions--what does this theory tell you the world looks like?--in the end it's so far out of your normal way of picturing things that you run into all sorts of conflicts. Now maybe that's the way the world really is. But you don't really know that there isn't another way of assembling all this information that doesn't demand so radical a departure from the way in which you intuit things."
"There's a fundamental presumption in physics that the way you understand the world is that you keep isolating its ingredients until you understand the stuff that you think is truly fundamental. Then you presume that the other things you don't understand are details. The assumption is that there are a small number of principles that you can discern by looking at things in their pure state--this is the true analytic notion--and then somehow you put these together in more complicated ways when you want to solve more dirty problems. If you can."
"In the end, to understand you have to change gears. You have to reassemble how you conceive of the important things that are going on. You could have tried to simulate a model fluid system on a computer. It's just beginning to be possible. But it would have been a waste of effort, because what really happens has nothing to do with a fluid or a particular equation. It's a general description of what happens in a large variety of systems when things work on themselves again and again. it requires a different way of thinking about the problem."
" . . . you're supposed to take the elementary principles of matter and write down the wave functions to describe them. Well, this is not a feasible thought. Maybe God could do it, but no analytic thought exists for understanding such a problem."
"One has to look for different ways. One has to look for scaling structures--how do big details relate to little details. You look at fluid disturbances, complicated structures in which the complexity has come about by a persistent process. At some level they don't care very much what the size of the process is--it could be the size of a pea or the size of a basket ball. The process doesn't care where it is , and moreover it doesn't care how long it's been going. The only things that can ever be universal, in a sense, are scaling things."
"In a way, art is a theory about the way the world looks to human beings. It's abundantly obvious that one doesn't know the world around us in detail. What artists have accomplished is realizing that there's only a small amount of stuff that's important, and then seeing what it was. So they can do some of my research for me. When you look at early stuff of Van Gogh there are zillions of details that are put into it, there's always an immense amount of information in his paintings. It obviously occurred to him, what is the irreducible amount of this stuff that you have to put in. Or you can study the horizons in Dutch ink drawings from around 1600, with tiny trees and cows that look very real. If you look closely, the trees have sort of leafy boundaries, but it doesn't work if that's all it is--there are also, sticking in it, little pieces of twiglike stuff. There's a definite interplay between the softer textures and the things with more definite lines. Somehow the combination gives the correct perception. With Ruysdael and Turner, if you look at the way they construct complicated water, it is clearly done in an iterative way. There's some level of stuff, and then stuff painted on top of that, and then corrections to that. Turbulent fluids for those painters is always something with a scale idea in it."
"I truly do want to know how to describe clouds. But to say there's a piece over here with that much density, and next to it a piece with this much density--to accumulate that much detailed information, I think is wrong. It's certainly not how a human being perceives those things, and it's not how an artist perceives them. Somewhere the business of writing down partial differential equations is not to have done the work on the problem."
"Somehow the wondrous promise of the earth is that there are things beautiful in it, things wondrous and alluring, and by virtue of your trade you want to understand them." |
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