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And that's, by the way, sort of why it's so hard to distinguish random radio noise from some kind of intelligent compressed encrypted signal.
And in the end the real challenge is to find a way to describe goals. OK, so let's talk about what it means to be a universal system. Well, there's another big place where I think computational irreducibility is very important, and that's in the foundations of mathematics.
And that has some very important consequences when it comes to thinking about nature. But at least in recent times mathematics has prided itself on somehow being very general.
Stephen Wolfram: A New Kind of Science | Online—Table of Contents
In a similar vein, Wolfram also demonstrates many simple scjence that exhibit phenomena like phase transitionsconserved quantitiescontinuum behavior, and thermodynamics that are familiar from traditional science. And that's in a sense why traditional theoretical science hasn't been able to make knd progress when one sees complexity. The theory of emergent order, fleshed out. And one will sort of get a whole tree of possible histories for the universe. Scott AaronsonProfessor of Computer Science at University of Texas Austin, also claims that Wolfram's methods cannot be compatible with both special relativity and Bell's theorem violations, and hence cannot explain the observed results of Bell test experiments.
If you can, then what it means is that you can figure out what the system will do with a lot less computational effort than scuence takes the system itself. But what the Principle of Computational Equivalence says is that when we don't see those kinds of regularities, we're almost always dealing with a process that's in a sense maximally computationally sophisticated.
A New Kind of Science
I want to talk about wolfrsm the Principle of Computational Equivalence says about sort of a big question: It's up to a paper a day being published about it. Well, linear Diophantine equations were cracked in antiquity.
And remember, each of these pictures is of the same universal cellular automaton, with the same underlying rules. But more important is that with our ordinary intuition there just didn't seem to be any reason to try the experiments; it seemed so obvious they wouldn't show anything interesting.
Well, the whole point of a model is that it's supposed to be an abstract way of reproducing what a system does; it's not supposed to be the system itself. I have some suspicions, based on thinking about generalizing basic notions of geometry to cover things like cellular automaton rule spaces. But what about the conceptual, theoretical foundations? Well, partly it's because one can only easily see it by doing lots of computer experiments—and it's only with computers, and with Mathematicathat those have become easy to do.
And that inhibits more ice nearby.
Wolfram Science and Stephen Wolfram's 'A New Kind of Science'
This happens to be our friend rule And it's exactly the same thing with models based on cellular automata: He posits that the computational process in the brain of the being with free will is actually complex enough so that it cannot be captured in a simpler computation, due to the principle of computational irreducibility. Wolfram argues that one of his achievements is in providing a coherent system of ideas that justifies computation as an organizing principle of science.
But I rather think that if rule 30 had actually been known in antiquity, a lot of ideas about science and nature would have developed somewhat differently. Starting now, in celebration of its 15th anniversary, Wolrfam New Kind of Science will be freely available in its entirety, with high-resolution images, on the web or for download.
And it was this realization that got me started building Wolfram Alpha. And in fact it's able to emulate absolutely any other cellular automaton, with any rules whatsoever.
A New Kind of Science
Then start from a seed and have a rule that says solid will be added at any cell that's adjacent to one that's already solid. NKS has also been criticized for asserting that the behavior of simple systems is somehow representative of all systems.
But I think that in an ultimate model one only needs space—and one doesn't need any other basic concepts. So the first question was: