Well yeah, ultimately the only complete mathematical representation of the universe IS the universe. We can only ever approximate or make generalisations... at least without building another universe for doing experiments in.
In truth it's not that fractals underlie everything, its that they're just another pattern that emerges from mathematical relationships. They emerge in nature because collections of plant cells, for example, are a really specialised computer for making plants that happens to have some recursive fractal parts to its algorithm.
Mathematics is a language, invented by humans to describe the universe. To say that the universe is a complete mathematical representation of itself makes as much sense as saying the universe is a complete English representation of itself (which would be silly, because everyone knows the universe is written in Latin, or possibly Hebrew...)
Saying cells are "really" computers using recursion is exactly as meaningful as saying cells are "really" a choir singing a round. One metaphor is more useful, yes, but both are equally meaningful.
My favourite example of this kind of thing is Richard Dawkin's claim that viruses are "really" biological chain letters.
I could, and in the past have, argued this point at length (Someone is wrong on the Intenet! Oh noes!) so I'll stop here. Feel free to go back to being wrong :-D
I think you have thought about this a lot harder than me and clearly have a lot more invested in the subject. :)
You're absolutely right, of course. I don't mind admitting I'm a lazy writer in LJ comments!
I'm sure you know what I was getting at--not that mathematics underpins everything, more that if you wanted to specify the universe entirely mathematically you'd been a universe's-worth of mathematics. It's incomplete. Just one useful way of looking at things, as you say.
Actually I didn't get that you were saying "if you wanted to specify the universe entirely mathematically you'd been a universe's-worth of mathematics." I pretty much agree with this, although I've spent a depressing amount of time arguing with mathematicians of a Platonic bent who would vehemently disagree with it. Thus my investment :-P
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In truth it's not that fractals underlie everything, its that they're just another pattern that emerges from mathematical relationships. They emerge in nature because collections of plant cells, for example, are a really specialised computer for making plants that happens to have some recursive fractal parts to its algorithm.
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Mathematics is a language, invented by humans to describe the universe. To say that the universe is a complete mathematical representation of itself makes as much sense as saying the universe is a complete English representation of itself (which would be silly, because everyone knows the universe is written in Latin, or possibly Hebrew...)
Saying cells are "really" computers using recursion is exactly as meaningful as saying cells are "really" a choir singing a round. One metaphor is more useful, yes, but both are equally meaningful.
My favourite example of this kind of thing is Richard Dawkin's claim that viruses are "really" biological chain letters.
I could, and in the past have, argued this point at length (Someone is wrong on the Intenet! Oh noes!) so I'll stop here. Feel free to go back to being wrong :-D
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You're absolutely right, of course. I don't mind admitting I'm a lazy writer in LJ comments!
I'm sure you know what I was getting at--not that mathematics underpins everything, more that if you wanted to specify the universe entirely mathematically you'd been a universe's-worth of mathematics. It's incomplete. Just one useful way of looking at things, as you say.
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Thanks for the clarification!