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Shut up and calculate: an extreme mathematical universe

arxiv.org

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Re: Shut up and calculate: an extreme mathematical universe

#21

Post by jxy »

vorg wrote: post #15

The author asks how we could use the mathematical equations which describe all of reality to compute the frog’s view of the universe — our observations — from the bird’s view. Wouldn't we also need to include our self-awareness in those observations, which would make the computation impossible according to Turing's halting theorem, or is it Godel's?

The basic result from Turing/Godel is that you don't need any self-referential. As long as it is an enough powerful formal system, it is always incomplete from day one. There will always be some mythical things you cannot compute. There will always be some universes you cannot prove their existence or nonexistence.

Re: Shut up and calculate: an extreme mathematical universe

#24

Post by Koshkin »

kaffeemitsahne wrote: post #20
jcmoscon wrote: post #14

Math works because reality works. Not the opposite. Reality is bigger than math. Math is just a tool that models reality.

..and with one simple statement, HN user jcmoscon forever put an end to this age-old debate.

Still, if you think about it, that statement is very deep (and is probably true).

Re: Shut up and calculate: an extreme mathematical universe

#26

Post by Avshalom »

_v7gu wrote: post #16
mazsa wrote: post #8

Earlier quoted context omitted.

"The complete proof of 2 + 2 = 4 involves 2,863 subtheorems including the 189 above. (The command "show trace_back 2p2e4 /essential" will list them.) These have a total of 27,426 steps—this is how many steps you would have to examine if you wanted to verify the proof by hand in complete detail all the way back to the axioms." http://us.metamath.org/mpegif/mmset.html#trivia

Oh jeez dude, just use ZFC and you'll get it done in ten minutes

Well that's precisely the point, if you pick your axioms/theorems/lemma's right you get to skip a lot of steps and only calculate a particular end result instead of every turtle along the way.

Re: Shut up and calculate: an extreme mathematical universe

#27

Post by teraflop »

kybernetikos wrote: post #19
js8 wrote: post #2

I like the idea that the reality is just mathematics, I thought about it independently, but it gets weird really fast. I yet to have read this essay, but I had two thoughts: 1. Similar to Turing completeness, actual physical reality might not matter for the internal observer. All Turing-complete machines can calculate same things and observer from the inside of the machine cannot determine on what hardware the machin…

> So there is no need for actual hardware on which the universe runs to exist You may enjoy the science fiction book Permutation City

If we're talking about Greg Egan, then I'll go ahead and also suggest his short stories Luminous and Dark Integers. They postulate a similarly tight connection between physical processes and mathematical theorems, and then go on to consider the implications of the fact that such a system can't prove its own consistency.

Re: Shut up and calculate: an extreme mathematical universe

#28

Post by ChrisClark »

kybernetikos wrote: post #19
js8 wrote: post #2

I like the idea that the reality is just mathematics, I thought about it independently, but it gets weird really fast. I yet to have read this essay, but I had two thoughts: 1. Similar to Turing completeness, actual physical reality might not matter for the internal observer. All Turing-complete machines can calculate same things and observer from the inside of the machine cannot determine on what hardware the machin…

> So there is no need for actual hardware on which the universe runs to exist You may enjoy the science fiction book Permutation City

And Egan's other book Distress. Which goes further into the OP's second point. Not actually needing to calculate everything.

Re: Shut up and calculate: an extreme mathematical universe

#30

Post by eref »

bitL wrote: post #23

Alright, why not go one step further and make Universe computational instead? (i.e. state is important in von Neumann-style instead of meaningless in classical math-style; or algorithms vs formulas)

Tegmark's Mathematical Universe Hypothesis (MUH) is actually pretty much a computational theory of the universe. It basically makes the assumption that there exists somewhere the simplest process imaginable: One that counts through and runs all mathematical formulas which includes all programs, which, in turn, includes all (computable) universes. This assumption explains why our universe is oddly complex: If all universes exist, then a weird one such as ours merely exists because all universes do. This line of reasoning is also called the "anthropic principle".

I think, Tegmark does not talk about computations per se because the hypothesis is agnostic about what kind of computer our universe is. It could, for example, be a geometric computer in which the position of objects can be determined with infinite precision (i.e. using the real numbers). Such a computer would be strictly more powerful than a Turing machine or equivalent (i.e. all computational models that can be described and run inside a Turing machine). Enumerating the space of all formalisms (i.e. the domain of mathematics) is as agnostic as you can be.

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