Comment by js8 | Hacker News Reader

Comment by js8 | Hacker News Reader

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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 machine is implemented. Similarly, this can be case for observers within our universe (humans). So maybe there is, at a high enough level of abstraction, only one mathematically possible universe.

2. The mathematical theorems are shortcuts to computation. For example, I can add two numbers on a computer, or I can use a theorem, which will tell me only some property of the result, but avoids having to do the actual calculation. So if universe is just a computation, one could avoid actually computing it in certain cases by having a theorem that would tell them enough information they need. In other words, you don't actually need to simulate anything in order for objects in simulation to "exist". So there is no need for actual hardware on which the universe runs to exist.

Addendum: So I read the essay, and I disagree that the multiverse is necessary. As I already stated in point 1, it can be the case that all possible mathematical realities are the same at high enough level of abstraction.

> 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.

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

If you like the idea of thinking about physical space in computational terms give one of the classics a try, Calculating Space by Konrad Zuse.

https://en.wikipedia.org/wiki/Calculating_Space

ftp://ftp.idsia.ch/pub/juergen/zuserechnenderraum.pdf

http://www.mathrix.org/zenil/ZuseCalculatingSpace-GermanZeni...

> I read the essay, and I disagree that the multiverse is necessary.

Read his book 'Our mathematical Universe', he explains there why these are actual separate multiverses.

I agree it's somewhat a matter of interpretation whether it's truly "multi" or a single entity. But if the domains are separate enough, it's probably best to think of them as different universes.

Imagine multiverse theory like this: there's a parallel universe just like this one in the same space and there is an infinity of these. And that's just how you would describe a geometric point, a singularity. Then it doesn't make any difference that there are an infinity of different variations in the near vicinity, so to speak, in different dimension. They just don't interact, the point stays a point. A fix-point, so to speak, and the metaverse is the Y-Combinator https://en.wikipedia.org/wiki/Kleene%27s_recursion_theorem

There is a seriously interesting question here about the information content of the theorem and whether it is equivalent to the Shannon limit on compressing the 'code' needed to run the computation. I bet someone published on this years ago but I haven't seen it phrased in quite the way you have it in point 2.

A concrete example of #2 is the hashlife algorithm. Intermediate steps can be skipped in a region of space, yet since it implements the CA nothing inside can notice that the computation didn't actually happen.

Can you expand on #2? I don't understand maybe the word "theorem" in this context, or perhaps the meaning of "property of the result."

"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.

Jesus! I guess I'll just stick to 2+2 holy shit this is well beyond me!

Thanks for the link.

EDIT: Nope guess not now I'm knee deep in this thing :P

For example, consider a rather trivial theorem which says that if you have two positive integers in decimal representations Aaa...aa and Bbb..bb, of same length, then the sum of them is larger than number (A+B)00..00.

So this theorem gives you a property of the result (the result is larger than some number) without you having to calculate the result (sum the numbers) explicitly.

The Hashlife algorithm mentioned in this thread is another case, basically it's an algorithm to prove certain "theorems" about the Game of Life and applying those theorems to speed it up.

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