If you believe in the axiom of choice (I don't) then you can imagine a process which has more degrees of freedom in in than anything at all.
On the (Small) Number of Atoms in the Universe
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Re: On the (Small) Number of Atoms in the Universe
#22Scott Aaronson's blog post on large numbers is also a very interesting read:
Re: On the (Small) Number of Atoms in the Universe
#23And it's even smaller if compared to Graham's number. Every time I try to imagine that one it feels like I'm going to mental asylum.
Related only to large numbers, but is there some theory about generalizing and extending our usual mathematical operators +, ×, and ^ (power)? + applied N times becomes ×N × applied N times becomes ^N ^ applied N times becomes ...? etcetera And would such a theory have any practical use?
> And would such a theory have any practical use?
Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function).
Re: On the (Small) Number of Atoms in the Universe
#24If you believe in the axiom of choice (I don't) then you can imagine a process which has more degrees of freedom in in than anything at all.
It is an axiom, it is not something that you believe in. You assume it and you prove stuff with it. You do not and you prove stuff without it. If you are really good, you show what theorems require it.
You can believe or not in some physical reality to math (I happen to), and then the axioms do become somewhat more than just logic statements, but that is different.
Re: On the (Small) Number of Atoms in the Universe
#25There is a theory that states the number of atoms (well, electrons) in the universe is exactly 1.
Re: On the (Small) Number of Atoms in the Universe
#26There is a theory that states the number of atoms (well, electrons) in the universe is exactly 1. https://en.wikipedia.org/wiki/One-electron_universe
But electrons interact with each other, don't they? How'd that work?
Re: On the (Small) Number of Atoms in the Universe
#27I think he's comparing apples with oranges: maybe 10^80 is not so big, but the number of configurations of the 10^80 atoms is huge.
That is entirely the point of the article.
It would be cool if there were more talk about properties we have observed of these configurations/which are more probable at any given instance, and efficient ways of computing such.
Re: On the (Small) Number of Atoms in the Universe
#28Earlier quoted context omitted.
Related only to large numbers, but is there some theory about generalizing and extending our usual mathematical operators +, ×, and ^ (power)? + applied N times becomes ×N × applied N times becomes ^N ^ applied N times becomes ...? etcetera And would such a theory have any practical use?
> And would such a theory have any practical use? Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function). [1]: https://en.wikipedia.org/wiki/Ackermann_function
Given that Graham's Number is an (over-)extension of the Ackermann-function that means that it's still computable, correct?
Given the series leading up to Graham's Number, G = g_64 (g_1, g_2, ...), BB(n) will outgrow g_n, right? If so what's the smallest n such that BB(n) > g_n?
Re: On the (Small) Number of Atoms in the Universe
#29I like how Ken Jennings dealt with the 'Go complexity' analogy:
"Go is famously a more complex game than chess, with its larger board, longer games, and many more pieces. Google’s DeepMind artificial intelligence team likes to say that there are more possible Go boards than atoms in the known universe, but that vastly understates the computational problem. There are about 10^170 board positions in Go, and only 10^80 atoms in the universe. That means that if there were as many parallel universes as there are atoms in our universe (!), then the total number of atoms in all those universes combined would be close to the possibilities on a single Go board."
http://www.slate.com/articles/technology/technology/2016/03/...
Re: On the (Small) Number of Atoms in the Universe
#30There is a theory that states the number of atoms (well, electrons) in the universe is exactly 1. https://en.wikipedia.org/wiki/One-electron_universe
But electrons interact with each other, don't they? How'd that work?
It's the same electron interacting with itself at different times is what I think Wheeler would argue
Disclaimer: I am not a physicist, but I love this theory