"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/...
In Go, the number of items is the number of pieces, and it's very small.
In the universe, the number of combinations of positions of all the atoms is, well, wonderful.
Compared with a googolplex (10^(10^100)) the entire Evrettian metaverse is negligible as (10^(10^100) - 10^80^2 * (average quarks in atom) * leptons(10^200) * dark multiplier(10^2) = ~1 googolplex
Has anyone ever used a googolplex for anything ?
[For ~ read approximately]
The real comparison would be the number of pieces on a Go board (19x19 = 361) compared to the number of atoms in the universe. And then to compare the number of possible board positions in Go, with the number of possible atom positions in the universe, and in this case I think the universe wins.....
However, other problems have even larger state spaces. Imagine writing an AI which read project Euler problem descriptions (in English) and output working code (in some given programming language). Keep outputs limited to 100-line scripts, max 80 characters per line.
There's roughly 100 usable characters in ASCII, so the possible space of 100-line programs is roughly:
(10^2)^(80 * 100) = 10^16000.
You could simplify this by having the AI work with predefined tokens rather than individual characters, but it's still a vast amount of combinations. Then consider 1000-line or 10000-line programs, and you see how high a mountain AI still has to climb. Humans are able to "compress" this state space via conceptual reasoning, which is much more complex than the "pattern recognition" many deep learning researchers are chasing.
(See "Introduction to Objectivist Epistemology" for more on how humans think in concepts - I'm planning to write more at some point on how this book shows where the practical limits of AI lie).
> And in Go even an amateur human can still rout the world’s top-ranked computer programs
However, I think I found a mistake:
"For example, ‘5 tetrated to the 3’ means 5 raised to its own power 3 times, or 5^5^5"
(I am paraphrasing slightly here because the essay uses an image to show 5^5^5 in normal notation (http://www.scottaaronson.com/cgi-bin/mimetex.cgi?5^{5^5}))
However, shouldn't this be 5^5^5^5, if we're raising 5 to its own power three times?
5 x 3 = 5 + 5 + 5
5 ^ 3 = 5 x 5 x 5
5 t 3 = 5 ^ 5 ^ 5
Where t is tetration. Each one counts 3 fives.
"..the number of atoms in a grapefruit is about equal to the number of blueberries you would need to fill up the entire sphere of planet Earth." [https://capitolhillscience8.wordpress.com/2012/10/03/just-ho...]
Edit: well, except that the Earth is shaped more like an oblate spheroid [https://en.wikipedia.org/wiki/Figure_of_the_Earth]
And not only do you live on a poppy seed, you live on a very thin layer on the surface of this poppy seed. Blow the poppy seed up to the size of a basketball and the habitable layer is about the thickness of a sheet of paper.
There's a few interesting things on there!
+ applied N times becomes ×N
× applied N times becomes ^N
^ applied N times becomes ...?
etcetera
And would such a theory have any practical use?
As for practicalities, the mathematician in me will let the scientists deal with that.
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).
How would we enumerate all these several gazillion image possibilities?
Well. Let's say number one is all black. Every pixels and every channel is all zero in its value. And let's say the last image to be enumerated is all white. 255 for each pixel and each channel.
Every conceivable image is created in between these two ends. For example, image two is all black, but the last pixel has a value of 1 instead of 0 for its value channel.
Image 1840274917 has pixel 27581 slightly reddish.
Hey, wait a minute, you've just created an image format for describing the data within the image! The only space you're saving is that (given this format) you save space on darker images, because they're likely lower in the sequence.
But that's only because this specification demands that each image be the same exact size and can make assumptions based on that. A lossless format like PNG would be able to perform much better over a wider range of images. (Eg all white will be huge in our system, but cheap in PNG)
The 'Everything' Formula - http://youtu.be/_s5RFgd59ao
Disclaimer: I am not a physicist, but I love this theory
This again reminds me how science today is no different than religion. Of course there is nothing wrong with having the number based on assumptions , but take it out of the field of study and suddenly it is a fact :)
Like in this article and the discussion in HN where its just the number and its name, and the fact that this him be is just somebody's wild guess is totally ignored. Same with Jesus , he exists and he loves you and the fact that it was just somebody's idea is totally left out.
This is a fun, thought-provoking story about a large combination of things.
Assuming the many worlds interpretation of quantum physics is true, then the number of atoms includes all combinations of locations, momentum, etc., and the real number of atoms is vastly vastly greater than combinations of just about anything else you might imagine. (Except for combinations of configurations of quantized spacial points!)
From this point of view this article is inappropriately comparing two scales. It's nothing more than saying "e^x >> x for big x".
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.
I'm not so sure. Is there any evidence that the value of any physical quantity is an irrational number?
Math was originally inspired by physical reality, but I'm not sure it's so closely related that the concept of the Axiom of Choice being "true" even means anything.
Interesting POV.
59! =~ 1.38 × 10^80
