But they don’t try repeated positions, shouldn’t it reduce the number of games has to be tried?
It does reduce the number of games they have to try, but there are so incredibly many games, that it doesn’t make even the slightest dent in it.
The computation is dividing 1010¹⁰⁰ by 1080, but that’s approximately 1010¹⁰⁰.
Similar to how adding 10 million to 1080 will not really change the size of the number; 1010¹⁰⁰ has a double exponent, which means that the effect of multiplying or dividing is similar to what the effect of addition and subtraction is compared to single exponents.
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This sounds eerily like mirror go. Does it mean if we can find a mirror go strategy that can guarantee win, then the game is solved?
Sadly, no, because in Go you can play moves that are not beneficial to your position (e.g. filling up an eye), so the mathematics doesn’t work.
Apart from that, some moves can’t be mirrored (such as the mirrored player capturing a group that’s needed to capture its own mirrored group)
I think this depends entirely on how one defines a “super-being”. To me, I would say that super-being should be completely able to play perfectly in any well-defined game with a finite game tree, which is the case for Go.
If we say that a super-being is limited in the size of game trees that they could handle, then I think it becomes a very arbitrary line to draw.
There are different concepts of solved. From Solved game - Wikipedia
A two-player game can be solved on several levels:[1][2]
Ultra-weak
Prove whether the first player will win, lose or draw from the initial position, given perfect play on both sides. This can be a non-constructive proof (possibly involving a strategy-stealing argument) that need not actually determine any moves of the perfect play.
Weak
Provide an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game. That is, produce at least one complete ideal game (all moves start to end) with proof that each move is optimal for the player making it.
Strong
Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides.
@Vsotvep’s discussion of Hex is an example of an ultra-weak solution
For Go without komi (and under some reasonable ruleset), we can also provide a partial ultra-weak solution, by saying that black should at least be able to force a draw (by jigo), by appealing to a strategy-stealing argument (black cannot have a disadvantage for playing first, since passing is always an option to effectively play second; if both players pass and refuse to play a stone, then that should naturally be a draw as well).
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Just to elaborate on this comparison a bit more…
10^48 hours is approximately 10^44 years, which is 10^34 times larger than the age of the universe.
A blink takes about 1/3 of a second, or 10^-8 years. The age of the universe is only 10^18 times longer than the duration of a blink.
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This is starting to fly over my head. But at some level, we limited human with limited understanding seems to comprehend the game of Go somewhat? (Or is it all just our collective delusion?)
We just pick a very limited number of shapes and local moves that look right or pretty for some reasons. However, stronger players indeed can beat others just a few ranks below with incredibly higher chance. What is the “human comprehension” all about? Is it just based on culture and accumulated history norms? If we build upon them along with the help of AI, will we even someday able to form very complicated strategies that constitute a weak solution?
I am confused.
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While we’re on the topic, I remembered this neat site which puts the size comparisons in perspective…
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It’s fascinating the largest scale is called observable universe, is there an unobservable universe? There is more stuff out there? And theoretically a super-being could exist (with the extremely large, but still finite list of all games memorized) and we just cannot see it?
The really crazy thought for me is that the “observable universe” is shrinking as everything accelerates away from everything else.
At some point in the very, very, very (huge number of very’s) distant future, we won’t be able to see any other galaxies in the night sky at all…
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Well, by definition we can’t ever observe it, so it’s not something we can ever determine.
If the super being exists, it consists of many parts that can never interact with each other.
Are there any theories on whether the unobservable universe contains a finite amount of matter? I assume yes, since it’s supposedly torus-shaped? Or could it be a torus of infinite volume?
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The question if the universe is spatially finite would determine if the amount of matter in it is finite.
Both the 3-sphere and the 3-torus hypothesis fall into the finite size category (among many other possibilities).
But I don’t think there is clear evidence yet for the universe being spatially finite, let alone for it being a 3-torus or having some other simply connected topology. It cannot be ruled out, but so can many other topologies. The part that is within our horizon (the observable universe) seems too small and flat to tell the difference.
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Anyway by that time, all stars will have burnt their fuel and humankind will have disappeared a long time before.
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I like this model of the universe 
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Well, it’s “classically” unobservable. It’s the part of the universe that we can never interact with because only radiation from the observable universe has reached us.
I wouldn’t say with confidence that we will never invent faster than light travel or communication, which would modify the definition of observable universe.
I also wouldn’t say that an infinite universe requires an infinite amount of mass. Space itself doesn’t obey laws that matter does. Space can expand faster than the speed of light for example. You can conceive of an infinite universe with a finite mass. Matter doesn’t have to be uniformly distributed in space, and we know it isn’t. It’s far too early to be assigning conditions on the universe, regardless of the topology which we don’t even know yet.
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It’s certainly mathematicallly possible, but it does not seem to be the case for the physical universe we live in?
I think the Big Bang model assumes an extremely uniform distribution of matter at the start (with a finite matter density that decreased over time by the expansion of the universe). If the universe is infinite, the amount of matter would then have been infinite all the time, wouldn’t it?
A cosmological model where the amount of matter was finite and concentrated in some local 3D volume of an infinite 3D universe when the expansion started, would be very different from the currently standard cosmological model I think.
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No, it wouldn’t modify the definition of observable universe, it would modify how much of the universe we consider to be observable: that part of the universe that we could observe would be larger than we expected; it would contain everything that we can currently observe, and then some more. But it wouldn’t contain anything unobservable, since by definition that is impossible.
The problem with finding out whether the universe is infinite or not, is also something we cannot decide, since infinity is unobservable in a strong sense. It’s impossible to put an infinite amount of information in a finite amount of time and space (i.e. our brains).
Asking what’s outside of the observable universe, or asking whether the universe is infinite, are unscientific questions: they can’t be answered by empirical observation.
The problem with models, is that we don’t know if it’s correct because that’s how the world works, or if it’s correct because it’s a great coincidence that the universe behaves exactly like we expect as far as we can test, but without our knowing behaves differently outside of what we can expect.
If I am thinking of a set of numbers, and I ask you to check whether it’s an infinite set or not, but the only thing you can do is ask for an individual number if it’s part of the set or not, then there’s no way you can ever determine whether my set is infinite in a finite number of steps.
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It is theoretically possible that as we try to work out the physics of the universe at more and more fundamental levels (e.g. finally figuring out what’s up with dark matter/dark energy, coming up with working quantum gravity theories and ways to actually test and validate them), we find that our final theory seems to requires for mathematical consistency that the universe be finite, or that it be infinite, or that it has a certain topology, even beyond the part of it that we will ever be able to observe.
If we find that our final theories become inconsistent or nonsensical when we try to assume the opposite, and that the literally only mathematically-feasible ways to achieve the opposite within any working theory that matches experiment involves postulating things like abrupt and discontinuous changes to the laws of physics beyond the region we can observe for no good reason, or other patchwork hacks, while the alternative is something clean and elegant and forever more passes every challenge and test we throw at it… then even if we would never be able to observe beyond the visible universe to rule out those ugly patchwork hack theories, I think that would be pretty suggestive.
Or it might not work out that way. But if it does, it’s possible that this would be the closest we would ever be able to get to determining finite/infiniteness of the universe beyond just the part in our light cone.
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