Go boards move > atoms.?

General Go Discussion
91

But that’s my point, this is impossible, mathematically speaking. For any theory that only works if the universe of discourse is infinite, if we take any finite part of said universe, we can create a different theory that behaves exactly as the infinite theory for the finite part, but does not require the rest of the infinite domain.

For example, if we take a theory like Peano Arithmetic, which requires an infinite universe since every natural number has a successor (which is different from all previous numbers), and we take a finite set of examples to check, say, we check that the number 10 exists, we check that it has a successor 11, we check that 5 * 8 is defined, yup, it does, it’s the number 40, we check that 2+3 is equal to 3+2, yes, once again seems to check out, then from only these examples we cannot conclude that we are working with the natural numbers: it might be that we’re working with the numbers 1 to 50, and just never bothered to check whether 50 has a successor.

Our theory may be requiring the universe to be infinite, but we cannot empirically determine that it is the correct theory, and not some other theory that works equally well, but only on a (large enough) finite part of the universe that the original theory requires.

This isn’t a constraint on humans, or on our physics, it’s a mathematical / logical constraint: it’s impossible to verify empirically whether something is infinite.

Nowhere in nature is it necessary that a theory is clean or beautiful. Quite the contrary, actually, there are a lot of things that appear to be true on first glance, but actually turn out to be a lot more ugly than expected. There exist extremely simple questions that can have incredibly complicated answers (think of Fermat’s last theorem, or the four-colour theorem).

Even proving consistency of a theory is something ugly: mathematics is either inconsistent (i.e. contains a contradiction), is relatively trivial and useless (i.e. cannot assume that all natural numbers exist) or cannot prove that it is correct and will always contain unprovable statements (by the incompleteness theorems).

There’s really no reason to assume that the truth is clean and elegant.

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92

Even in a finite but an extremely large set of Go positions, can we even verify some assumptions to be true? Like are there board positions that will never be part of a normal game play (think of most tsumego problems don’t appear in normal game plays? Since they usually have different amount of black and white, if pass is not added into part of the build up sequences, they would not be part of a normal gameplay to reach at that point), and how many of them are there? Do we have to check every single positions against all game plays to be sure? If so, then although it is theoretically possible, but we will never know.

Or what’s the longest cycles for positional super-ko to be forced to repeat? There are pretty huge and complicated problems that no one is ever sure the answers to the mainline and their variations that would lead to. We will never be sure it is the longest unless we check them all. Hence, if something is not feasible to be checked, then it would remain unknown for us.

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93

For now. I’m reminded of when we thought we pretty much had physics “solved” and just needed to work out a few small issues like the ultraviolet catastrophe.

Personally I refrain from saying things like “we will never know what it outside of the observable universe today” because we might know one day…

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.

That’s what I said, just that you disagree on the semantics.

We also don’t have to limit our imagination to the “regular” topology that we can easily conceive of. The universe could be finite and unbounded. It could be infinite, but bounded (makes more sense to me but making sense is hardly evidence let alone proof)

94

But how? Something that’s unobservable is literally impossible to observe, so how can we ever find observational evidence of its existence? If there is no observational evidence, then how can it be empirically justified? Observing the unobservable universe is a contradiction in terminis, not a lack of understanding or of technology.

I’m not saying that we’ll never be able observe something outside of what we currently think we can observe, I’m saying that that part of the universe that actually is unobservable will always remain unobservable. I’m also saying that an infinite amount of things is unobservable, since observations are finite events; we cannot observe an infinite amount of things with only finite resources.

95

Well, physically it’s impossible that we’ll play all the Go positions, yet if you give me any board position, I can play a game that results in it. If pass is not included, we restrict the possibilities, and it’s once again hard to count those board positions, since stones can be captured, thus simply counting the number of black and white stones is not going to be of much help.

It’s a bit like the question if there’s any tree in the world that I’ll never see. I’m sure that’s the case, but if you’d tell me right now which tree it is, I could just go there and see it :slight_smile:

A lot will remain unknown, but we can’t know beforehand what it will be, just that it will be.

96

Okay let me be more precise than “clean” or “elegant”: regardless of how complex or simple or human-understandable the laws of physics are, “the most fundamental laws of physics are the same everywhere in the universe, the only reason things are different in different places is because the stuff there is different, not because the laws governing how the stuff evolves are different” is by far one of the most powerful principles that has driven our ability to predict the universe and proven time and time again to be effective, and which we see no convincing violation of pretty much anywhere in the observable universe as far as we know.

I’m making the simple observation that in principle it is possible that we may find that our final theories of physics, plus the additional supposition “the laws of physics are the same everywhere in the universe” - note truly a supposition that you are free to reject since we could never actually prove it - imply that the universe is finite. Or that it is infinite. As you say, we could of course never rule out the laws of physics suddenly changing beyond the edge of the observable universe in completely arbitrary ways. So we can never “know” for sure. But if you do grant that supposition, it becomes possible, although I don’t know how likely it will turn out that way! If you don’t, of course it doesn’t.

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97

Ok, let me clarify my point as well, then.

I think that the realm of physics is describing laws and gathering empirical evidence in an effort to support those laws or to contradict them. If a law is contradicted, it cannot be true, but if a law is supported, that only makes the law a good predictor for reality, but not reality itself.

Claims that something is working a certain way need to be verifiable in the sense that it must be possible to find a counterexample if the law turns out to be false. The claim that a universe has a certain fixed finite size is something that we can show to be incorrect, simply by finding stuff that’s larger than the finite size permits. However, the claim that a universe is finite (so an unfixed finite size), or alternatively that it is infinite, is not something that we can verify: there is no observation that can tell us which of the two it is (again, since observations are finite things).

Claims that are not verifiable are not part of physics, but of philosophy, or religion, or idle speculation. The claim that the laws of physics are uniform is not something we can verify, thus it’s not a law of physics.

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98

This is where I get this question from

It looks like a special type of positions that required some pass moves to reach (with several different ways), so in a sense there are definitely positions that can be categorized as “normal”, and some that required multiple pass moves to reach (including most tusmego positions). They can even be further divided into at least 1, 2, 3, etc. pass moves to reach. Although they are definitely finite, but not feasibly possible to give numbers to. Can we even reasonably estimate the portion of them?

And my intuition would say the “normal game play” positions are comparably very few to all possible positions, but what’s the portion be like? I have no idea. Not even an educated guess in range.

The positional super-ko repeating cycle length problem is even harder, I don’t even know where to begin.

99

Not without making precise what exactly is meant as “normal”. Can we count the number of “rich” people in the world? We’d need to make precise what exactly is meant by “rich”. Do we just make a threshold? If so, and some person in Bangladesh is making 5 times the average income of the country, doesn’t that count as rich comparably? What about a millionaire who is stranded on an island somewhere without access to money, is that still a rich person? How about someone who won the lottery but doesn’t know it yet?

To count things it needs to be precise what exactly is counted. Intuitively “normal” positions aren’t exactly precise, and furthermore can be subjective (like a 30k will find very different things normal than a 9p). Does the game where Shin Jinseo misclicked because of a malfunctioning mouse count as “normal”? They continued the game for quite a while after that.

Suppose each atom in the universe was a game of Go, normal or abnormal, would you estimate that there would exist at least one atom that was a normal game?

If you’d answer this with “yes”, then still there would be more “normal” Go games than atoms in the universe. :slight_smile:

Let’s look at it alternatively. On average, in a game, how many moves would you consider to be reasonable moves? This is of course subjective, but suppose we aim low and say that there are on average somewhere around 3 moves that could be considered reasonable.

A (short-ish) Go game lasts roughly 250 moves, assuming the players continue until endgame. So that means that as a very rough estimate, there would be about 3250 ≈ 10119 many “reasonable games” that can be played.

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100

I would say the simplest definition is a position that can be reached without any pass moves, and each side has to play one after another. And I don’t think the total number of moves is the issue, but only categorize all positions into which can be reached with 0 pass, at least 1 pass, 2 passes, etc.

In this case, most handicap games would be categorized into one of the requiring pass categories from the start (black play, passes till white). This is why I feel normal “0 pass” games are proportionally very few. And they don’t just include finishing end game positions, but all in between from the first move.

101

The size comparisons site, loved this, thanks for sharing!

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102

Here is a similar video with the orders of magnitude explicitly marked

Note that Earth is about 10^7 meters, while the observable universe is about 10^27 meters.

Saying that there is only 20 orders of magnitude between the two almost makes the difference seem small, but what it really should indicate is that going up in orders of magnitude grows unfathomably fast.

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