My (vague) intuition is that infinity is not a well defined unique number, so you can’t really use it in arithmetic.
But +infinity and -infinity are distinct from NaN, because they have different constraints. NaN does not have a +/- sign, but it is limited to finite values. The opposite goes for +infinity and -infinity.

So my intuition about categorization of various quantitative entities is something like this:

Does there exist some formal categorization like this?

The word “infinity” has several meanings, which may cause confusion.

Infinite cardinals: a cardinal is the “number of elements” of a set. There is an infinity (!) of infinite cardinals.

Infinite ordinals.

Limits of sequences of real numbers: there are only two infinities, +∞ and -∞.

Limits of sequences of complex numbers: there is only one infinity (adding a point to a plane gives a sphere).

Points at infinity in projective geometry. For instance the projective plane is the union of an ordinary plane and of a projective line, the “horizon”. Points of the “horizon” are called points at infinity.

So before categorizing, you need to say which kind of “infinity” you are talking about.

I was mostly thinking about the integers (but perhaps also rational numbers or p-adic rationals like in games and numbers, I don’t know if that would make things very different).

Generally when you think about integers and write 1/0 = ∞, you are thinking of infinity in the sense 4 (there is no reason to put a sign in front of ∞ since 0 has no sign), so there is only one infinity here.

Do I understand correctly then that -1/0 = ∞ if -1 is interpreted as a complex number or integer (case 4), but -1/0 = -∞ if -1 is interpreted as a real number (case 3)?

No I wasn’t clear. You can’t write things like -1/0 = -∞ either. Signed infinities (+∞ or -∞) are only used to write limits of real valued sequences or of real valued functions, like lim (1/sin(1/n)) = +∞ or lim (-1/sin(1/n)) = -∞.

The idea is that you are adding two extra points to the real line, and a sequence may converge to one of these points since each term of a sequence has a well-defined sign.

But an expression like 1/0 doesn’t represent a sequence, so you may view it either as a “meaningless expression” (NaN as you said), or you may want to view it as the limit of 1/u_{n} for any sequence u_{n} of nonzero real numbers which converges to 0. That makes sense only if you only add one point at infinity to the real line.

People tend to grab a random number of stones, in the absence of magic. There is no natural probability difference between an odd and even number. On the other hand, if a player is allowed to choose only a small number k of stones, such as K<4, then the drawing is clearly unfair.

It’s a pretty good way of choosing who gets black, and more pleasing to a go player than flipping a coin or rolling a die.

In the history of AI, Perceptrons were the first version of neural networks, and it proved mathematically impossible to train a Perceptron to distinguish the parity of a number of objects. Humans, also, cannot distinguish the parity of a large number of objects without some sort of counting or pairing procedure, with the possible exception of certain savants.

But they can be paired, there’s just no pairs If they couldn’t be paired, there would always be something left over, but there’s nothing in zero to begin with, so nothing to be left over.

But this begs the question why -1 is odd. If the even-ness of 0 is debatable, so is the odd-ness of -1.

In mathematics: if it makes whatever you do more easy, then yes, if it makes whatever you do harder, then no

To be precise, it’s not generally defined, but there’s nothing against defining it on the spot to make things work for you.

Second question is how to define it: clearly it’s the limit of 0=0/x with x getting smaller and smaller, so the value of 0/0 should be 0. But it’s also the limit of x/(x^2) = 1/x which grows infinitely large from the positive side so 0/0 should be positive infinity. But it’s also the limit of x/x = 1, so it must be 1. !?

A prime number is a number that is divisible by exactly two positive integers. 0 is divisible by every positive integer, so it’s more like an anti-prime.

Positive numbers are defined as numbers larger than 0, negative numbers as numbers smaller than 0 (or as additive inverses of positive numbers). That makes 0 neither positive nor negative. When including 0 with all positive numbers, in mathematics these are called “non-negative numbers”.

It really depends on definitions (as usual). Any high school teacher as well as any calculus teacher will repeat time and time again that infinity is not a number. The reason being that the standard arithmetic that we know from real numbers don’t work with infinite values.

You could define + / - infinity as some sort of number and define separate arithmetical rules, for example in the extended real line, but it’s crucial to realise that some arithmetical rules will fail. For example, A+B = A+C does not imply B=C anymore (take A infinite), there’s no unique additive or multiplicative inverses, and certain terms are still undefined, such as 0 * infinity.

You could consider infinite numbers that are more suitable to arithmetic, such as ordinal numbers, cardinal numbers, or surreal numbers. Especially the latter behave very nicely. However, these kind of infinities are very different from the type that we use in high school mathematics (e.g. the type where 1/x tends to + infinity when approached from above): limit infinities are not numbers.

I see you put Games in there, which is interesting, since the concept of a (combinatorial) game is closely related to the concept of a surreal number, except that in surreal numbers we kind of define a new number by filling a gap between two sets of “small” and “large” numbers (where each small number is smaller than each large number). In games we fill the gaps between any sets of games, so there’s no “small” / “large” distinction.

An even more fundamental question is, what result are you hoping for when you do nigiri? Depending on your personal preferences, whether or not you belief komi to be fair, which color you think komi might be biased toward, etc., I think there could be a diverse range of preferences, including (but not limited to):

You want to play black

You want to play white

You want the colors to be assigned with 50-50 odds

Other possibilities? (like 30-70 odds, etc??)

Maybe these preference might also depend on the context and who your opponent is? Maybe one opponent you perceive to be particularly weak as playing fuseki as black, while another you think is weaker in fuseki as white>

Let’s not forget that whilst “winning” the nigiri in most rulesets means that you play Black, in the Ing rules it means that you have the choice of playing Black or White.

If anything, at the current value of komi professionals seem to prefer White – see, for instance, that Ke Jie chose to take White twice against AlphaGo in 2018.

I think that preference for white is only the case with a “large” komi of 7.5+ (common with Chinese rules, AGA rules, ING rules etc), not with 6.5 komi (common with Japanese rules, Korean rules).

As far as I know, pro game statistics and KataGo give white about 51% win rate for 6.5 komi under territory scoring rules. Such a small advantage for white doesn’t seem to justify a clear preference from pros.

I don’t know of any pro game statistics for 7.5 komi under area scoring rules, but KataGo gives white 59% win rate, which is a significant advantage for white.

Note that black usually has to win by 9 points on the board to win with 7.5 - 8.5 komi under area scoring rules. With 7 komi under area scoring rules, KataGo gives white 51% win rate.

I tried to find out if white or black favors me but the tool https://avavt.github.io/gotstats doesn’t say anything about OGS games. So I looked at my IGS games (net social plaza): I have played 230 even games as black with winrate 58% and 237 even games as white with winrate 57%. The conclusion is that the color is indifferent.