In your example, I would say that it's
addition that's failing.
Addition is defined as an operation with two inputs. You can't add more than two things, unless there is some particular rule that lets you.
If you have finitely many things, then this rule is the associative law: add them pairwise in whatever order, and you are guaranteed to get the same result.
To add infinitely many numbers, you need to talk about limits. Formally, when you say something like
1 - 1/2 + 1/4 - 1/8 + 1/16 - ...
you mean: look at the sum of the first two; then, look at the sum of the first three; then, look at the sum of the first four; and so on -- this sum converges to a limit, which is 2/3.
This sum is "absolutely convergent", which means you get the same result no matter how you order the summands, but some infinite sums change if you reorder things!
With points on the sphere the situation gets even worse, as there is no way to "list them in order". These sets are "uncountable", which means don't even try to sum any function defined on them.
To say approximately the same thing using technical jargon, one has countable additivity for Lebesgue measure on the reals, but uncountable additivity does not hold.
https://mathworld.wolfram.com/CountableAdditivity.html