My problem with this is how do you know the proper way to take the limit? The scenario isn’t a limit. It’s just something with infinities in it. The limit is something we force onto it.
And as such, it is ambiguous. The limit is how we resolve the ambiguity.
If you take a particular converging limit, that gives you a resolution to the ambiguity, but the idea of taking a limit does not. You’re just picking your answer when you pick your limit.
Deciding what to do in situations involving infinities by invoking limits is not a well-defined decision theory.
Deciding what to do in situations involving infinities by invoking limits is not a well-defined decision theory.
Well yes, because those situations are not well-defined without some additional structure capturing something which also describes the limiting process.
There is no additional structure. It’s not as if we can come upon two pairs of spheres, and notice while the end result is the same, they’re the limits of two different processes, and therefore different choices are better. There is only the infinite case. If you want to consider sequence that converge to it, there’s no clear way to decide which sequence to look at.
Limits help you if you’re looking at an extreme value. If the limit as the population goes to infinity is that a random sample of X of them will give you Y confidence on a poll, then you can just use that if there’s a large population. If you’re dealing with the limit itself, it doesn’t always help. You can start with a square, and then cut little squares off of the corners, and then more squares off of those corners etc. until you approach a circle. The perimeter will always be four times the length, but this won’t be true of the circle.
In this problem, you can get literally any answer if you take the limit appropriately, so once you’ve decided on the right answer, there is some way to get to it with the limit, but deciding that the right answer is one that a limit converges to helps you not at all.
The problem isn’t a sequence of finite cases. It’s just the infinite case all by itself.
You’re completely right! As stated, the problem is ill posed, i.e. it has no unique solution, so we didn’t solve it.
Instead, we solved a similar problem by introducing a new parameter, \alpha. It was useful because we gained a mathematical description that works for very large n and s, and which matches our intuition about the problem.
It is important to recognize, as you point out, that that taking limits does not solve the problem. It just elucidates why we can’t solve it as stated.
And as such, it is ambiguous. The limit is how we resolve the ambiguity.
If you take a particular converging limit, that gives you a resolution to the ambiguity, but the idea of taking a limit does not. You’re just picking your answer when you pick your limit.
Deciding what to do in situations involving infinities by invoking limits is not a well-defined decision theory.
Well yes, because those situations are not well-defined without some additional structure capturing something which also describes the limiting process.
There is no additional structure. It’s not as if we can come upon two pairs of spheres, and notice while the end result is the same, they’re the limits of two different processes, and therefore different choices are better. There is only the infinite case. If you want to consider sequence that converge to it, there’s no clear way to decide which sequence to look at.
Limits help you if you’re looking at an extreme value. If the limit as the population goes to infinity is that a random sample of X of them will give you Y confidence on a poll, then you can just use that if there’s a large population. If you’re dealing with the limit itself, it doesn’t always help. You can start with a square, and then cut little squares off of the corners, and then more squares off of those corners etc. until you approach a circle. The perimeter will always be four times the length, but this won’t be true of the circle.
In this problem, you can get literally any answer if you take the limit appropriately, so once you’ve decided on the right answer, there is some way to get to it with the limit, but deciding that the right answer is one that a limit converges to helps you not at all.
The problem isn’t a sequence of finite cases. It’s just the infinite case all by itself.
You’re completely right! As stated, the problem is ill posed, i.e. it has no unique solution, so we didn’t solve it.
Instead, we solved a similar problem by introducing a new parameter, \alpha. It was useful because we gained a mathematical description that works for very large n and s, and which matches our intuition about the problem.
It is important to recognize, as you point out, that that taking limits does not solve the problem. It just elucidates why we can’t solve it as stated.