4 ms·

But with mathematics, nitpicking is ~kind of~ the point. For example when dealing with real numbers, you can get away with defining Compact sets as bounded and

by ulucs 10y ago

But with mathematics, nitpicking is ~kind of~ the point. For example when dealing with real numbers, you can get away with defining Compact sets as bounded and closed sets; however when you enter a different Topological space, that definition will surely fall apart. So in retrospect, defining compact as bounded and closed was working in your limited test cases (real numbers) but in the real world you really have to embrace that it means there exists an open cover that a finite subcover covers the set. It is not pretty or clever as the limited case, but it is what you need to do if you want applicable flexibility. Same with fitting polynomials to data. Statistics and Machine Learning specifically warn you of overfitting. You can get 0% training error if you try to fit MNIST with a 60,000th order polynomial, but can you imagine the test error rate?

belovedeagle 10y ago
> that definition will surely fall apart In what sense could that even conceivably be a true statement? I could define compact sets as sets which make me think of the color purple and it would be a completely valid definition. Nothing will fall apart. I think you are vaguely referring to the fact that proofs concerning compact sets in general topological spaces are not valid if you replace compact sets with bounded and closed sets, but that's just confusing the term "compact set" with what it refers to; the proofs aren't about the term but about the thing, which doesn't change. If "compact set" meant "bounded and closed" and, say, "cover-compact set" referred to sets where (ahem) every open cover has a finite subcover, then the concept of "cover-compact" will have certain general proofs associated to it and the term "compact" will have a different collection of general proofs associated to it. So in some ways you're making the same mistake the author did: carefully selecting a certain way to generalize a concept (real -> topological) and a certain ill-conceived claim about the original concept (compact=bounded and closed), and then acting all surprised when, oops!, the claim doesn't generalize.

by ulucs 10y ago

belovedeagle 10y ago

> that definition will surely fall apart In what sense could that even conceivably be a true statement? I could define compact sets as sets which make me think of the color purple and it would be a completely valid definition. Nothing will fall apart. I think you are vaguely referring to the fact that proofs concerning compact sets in general topological spaces are not valid if you replace compact sets with bounded and closed sets, but that's just confusing the term "compact set" with what it refers to; the proofs aren't about the term but about the thing, which doesn't change. If "compact set" meant "bounded and closed" and, say, "cover-compact set" referred to sets where (ahem) every open cover has a finite subcover, then the concept of "cover-compact" will have certain general proofs associated to it and the term "compact" will have a different collection of general proofs associated to it. So in some ways you're making the same mistake the author did: carefully selecting a certain way to generalize a concept (real -> topological) and a certain ill-conceived claim about the original concept (compact=bounded and closed), and then acting all surprised when, oops!, the claim doesn't generalize.