3 ms·

These mathematics seem highly dubious to me. It's like they just found a bunch of different ways to describe the same thing, Infinity. Yes 0 is 0. I could also

by SeoxyS 16y ago

These mathematics seem highly dubious to me. It's like they just found a bunch of different ways to describe the same thing, Infinity. Yes 0 is 0. I could also describe it as (5 * 4 * 3 * 0), but how does that help? When you think about it, there is no such thing as infinity. It's just a word and concept used to describe an amount bigger than you're able able to imagine or comprehend. It's something we use because we need to give a name to the "limit" to the far end of a spectrum.

simias 16y ago
Well, I'm not a mathematician but I've always thought there were different "strengths" of infinity. Consider for instance that both lim(x -> ∞) x and lim(x -> ∞) x^2 are infinity, however lim(x -> ∞) x / x^2 and lim(x -> ∞) x^2 / x are not the same thing. That's of course a silly example, but I hope it gets my point across.
dexen 16y ago
Back at the uni, I was shown a nice proof that one infinity (the (infinite) count of natural numbers) is not the same as another infinity (the count of real numbers, IIRC). Basically, it was shown no mathematical function can be constructed that maps one onto another, IIRC. Also, please note the Cantor's theorem [0] -- this bears some relation to programming, as considering subsets of a set is an important problem. In short, it makes sense to discuss different infinities. All that may sound purely academic theory, but please remember current cryptography grew from what once seemed to be just purely academic theory. ---- [0] http://en.wikipedia.org/wiki/Cantor_theorem http://en.wikipedia.org/wiki/Cantor_theorem
radq 16y ago
> When you think about it, there is no such thing as infinity. It's just a word and concept used to describe an amount bigger than you're able able to imagine or comprehend. You said that there is no such thing as infinity, and then you say that it describes a concept. All words describe concepts, don't they? Similarly, I could say, "there is no such thing as node.js, it is just a word and concept we use to describe an event-driven I/O framework for the V8 JavaScript engine on Unix-like platforms". It proves absolutely nothing.
cdavid 16y ago
Those are very conventional mathematics, and most core concepts have been settled for a century now. Cantor in particular paved the way toward a better understanding of many related concepts in that area. As for the significance of those differences: I can think about two areas where this is fundamental (but there are many more). First, the natural numbers (0, 1, ...) vs rationals vs real numbers is really big in terms of "how many" numbers you have. You can prove than the set of natural numbers and the set of rational numbers is roughy the same size (they are both countable). Now, as you may know, there are some irrational numbers, like square root (2), pi, e, etc... But how many ? The answer is "many more than rationals". One way to define the real numbers (rationals + irrationals) is as limits of a set of rational numbers - i.e. for any real number, you can find a set of rationals which get arbitrarily close to an irrational (we say that rational numbers are dense for the set of real numbers). Another field where you see the difference: topology vs probability. Roughly speaking, probability is a function which for a set gives you a number between 0 and 1. The key property of probability is that P(A U B U ....) = P(A) + P(B) + ... where A, B, ... are disjoint sets. Another way to look at it is that probability is a special case of the notion of set measure, which is the mathematical way of talking about volume, length, etc... There is a key restriction when defining measure, that is you cannot take arbitrary unions of sets, like say as many as real numbers - actually, you can prove that you cannot build measures if you want arbitrary (uncoutable) unions. This is a naive way to describe the Banach-Tarski paradox, which says that you can split a ball into two new balls which are identitical to the original one.

by SeoxyS 16y ago

simias 16y ago

Well, I'm not a mathematician but I've always thought there were different "strengths" of infinity. Consider for instance that both lim(x -> ∞) x and lim(x -> ∞) x^2 are infinity, however lim(x -> ∞) x / x^2 and lim(x -> ∞) x^2 / x are not the same thing. That's of course a silly example, but I hope it gets my point across.

dexen 16y ago

Back at the uni, I was shown a nice proof that one infinity (the (infinite) count of natural numbers) is not the same as another infinity (the count of real numbers, IIRC). Basically, it was shown no mathematical function can be constructed that maps one onto another, IIRC. Also, please note the Cantor's theorem [0] -- this bears some relation to programming, as considering subsets of a set is an important problem. In short, it makes sense to discuss different infinities. All that may sound purely academic theory, but please remember current cryptography grew from what once seemed to be just purely academic theory. ---- [0] http://en.wikipedia.org/wiki/Cantor_theorem http://en.wikipedia.org/wiki/Cantor_theorem

radq 16y ago

> When you think about it, there is no such thing as infinity. It's just a word and concept used to describe an amount bigger than you're able able to imagine or comprehend. You said that there is no such thing as infinity, and then you say that it describes a concept. All words describe concepts, don't they? Similarly, I could say, "there is no such thing as node.js, it is just a word and concept we use to describe an event-driven I/O framework for the V8 JavaScript engine on Unix-like platforms". It proves absolutely nothing.

cdavid 16y ago

Those are very conventional mathematics, and most core concepts have been settled for a century now. Cantor in particular paved the way toward a better understanding of many related concepts in that area. As for the significance of those differences: I can think about two areas where this is fundamental (but there are many more). First, the natural numbers (0, 1, ...) vs rationals vs real numbers is really big in terms of "how many" numbers you have. You can prove than the set of natural numbers and the set of rational numbers is roughy the same size (they are both countable). Now, as you may know, there are some irrational numbers, like square root (2), pi, e, etc... But how many ? The answer is "many more than rationals". One way to define the real numbers (rationals + irrationals) is as limits of a set of rational numbers - i.e. for any real number, you can find a set of rationals which get arbitrarily close to an irrational (we say that rational numbers are dense for the set of real numbers). Another field where you see the difference: topology vs probability. Roughly speaking, probability is a function which for a set gives you a number between 0 and 1. The key property of probability is that P(A U B U ....) = P(A) + P(B) + ... where A, B, ... are disjoint sets. Another way to look at it is that probability is a special case of the notion of set measure, which is the mathematical way of talking about volume, length, etc... There is a key restriction when defining measure, that is you cannot take arbitrary unions of sets, like say as many as real numbers - actually, you can prove that you cannot build measures if you want arbitrary (uncoutable) unions. This is a naive way to describe the Banach-Tarski paradox, which says that you can split a ball into two new balls which are identitical to the original one.