Hacker News Reader

Comment by impendia | Hacker News Reader

The problem is that "a number" is meaningless.

Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, you can define the "extended real number line" (cf. the Wikipedia article of that title) which includes infinity and satisfies a list of axioms which you can write down.

This article is dangerously misleading. "Numbers" is not a well-defined set and there is no way to say that infinity does or does not belong to it. As is often the case in mathematics, you need to make the discussion more precise before you can reasonably answer questions.

View on HN

The problem is that "a number" is meaningless.

No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of. You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently.

A lot of the comments here are getting hung up on trying to pin down the mathematics of what you can and can't do with infinity. That's fine, it's been keeping mathematicians busy for centuries, but this article is for the layman who doesn't know about rings, groups, algebras or any other mathematical structure which you might call 'numbers'. It's for someone who thinks the obvious when someone says 'number'.

In this context, infinite limits, I think the distinction is beginning to become important. I agree that colloquially it often doesn't matter, but I think it does here.

"Something which you can have that many of" is a terrible definition: You can have sets of infinite size. Do you mean physical things? What, then, is a "thing"? I can have infinity intervals of different length on my arm. So are intervals not a "thing"? You appear to think irrationals are numbers, so what physical thing can you have an irrational number of?

This is a dangerous path to walk down, but that doesn't make it an unimportant one. And that's why this article is ultimately flawed. The idea is important though. We need to understand how to operate formally on mathematical objects. If there's one thing I learned in my Philosophy of Mathematics class as a math undergrad, it is that we shouldn't struggle over defining what it means to be a number. We can just use them. I'll leave the philosophy to the philosophers.

I pretty much agree with you. I'm just worried that with all the comments about 'what is a number?', and 'sure, we can make infinity a well defined number', it will dilute the message of the article. Which is that infinity is not a 'number' (in the sense of real numbers). It's important to understand why it's not a number and what it means when we write it on one side of an equality.

You clearly can't have 2i + 3 dollars, so by that logic, complex numbers aren't real numbers.

> Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently.

Well, if your definition doesn't work for irrational numbers, then your argument isn't good enough: you exclude infinity on criteria that also exclude "good" numbers. In the context of the article, π is a number. And you cannot count things with an irrational number,　by definition.

Understanding what mathematical objects are can lead to insanity. Much better to focus on what you can do with them.

Or to quote John von Neumann, "In mathematics you don't understand things, you just get used to them."

You have to know whether your definition of "numbers" makes them into a field or not, before you can go applying field operations to them and expecting usual results.

Including infinite ordinals in the set of numbers is legitimate, as parent points out that "number" is not strictly defined, but if you include infinity, numbers are no longer a field, and you have to cut someone off when they try to use field axioms in theorems, the uniqueness of multiplicative and additive inverses, which must be how people end up with nonsense like 1=0.

Reply on news.ycombinator.com