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Comment by david927 | Hacker News Reader

I don't believe in infinity. And most probably the reason is because it doesn't exist.

Every time we expect something infinite in physical phenomenon, it turns out to have some sort of Plank-like constant that compartmentalizes the effect at different levels to avoid that result.

Which is good, because a universe where infinity existed would most likely not be stable enough to support life. I understand that mathematics isn't simply about physical phenomenon. I understand the beauty of i, for example. But I think that screwing around with the bizarre aspects of a concept like infinity is a huge waste of time.

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I don't believe in infinity. And most probably the reason is because it doesn't exist.

How about the number three? Does it exist? Or the square root of 2? I can't touch it, that's for sure.

People don't want infinity per se. We want a number system where you can do nice things to numbers and get out other numbers... and have nice theorems like "a continuous graph that goes from negative to positive must cross the zero line somewhere"... and then it turns out that the number system that fulfills our need consists of infinite sequences of digits, and finite sequences won't suffice.

Rider has set me straight, but here's my original beef:

I said, quite clearly, "I understand that mathematics isn't simply about physical phenomenon. I understand the beauty of i, for example."

You mention 3 and the square root of 2. You plug both of those in to: (x == x+1) and you'll get an inequality. Not so with infinity.

Again, Rider has set me straight that I need to learn more. I can believe that, but when I see how often in the physical world we creep up on infinity only to have it not exist, I'm definitely skeptical about plugging in infinity into equations like x == x + 1. It seems it doesn't make any sense at all.

It's true that you can't extend the number system to include something called "infinity" that would behave like a number. Just like you can't extend the number system to make division by zero make sense while keeping all the usual laws of arithmetic. But this doesn't lead to any philosophical implications like "infinity doesn't exist". In every context where it makes sense to talk about infinities without leading to contradictions, we want to do that, because it enriches our vocabulary and allows us to solve more problems.

Allow me to illustrate. Take the following problem stated in school geometry terms: A square is cut into triangles of equal area, now prove that their number is even. Interestingly, we don't yet know any elementary solution to this problem that can be stated in school math terms. The only solution known was discovered in the 1970s and relies on something called "p-adic numbers". What are those p-adic numbers you ask? They are weird number-like things that have infinitely many digits to the left of the decimal point. Freaky constructs without any counterpart in the real world, but you can still add them, multiply them and all that. Do such things "really exist"? Don't know, don't care. But they helped us solve a difficult problem and that's all that matters.

Cool. Thanks.

Edit: but that still seems to me to reflect infinitely, the adjective, as opposed to infinity, the noun. Still, thanks.

When we talk about infinity, we never really talk about it as though it were an actual value. In calculus, we really only talk about limits approaching infinity (sometimes using "infinity" as a shorthand). In set theory, we really only talk about things having infinite cardinality (sometimes using "infinite" as a shorthand). In geometry, we talk about things being "infinite" not to suggest that there's some endpoint that is simply unreachable; we mean something more along the lines of "unbounded."

Hilbert's hotel is more of a set-theory problem than a calculus or geometry problem, so it's best to think of the set of rooms having infinite cardinality. A set of countably infinite cardinality (the guests) is of the same cardinality when another (finite or) countably infinte set is added, so we can develop a one-to-one correspondence between it and another set of countably infinite cardinality (the rooms). Thinking about it this way demystifies the problem a little.

Nice answer, thanks.

Calculus is pretty tough without infinity. Most of the theory behind fluid dynamics is pretty tough without infinity.

Not knowing that there is an unlimited number of numbers becomes a problem. Trying to do any math without allowing that between any two numbers there's another number is pretty difficult.

There are people who derive a mathematics without the axiom of infinity, but it's weird, distorted, unnatural and doesn't turn out to be as useful as often as the regular variety of math.

Feel free not to believe. Just don't expect to do much advanced math.

Oh, and if you use proof techniques in programming, such as loop invariants or transformation theory, you are using infinity, you just didn't realise it.

Sorry for the confusion. I meant infinity, not the limit of infinity.

I have no problem with the concept of infinity in terms of limits, but that's not what this article is about. It's about the concept of infinity itself. And that is what I disagree with.

I didn't down-vote you, by the way, even though I think you are mis-guided, and probably have been badly taught. From other things you've said I think you could, if it were presented well, find the whole topic intriguing.

Because, in part, you can't have the one without the other. You can't talk about limits without having a model for infinity. They are inextricably entwined, each idea leading to the other.

It's sounds to me like you've simply been put off by loads of badly written Pop-Math articles that attempt to boggle and astound, without showing the real truths as to why the whole subject is really, really cool, and simultaneously, useful and practical.

I have no problem conceding that. Do you have anything I could read that might enlighten me on this?

Oh, perfectly reasonable but rather tough question. My thinking about teaching infinity is now at odds both with everything I've read, and how I was taught. It also depends on your own inclinations, strengths and background.

Brian Clegg's book on infinity gets good reviews, but I haven't read it. I'm thinking of starting a blog series on it, but with everything else I have on I may never really get to it.

But as I say, in your specific case it really depends on your background. If you're really interested perhaps you could email me more details on what you've studied and what you've read, and I can write some brief articles by way of reply.

Try _Everything and More: A Compact History of Infinity_ by David Foster Wallace.

Not to express an endorsement in either direction, but you may be interested in the finitist school of thought: http://en.wikipedia.org/wiki/Finitism

or its even stricter cousin, which not only denies the meaningful existence of infinity, but also the meaningful existence of technically finite but infeasibly large numbers: http://en.wikipedia.org/wiki/Ultrafinitism

See also Constructivism: http://en.wikipedia.org/wiki/Constructivism_(mathematics) . And the concept of "computable numbers": http://en.wikipedia.org/wiki/Computable_number

That said, I will point out that math is essentially a game played by certain rules known as "axioms", and there is no one true set of axioms. If you choose one set, you get infinity; if you choose another, you don't (but you will pay for that). You can't say which is "more right" or "more wrong" without bringing other subjective standards into it, each of which may have their own utility. While "corresponds to the physical universe" is definitely one useful such standard, it is still not the only one.

Do read up on the limits of computable numbers though; they are interesting, but it's not a free lunch.

> But I think that screwing around with the bizarre aspects of a concept like infinity is a huge waste of time.

Maybe for you, maybe not for someone who enjoys doing it.

Fair enough. But I'm saying, let's get rid of it. Let's kill the concept of infinity, and we'll all be better off for it.

But that's like saying let's get rid of zero. You can make the argument that zero doesn't exist as equally convincingly as you can that infinity doesn't exist.

That's like saying "let's get rid of recursion".

Imagine defining a function in terms of itself! That's circular and you can always get away with just using a loop instead.

Well, it turns out recursion often works and is very useful. Just like infinity.

Saying that a function is defined "in terms of itself" is a useful mental shorthand for what's going on, but it only works if you deliberately confuse the function at compile time with the function at run time. If you think about it as a function calling itself, you don't run into this difficulty, since "calling" implies "at run time". There are a lot of things in how computer science is taught that have this "and then some magic happens; just accept it" flavor, when there's really no magic -- everything is ultimately and really imperative.

I'll repeat what I said above here:

I meant infinity, not the limit of infinity.

I have no problem with the concept of infinity in terms of limits, but that's not what this article is about. It's about the concept of infinity itself. And that is what I disagree with.

> But I think that screwing around with the bizarre aspects of a concept like infinity is a huge waste of time.

It's playing around at the limits of human reason. I don't see it as any different to most sports, music, and art. All are appreciated by different groups of people but, ultimately, "useless" if you do not value the enjoyment gained from exploration for its own sake.

What makes you think the universe is finite? As far as I am aware, the prevailing view of cosmologists is that it is finite, but there's certainly plenty who believe it could be infinite.

well the prevailing view of cosmologists is pretty much all we have on that..

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