Infinity...

[Adso de Fimnu]

There's a question that's been bothering me for quite a long time, and I was hoping the people at this forum could help clear it up. I must warn you in advance, though: it's one of those thoughts that can give you a headache.
How can we [humans] even have the idea of infinity in our heads?
I certainly can't comprehend infinity - if anyone can, please explain it to me. But my question is about how we got the idea of something without end in our minds at all. (I realize that some explanations could be religious, but please no arguing about that here. I'd just like to know what other people think about this.)

 

[Perfection]

Well it is an immensely useful concept in mathematics.

 

[Azadre]

Infinity, as we know it, is just a concept and not a number. Does eternity bother you to? How about a line? It's all the same.

 

[Halcyon]

Infinity's not all that difficult to comprehend. It's just hard to visualise.

 

[Azadre]

Infinity's not all that difficult to comprehend. It's just hard to visualise.

Nice avatar

 

[Erik Mesoy]

Imagine a stainless steel ball the size of our planet. Every thousand years, a fly lands on it, and flies off again.
When the ball is worn down to nothing, infinity (eternity) hasn't even begun.

Helps a great deal with visualising it. I often play around with Hilton's Hotel for some fun with infinity.

Hilton's Hotel has an infinite amount of rooms and a very happy owner.

 

[WillJ]

I don't see what's so hard to comprehend about the concept of a quantity that's larger than any definite number you can name. But what I DON'T understand is the non-existance of the concept of "infinitely close." Supposedly you can't say that something is infinitely close to 5, for example. Or that .9 repeating is infinitely close to 1 (rather you must accept that it IS one). I don't understand this. I suppose I could read up on it, but I'm too lazy to until I have to.

Well it is an immensely useful concept in mathematics.

Would you say it's infinitely useful?

 

[JohnRM]

I have thought, often, about infinity. Where does the universe end? Does it end? If it does, what is outside of the universe? These questions, while unanswered, bolster my belief in God.

 

[Halcyon]

Nice avatar

Great minds, eh?

But what I DON'T understand is the non-existance of the concept of "infinitely close." Supposedly you can't say that something is infinitely close to 5, for example. Or that .9 repeating is infinitely close to 1 (rather you must accept that it IS one). I don't understand this.

A number infinitely close to 5 would be 5 +/- 1/infinity. Unfortunately, one divided by infinity might as well be zero, so anything infinitely close to 5 is 5. You'd have to multiply almost-5 by infinity before you noticed a difference, and the product would still only be only be one greater.

 

[Erik Mesoy]

The reason for the non-existence of "infinitely close" starts here:
Divide 1 by infinity.
This cannot be resolved, only approximated by the function:
As N->Infinity then (1/N)->Zero.
You'll never reach either of them. So you can't get an infinitely small difference. Either there is a finite one, or nothing.

EDIT, Lol, Halcyon posted before me, saying something along the same lines.

 

[Adso de Fimnu]

I don't see what's so hard to comprehend about the concept of a quantity that's larger than any definite number you can name.

Infinity's not all that difficult to comprehend.

How can we comprehend something unlimited in space and time, when we ourselves are beings based (could I say, 'grounded'?) in both space and time? I just don't understand how you people can wrap your heads around that...

 

[Adso de Fimnu]

I guess I was looking for a philosophical answer, not a mathematical one...

 

[Erik Mesoy]

We can't. We only delude ourselves into thinking we can, and what we are comprehending is actually just the different tools available for different areas where we work with infinity.
It's impossible, so we do what we can, and declare that a success.

 

[Halcyon]

What exactly is there to fail to comprehend, here? We can say what infinity is, and explain the concept. We can use infinities in mathematics. We can't count to infinity or make an infinitely large casserole, but neither of those much counts as comprehension, being 'performing infinity' at best, which is notably tricky.

Are we only deluding ourselves into thinking that we understand, say, the colour blue?

 

[WillJ]

A number infinitely close to 5 would be 5 +/- 1/infinity. Unfortunately, one divided by infinity might as well be zero, so anything infinitely close to 5 is 5. You'd have to multiply almost-5 by infinity before you noticed a difference, and the product would still only be only be one greater.

1/infinity "might as well" be zero? Says who? It seems to me like they're different concepts. Aren't you sacrificing exact precision when you say that?

The reason for the non-existence of "infinitely close" starts here:
Divide 1 by infinity.
This cannot be resolved, only approximated by the function:
As N->Infinity then (1/N)->Zero.
You'll never reach either of them. So you can't get an infinitely small difference. Either there is a finite one, or nothing.

Huh? First of all, I always thought "->" means "yields." What does it mean here? "Approaches," perhaps? And does N represent a number?

Oh, and let me ask you this. If 1/infinity equals zero, since cross-products are always equal, does that mean zero times infinity equals anything you want it to?

 

[Loaf Warden]

I think there's a difference between knowing what the word 'infinity' means and truly comprehending infinity. Infinity, as a word, is easy to define and easy to understand. I know what it means. You all know what it means. We can talk about it, use it in calculations, and base entire philosophical discussions on it. But I don't think that proves that we truly comprehend it.

Similarly, we know how much a trillion is, and we know how much a quadrillion is. Again, we can talk about them and make calculations with them. We know what the words mean. But we have no way to really grasp them on an intuitive level. Once you get to that point, they're just numbers. Just longer strings of zeroes to work into your calculations. Unlike smaller numbers like five, nineteen, or thirteen thousand, we have no frame of reference from our corporeal existence to really grasp the staggering enormousness of a number like a quadrillion. We know it's a high number, and we know exactly how high it is in relation to other numbers. But we still have to compare it to other numbers to make the concept meaningful. Without that, it's just another word for 'a hell of a lot'.

And a quadrillion is nothing compared to infinity. Infinity isn't even a number. We can't even compare it to anything.

We know what the word 'infinity' means. I don't think any mortal being can truly comprehend what infinity is.

 

[Halcyon]

1/infinity "might as well" be zero? Says who? It seems to me like they're different concepts. Aren't you sacrificing exact precision when you say that?Huh? First of all, I always thought "->" means "yields." What does it mean here? "Approaches," perhaps? And does N represent a number?

The difference might as well be zero because there is only one way you could tell the difference between 5 and 5+1/infinity, and that is to multiply by infinity, which is impossible, since the answer is always infinity. As such, no matter what you do with the numbers, the answer is the same. They therefore might as well be the same. If someone offered you 1/infinity of a pint of beer, you'd probably not feel too grateful. That's not even a molecule - it's not even a quark. It's an infinitely small fraction of the smallest thing there could possibly be, which is already pretty small.

So, 1/infinity is zero. Any integer divided by infinity might as well be zero. You can't divide by infinity. How many ones make one infinity? Might as well ask how many ones make the Second Boer War.

 

[Azadre]

1/infinity is zero. Just as .9999999999999999999.... is 1. There was a thread a while ago on it.

 

[Amenhotep7]

I was also dwelling on something similar to this. In theory, with our concept of time, you could measure it in infinitely smaller parts. So that means and infinitely smaller number has to hit 9 before the number preceding hits 9, until it reaches the next second, minute, what have you. But this means that because it is infinite, technically it should forever be stuck on 0.0000000000000000000000000000000000000000000000.... seconds. As there are infitesimally smaller numbers that must hit 9. See?

 

[Perfection]

Well, the use of talking about things that are infinitely close to something is essential in calculus. For example

x/x is undefined at zero, but infinitely close to zero it is one.