How do you comprehend infinity?

[gnarledreaper]

This may be a question for the philosophy section, but i do not have the privilege to post there so it seems to be best placed here.

 

Does trying to understand infinite space ever bother you, especially as physicists, for instance if you take a box, it has an inside and an outside, everything seems to have boundaries but apparently the matter and energy in the universe is expanding into....nothing?, like we exist inside a space without a boundary

 

I get the same mind boggling confusion if i try to comprehend cause and effect, something has to trigger an event, (take the big bang for instance) but some action has to cause the trigger and so on, chicken and the egg etc.

 

I may well be ignorant of a theory that has been put forward but to me the reality of infinity seems to be impossible yet it cannot be denied, yet.

 

I have listened to lectures on youtube and read up in the topic but it seems there is no way to understand so do you prefer to brush it aside as not yet explained, conveniently forget or just ignore?

[ajb]

Does trying to understand infinite space ever bother you, especially as physicists, for instance if you take a box, it has an inside and an outside, everything seems to have boundaries but apparently the matter and energy in the universe is expanding into....nothing?, like we exist inside a space without a boundary

 

Infinite dimensional spaces are required in physics, so one should take care. Many ideas from the finite case do not go over to the infinite case very directly. (Identifying spaces and their duals for example)

 

However, quite often in physics we are not worried about describing the whole space in one go. We can work locally and pick local coordinates. You can often simply use local coordinates and not worry too much. You can "pretend" the space is finite dimensional for a lot of calculations. If you require a bit more rigour finite dimensional subspaces could be used instead.

 

Another approach is to model model infinite dimensional manifolds on Banach spaces. I don't know much about this other than it is hard and does not fully work as compared to the finite dimensional manifolds.

 

 

 

 

I may well be ignorant of a theory that has been put forward but to me the reality of infinity seems to be impossible yet it cannot be denied, yet.

 

Infinity is not a real number and thus no experiment or observation will produce a result of infinity. In that sense it is impossible. However, infinity is mathematical concept that arises in the mathematics of the theories used to describe nature. The calculation of something that should in principle be observable to be infinite is seen as a "sickness" of the theory. It signifies trying to apply the theory to something it just cannot cope with.This signals new physics is needed.

 

I have listened to lectures on youtube and read up in the topic but it seems there is no way to understand so do you prefer to brush it aside as not yet explained, conveniently forget or just ignore?

 

Personally, I don't loose sleep over (plus or minus) infinity.

 

You can "tack them onto the end" of the real numbers to give the extended reals. You can then deal with infinity in a consistent way. Note infinity is still not a real number in this set-up

 

Or you can confine yourself to think in limits.

 

I guess depending on the situation, I am happy to think either way.

[CaptainPanic]

Infinity is like asking "are we there yet?", and always getting "no" for an answer.

 

The concept doesn't bother me at all.

[granpa]

i prefer to think in terms of what is called the 'arbitrarily large' and the 'negligibly small'.

 

infinity is not the largest number. infinity^2 is much larger.

1/infinity is not the smallest number. 1/infinity^2 is much smaller.

 

1/0 is not infinity. Would it be positive or negative?

[ajb]

infinity is not the largest number.

 

True as infinity is not a real number. The real line has no maximum or minimum element.

 

1/infinity is not the smallest number.

 

As infinity is not a number you will have to define this. A good choice is to set it to zero. So it is not the smallest number, by this convention.

 

 

1/0 is not infinity. Would it be positive or negative?

 

Zero as a real number is not invertible. 1/0 is not defined.

 

Things like infinity - infinity and 0 x infinity are also often left undefined.

[michel123456]

(...) for instance if you take a box, it has an inside and an outside, everything seems to have boundaries but (...)

 

The problem you raise is about the concepts of inside and outside. When you make a closed boundary, a box, you are making a division of the Universe in 2 entities: inside of the box, and outside of the box.

Simple observation makes us formulate 3 objective statements:

1. the inside part of the box (the content) is always smaller than the outside.

2. the inside part (the content) is finite.

3. the outside part is ...[insert here either "finite" or "infinite" following your convictions]

 

IMHO the problem of point 3 is not the question of finite versus infinite, the problem is our "objective statements" under "simple observation". Mother Nature is tricky.

Edited December 2, 2010 by michel123456

[Anura]

Infinity used to scare me. But now i think of it as a concept.

[gnarledreaper]

I am more perplexed by the fact our universe, the observable part of it anyway, seems to be expanding into empty space or an absence of matter and energy than i am by the maths part.

 

Here's a poser, as i understand it an object that has a high density can curve the space and time around it does that mean that space is made of something tangible that could have a boundary?

 

I feel a bit dumb for asking that as i can barely get my head around three dimensional space being curved

[md65536]

1. the inside part of the box (the content) is always smaller than the outside.

I don't think this is true when relativity comes into play. Specifically I think that the inside of a black hole is "bigger" than the outside.

 

 

But regarding the original question...

 

Here are some concepts that I think are interesting when trying to understand infinity:

- Infinity is not a value (infinity + 1 etc). If you treat it as a value, I think it's possible to say that infinity^2 == infinity. Google "infinite hotel" for seeing how you can expand an infinite value and not change its value. Or whatever.

- There's a difference between countably infinite and uncountably infinite. The real numbers are uncountable but the rational numbers are countable (to my great surprise when I learned it). When speaking of infinite variability (such as the idea of multiple universes for every possible reality) the pigeonhole principle is also interesting. Example puzzle: If you have an infinite long string of digits that never repeats, will you eventually find any given finite string of digits within it?

- An infinite number of things does not necessarily need to take up an infinite amount of space. Consider the infinite sum 1/2⁰ + 1/2¹ + 1/2² + 1/2³ + ... = 2. An infinite sum of positive numbers can be finite and in fact quite small. So an infinite number of monkeys could fit in a small room if the first monkey was normal sized and each next monkey was half the size of the previous monkey.

 

Ask, or just google, for more info on the concepts. Probably the most fun way to understand infinity is to work through the puzzles and paradoxes involved, and build up not just one universal understanding of "infinity", but a whole set of tools for rationalizing about infinite values or quantities.