Well I’m both a little surprised and impressed with the reception my last post received and am glad to continue with a topic with many similarities with the last one. This post will look at the concept of the infinite by looking at a couple of paradox’s and in the end it will show how and idea can work in theory, and in reality. Now as a forewarning I am firstly a philosopher and only further down the line a mathematician/physicist, but I’m going to play those roles for this post. If there are any mistakes or clarifications that are needed, feel free to post the information in the comments and I’ll make the corrections as soon as time permits.
First, I’ll just introduce the subject matter through an illuminating argument I had with a former professor a few years ago. He brought up the topic of the infinite and how it worked as a mathematical concept, and in doing so he pointed out that in math you could have a number that was infinite ( Z for example) and also a number that was double that infinite (2Z). Fair enough it does make sense within the framework of mathematics, but a problem arose when I was asked to define what the infinite meant.
Now when asked I said that the infinite is really a concept that is irrational when applied to the world around us (Look back that may be going a bit to far, but it points out how much the meaning changes when applied to real world issues). The infinite really breaks down and can’t be used in real applications, except as a concept that doesn’t exist in reality. Take for instance the simple equation above, and try to put that into an example that makes sense.
There is an infinite number of apples, but there are twice as many oranges as apples. Try to prove that statement is true. It makes sense as an idea when the sentence is read, but to really prove it all the apples and oranges would have to be counted, which cannot be done. Yet there is also a problem with being double the infinite of another infinite, because as soon as you hit infinity you’ve reached the limit, there can no longer be anything more. It is the equivalent of saying that something is more unique than another unique item. Unique means one of a kind, and once you’ve hit the status of one of a kind there can no longer be anything more unique. If there was something more unique it wouldn’t exist.
So, what if someone is running on a circled track and there is another person running twice as fast as the other person. Couldn’t it be said that the one person is traveling twice the distance of the other, while they are both in an infinite loop? Well I think that would be a fair but inaccurate description of that is going on. The relative speed is the important part, while noting the infinite distance, just implies that it isn’t known when or if they will stop. It doesn’t add any information.
Now this seems to be an example of the infinite, and in a way it is. The problem comes with trying to find a home for the infinite within out observable universe, without there being a loop. The earth, due to it having a curved surface, has no endpoint. A person could fly around the earth and end up in the exact same spot they were in, but it wouldn’t be fair to characterize the earth as having a surface of infinite size.
This brings up how the infinite is a supernatural answer to some questions. How far can the universe expand? How long will space/time exist? How dense is a black hole? If the answer to those questions was given as infinite, it just means that the answer really isn’t known, but there is no reason to think that there is an upward limit.
The infinite is a placeholder when used to talk about most real things, and an effective tool when used conceptually.
Sorry I didn’t get to some of the interesting paradoxes, but I will in my next post.
Thanks for reading
-the moral skeptic
(Redit comment) Hmmm. Interesting piece, but there are a couple of mistakes/misconceptions here.
ReplyDeleteThere is an infinite number of apples, but there are twice as many oranges as apples. Try to prove that statement is true. It makes sense as an idea when the sentence is read, but to really prove it all the apples and oranges would have to be counted, which cannot be done.
Prove which statement? Say 'apples' are the even numbers and 'oranges' are the integers. It's easy to see that there are twice as many oranges as apples, but as the set of oranges and the set of apples can be put into 1:1 correspondence then they are both the same type of infinity.
Yet there is also a problem with being double the infinite of another infinite,
This isn't a problem - double infinity is still just infinity.
because as soon as you hit infinity you’ve reached the limit, there can no longer be anything more.
This isn't true. There are different sized infinities, some larger (infinitely larger, in fact!) than others.
Take the integers. Start counting them: 1,2,3,4,5... there are infinitely many, right? So we call the number of integers 'infinity'. So, how many prime numbers are there? Start off: 2, 3, 5, 7, 11,13... looks like it's going to go on forever. And you can match up the lists:
1 -> 2
2 -> 3
3 -> 5
4 -> 7
5 -> 11
And so on. Because you can line up the groups, this means that mathematically they are the same size. We call this 'countable infinity', because by matching the set up with the integers you are just counting them.
OK so far; now, a more tricky question. How many real numbers are there? Numbers like 0.234723423. Or pi. Or e. What is the first one? Tough one, huh.
There is no way to 'count' the real numbers by matching them up, one-to-one with the integers (Cantor proved this ingeniously, http://en.wikipedia.org/wiki/Cantor's_diagonal_argument). So while there are still 'infinite' real numbers, this infinity is formally larger than the number of integers. It's so big you can't, even in theory, start to count it! This is a second-order infinity, called aleph-1. And there are more... a whole infinite-tree of infinities.
Oh, and a couple of your rhetorical questions at the end are actually simple:
How long has space/time existed?
13.7 billion years.
How much mass is in something infinitely dense?
Density isn't the same as mass. A black hole is infinitely dense, but just has the same mass as its parent star. If the sun became a black hole, we'd still happily orbit around it.
(My Response)
ReplyDeleteWell there is a line of argument stating that space time existed before the big bang, and that was the question how I answered the question, but if you used like Stephen Hawking, the big bang as the start of physic's and therefore the start of space time I guess your answer would be correct.
Also the point about the black hole is that it is infinitely dense, but only because physic's breaks down when it try's to calculate it...this Hawking to has talked about, and would be a good place to look at that. So the black hole is a place where physic's breaks down, but it can't actually be proven that it is infinitely dense, unless my understanding is outdated, which it might be.
As for the other corrections...
I understand that the apples and oranges thing works in integers, but try to put that in a real world example where there are infinite apples and oranges. To prove that the integers were correct at some point the apples and oranges would have to be counted.
Also saying that there being double infinity isn't a problem it is just infinity creates a problem for me. I'm not sure how it works. Say you have a car and then double a car (2 cars), so you have created a difference to be a marker, but with the infinite you have the infinite and double the infinite, but they are the same thing, so how can the distinction between the infinite and double the infinite be created. I understand it works mathematically, but thinking about it logically like that causes, at least myself, some problems.
"How far can the universe expand?"
ReplyDeleteI like the idea of treating the development of the universe as an isomorphic relationship to the Mandelbrot Fractal. The further we zoom into the fractal the more variations we get, similarly it could be that the further our time passes within the universe the more variations of it we get.
I sort of like the idea which you also states that there is no upward limit, why does everything need to have an end? Should the concept of infinity rather be understood as something without limit than the limit?
In the end, I think the beginning of our universe to be the more puzzling question than how far it can expand, unless of course our universe is a loop that has always existed, and will always exist.