Let's have some personal background to this logical exercise first:
A year ago this September 19th, the Society of Non-Theists at Purdue University
Throughout the day, we drew larger and larger crowds. The various people were eliciting emotions ranging the gamut from "lol" to "Alright!" to "What on Earth are they doing?" to "OH LAWD BABEH JEEBUS HELP ME!!!1!1!eleven"
In the corner, however stood two people. One was holding a video camera and the other was talking, perhaps if narrating.
After I read a chapter of the Gospel of the Flying Spaghetti Monster in my best "Preacher Voice" I went to go get some water and was stopped by these guys to talk.
It became clear very quickly that these people were not here to get the poop on what we were doing and why we were doing it, but I talked to them civilly. They asked me how I felt about God and what I, myself believed.
I told them. As a scientist and, much more specifically, a Math major, education classes be damned, I had very specific ideas of what I would need to be persuaded to the side of the believers. That is to say, PROOF. Logic. In order for me to stand up and say "There exists a higher power." I need to see a written proof with QED at the end (although now, I think I'd also accept the heavens parting and having God himself send me on a quest a-la Monty Python).
"How about infinity?" was the response. "Can you prove infinity?"
"Well...prove infinity itself? Hmm...I'm not sure I, myself know how to do that. I suppose you could go through the route proving that the integers have no upper bound and are therefore infinite."
This, of course, didn't help. Eventually, they moved away from Math onto subjects that they were more properly coached on and that I didn't have enough real training in to properly bullshit mine their arguments. I made eye contact with other people from our group in the traditional, "Shit, shit, help me! They won't stop throwing bullshit at me!" fashion, allowed someone else to get caught up in the argument and then bowed out claiming various excuses.
But, it got to me. Infinity. What does infinity mean? We use it in math all the time, don't we? Calculus is basically built around the concept of infinity, isn't it? Differentials...integrals, Reimann sums...The infinite and the infinitesmal are all around...well...sort of.
Infinity is a somewhat wooly concept...and by somewhat wooly, I mean completely incomprehensible. To have an infinite quantity of something is physically impossible. It's a contradiction of terms, really. If you have a quantity, you have quantified it. How can you quantify something that is, by definition, inquantifiable? Well, that's precisely it. You can't.
Infinity is not a thing. It is not a measurement. It is not, really, even a state of mind.
Infinity is, in all senses, the impossible we can never and WERE NEVER MEANT TO reach simply by the very nature of the concept! It's not even a benchmark that is merely set too high.
"But, Mark," You say. "There are so many other concepts that we can't actually see that we use all the time, too!"
"Well, yeah. Sorta."
"I mean, you have imaginary numbers, transendental numbers, even, perhaps, NEGATIVE numbers are also abstract concepts that we are surrounded by in math that we don't actually argue with."
This is very true. You probably couldn't find -1 apple, or 2i dollars in your wallet...and I'd love to see someone come up with exactly pi of something.
However, in each of these situations, regardless of their abstractness, we use them because they appear in nature. Even imaginary numbers have a very useful practical application that translates into something tangible. Just ask your friendly neighborhood electrical engineer. I'm sure he'd be glad to point you in the right direction.**
The point is that infinity is really the only one of these that doesn't get a real, practical analog because it doesn't exist on its own.
Infinity is a tool, certainly, but not something that can be proven.
*Yes, they're bowing down to me. It seems that, on the same day, there was a flash mob. They were going through and taking showers and brushing their teeth in all the fountains on campus. There is a small fountain right by where we were holding our event.
**Just make sure he's bathed recently.***
***I'm just kidding. I love you all. Please don't kill me with your trebuchets.