Infinity in mathematics
This will be the trickiest part to talk about in a way that can hold the attention of the moderately cool person. I'll do my best.
The concept of infinity is so intuitive that I don't think I need to say anything about it. It's actually kind of hard to give it a good definition without specifiying what type of thing we're calling infinite (i.e., an infinitely long span of time and infinitely many Blu-Rays of LOST season 6 are two different beasts -- though related, since the show is so addicting you could probably watch it for eternity). So I'm just going to assume that when I say "infinity," you roughly know what I'm talking about.
But I will point out that in mathematics, the concept of infinity is extremely important. In grad school, we literally talked about infinity every single day in class. It played many different roles -- the limit of a function, the number of elements (e.g. numbers) in a set, etc. I sense I'm losing most of you, so just suffice it to say that literally every day as a math grad student, you use infinite things to understand mathematics.
A few examples of infinity:
- The set of all natural (or "counting") numbers is infinite in size: {0, 1, 2, 3,...}
- The set of all real numbers is infinite in size, and in fact is more infinite. I won't go into this, but it's really interesting to you if you're legendary in your nerdiness.
- One of the first proofs you learn as an undergrad math major is how to show that there are infinitely many prime numbers.
- The limit of the graph of f(x)=1/x as x approaches 0 from the right is positive infinity:
Infinity in our imaginations
Furthermore, as I mentioned before, we have this intuition about infinite things. The author of Tuck Everlasting didn't have to explain what eternal life is -- we can all imagine living forever. Most of us naturally assume the universe is infinitely large, that space just "goes on forever." Science fiction (and many an actual scientist) posits that there are infinitely many universes.
But the crazy thing is -- and this will serve as our punchline --
We haven't discovered anything infinite in nature.
Yeah, that's right.
The number of atoms in the universe: less than 4 x 10^81
The age of the universe: less than 14 billion years
The number of mudslinging Senator commercials: somehow less than 70% of all Senate commercials, which is itself (surprisingly) a finite number
The three things you would most naturally guess to be infinite turn out to be finite. Part of the reason we don't find actual infinites anywhere is that it would be impossible to measure one. Think about it: if something had infinite weight, it would exceed the capacity of any scale we could build. We'd just keep building bigger scales, and it would break every one. And the same is true for length, age, etc. -- our ability to observe/measure will always be finite, so we could never conclusively say that anything is actually infinite.
But it's more than just a measurement problem. We know the age of the universe and number of total atoms are finite.
There are two possibilities for infinitude in our natural universe, I think -- black holes' density and the overall volume of the universe. The latter is still an "open question" in cosmology, and we don't know yet the overall structure or size of the universe. And the thing about density is that it's sort of a pretend quality -- you take two measurable things (mass and volume) and divide them. So a black hole gets its infinite density since its volume is zero, and when you divide by zero you (sort of) get infinity. If the only way to get an infinite thing in reality is to divide by zero, I think the point still stands.
Why is this true?
Recap: we all intuitively understand and can easily call upon the idea of infinity, and yet we don't know of any actual instances of infinity in the natural world.
Why? What does this tell us about the world? About ourselves?
I don't know. That's why it's written in question form. Please tell me what you think.
I'll give you my one initial thought on this mystery -- we're a bunch of beings who are finite in every way, surrounded by finitely many things that are each finitely sized and finitely old, and yet we think and dream of infinity. To me, it seems like it could be an innate comprehension of and yearning for something or someone infinite, outside our universe and its finiteness.
Abraham Lincoln thought the same thing: “Surely God would not have created such a being as man, with an ability to grasp the infinite, to exist only for a day! No, no, man was made for immortality.”
Sorry this was so long. But imagine if it had been infinitely long...
Please feel free to take a stab at the above questions or any other thoughts, comments, questions, or ideas you have. Thanks!
Jon
42.
ReplyDeleteInfinity describes not a measure of existence but rather a state of continuation. It thereby exists only as an artifact of language. This is why, as example, the proposed question of, "Can God make a stone so large that he cannot move it?" is intrinsically invalid. The language of this "question" is semantically malformed and thereby meaningless. I posit that to ask whether an instance of infinity exists, in the natural world, is likewise invalid. Furthermore, the mere perception of infinite and unbounded in a given universe does not preclude the possibility that finite bounds are merely expanding infinitely at a rate which cannot be perceived by a being confined to the dimensionality of that universe. (eg. if light has a head start, forget about catching up!) Curiously, a being of higher dimensionality may view that same universe as being in stasis. To be clear, the preceding is not intended as a theological statement, but rather a useful abstraction.
ReplyDeletehm.. Why do I post blog comments this late at night? OK, back to reading this boring research paper.. :-)
I don't know if this is where you're going, but it's quite interesting that it was just published in the past couple days. :)
ReplyDeletehttp://biologos.org/blog/id-like-to-have-an-argument-please/