The Problem of Infinity: Our Paradoxical Reality

(Note: Everything here defies what I was taught in Calculus. Thus, it could all very well be completely wrong.)

How do you define 0?

Obviously, there are a few different ways. The simplest I can think of is to define it as the result when a number is subtracted from itself (x - x = 0). But 0 is much, much more interesting than that; 0 is also the reciprocal of infinity.

Why is that interesting? Because infinity is not truly a number, but an abstract concept. Therefore, 0 is not only a real number, but also an abstract concept. (Of course, numbers themselves are abstract concepts...) Therefore, 0 is not only that which is non-existent; it is also that which is infinitely small.

Basically, infinity, as a concept, is pretty frickin' important.

Mathematics, of course, has incorporated both zero and infinity. In such an abstract system, infinity flourishes; without the concept of infinity, we would not have calculus, God (Who is infinite in many ways), and many other extremely fun things.

The difficulty arises when infinity is incorporated into our physical reality. More than merely incorporating it, we have used infinity to define the beginning of the known physical universe; what is the Big Bang if not the universe divided by infinity?

According to today's accepted scientific theory (which is a pretty useless standard when you consider what "today's accepted scientific theory" meant in centuries gone by), the universe began as a singularity - a point of infinitely small size and infinite density, temperature, and space-time curvature. The size of this point was not merely extremely small; it was, in fact, 0. Regardless of the actual amount of matter and energy present in the singularity, its size, temperature and density would remain the same. Does that make sense?

Think about that for a second. (Eventually stop, because thinking about the universe's being compressed into nothingness should blow your mind.) According to this theory, the universe existed as a dimensionless single point at its beginning (t = 0). Now, of course, its energy, density, and other characteristics are finite. At some time t between t = 0 and t = now, the universe became finite. The question is: When?

The only answer that could possible make sense (to me) is that this transition from infinite to finite occurred instantaneously - but that explanation is just as inconceivable as the question it purportedly answers!

An instantaneous change implies continuous space-time. Continuity is something that is very nice in calculus; in reality, is is nothing short of awe-inspiring and perplexing. (For one, it is hard to reconcile strictly classical conceptions of continuity with quantum tunnelling.) You can keep dividing any interval of time in half and still get an interval of time that is greater than zero. In reality, you can reiterate this process an unending number of times and still have an interval greater than zero.

But wait! Isn't any constant divided by infinity zero? Technically, yes. But don't we define infinity as "the divisor for which the quotient will be 0, regardless of the dividend"? Infinity is sort of like i, which we arbitrarily define as the square root of -1. This does nothing to explain how a number squared can be negative; it only creates an "imaginary" number (Aren't all numbers imaginary?) to fill the void in our mathematical knowledge. We have done the same thing with infinity; we've used it as a plug-in for things we don't really understand.

Though we try to use infinity as a number to express real physical quantities, we should eventually realize that infinity doesn't behave like a number. Divide infinity by any number, and you'll get infinity (That also works with zero...). Add or subtract any number to it, and you'll still get infinity (That definitely doesn't work with zero...).

For me, this means describing the beginning of the universe as "infinite" - in any way - is breathtaking. And we should probably think about it more.

(Also, if you believe in God...a singularity is a great way to get the universe going.)