I’ve been doing a lot of reading about math lately — calculus, complex numbers, the discovery of the transcendental functions, and so on. I also occasionally catch a random article discussing string theory. Now, string theory I don’t really understand. It discusses ideas like the possibility that our universe actually exists in many more than 3 dimension, but for some reason, the 3 that we see dominates.
I’ve noticed a few things that could be related to that. First of all, there is Fermat’s theorem. Basically, it says that there is no solution for x^n+y^n=z^n for any n greater than 2. Could this have anything to do with our ability to only perceive the world in 3 dimensions? (it’s only one more than the magic 2… you see off-by-one sorts of behavior all the time.)
There’s another place where I’ve noted the the 2nd dimension is some sort of fundamental limit. Consider real and complex numbers. The real numbers end up with holes for some operations… namely, the roots of negative numbers. However, it turns out that mathematicians got around this by simply “inventing” a new number, i, which is the square root of negative one. This was generalized into the concept of a complex number, which is the sum of some real part, and some imaginary part. So, whereas you can think of the real numbers populating a single dimension, you can think of the complex numbers as populating two dimensions.
As it turns out, this is “enough.” There are no more holes once you reach the two-dimensional complex plane. At least, for the mathematical operations that we know of. For example, you might say “Hey, wait a second, what is the square root of -i?”) Well, there is a square root of i that is a complex number. To find it, just solve the equation (a+bi)^2=-i. It’s there. It’s trivial.
Perhaps there is some additional weirdness to be discovered in some mathematical function that we have not thought of, that would push things to the third dimension. But then, the question would be, can we always find such holes, and require the addition of another dimension, onwards into infinite dimensionality?
At this point, I back away slowly, because my mathematics knowledge is simply not complete enough to seriously tackle this question.