Friday, February 29, 2008

Numbers

It's funny because I was thinking about the difficulty with math that many people have, and then PoMoCon writes about it. He links to a really good analysis of the levels of mathematical understanding here.

The root of this problem, as far as I can determine, is that math is a completely abstract realm that implies existence.

To clarify what I mean about implying existence:
Let's discuss the simplest number, 1. We learn that it means a lone thing. However, what that thing is doesn't matter. It could be one orange, one person, one boat, etc. When dealing with counting or addition, teachers say that if you have 1 orange and another orange, then you have two oranges. This implies the existence of two oranges.

Numbers exist as concepts, which imply the existence of some thing or things. In this way I think the math and God analogy is relevant. When you think of any number, it is the idea of an amount of undetermined things that exists in some abstract realm. This realm has rules and laws that we can discover and understand. However, we have the perception in our mind that we are creating this realm. Therefore, many philosophers are drawn to math because it creates the illusion that we are creating another world, which has rules, and these rules can be used to understand the Truth.

As you move from addition and subtraction, which can be intuitively understood with theoretical objects that exist, to multiplication and division this starts to fall apart. The numbers themselves begin to interact in ways that just don't make sense when you are talking about oranges.

As you move to limits, matrices, etc, this moves even further from actual existence. Infinity is not a number, but a concept. Zero is a number, but signifies the absence of existence.

One concept from calculus can be understood intuitively. Functions are much like a machine with inputs and an output. 