On Mathematics and Mathematical Proofs

Mathematics is a language like any other language. It uses symbols to represent ideas, and those ideas attempt to represent reality. Like any language however, the symbols, letters, or words used are an abstraction, they are meaningless by themselves.

The beauty of mathematics is that new meanings fall from the symbols which cannot always be explained. One could say the same thing for words, a certain arrangement of words, in a novel or poem might enlighten us to new meanings we had not before perceived, and in mathematics there exist certain inherent patterns and meanings within the system, waiting to be explained or discovered.

The symbols matter. The better the symbols, the more efficient the phrases, and the more profound the thoughts. Let's consider for a moment all the different ways to write division:

1. In English: Six divided by four.
2. On way in math: 6 ÷ 4
3. Another math way: 6/4

Which method of representing the idea of six divided by four best leads you to realize that six divided by four is that same idea as three divided by two?

Or that is to say:

6/4 = 3/2

Thus in mathematics, or all languages, we are presented with the interesting problem that the symbols we contrive to use have a direct impact on the conclusions we make and how clearly we can express them.

For the purposes of mathematical proofs we will use the following generally accepted standard symbols:

Λ as "and"
V as "or"
~ as "the opposite"