How to (and Why) Prove a Mathematical Theory

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When it comes to the world of our sensory experience, the idea of "proof" is somewhat misplaced, since there is always the possibility of new evidence contradicting something we previously had every good reason to believe, not to mention the fact it could all be a dream or a simulation in The Matrix!

But in the world of abstract ideas, especially numbers, the kind of reasoning used there does lend itself to proofs because you get to specify exactly what you mean by a certain concept, and then all you have to do is follow a few rules of inference to deduce the logical consequences of your idea.

And we owe a debt of gratitude for the whole idea of theory and proof to the ancient Greeks, people like Pythagoras, Plato and Archimedes. But today we get to learn a little bit about the man who is widely considered the father of geometry, a title appropriate to the scope and importance of his work: he formalized the rules of geometry that mathematicians have relied upon for over two thousand years. That man was Euclid.

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