Regardless of whether or not God exists, God has no place in mathematics, at least in my book. - Doron Zeilberger

Many of the successes of `fancy' mathematics are due to sociological and linguistic reasons. They are really high-school-algebra arguments in disguise. Once you strip the fancy verbiage off, what remains is a bare (and much prettier, in my eyes) argument in high-school mathematics, or at most, in Freshman linear-algebra and (formal!) Calculus. For example, I am (almost) sure that Wiles's proof would be expressible in simple language, and if not, there would be a much nicer proof that would.

The best way to learn a topic is by teaching it. Similarly the best way to understand a new proof is by writing an expository article about it.

Programming is much much harder than doing mathematics.