I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions -- they are not just repetitions of each other.
Algebra is the offer made by the devil to the mathematician. The devil says: "I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine."
The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial.
The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics.
But for most practical purposes, you just use the classical groups. The exceptional Lie groups are just there to show you that the theory is a bit bigger; it is pretty rare that they ever turn up.
Scientists, outside of religion, have their own faith. They believe the universe is rational. They're trying to find the laws of nature. But why are there laws? That's the article of faith for scientists. It's not rational. It's useful. It's practical. There's evidence in its favor: The sun does rise every day. But nevertheless, at the end of the day, it's an article of faith.