How to explain that proof is important

I'm sure I could write more about why proofs are important, but I believe it all boils down to this. Proofs are important because some very intuitive results turn out to be false, and if we just accept results with no proof because something 'sounds correct', then we could be led down false pathways. Who knows what horrors that could reap?

An idea on why learning proofs (not just theorems) is important: One could say that it is unimportant to know how to prepare food because there is a McDonald's down the street. But, if a person becomes strictly reliant on McDonald's for preparing food, then we can be assured that (s)he will never be able to produce a (worthwhile) dish of their own creation.

Likewise with proofs--one could say it is unimportant to know how to "prepare" the "food" of a theorem via proof because there is the "McDonalds" of the math book nearby. But, after years of just relying on memorizing theorems, a person will never be able to come up with a sound theorem of their own.

Being able to prove something makes it much more solidified in one's mind, and gives you a tool that is applicable to many circumstances, not just a single instance. For example, my double angle formula may not be useful when I need a triple angle formula, but I could use the proof/derivation of the double angle formula to find my own triple angle! This is where proof is much more powerful than memorization.