I read a book of logical puzzles once and there was a little story in it (fictional) about some boy who received an F on his geometry test because his professor said his proof that all angles of a triangle add up to 180 degrees was wrong. The boy countered with something like, "You've defined for our class everything about parallel lines, angles, etc., but you haven't defined 'proof'. How am I supposed to know how to prove something if you haven't defined what it means to prove something?".
I thought this was very interesting, so I looked up the Google definition:
Evidence or argument establishing or helping to establish a fact or the truth of a statement.
Merriam-Webster defined it as:
a : the cogency of evidence that compels acceptance by the mind of a truth or a fact
b : the process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning
Why don't the majority of teachers concern themselves with defining "proof" for a class?
Furthermore, is it possible to prove that there are a certain number of ways to prove a statement?