The Pythagorean Theorem, for instance, has many, many proofs which are fundamentally different and use what appear to be very different ideas. These range from static geometric constructions to similarity factor arguments.
My question is: are there any provable statements which do not have more than one way to prove them (up to proofs being "effectively the same" in some way)? If so, what is this notion of "the same", and how do we know there is only one way to prove something, or that no such provable statements exist?
Equivalently, can everything which is provable be proved in more than one way? We know some true statements cannot be proven at all by Godel, but do some have just one proof? (or finitely many?)
EDIT: The notion of "effectively the same" and what that means is designed to be a part of the question; asking whether such a notion formally exists and, if so, what that notion is.
Derik Elkins does a very good job of answering with similar level of detail to what was asked. The answer was that, yes, there are some statements which only have one "normal form", and thus all their proofs are equivalent.