Why do we not have to prove definitions? I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? It seems like some definitions should have a foundation based on proof. How simple (or intuitive) does something have to be to become a definition? I mean to ask this and get a clear answer. Hopefully this is not an opinion-based question, and if it is will someone please provide the answer: "opinion based question."

Frequently, a definition is given, and then an example or proof follows to show that whatever has been defined actually exists. Some authors will also attempt to motivate a definition before they give it: for example, by studying the symmetries of triangles and squares and how those symmetries are related to each other before going on to define a general group.
A definition is distinguished from a theorem or proposition or lemma in that a definition does not declare some fact to be true, it merely assigns meaning to some group of words or symbols. The statement of a theorem says that "such-and-such" thing is true, and then must back up the claim with a proof.
