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."

If you want to prove a statement, you need to first tell me what the objects involved in the statement are. For example, if you want to prove that the product of two even numbers is even, you first need to tell me what an even number is, what a product is, ... even what a "number" is. Mathematics is about finding out what relationships/results hold after starting with certain objects that you define. Yes in some sense, the definitions seem to come from nowhere (why do we need imaginary numbers? they're just "made-up"?), but are usually well-motivated (we want roots of negative numbers, solutions to quadratics, etc.).
