Rules of Inference and Logic Proofs A proof is an argument from hypotheses (assumptions) to a conclusion. Each step of the argument follows the laws of logic. In mathematics, a statement is not accepted as valid or correct unless it is accompanied by a proof. This insistence on proof is one of the things that sets mathematics apart from other subjects.
We work backwards to know where we're going, but we write forwards to make sure everything actually works. However, it is not always the case that proofs proceed from assumptions to goals. Here are two typical exceptions to the rule of start at the beginning and end at the end: Theorem: XXX. proof.