Proof

A mathematical proof is a deductive argument that shows a statement must be true in every possible case, because its stated assumptions logically guarantee its conclusion. It differs from empirical evidence: presenting many confirming examples is not enough for a proof, which reasons instead by exhaustive deductive logic from axioms and previously proved statements through valid steps of inference to reach a conclusion with certainty rather than reasonable expectation. This axiomatic method was formalized by Euclid in the Elements around 300 BCE, beginning from a small set of postulates and common notions and building the whole of ancient geometry from them by explicit, checkable steps, and it remains the standard by which mathematical knowledge is distinguished from mere plausibility today.