proof
A proof is an airtight chain of reasoning that shows a mathematical statement is true — not just probably, not just for every case anyone has checked, but with total certainty, forever. Each link follows from the one before by pure logic, like a row of dominoes where knocking the first guarantees the last must fall. Once a proof is sound, the result is settled for all time; no future experiment or counterexample can ever overturn it.
This is what makes mathematics unlike almost everything else. A scientist can test a claim a million times and still be proven wrong on the million-and-first; a mathematician who has proved that there are infinitely many prime numbers knows it the way you know two plus two is four. The whole edifice is built by starting from a few plain assumptions and deducing everything else, step by careful step.
A common mistake is to think that checking lots of examples counts as proof. It doesn't. A pattern can hold for the first billion numbers and then quietly break — and history is full of such traps. Examples can convince you a thing is likely true and hint at why, but only a proof can promise it holds in every single case, including the ones too large to ever check.
To prove √2 is irrational, assume it equals a fraction in lowest terms — then show that very assumption forces the fraction to not be in lowest terms. The contradiction proves no such fraction can exist.
A proof by contradiction: assume the opposite, then watch it self-destruct.
The systematic axiomatic style of proof still used today — start from explicit axioms, then derive theorems step by step — was brought together and made canonical around 300 BC by the Greek mathematician Euclid in his Elements, building on earlier Greek geometers. It became the most influential textbook ever written. The closing flourish "Q.E.D." comes from the Latin quod erat demonstrandum, "which was to be demonstrated."