Many mathematicians spend their time proving results. The (very old) joke is that they are machines for turning coffee into theorems. A *theorem* is a statement that has been shown, by a sequence of irrefutable steps, to follow logically from a set of fundamental assumptions known as axioms.

These axioms themselves may be self-evident, or may simply be assumed to be true. Given this, the statement contained in a theorem is known with certainty to be true.