the axiomatic method

Suppose you want to build a body of true statements you can fully trust. You cannot prove everything, because every proof leans on earlier statements, and that chain has to stop somewhere or run forever. The axiomatic method is the honest way out: lay down a short list of starting assumptions, agree on what they say, and then prove every other claim from them by pure logic. Everything you assert afterward is either an axiom you chose openly, or a theorem you derived — nothing sneaks in unannounced.

Concretely the method has four parts. First, undefined terms — basic words like 'point', 'line', 'lies on' that you do not try to define, because definitions also have to stop somewhere. Second, axioms (or postulates) — the statements about those terms you accept without proof. Third, definitions — new words introduced as shorthand for combinations of the old ones (a 'segment' is defined from points and betweenness). Fourth, theorems — everything else, each one proved by a deductive argument from axioms, definitions, and previously proved theorems. Run an example in your head: from the axioms 'any two points lie on exactly one line' and 'there exist at least three points not all on one line', you can already prove the theorem 'two distinct lines meet in at most one point' without drawing anything.

This is the method Euclid pioneered around 300 BC and that Hilbert perfected in 1899; it is the shape of all modern pure mathematics, not just geometry. Its great virtue is that it makes your assumptions visible and your reasoning checkable. Its honest limit, made precise by Godel in the twentieth century, is that a single rich system of axioms can never prove every true statement about itself nor prove its own freedom from contradiction — so the method gives certainty relative to your axioms, not certainty from nothing.