Definitions, Postulates, and Proof

Point, line, and plane are undefined terms. Everything else is built from them.

Some statements are accepted without proof.

Postulate is a statement accepted as true without proof, used to prove other things.

A theorem is earned

A theorem is proved from definitions, postulates, and theorems already established.

What a proof is

Each statement follows from what came before, with a reason given for every step.

What counts as an argument

A proof is a chain of claims where every link follows from a definition, a postulate, or something already proved. Measuring a drawing is not proof — the drawing might be inaccurate, and one accurate case says nothing about all cases.

Some terms are left undefined

Point, line and plane are taken as primitive. Every definition must be built from something, and if everything were defined the definitions would circle back on themselves. Starting from a few undefined terms is what stops the regress.

Postulates are assumed, not proved

A postulate is a statement accepted without proof, chosen because it is obvious and because useful mathematics follows from it. Changing one postulate — the parallel postulate — produces entirely different, and entirely valid, geometries.

One counterexample kills a claim

To disprove "all quadrilaterals with equal diagonals are rectangles", exhibit one that is not. Proving a general claim needs an argument covering every case; disproving it needs a single example. The asymmetry is worth internalising.