Postulates

Postulates are statements assumed to be true without proof that form the basis for deductive reasoning within a theory. In mathematics and logic, postulates (often called axioms) are foundational assumptions from which theorems are derived. They differ from theorems, which are propositions proved from the postulates using rules of inference.

In geometry, Euclid presented a set of postulates used to derive its theorems. The famous five postulates include: a straight line can be drawn between any two points; a finite segment can be extended indefinitely; a circle can be drawn with any center and radius; all right angles are equal; given a line and a point not on it, there exists exactly one line through the point parallel to the given line. The fifth postulate, the parallel postulate, has been central to many developments and variations, leading to non-Euclidean geometries when altered.

Historically, postulates were first systematically used by Euclid in the Elements. In the 19th and 20th centuries, mathematicians like Hilbert reformulated foundations in terms of explicit axioms, improving rigor and scope. In modern usage, "axiom" is more common for the foundational statements of a theory, while "postulate" remains common in geometry and in descriptions of physical theories.

Postulates serve to define the scope and structure of a theory; they are not proven within that theory but tested against models and observations. Theorems are the conclusions drawn from these postulates under logical deduction.