A postulate is a statement that is assumed to be true without proof. Euclid, the father of geometry, based The Elements on ten such statements, divided into five "axioms" and five "postulates." These are used as foundation for geometry and occasionally applied to other sciences, such as special relativity.
In ordinary discourse about a subject in mathematics or science, the terms "axiom" and "postulate" are often not distinguished—they are the statements that one takes as true from the outset, that one does not need to prove. One then uses those to prove other statements, which are then called theorems. The difference is that postulates are initial statements relating to the subject matter, such as the geometrical postulates (including Euclid's fifth postulate, below) about angles and lines. Axioms, on the other hand, tend to be more universal statements about logic, such as "if X=Y, then Y=X".
Euclid's fifth postulate
Most of Euclid's axioms and postulates do seem to be "true without needing proof;" for example, "Things which are equal to the same thing are also equal to one another."
Euclid's fifth postulate, the parallel postulate, however, is a little different. It says:
- It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
This is sometimes stated in an equivalent form, known as Playfair's axiom:
- Given a line and a point not on that line exactly one line can be drawn through that point which does not intersect the original line.
That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you negated the parallel postulate, you ended up with a perfectly logical, consistent system, a non-Euclidean geometry that simply happened to describe a kind of geometry different from plane geometry.
Equivalent statements of the fifth postulate
Without the fifth postulate many familiar geometric ideas are no longer true, these include,
- The sum of the interior degrees of a triangle are two right angle (180 degrees)
- Euclid. The Elements.
- "1.1: Definitions, Theorems, and Postulates." Beauty, Rigor, Surprise.
- "Geometry." The New Encyclopedia Britannica, 15th ed.
- "4.4: The Playfair Postulate and its Consequences." Beauty, Rigor, Surprise.