Are Postulates Accepted As True Without Proof?

Postulates are mathematical propositions that are assumed to be true without definite proof. In most cases, axioms and postulates are taken to be the same thing, although there are some subtle differences.

The difference between axioms and postulates is that axioms, or algebraic postulates as they are sometimes called, are generally about real numbers, whereas postulates relate more to geometry.

There are five key postulates that form the basis of Euclidean geometry that are known as Euclid’s postulates. Euclid laid these postulates out in “The Elements.” Euclid’s postulates have been corrected slightly over the centuries, but they still remain basically sound. From these postulates, mathematicians are able to form theorems and geometric proofs.

Euclid’s basic postulates are that a straight line can be drawn to connect any two points, any line segment can be extended into a line that goes on forever, any straight line segment can be transformed into the radius of a circle with the centerpoint of the circle on the segment, all right angles are congruent, and if two lines are drawn so they intersect with a third and the sum of the inner angles is less than 180 degrees, then those two lines eventually intersect if they are extended.