Foundations of Mathematics

Axiomatic System

What are axioms and postulates?

These two words are often treated as the same: in fact, some mathematicians consider the word axiom a slightly archaic synonym for postulate. Although both are considered to be a proposition (statement) that is true without proof, there are subtle differences.

An axiom in mathematics refers to a general statement that is true without proof, and is often related to equality, such as “two things equal to the same thing are equal to each other,” and those related to operations. They should also be consistent—or it should not be possible to deduce any contradictory statements from the axiom.

A postulate is also a proposition (statement) that is true without proof, but it deals with specific subject matter, such as the properties of geometric figures. Thus, it is not as general as an axiom. For example, Euclidean geometry is based on the five postulates known, of course, as Euclid’s postulates. (See below; for more about Euclid, see “History of Mathematics” and “Geometry and Trigonometry”).


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