What is the Difference Between Axiom and Postulate?

The difference between an axiom and a postulate lies in their application and specificity within the field of mathematics:

1. Axiom: An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true on which an abstractly defined structure is based. Axioms are true assumptions used throughout mathematics and not specifically linked to geometry. They are considered to be self-evident and are common to all branches of science. An example of an axiom is the statement that "Things which are equal to the same thing, are equal to one another".
2. Postulate: Postulates are true assumptions that are specific to geometry. Euclid used the term "postulate" for assumptions that were specific to geometry, whereas axioms are used throughout mathematics and are not specifically linked to geometry. Postulates aim to capture what is special or particular about a specific field or science. An example of a postulate is Euclid's statement, "It is possible to produce a finite straight continuously in a straight line".

In summary, axioms are general statements that apply to all branches of mathematics and are self-evident, while postulates are assumptions specific to geometry or a particular field of science.

Comparative Table: Axiom vs Postulate

The terms "axiom" and "postulate" are often used interchangeably in modern mathematics, as they both refer to statements that are assumed to be true within a specified context. However, there are some subtle differences between the two:

In summary, axioms are universal, general, and self-evident truths, while postulates are more specific to the context at hand. Both axioms and postulates serve as the foundation for building theorems and theories in various fields of mathematics and other sciences.