An axiom is a statement that is considered to be true in order to serve as a premise or starting point for reasoning and arguments. The word axiom comes from the Ancient Greek word axioma meaning ‘that which is considered worthy or fit’ or ‘that which presents itself to be evident’.
The term has subtle differences in definition when it comes to different fields of study. In classic philosophy, an axiom is defined as a statement that is so obvious, evident or well established that it is accepted without question. As used in modern logic, an axiom is a premise or starting point for reasoning.
As used in mathematics, the term axiom is used in two related but distinguishable senses:
- Logical axioms – These are generally statements that are considered to be true in the system of logic that the axioms define. They are usually represented in a symbolic form.
- Non-logical Axioms – These are axioms that where there are substantial assertions regarding the elements within a domain of a certain mathematical theory
In a sense, the terms axiom, assumption, and postulate can be used interchangeably. More often than not, a non-logical axiom is simply a formal logical expression that is used in deduction to construct a mathematical theory, which may or may not be self-evident in nature. When a system of knowledge is said to be axiomatized, what it means is that its claims can be derived from a small set of sentences, and there may be multiple ways to make a given mathematical domain axiomatized.
Any axiom is a statement that is a starting point from which other statements are logically derived. Whether it is meaningful for an axiom to be true is a subject of debate within the philosophy of mathematics.