The dictionary definition of axiom is "a rule or statement that is accepted as true without proof". In simple terms, an axiom is something we know to be true without questioning it.
There are two different types of axioms, logical axioms and non-logical axioms.
Logical axioms are axioms that are taken to be universally true, such as (A and B implies A). Non-logical axioms are properties we use to define specific mathematical fields, such as a + b = b + a.
An example of an addition and multiplication axiom we know to be true:
Let x and y be real numbers.
Then x + y is a real number and xy is also a real number.
What is the role of axioms in math?
There are axioms that we assume to be true for a variety of mathematical content areas, and these axioms help us develop important theories. Axioms are the building blocks of mathematical proofs, as they are helpful when processing information and assess what we already know to prove mathematical conjectures. Axioms also define unknown terms, such as line, point, etc. Axioms play a crucial role in mathematics, as they give us a basis for proofs, and give us information to use when solving mathematical problems. When people discuss using "logic and reasoning" to determine the results of a problem, it is my opinion that they are using axioms to determine those results. Axioms are directly associated with the logic and reasoning we use in many different subject areas today, not just mathematics.
**To create this post, I used my own knowledge of axioms as well as information published on Princeton University's webpages.**