A mathematical system of logic seeks to prove statements, or theorems, by combining other statements according to the inference rules defined as part of the logical system. But in order to get started, it is desirable to have statements that don't need to be proved. These are called axioms, and form part of the definition of the system of logic. (It is possible for a system of logic to have no axioms, but most do.)
For example, if we want to devise a system of logic to prove statements about arithmetic, we might define rules of inference like "if a is an integer, then a+1 is also an integer" and "if a and b are both integers, and a = b, then a+1 = b+1". But this does not get us very far unless we also define some axioms as starting points, e.g.:
- 0 is an integer.
- 0 = 0
By combining these axioms with the rule of inference, we can prove theorems like
- 0+1 = 0+1
- 0+1+1 = 0+1+1
- 0+1+1 is an integer
and so forth.
Note that an axiom need not be true in any obvious sense. Rather, axioms are accepted for the sake of argument. If a mathematician demonstrates a theorem in a given system of logic, she is really saying, "if these axioms are true, and if these rules of inference are valid, then this theorem must necessarily be true as well." To the extent that the axioms and rules of inference are true in real life as well, the theorems allow us to make deductions that are also true in the real world.
In philosophy, axioms are the fundamental assumptions upon which a philosophical system or set of arguments is based. For example, a system of ethics might be based on the axiom "everyone has a good idea of what other people like and dislike."