Axioms as Irreducible Primaries
The term “axiom” as used in philosophy has a distinct meaning than “axiom” as mathematicians often use it. The mathematician (at least the modern one) normally uses “axioms” to denote a set of inter-consistent but arbitrarily selected rules for a given arbitrarily selected “system of logic.”1 These are assumptions that are taken to be true in order to allow for further reasoning within the given domain, but that are not (or perhaps cannot) be validated.
The philosopher (at least a good one) does not use “axiom” to mean some arbitrarily selected starting point. In philosophy an axiom means an irreducible primary—a starting point that cannot be escaped in any process of cognition. Axioms themselves are not proven, there is no proof that A is A, as “proof” itself relies upon axioms. An axiom is rather validated.
An axiomatic concept is the identification of a primary fact of reality, which cannot be analyzed, i.e., reduced to other facts or broken into component parts. It is implicit in all facts and in all knowledge. It is the fundamentally given and directly perceived or experienced, which requires no proof or explanation, but on which all proofs and explanations rest.2