AXIOM (ἀξίωμα, from ἀξίω “I esteem, judge well”). — In the current sense, an axiom is a general principle of immediate evidence that does not require proof; in particular, the first principles of mathematics. In Aristotle the meaning is somewhat narrower: an axiom is a common immediate universal principle, necessary for every science, for example the principle of identity (Anal. Post., I, 2, 72 a, 16). Among the Stoics, axiom becomes synonymous with general proposition (Aristotle’s ἀξιόπαντος); a sense substantially resumed by B. as “axioms” are all general propositions (“most general axioms” and “mediate axioms”) derived from the senses and particular facts (Novum Organum, I, 19). In Kant, axioms of intuition are called synthetic a priori judgments derived from pure sensible intuition of space and time: for example, “two straight lines do not enclose a space.”
Axioms, understood in the common sense as self-evident general principles, claim, against every empiricist attempt at devaluation, an absolute noetic value, because they are at once expressions of the laws of thought and of the objective relations of being. Sprouting in the intellect at the first contact with experience, so that they cannot be said to be absolutely a priori, whether they are most universal and therefore irrefutable, or derived from these as their particular expression, axioms remain, by virtue of their character of universality and necessity, one of the firm foundations of certainty and of science.