Axioms are usually considered to be propositions identifying a fundamental, self-evident truth. But explicit propositions as such are not primaries: they are made of concepts. The base of man’s knowledge—of all other concepts, all axioms, propositions and thought—consists of axiomatic concepts.
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.
The first and primary axiomatic concepts are “existence,” “identity” (which is a corollary of “existence”) and “consciousness.” One can study what exists and how consciousness functions; but one cannot analyze (or “prove”) existence as such, or consciousness as such. These are irreducible primaries. (An attempt to “prove” them is self-contradictory: it is an attempt to “prove” existence by means of non-existence, and consciousness by means of unconsciousness.)
[The] underscoring of primary facts is one of the crucial epistemological functions of axiomatic concepts. It is also the reason why they can be translated into a statement only in the form of a repetition (as a base and a reminder): Existence exists—Consciousness is conscious—A is A. (This converts axiomatic concepts into formal axioms.)
Epistemologically, the formation of axiomatic concepts is an act of abstraction, a selective focusing on and mental isolation of metaphysical fundamentals; but metaphysically, it is an act of integration—the widest integration possible to man: it unites and embraces the total of his experience.
The units of the concepts “existence” and “identity” are every entity, attribute, action, event or phenomenon (including consciousness) that exists, has ever existed or will ever exist. The units of the concept “consciousness” are every state or process of awareness that one experiences, has ever experienced or will ever experience (as well as similar units, a similar faculty, which one infers in other living entities).
Since axiomatic concepts refer to facts of reality and are not a matter of “faith” or of man’s arbitrary choice, there is a way to ascertain whether a given concept is axiomatic or not: one ascertains it by observing the fact that an axiomatic concept cannot be escaped, that it is implicit in all knowledge, that it has to be accepted and used even in the process of any attempt to deny it.
For instance, when modern philosophers declare that axioms are a matter of arbitrary choice, and proceed to choose complex, derivative concepts as the alleged axioms of their alleged reasoning, one can observe that their statements imply and depend on “existence,” “consciousness,” “identity,” which they profess to negate, but which are smuggled into their arguments in the form of unacknowledged, “stolen” concepts.
It is worth noting, at this point, that what the enemies of reason seem to know, but its alleged defenders have not discovered, is the fact that axiomatic concepts are the guardians of man’s mind and the foundation of reason—the keystone, touchstone and hallmark of reason—and if reason is to be destroyed, it is axiomatic concepts that have to be destroyed.
It is only conceptual awareness that can grasp and hold the total of its experience—extrospectively, the continuity of existence; introspectively, the continuity of consciousness—and thus enable its possessor to project his course long-range. It is by means of axiomatic concepts that man grasps and holds this continuity, bringing it into his conscious awareness and knowledge. It is axiomatic concepts that identify the precondition of knowledge: the distinction between existence and consciousness, between reality and the awareness of reality, between the object and the subject of cognition. Axiomatic concepts are the foundation of objectivity.
It is only man’s consciousness, a consciousness capable of conceptual errors, that needs a special identification of the directly given, to embrace and delimit the entire field of its awareness—to delimit it from the void of unreality to which conceptual errors can lead. Axiomatic concepts are epistemological guidelines. They sum up the essence of all human cognition: something exists of which I am conscious; I must discover its identity.
Since axiomatic concepts are identifications of irreducible primaries, the only way to define one is by means of an ostensive definition—e.g., to define “existence,” one would have to sweep one’s arm around and say: “I mean this.”