In discussing Objectivism’s view of axioms, it is helpful (essential, actually) to remember Objectivism’s overall view of how *all* concepts are formed, including axiomatic concepts.
In ITOE Chapter 1, Ayn Rand writes: “Although, chronologically, man’s consciousness develops in three stages: the stage of sensations, the percpetual, the conceptual – epistemologically, the base of all of man’s knowledge is the *perceptual* stage. … Discriminated awareness begins on the level of percepts.”
On the role of sensations, Ayn Rand explains [ibid.]: “Percepts, not sensations, are the given, the self-evident. The knowledge of sensations as components of percepts is not direct, it is acquired by man much later: it is a scientific, *conceptual* discovery.”
Concepts, according to Objectivism, are integrations of percepts – specifically, percepts of *existents*, with the existents regarded as *units* of a type or kind. The type or kind is an *integration* (essentialization) of the units. [iTOE, Chap. 1]
So, if this applies to axiomatic concepts as well as non-axiomatic ones, what is the purpose or function of classifying certain concepts as axiomatic? The term “axiomatic” in Objectivism pertains to the process of *validating* concepts, i.e., of establishing a concept’s relation to reality. Axiomatic concepts differ from other concepts in that axiomatic concepts are *perceptual self-evidencies*. [OPAR, p. 8] No “proof” of an axiomatic concept is needed or possible beyond pointing to abundant examples of it (ostensive definition), preferably along with representative “near misses” (instances that closely resemble the concept in question but don’t quite qualify), and showing that the concept is axiomatic, i.e., that it identifies “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 all knowledge.” [iTOE Chap. 6]
OPAR emphasizes the validation issue very succinctly [p. 11]: “The foregoing is not a proof that the axioms of existence, consciousness, and identity are true. It is a proof that they are *axioms*, that they are at the base of knowledge and thus inescapable. This proof itself, however, relies on the axioms. Even in showing that no opponent can escape them, Ayn Rand too has to make use of them. All argument presupposes those axioms, including the argument that all argument presupposes them.”
In what sense, then, are axiomatic concepts “starting points” for knowledge? Are they to be taken as the exhaustive “givens” for all sorts of deductive chains purporting to derive all other concepts and propositions from the axioms by deductive logic? Clearly not, according to Objectivism. Knowledge – propositions and concepts – is derived from observation of reality (ultimately percepts) and integration of those observations into abstractions. In essence, we look at reality and report what we see. It’s only when we have sufficient knowledge to begin *proving* various propositions that we discover that all our proofs depend on certain fundamental concepts that we necessarily must accept as axiomatic, inescapably and ostensively verifiably so.
The idea of simply looking at reality and reporting what one sees may seem fraught with pitfalls. Prof. John Ridpath once commented to a live audience at a talk on a college campus [i’m quoting approximately from memory]: “As children you probably had total confidence in your sensory-perceptual apparatus; it took a sophisticated 20th Century argument to talk you out of it.” Objectivism, however, navigates all the pitfalls astoundingly skillfully. See especially ITOE and OPAR.
In defense of Reality and Reason, Life and Value, Freedom and Rights, Spirit and Happiness, through epistemology and individualism – RReLV_FRiSH.