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Question: What is the significant difference between "a priori concepts" in Kant's terminology and "axiomatic concepts" in the vocabulary of Ayn Rand ? In other words, when Kantians say "independent of experience," do they mean that the concept is not derived from experience, or merely that no experience can prove it otherwise (that it's objective)?
Answer: I am not a Kantian, nor an expert on Kantianism. But the common understanding of "a-priori" is knowledge that can be established prior to experience. As Bertrand Russell puts it in discussing Kant in his "History of Western Philosophy": "An 'a-priori' proposition... is one which, though it may be elicited by experience, is seen, when known, to have a basis other than experience." (pp. 706-707) Traditionally, it is thought that the a-priori has no necessary bearing on experience, which is radically "contingent."
Rand's axiomatic concepts and her axioms are self-evident on the basis of the content of any particular experience. Thus, the Objectivist axioms are known by experience, but can be known from any experience. They cannot be known prior to or independent of experience, however. It is a contradiction in terms to posit consciousness with nothing to be conscious of. Consciousness is an active faculty of awareness: it exists to be aware of objects, and has no contents without them.
One sign of the difference between Objectivist axioms and Kantian "a-priori" is that Kantian "a-priori" concepts are claimed based on characteristics of language, mathematics, and logic. They are described formalistically or based on appeal to an intuition about the necessary. This basic approach thus examines the contents of a well-educated mind, ignoring the world around it, to try to establish the "apriori."
The Objectivist axioms, by contrast, are established based the characteristics of the objects of experience. That they exist, that they are what they are, that one is aware of them, (which is the basis of the axiomatic concepts of existence, identity, and consciousness) are all statements that presuppose a relation between reality and the mind of the knower, a relation that is self-evident in every experience. Rand's most fundamental axiom is "Existence exists." Non-contradiction, the point of logic that features prominently in theories of the "a-priori," is a kind of corollary of what we observe in existence. Thus logic is derived from our experience of existence.
See Ayn Rand 's Introduction to Objectivist Epistemology, and Leonard Peikoff's essay therein on the analytic-synthetic dichotomy, for further details. Tibor Machan also has a nice discussion of the Axioms in his "Ayn Rand." See also Chapter 1 of "The Logical Structure of Objectivism" beta draft.