Lennox on Axioms

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Submitted by seddon on Tue, 2006-02-14 15:16

In this article I would like to raise five points about Jim Lennox’s paper, “Ayn Rand as Aristotelian: Axioms and their Validation,” which he delivered at the American Philosophical Association meeting in New York City on Dec. 29, 2005.

1. THE NECESSITY OF AXIOMS: On p. 3 of his paper Lennox quotes Rand approvingly, (All four speakers ALWAYS quote Rand approvingly) “But what will come out of this is an arrangement of the whole in a logical system, proceeding from a few axioms in a succession of logical theorems. The axioms will be necessary—even mathematics has them—[because sic.] you can’t build something on nothing. . . .(Harriman 1997, 72) Lennox then gives the “you can’t get something from nothing” argument. You need axioms because you can’t get something from nothing. But he gives no argument for identifying the “something” as “axioms.” He also gives no argument for equating a non-axiomatic foundation with nothing.
Non-axiomatic propositions could be the “foundation” of other non-axiomatic propositions in two ways. One could have an implicit regression to infinity; or one could have a circle of non-axiomatic propositions in a web of mutual support. (An example of the latter, albeit on the level of concepts rather than propositions would be a dictionary—all words are defined in terms of other words which are defined in terms of other words etc. An example of the former would be division of rational numbers; you never run out. Now don’t misunderstand. There are arguments against both of these alternatives I have mentioned. But there are also arguments against axiomatic foundationalism. A lot of 20th century philosophy consists of arguments against foundationalism. I just would like to see Lennox’s arguments for foundationalism.

Strictly speaking, even Aristotle has no argument, in the sense of a demonstration, against the non-foundationalist. But he does have some arguments by refutation. I have written a book about this and the interested reader my consult chapter 11, “Demonstration vs. Refutation” in my ARISTOTLE AND LUKASIEWICZ ON THE PRINCIPLE OF CONTRADICTION. In his METAPHYSICS, IV, 3, Aristotle offers a “refutation” of those who would deny the principle of contradiction, which is an anticipation of Rand’s “accept in order to deny” argument, to which I now turn.

2. LENNOX ON THE “ACCEPT IN ORDER TO DENY” ARGUMENT: Here is Lennox’s statement of the argument. “These most fundamental of all truths, are incapable of proof, for all proof [here equivalent to Aristotle’s concept “demonstration”] rests on them. Any attempt to establish them by a proof, Rand argues, would be self-contradictory (Rand 1990, 55). What one CAN prove is precisely that they are axiomatic, by showing that they are presupposed in any attempt to deny them.” (4) Lennox rightly notes at the top of the page just quoted that Rand is not talking about axioms, but rather about axiomatic concepts. But wait. Why isn’t that his title? His title is “Ayn Rand as Aristotelian: Axioms and their Validation” and not Ayn Rand as Aristotelian: Axiomatic Concepts and their Validation.” Is there a little sleight of hand going on here? Lennox seems a bit confused because in the first sentence just quoted he refers to the “most fundamental of all truths.” But truth is a property of propositions, not concepts. We expect him to be talking about axioms, not axiomatic concepts. Not only that, proofs deal with propositions, not concepts, and Lennox is talking about “proofs.”

3. EINE KLINE CONFUSION: Rand also seems confused in her discussion on p. 55. After telling us that axiomatic concepts are “implicit in all facts and in all knowledge,” she then goes on to say that any “attempt to ‘prove’ them is self-contradictory.” But this is wrong. If they are implicit in all knowledge, any attempt to prove them would be question begging, not self-contradictory.

4. RAND ON THE “ACCEPT IN ORDER TO DENY” ARGUMENT: Rand states the “accept in order to deny” argument on p. 59. “…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.” But must we not ask if this criterion or test of axiomaticity is itself axiomatic? If it is not axiomatic, why use the “accept in order to deny” criterion instead of say, self-evidence? By self-evidence is meant, following Aquinas, “A proposition is self-evident because the predicate is included in the essence of the subject, as "Man is an animal," for animal is contained in the essence of man.” (ST, I,2) Rand’s opening sentence of chapter 6 would lead on to believe that she should use this criterion rather than the “accept in order to deny” test. Here are her words: “Axioms are usually considered to be propositions identifying a fundamental, SELF-EVIDENT truth.” (55 Emphasis mine.) We are given no argument for why we should accept the “accept in order to deny” criterion instead of the self-evident criterion. Maybe she prefers the “accept in order to deny” criterion because it is the one Aristotle uses it in the METAPHYSICS, IV, 4. But what is fascinating about this is the fact that Aristotle uses the “accept in order to deny” to establish a different axiom than Rand does. He uses it to establish the principle of contradiction; she uses it to establish the axiomatic concepts of existence, consciousness and identity. (Aristotle didn’t even have the concept of “consciousness.”)

5. AXIOMATIC CONCEPTS: Finally, does Rand’s move from axioms in ATLAS SHRUGGED to axiomatic concepts in ITOE succeed? Recall the opening paragraph of chapter 6: “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…consists of axiomatic concepts.” (55) From this we may conclude that concepts are more fundamental than propositions, and a fortiori, that axiomatic concepts are more fundamental than axioms. (Forget about the fact that in ATLAS SHRUGGED we are told nothing of axiomatic concepts, there everything depends on axioms, viz., Existence exists and consciousness is conscious.) Here the problem seems to arise from the fact that Rand has forgotten what she wrote just seven pages earlier, to wit, “Every concept stands for a number of propositions.” This would seem to make propositions more fundamental. (48) So what is it? Are concepts primary or are propositions? Perhaps one can solve this problem by distinguishing the order in which one learns concepts from their logical sequence. One may know lots of propositions before forming a concept, but once one has lots of concepts, one can then organize them in a logical manner. This could be what Rand had in mind in the Harriman quotation in section 1 above. Let me give Rand the last words by quoting the entire paragraph from p. 72 “

“(All these things are ONLY for my use. They are pretty disjointed and not in any logical sequence. But what will [ultimately sic] come out of this is an arrangement of the whole in a logical system, proceeding from a few axioms in a succession of logical theorems. The axioms will be necessary—even mathematics has them—[because sic.] you can’t build something on nothing. The end result will be my ‘Mathematics of Philosophy.’)”

These are some of the questions and problems I had with Lennox’s paper.

Fred


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