RoR: Forum

Post 0

Monday, January 9, 2006 - 11:33am

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Stephen, thanks for the report. Interesting topic.

For me to understand why they might select the axioms they did, or what differences there might be between them, I'd need to know what each thought the purpose of these axiomatic concepts was. What did each think was the significance?

For instance, an unfortunately common view of the axioms is that you deduce all knowledge from them. That's obviously not Rand's view, as she strongly grounds her knowledge via induction. So what is the purpose of the axiomatic concepts for Rand? And was it the same as Aristotle's?

Was any of this presented or discussed?

--Joe

Post 1

Wednesday, January 11, 2006 - 7:43am

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Stephen, I am a contributor to ARS and they sent me the 2005 papers, so I appreciate your good comments on them. I may chime in from time to time on these discussions. For now, I just want to point out a noteworthy essay in vol. 2, no. 2 of Stephen's own (former) journal, Objectivity. Although (I believe) the essay does not mention Aristotle's view of the axioms, it deals at considerable length with Rand's axioms. It's certainly "grist for the mill." Here is the abstract from the Objectivity website:

"Axioms: The Eightfold Way" by Ronald E. Merrill
Proposes a new organization of Ayn Rand's axioms, which were three: Existence, Consciousness, Identity. In the new organization, there are eight axioms. There are three logical axioms, which identify the rules of reasoning; three metaphysical axioms, which root our knowledge of reality; and two epistemological axioms, which are presumed when we assert anything to be known.

REB

Post 2

Wednesday, January 11, 2006 - 10:50am

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Rand introduces her most fundamental axiom on page 1015 of Atlas Shrugged (hb). That is the assertion existence exists. As she introduces the axiom, she says that the moral code that she is overturning and replacing attempts to escape the axiom existence exists. She has already said that the code she means to overturn comes in a variety based on dictates of a supernatural being known as God (1011-12). One of the purposes of Rand's axiom existence exists is to foreclose the possibility of the existence of God.

In her later essay "The Metaphysical v. The Man-Made" (1973), she tells us that her axiom "existence exists" means that the universe exists independently of consciousness (24) and that the universe as a whole "cannot be created or annihilated, that it cannot come into or go out of existence" (25). She says that her fundamental axiom invalidates the question "If there is no God, who created the universe?"

Immediately after introducing he axiom existence exists in AS, Rand introduces axioms concerning consciousness, which are corollaries of grasping the statement existence exists. These are "that something exists which one perceives and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists" (1015). This characterization of consciousness and self-consciousness rules out the possibility of God as a mind existing before the existence of anything else. "A consciousness conscious of nothing but itself is a contradiction in terms: before it could identify itself as consciousness, it had to be conscious of something" (1015). In these strokes, not only God is swept away. Idealism (Berkleyian, transcendental, or metaphysical), Cartesian skepticism, and materialism (non-existence of mind) are also out of court. (See also, page 1027 and "For the New Intellectual" [1960].)

Readers here know that Rand articulated another axiom:

"To exist is to be something, as distinguished from the nothing of non-existence, it is to be an entity of a specific nature made of specific attributes. Centuries ago, the man who was---no matter what his errors---the greatest of your philosophers, has stated the formula defining the concept of existence and the rule of all knowledge: A is A. A thing is itself. You have never grasped the meaning of his statement. I am here to complete it: Existence is Identity, Consciousness is Identification." (1016)

With this further statement of her axioms, Rand can go on to rule out radical indeterminacy of human nature and to portray the applicability of the law of non-contradiction to the real world known by ordinary experience and science (1016, 1037, 1040-41).

That is not all. With her full complement of axioms on the table, Rand puts them to the purpose of refuting the method of faith and revelation (1018, 1035-36), radical separation of human values from matter or mind (1029-30), supremacy of will or feeling over rational perception of reality (1036-37), skepticism concerning sensory perception (1036, 1040-41), skepticism concerning causality (1037), and skepticism concerning knowledge (1039-40).

On page 1040 Rand says that "an axiom is a statement that identifies the base of knowledge and of any further statement pertaining to knowledge, a statement necessarily contained in all others." She contends that anyone who denies the "axiom of identity" will be unable "to present his theory without using the concept of identity or any concept derived from it." Oh, I almost forgot another purpose to which Rand put her axiom of identity. She used it to bar the "negative way" of approaching God (1035). In Christianity that was an approach going back to Pseudo-Dionysius (c. 500).

Ten years later, Rand explicitly stated the general purposes of philosophical axioms. "Although they designate a fundamental metaphysical fact, axiomatic concepts are the product of an epistemological need---the need of a volitional, conceptual consciousness which is capable of error and doubt" (IOE 58). James Lennox reminds us that this was one of the purposes Rand gave for identifying philosophical axioms. I have compiled above several specific cases of Rand using her axioms for this purpose.

"Both Ayn Rand and Aristotle see axioms as grasps or explicit identifications of fundamental facts about being or, to use Rand's more typical wording, reality" (AV 13). Do Aristotle and Rand see philosophic axioms as serving the same purposes?

That is one of Joseph Rowland's main questions. I hope that thinkers reading here at RoR will contribute to this thread and supplement or correct that Jim and I are saying concerning Aristotle and Rand on axioms and to help find answers to the questions raised by Joe.

One of Aristotle's purposes in searching for axioms was to find out what is the most fundamental character of being (Metaphys. 982b11-28). I've always suspected that one purpose served by Rand's quests for fundamental elements in each area of philosophy was simply to satisfy her own curiosity. But unlike Aristotle and many other philosophers, I don't recall Rand ever acknowledging such a reason for doing deep philosophy. Could this difference concerning axioms contribute to Rand having arrived at axioms different from Aristotle's? I don't know, but see Fred Seddon's exchange with Roger Bissell here http://rebirthofreason.com/Forum/ArticleDiscussions/1339_11.shtml

I gather from Lennox's paper and more so from Tibor Machan's writings (those listed in my bibliography to this RoR article) that Aristotle used his axiom of non-contradiction to refute the Monism of the Eleatics, the Fluxism of Heraclitus, and the Paradoxes of the Megarians. I'm pretty sure Aristotle did not use his axioms to refute theism. Could someone help us out in this area?

It seems then that Aristotle, like Rand, used philosophic axioms as correctives for some big errors in philosophy. Lennox mentions a second purpose that Rand states for axioms. "It is axiomatic concepts that identify the precondition of knowledge: the distinction between existence and consciousness, between reality and awareness of reality, between the object and the subject of cognition. Axiomatic concepts are the foundation of objectivity" (IOE 57). The nature of consciousness and the distinction of subject and object seem to be not major concerns in ancient philosophy. Am I right in that impression? Could this be part of the reason Rand's axioms differ from Aristotle's?

Lennox draws attention to Rand's (1957) conviction that by welding identity to existence and identification to consciousness, she was completing the law of identity stated by "the greatest of your philosophers [Aristotle]." But I thought Aristotle's axioms were the principle of non-contradiction and excluded middle. Aristotle knew something of Rand's idea that to be is to be something specific, but did he know the traditional law of identity as it was received by Rand in her logic texts? When did the law of identity come into the texts? Leibniz said that the laws of identity and non-contradiction are the same law. Is that right?

Post 3

Wednesday, January 11, 2006 - 3:03pm

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I've always wondered if Rand attempts to prove too much (or derive too much) based on her axioms. Do these axioms really mean that God doesn't exist? Peikoff goes so far as to say that ESP can't exist.

Post 4

Wednesday, January 11, 2006 - 4:15pm

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"I've always wondered if Rand attempts to prove too much (or derive too much) based on her axioms. Do these axioms really mean that God doesn't exist? Peikoff goes so far as to say that ESP can't exist."

Axioms themselves cannot be used as the basis for proof that something doesn't exist. They are the basis for positive claims of evidence that something does exist. One cannot make a claim like "only existence exists and therefore God doesn't exist" . But we can say based upon the axioms inherently assumed in any valid claim to truth that arguments for the existence of God are arbitrary. Rand never claimed that the axioms are the jump off point with which we can rationalize all of our claims to knowledge.

- Jason

Post 5

Saturday, January 14, 2006 - 8:20am

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I should say something more about Aristotle's view of axioms. Joe had asked what Rand and Aristotle each thought to be the significance of axioms or axiomatic concepts, and in Post #2 I referred to Aristotle's principle of non-contradiction as an "axiom" and mentioned some refutation-purposes to which he put it.

In the Posterior Analytics, immediately after saying that a contradiction is an opposition of propositions "which of itself excludes any intermediate," Aristotle defines an axiom thusly: an immediate deductive principle, unprovable, but necessary to grasp if one is to learn anything (72a12–18).

The various special disciplines, such as geometry, optics, or harmonics, all use principles of deduction such as the principle of excluded middle or the principle that equals from equals leave equals. Those are principles used but not at issue within the various special disciplines (77a26–79a15). Beyond the study of logic itself, disciplines of study require consideration of additional principles.

(As I recall modern logic, set theory, and geometry, I believe that the principle that equals from equals leaves equals is actually an extralogical principle, not a logical principle. But let that go just now.)

Later in his development, in the Metaphysics, Aristotle says that the principles of excluded middle and non-contradiction are not only principles of deduction, but subject-matter axioms for the study of the essential attributes of all being. That study is metaphysics (996b26–997a25).

"There is a science which investigates being as being and the attributes which belong to this in virtue of its own nature. Now this is not the same as any of the so-called special sciences; for none of these others deals generally with being as being. They cut off a part of being and investigate the attributes of this part—this is what the mathematical sciences for instance do." (1003a22–26)

Metaphysics inquires "into the truths which are in mathematics called axioms [those common to all disciplines], and into substance" (1005a19–20). The false member of a contradiction is "a privation of substance; and privation is the denial of a predicate to a determinate genus"(1011b18–20).

It seems to me that Rand's conception of her axioms and their significance is basically the same as Aristotle's conception. For Rand, as for Aristotle, the axioms are not only deep principles of logic, but of metaphysics. Rand put entity in place of substance on the metaphysical stage. But that does not seem to change much the legitimate purposes for which such axioms could be used.

I agree with Joe's picture that Rand did not think one could deduce all knowledge from such axioms. Aristotle also rejected such a possibility. However, Joe had indicated that one reason such axioms cannot generate all knowledge by deduction from them is that they themselves rest on induction. I think that their being established by induction (broadly speaking) is not what bars fantastical all-of-knowledge deductions from them. I say that because it seems plausible that axioms distinctive to geometry are or could be based on inductions from experience, yet all of geometry is deducible from them.

Metaphysical axioms could be based on induction, yet have some deductive knowledge-producing power (such as tried by Spinoza). Rand doesn't see hers having any such power at all—only corrective and integrative power—but their inductive base would not seem to preclude them having some amount of such deductive power.

Thank you, Joe, Neil, Jason, and Roger, for your inputs. Joe asked if any of the issues in his post had been discussed at ARS 2005. As best I recall, all of the raised hands that were chosen posed questions concerning the other three papers, none concerning this one.

Post 6

Sunday, January 22, 2006 - 2:41pm

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Sorry, but there's a dumb error I made in post #5, which should not go uncorrected. I indicated there that the sections of Metaphysics I was discussing were composed later in Aristotle's development than the sections of Posterior Analytics I had been discussing. That was not an expression of knowledge, but ignorance or forgetfulness. The scholars tell us that the course of Aristotle's development is not really known in anything like that detail. Please erase my error from your world of truth.

On the topic of the legitimate uses of philosophical axioms, some readers may be interested to know of another corrective use to which Aristotle put his axiom of non-contradiction, in addition to his corrective uses I mentioned in post #4. Protagoras had held that all appearances are true. Aristotle argues that this implies that contradictions can be true. Therefore Protagorean relativism is false.

Aristotle proceeds this way:
If everything that appears true is true, then everything that is true is as it appears true to someone. Then all truths are relative to individuals. A truth is then always true for someone, yet possibly not true for someone else. If all appearances are true, then everything that is is relative to perception and appearance. Then nothing is anything in itself, but is only what it is relative to someone who perceives it as such.

Then Aristotle draws out the contradiction:
Protagoras believes, for example, that virtue can be taught. Socrates believes it cannot. Then Protagoras must believe that the view of Socrates is false, yet, for Protagoras, because Socrates believes it true, it is true. Then, for Protagoras, virtue is both teachable and not teachable. And so for every truth Protagoras holds, it both is and is not so. If all appearances are true, contradictions can be true.

But it is established as an axiom that there are no contradictions in reality. Therefore it is false that all appearances are true. Here the falsehood of Protagorean relativism is corrected by Aristotle's axiom of noncontradiction. (Metaph. 1011a-b, 1109a; cf. Peikoff's The Ominous Parallels, p. 61)

For the present post, I should acknowledge the admirable work: Epistemology After Protagoras: Responses to Relativism in Plato, Aristotle, and Democritus by Mi-Kyoung Lee (OUP, 2005).