Objections to the Axioms (Part 3)

Previous: Objections to the Axioms (Part 2)

Question: “Are Axioms Proven or Merely Assumptions?”

“Are first principles or the axioms of logic (such as identity, non-contradiction) provable? If not, then isn't just an intuitive assumption that they are true?[...]”[1]

The axioms are neither “proven” nor “assumed.” 

(In the Objectivist view of axiomatic corollaries, Aristotle’s “Laws of Thought” are corollaries of the Existence axiom.  And more specifically, the Law or Principle of Non-contradiction and the Law of the Excluded Middle are restatements/corollaries of the Law of Identity, which is a corollary of “existence exists.”[2] So I’ll consider this question as broad enough to encompass any first principle, including the Objectivist axioms.)

I’ll make several points about why this can’t be the case when speaking of actual axioms.

Infinite Regress of Proofs

The first point is that there’s a logical implication to the question: if the axioms are provable, doesn’t that mean that there’s something antecedent to the axioms that prove them?  And if so, how are these antecedent axioms proved?  There must be something else that proves them.  And so on for infinity.

It is impossible to prove everything.  The attempt to prove everything including philosophical axioms would go on forever and so the proof would never be finished (if the requirement really is to prove everything).  This was Aristotle’s original response to the objection that everything needs a proof. (The objection was directed towards his Law of Non-contradiction, but could be applied to any axiom.  See his Metaphysics, Book IV.)

Some observations and principles must be at the foundation, the starting points which cannot be proved but are relied upon by all proofs.

No Proof from a Void

There is another reason why real axioms cannot be proved. 

This question being raised shows a kind of uncertainty about what constitutes a “proof.”  As Objectivism defines it in its broadest sense, it “is the process of deriving a conclusion step by step from the evidence of the senses, each step being taken in accordance with the laws of logic.”[3] This applies to both inductive and deductive proofs: both are a series of principles/premises which reach a certain conceptual product (an inductive generalization or deductive conclusion) using the laws of logic and our sense-perception as the base and ultimate proving ground.

The following description of “proof” is a bit more relevant to my response, however: “Proof is the process of deriving a conclusion logically from antecedent knowledge.”[4] Taking more basic knowledge you already possess, you logically infer a series of principles or premises and end with a more technical conclusion.  But axioms and their axiomatic concepts are the most basic conceptual knowledge; there is no knowledge antecedent to the axioms. 

Rand once described the axiomatic concepts as “irreducible primaries”: they are the very foundation for cognition and philosophy, and literally cannot be “analyzed” or broken down into component parts.  Such an analysis or proof would be futile.  Rand notes that, “([a]n attempt to ‘prove’ them [axiomatic concepts] is self-contradictory: it is an attempt to ‘prove’ existence by means of non-existence, and consciousness by means of unconsciousness.)”[5]

There is no antecedent knowledge or more basic alternative facts with which to derive “existence,” “consciousness,” or “identity.”  Without them, there is only a void.  Nothing can be proved from nothing.

(A follow-up point is that you cannot assume what you are trying to prove.  Even a purported proof of an axiom must rely on the axiom.  In a future post, I will reply to an objection about the “circular” nature of axioms, in that the axioms rely on other axioms for their validity, and I will discuss this point in more detail there.)

Starting Points as Not Merely Assumptions

This all being said, the axioms are not assumptions, if this implies that they are baseless, arbitrary starting points.

Without these starting points, there is nothing to consider or discuss, and no one conscious to even consider or discuss these (non) things.  There is a “rhyme and reason” to the discovery of axioms, and it is sense perception or direct introspective experience.  Do you want to know if something exists?  Open your eyes and/or ears (and/or your other senses) and experience existence for yourself.  Do you want to know if you are conscious?  Reflect on your decisions and dreams and memories and realize that only a consciousness could be aware right now to presently do all of these things.

Another point to consider is the common conception of “assumption.”  Dictionary.com defines “assumption” as “a statement or formula that is stipulated to be true for the purpose of a chain of reasoning: the foundation of a formal deductive system.”

This definition relies heavily on the philosophical rationalist’s position on axioms: the axioms are used as the jumping off points to begin spinning off deductive arguments and theories.  But as I said in my previous post, the Objectivist axioms do not serve as a founding platform for a deductively constructed philosophy.

It is true that the theoretical (i.e., provable) principles of Objectivism take the axioms for granted.  (In fact, I would argue that all theoretical principles of all disciplines must take these axioms for granted.)  As far as discussing or proving the non-axiomatic principles of Objectivism, the axioms are assumed as the basic framework of the proof/discussion.  But the validation of the axioms is not taken for granted or an arbitrary choice: it is metaphysically given as the phenomena of sense-perception and introspection.

Conclusion

My point is that we’re not faced with two choices: (1) either find a way to prove the axioms, or (2) admit that they are baseless claims that we conveniently take for granted.  Proof is not the only way to establish an idea’s relationship to reality, according to Objectivism.  And the way to validate or show a proper axiom’s connection to reality is direct experience.  If anyone seeks to object to the self-evident, then it is possible to show that even the objector must rely on the axioms in his attempt to refute them: his capacity to gain and discuss knowledge depends on the axioms.  So the case is that the axioms cannot and should not be proved, but we do not randomly adopt the axioms, or take them for granted without grounds.

References and Notes

[1]: Part of a question at the Ask Philosophers site: http://www.askphilosophers.org/question/4821

[2]: In lecture 3 of Nathaniel Branden’s Basic Principles of Objectivism course, he said: “The three Aristotelian laws of logic are: the Law of Identity, the Law of Contradiction, [and] the Law of Excluded Middle. The last two are merely corollaries or restatements of the first.” (Vision of Ayn Rand: Basic Principles of Objectivism, Lecture 3: Logic and Mysticism, p. 66.)  As Rand considered Branden’s collaborations before their split in 1968 as consonant with her philosophy, his lecture course is still Objectivist material.

[3]: Leonard Peikoff, Introduction to Logic lecture series, Lecture 1.

[4]: Leonard Peikoff, The Philosophy of Objectivism lecture series, question period, Lecture 3.

[5]: Ayn Rand, Introduction to Objectivist Epistemology, Chapter 6: Axiomatic Concepts, p. 55.