Reaching the Axioms
All topics and all fields of research have a beginning or starting point. Philosophy may be the most abstract field that we study, but it is no different. Whether they admit to them or deny them, all philosophies rest on a set of axioms, or starting points. Axioms are self-evident propositions that indicate the bases of all knowledge and are at the base of all statements and claims. Philosophical axioms must be accepted in order to make any statement or claim to knowledge of any subject, because philosophy is the backdrop for all other areas of study. Aristotle was perhaps the first individual to discuss the importance of axioms, and Objectivism is the most recent philosophy to emphasize their role in knowledge.
The axioms of most fields are the assumptions that underlie the entire field, and must be accepted in order to learn anything else about that subject. But philosophical axioms are not merely assumptions. They are not accepted simply for convenience, or to easily reach point B from point A in some logical argument. They should be accepted because they state important and fundamental truths about reality and our relationship to it, truths that cannot be avoided or ignored without tossing out the field of philosophy and all knowledge as such. All knowledge has to be validated or verified in some shape or form, and axioms are not an exception.
If axioms are not to be automatically accepted, what are the axioms of the philosophy of Objectivism, and how does a person go about reaching and understanding them? To answer these questions, we have to discover how to reach the components of the axioms themselves: axiomatic concepts.
The Method of Reaching Axiomatic Concepts
Here are the axiomatic concepts that Objectivism recognizes. I will note that Objectivism's creator, Ayn Rand, did not believe that this exhausts the field of what counts as an axiomatic concept:
7. Validity of the Senses
8. Volition/free will
9. Self 
Axiomatic concepts specify a primary or fundamental fact about reality, a fact that cannot be reduced to other facts or broken into constituent parts. One example is the axiomatic concept of “existence”: there is no more basic thing to say about anything other than the fact that the thing exists, whatever it is. On the other hand, consider the concept of “water”: there are many things to say about it, like that it can boil, that it can be mixed with chemicals, that it is a liquid, or even that it is an entity, but the basic fact about water is that it exists; it is a part of reality.
There are 4 very important characteristics of axiomatic concepts:
1. They are irreducible: they cannot be broken up into more basic facts.
2. They are implicit in all facts and in all knowledge.
3. They are directly perceived or experienced.
4. They require no proof or explanation, but all proofs and explanations rest on them.
I've already covered point #1 above. Point #2 concerns the fact that axiomatic concepts are implicit in all cognition. The “implicit” is that which is available to your consciousness but which you have not conceptualized. As soon as you know any fact or are aware of something, the material to form the concept “existence” is present, but it will take a separate act of consciousness to explicitly grasp it. Other concepts are implicit when you focus on a specific set of facts, but axiomatic concepts apply to everything in your awareness of reality, and thus are implicit in all of your knowledge and in all of the facts that you know.
Now that I’ve given a background of sorts on axiomatic concepts, I’ll cover exactly how we reach them, point #3.
In one sense, axiomatic concepts are the easiest of concepts to reach because the material to form them is always there; they are involved in all of our knowledge. But in another sense, they are among the most difficult: they are concepts that cover the entire human experience, and so can only be formed by thinking of the totality of things, instead of everyday events.
The method to reach axiomatic concepts is direct perception or experience. As in, you literally see birds, cars, mountains, and people, you perceive them existing.
For the concepts “existence,” and “identity,” you see things and know that they exist rather than not existing, and they are whatever they are, instead of something else that could exist. For “consciousness,” you introspect and observe yourself having different states of consciousness, such as watching a show, imagining a shirt you want to buy, or remembering an old friend. “Existent” is formed by considering individual things that exist, whether it’s a thing, an action, a relationship, or an attribute or feature. We form “entity” by observing the things that act and have qualities and features in our experience, such as horses, TVs, boulders, and oranges. “Action” is formed by observing entities that act, as in a bear eating fish, a person running in the street, or a cloud floating in the sky. “The validity of the senses” is formed by introspecting on the fact that you perceive external things and in doing so, you have to acknowledge the process by which you are conscious of other things. “Free will” or “volition” is created by observing your own mind focusing and unfocusing, thinking and not thinking, making choices and not making other choices. Lastly, we form the idea of “self” by introspecting and realizing that any act our individual consciousness’ carry out or experience amounts to: you – your "self" – experiencing something.
Lastly, for #4, I said earlier that axiomatic concepts do not require proof. Proof is a complex mental activity of tracing a conclusion back to antecedent knowledge, back to more basic knowledge that someone has. But there is nothing more basic than axiomatic concepts, and nothing antecedent to them. Without existence, entity, identity, or existent, there is nothing that exists which requires proof. Without consciousness or self, there is no power to make a proof, and no one to judge the merits of the proof. Without action, the proof cannot be gathered, it cannot be communicated, and it cannot be accessed, all of these are actions. Without sense-perception, there is no way to begin the proof, and no irrefutable data to trace the proof back to its conclusion. And without free will/volition, there is no point of examining arguments or convincing yourself or others of conclusions, because you aren’t in control of yourself and you would have had to accept or deny them anyway. A proper proof cannot rely on the very object that has to be proved. What this means is that the only way to validate that the axioms are true, that they are axioms at the base of knowledge, is by either direct perception or experience.
Now that I’ve covered how we reach axiomatic concepts, we can move on to the next step: understanding how to form axiomatic statements, the starting points or axioms of philosophy.
Forming Axioms and Their Role in Knowledge
Objectivism holds that axiomatic concepts are very special, and have many peculiarities that they either share with very few other concepts or are exclusive to them. For instance, most concepts have definitions, but axiomatic concepts and concepts of sensations cannot be defined; for instance, it is impossible to communicate exactly what sound is to someone who was born deaf. (Sensations and axiomatic concepts can be ostensively defined, by pointing to particular objects, however.) Sensations are primarily experienced, not described in words. Similarly, there are no words or ideas that can explain or define “existence” or “identity.”
Another characteristic of axiomatic concepts is that they are among the most open-ended concepts of the human mind. For instance, the concept “wallet” delineates a very specific kind of existent, with very specific measurements and attributes (such as its size, its purpose in our lives, and the material it can be made from), but the axiomatic concepts have very little restraints. The “existence” concept does not specify what existents are included in the concept, it merely stresses the primary fact that they exist. The “identity” concept doesn’t indicate the particular natures of whatever exists, it just stresses that they are what they are. And the concepts of “consciousness” and “self” do not specify what particular existents one is conscious of or what exactly the conscious being is experiencing at any given moment, they just highlight the fact that the conscious being is experiencing something. This all-inclusive nature of axiomatic concepts leads to an important implication for their conversion into axioms.
Axiomatic concepts cannot be defined normally, as I said earlier, but they can still be changed into statements. Axioms are statements of axiomatic concepts, disclosing a fundamental truth contained in the respective axiomatic concept. Rand’s view is that the form of this statement must be that of “a base and a reminder.” The fundamental fact that is implicit in other statements and other concepts is repeated, stressed, and emphasized in axioms. Existence exists. Consciousness is conscious. The Law of Identity: A is A ("A is A" is philosopher Gottfried Leibniz’s formulation of this law). To my knowledge, Rand never converted all of the axiomatic concepts that she recognized into axioms, but I will make my own list here:
1. Existence: Existence exists.
2. Consciousness: Consciousness is conscious.
3. Identity: A is A.
4. Existent: "The building-block of man’s knowledge is the concept of an 'existent'—of something that exists, be it a thing, an attribute or an action."
5. Entity: Entities are the primary existents. (Or: Entities constitute the content of the world men perceive; there is nothing else to observe.)
6. Action: Actions are actions of entities. (Or: “Action” is the name for what entities do.)
7. Validity of the senses: A means of awareness is a means of awareness.
8. Volition/free will: "Man's volition is an attribute of his consciousness (of his rational faculty) and consists in the choice to perceive existence or to evade it."
9. Self: Every awareness is: me experiencing something. 
Having stated the axioms, we should now look into the purpose of axioms.
Axioms lay the foundation for a philosophical system, by articulating the beginnings of human experience. Philosophies lay down principles and assertions that have to be proved to be true, but ultimately, they all rest on the kinds of principles that cannot be proved, because these axiomatic principles serve as the basis of proof. Aristotle once termed these sorts of principles as “unhypothetical,” as in we can’t be wrong about them, and they are the most intelligible of principles (because they are immediate, self-evident). Axioms serve as the base for the system of principles that comprise various philosophies, the set of unhypothetical principles on which rests the hypothetical, provable principles.
Are Axioms Formed by Induction Then?
Since axioms serve such an important role in philosophy, it is worth noting whether they are formed through induction, deduction, or some other process.
Axioms are certainly not formed through deduction. Deduction as such is based on the idea that, supposing that certain other principles or statements are true, a conclusion follows logically specifically because those prior statements were true. But the truth of axioms is prior to all other principles and statements, and cannot be deduced from antecedent knowledge. In other words, the axioms have to be true for any other claim to be true. Thus, axioms are at the base of deduction as a science, just as they are at the base of all knowledge.
A more technical issue is whether or not axioms are formed inductively. Induction is generally a process of reasoning that moves from particular observations to universal statements. On just the definition alone, it seems that axioms would qualify as inductions. They are all formed by observations, the axioms are self-evident, either by direct perception, or by experience (such as exercising one's will and directly experiencing "free will"). And the axioms themselves are all universal statements; e.g. “Whatever exists, exists,” “All things are whatever they are,” “Whomever is conscious, is conscious.” So what is the difference?
Induction and deduction are the two primary methods of reasoning. The resulting inductive or deductive conclusion is based on evidence and reasoning, the product of a chain of logical connections. But the axioms are the beginning of the chain, they are the foundation on which the chain of reasoning can be connected. Axioms are the foundation of all reasoning. The basic facts that axioms identify are immediately, directly available to us, and as a result, we cannot be in error about them, as Aristotle was the first to tell us. It is not necessary to construct a proof for something that we can never be wrong about, and it is not necessary to prove something that is right in front of you, or that has to do with the fact of your awareness. More importantly, it is not possible to create such proofs, either: a proof cannot rely on the object of the proof. And the axioms would have to be relied on in any attempted proof, inductive or deductive, defeating the purpose.
What process then is responsible for the formation of axioms, if not induction or deduction? According to Objectivism, it is the process of differentiation, one of the two essential actions a consciousness can take (the other being its opposite and complement, integration). Differentiation is the process of noting differences among existents, and integration is the process of putting or blending things together. (The actions of differentiation and integration function in many different ways, and operate in other mental activities outside of reasoning, such as the brain integrating sensory data into perceptual awareness.) On forming axiomatic concepts, Rand writes: “Epistemologically, the formation of axiomatic concepts is an act of abstraction, a selective focusing on and mental isolation of metaphysical fundamentals; but metaphysically, it is an act of integration—the widest integration possible to man: it unites and embraces the total of his experience.” Stating the axioms is an act of differentiation, picking out and isolating the fundamental fact of an axiomatic concept, and restating it, stressing it. Just as the “existence” concept is an integration of all things that have existed, exist presently, or will exist, the statement “Existence exists” is a differentiation, a highlighting of a metaphysical point already contained in the concept of “existence.”
True to everything I’ve discussed, in no way is this essay a proof that the axiomatic concepts and axioms that I’ve discussed can be proven true. It merely discusses how and why they are axiomatic concepts and axioms, and are at the base of one particular philosophy, Objectivism. So concludes an essay concerned with the starting points of philosophy.
 Here are the references for these lists:
1. Leonard Peikoff (LP), "Existence, Consciousness, and Identity as the Basic Axioms," Objectivism: the Philosophy of Ayn Rand (OPAR), p. 4.
2. Ibid., p. 6.
4. Ayn Rand (AR), "Cognition and Measurement," Introduction to Objectivist Epistemology (IToE), p. 5-6. AR, “Art and Cognition,” The Romantic Manifesto, p. 46.
5. LP, "Causality as a Corollary of Identity," OPAR, p. 13. AR, “Concept-Formation,” Introduction to Objectivist Epistemology, p. 15.
6. LP, Volume 1, Disc 1, Lecture 1, Track 3 of “Objectivism Through Induction.” Also: LP, "Objectivism: The State of the Art," Lecture 2, around 16:04. LP, "Causality as a Corollary of Identity," OPAR, p. 14.
7. LP, "The Senses as Necessarily Valid," OPAR. AR, “Cognition and Measurement,” ItOE, p. 5.
8. AR, "The Metaphysical Versus the Man-Made," Philosophy: Who Needs It, p. 25.
9. Ibid., p. 50. AR, sub-section "Self" in the Epistemology Workshop Q&A, ItOE.
 AR, "Axiomatic Concepts," ItOE, p. 56.
Next in the series: The Law of Causality (Cause and Effect)