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Wednesday, August 27th, 2008, 07:59 AM
Axioms: The Eight-fold Way
by Ron Merrill (ronmerrill@bix.com)
Copyright (C) 1994, Ron Merrill, All Rights Reserved

The notion of an "axiom" plays an absolutely crucial role in any Aristotelian philosophical system, including Objectivism. Because axioms constitute the ultimate tap-root of our entire system of knowledge, we ought to be very clear on what we mean by the term "axiom," what axioms there are, and how we know them to be true.

From Axioms to Postulates

In any logical structure of propositions, there must be some starting point. Not every true statement in the system can be proved by deductive reasoning from other statements, or we would have an infinite regression. These primary or source statements, on which the others are based and from which the others are proved, are known as axioms. This is the most general sense of the term.

The question instantly arises: How do we know that the axioms we wish to use are true? Historically, the ancient Greeks conceived of axioms as being self-evident truths. Aristotle, as we shall see presently, refined and extended this view in a way that made the concept of "axiom" much stronger. However, the looser view was retained by Euclid (or at least his followers; see Kline 1972, 59-60) in his systematic organization of geometry as an axiom-based set of deductive proofs.

Euclidean geometry was based on a series of assertions about geometry that were considered unquestionably true or "self- evident." But "self-evident," in practice, generally is interpreted as "obvious." This invites the obvious rejoinder, "Well, it isn't obvious to me." This was a problem, right at the start, for Euclid. Certainly it seemed self-evident or obvious that, for instance, "it is possible to describe a circle with any center and radius". But another axiom of his system struck him, and subsequent geometers, as not quite self-evident: "That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles."

Modern mathematics may be said to have begun with the practice, stimulated by the discovery of non-Euclidean geometries, of regarding axioms as arbitrary postulates. Mathematicians no longer appealed to intuitive or obvious truths as the basis for their systems. For, they realized, they could not even rigorously define the entities (such as "point" or "line") to which these "self-evident" propositions referred! So, in at least one view, mathematical entities such as geometrical "points" had no necessary correlation to reality. A mathematical system should be just the logical consequences of a set of chosen postulates, about entities which are defined simply as that to which the postulates apply (Kramer 1970, 42-60). Of course, mathematicians do not choose postulates at random; they seek axiom sets that will result in meaningful systems that can, however indirectly, be related to the physical world. Nonetheless, mathematical axioms are no longer regarded as having inherent truth; their validity, if any, can only be ascribed to the usefulness of the structures they generate.

The epistemology of mathematics is beyond the scope of this essay (see Jetton 1991 and Kline 1972, 1192-1210, for an overview of some of these issues). What is relevant is that this notion of axioms as arbitrary has diffused back from mathematics into philosophy. If there is a unifying principle to modern philosophy, it is the conviction that knowledge cannot be grounded on any set of propositions that are indisputably true.

The explicit denial of the possibility of axiomatic knowledge is of course ultimately equivalent to philosophical relativism. It is generally conceded to be indefensible, though philosophers such as Feyerabend seem, at least to their critics, to take this position (Laudan 1990). More commonly it is taken that some propositions are irrefutably true, but that no knowledge about reality can be deduced from them; they are "analytic" and tell us nothing beyond the definitions of the terms used in them. (For a refutation of this viewpoint, see Peikoff 1990.) There is also the strict Kantian viewpoint, in which synthetic a priori judgments are presupposed by all knowledge--but which cannot prove anything about reality; they merely express the "categories" that limit the operation of the human mind (Machan 1985, 105-106).

The foundationalist program cannot be executed at all if axioms are regarded as arbitrary postulates with no inherent truth. It cannot be executed effectively if axioms are taken to be intuitively true--merely "obvious" or "self-evident". We require a stronger and stricter notion of what axioms are; and this is just what was provided for us by Aristotle.

What Are Axioms?

The axioms we seek are statements of inherent truth, that is, something that we know to be true without demonstration from some other, prior, statement. Rather than "self-evident," the term we are looking for is, perhaps, "undeniable." In this sense, an axiom is a statement which must be true because it cannot be denied. More than this, an axiom is "inescapable." It cannot be denied because its truth is assumed in making any statement. This is not a matter of some particular reductio ad absurdem which can be demonstrated on the specific denial of the axiom. (Otherwise, the irrationality of the square root of two, for instance, would be axiomatic.) As usual, this issue has been anticipated and dealt with by Aristotle.

However, the impossibility of a thing both being and not being can be proved by refutation, if only one's opponent says something. If he says nothing, it is absurd to seek to give an account of the matter to a man who cannot himself give an account of anything; for insofar as he is already like this, such a man is no better than a vegetable. . the right way to start is not to ask one's opponent to say that something is or is not so (since this might be thought to be begging the question), but rather to ask him to say something that has meaning both for himself and for someone else. For this he must do if he is to say anything at all. Otherwise, he could not engage in discussion either with himself or with anyone else. But if he grants this request . . . [Aristotle 1963, Metaphysics IV]

It will be seen that Aristotle regards this axiom (the principle of non-contradiction) as being not merely undeniable in and of itself, but inescapable in asserting any proposition whatsoever: the opponent cannot "say anything at all" without assuming the truth of this axiom. As Machan puts it, "axiomatic concepts identify facts which ground the possibility of all thought." (Machan 1992, 44; compare Rand 1990, 59.)

Note also that Aristotle here brings out the point that the undeniability of axioms is not merely a matter of a debating position. Just as one cannot argue against an axiom with another person without being immediately enmeshed in self- contradiction, one cannot deny an axiom even privately, in one's own thoughts. For to do so is to destroy one's own ability to think. Aristotle reaffirms and expands on this in the Posterior Analytics.

Axioms and Axiomatic Concepts

In Aristotelian philosophies, the correspondence idea of truth is used; that is, truth is congruence to reality (cf. Jetton 1992a, 1992b, 1992c). In the context of discussion of axioms, this means that they are more than subjectively true. That is, axioms are not merely necessary in order for us to debate or to think about reality (as with Kant's "categories"); they are inherently true of reality itself, independent of human thought.

Immanuel Kant rejected the pre-Aristotelian view of philosophical axioms as intuitively true propositions. (He accepted, though, mathematical "axioms" of this sort.) He replaced axioms with "principles" and "judgments" that are "true" only as an artifact of epistemology. In his view, we must, for instance, take it as true that contradictions cannot exist because this is necessary for our minds to operate. It does not, however, imply that contradictions cannot exist in the ultimate, "noumenal" reality. (See for instance Kant 1958, 18.) It is simply that we cannot think or perceive without organizing our thoughts by means of "categories" which preclude the possibility of contradiction. (Durant 1926, pp. 202-207)

What objects may be in themselves, and apart from all this receptivity of our sensibility, remains completely unknown to us. We know nothing but our mode of perceiving them--a mode which is peculiar to us, and not necessarily shared in by every being, though, certainly, by every human being. (Kant 1958, p. 54)

Ayn Rand broke through this position by recognizing an axiom that Kant denied: the axiom of consciousness. By taking it as axiomatic that things-as-we-perceive-them are things-as- they-are, she re-validated Aristotle's conception of axioms as objective facts about the organization of reality, rather than subjective facts about the organization of the human mind.

Nonetheless, Rand focused primarily on the epistemological function of axioms. In particular, her theory of concept- formation had to have a role for them, which is filled by the notion of "axiomatic concepts." Specifically, she cited the concepts of existence, identity, and consciousness as axiomatic. These concepts are epistemologically inescapable, because they must be used in any act of concept-formation. Thus, for instance, the method of measurement omission already requires the concept of identity for its operation. One cannot grasp "relationships among these entities by grasping similarities and differences of their identities" (Rand 1990, 6) without already having the notion of identity. Thus Rand asserts that "identity" is an axiomatic concept, "perceived or experienced directly, but grasped conceptually" (Rand 1990, 55).

I want to stress, though, how crucial it is to recognize that axiomatic concepts arise because, and only because, the underlying axioms are true in reality. Kant viewed axioms (which he called "principles of judgment") as essentially epistemological; axiomatic concepts (which he called "categories") reflect only the structure of the human mind. Rand views axioms as essentially metaphysical; axiomatic concepts exist because they reflect the nature of reality, to which human thought must conform if it is to understand reality.

It will be seen that I am disagreeing--though I think more in emphasis than on fundamentals--with Tibor Machan, who states: "It is axiomatic concepts, not propositions, that serve as the first principles of Rand's philosophy." (Machan 1992, 32.) It is the axioms as metaphysical propositions that are really fundamental; if axioms were merely conceptual constructs, we would be back with the Kantian categories.

An Organization of Axioms

Very well, then. What statements are axioms? I am going to suggest that there are eight axioms, falling into three categories. (Of course other ways of organizing these fundamental truths could be proposed.)

1. First, there are three logical axioms (cf. Johnson 1987, 4). These are the basic laws of Aristotelian logic.

The Law of the Excluded Middle is Rand's "either-or". Every statement is either true or false. Attempts have been made to construct "non-Aristotelian" or so-called "multi-valued logics." (Cf. Kramer 1970, 132-133.) But no such structure is truly assertable; to make an assertion is to claim that something is true rather than false. Note the absurdity of attempting to claim that it is true that a "three-valued" logic is valid, and therefore Aristotelian logic is invalid.

The Law of Contradiction requires that no statement may be simultaneously true and false. As Aristotle points out, in the passage quoted above, it is impossible to argue, or even to think, without accepting this principle.

The third law of logic is usually given in the form that "the denial of a true statement is false, and the denial of a false statement is true". Here again we have a proposition that is quite literally undeniable; for to deny it is to assert that it is not necessarily false!

The three logical axioms are the necessary and sufficient conditions for the existence of truth. Taken together, they may be formulated as the axiomatic statement: There is such a thing as truth. That this is undeniable and inescapable is clear; for if there is no such thing as truth, there is no such thing as falsehood, and one cannot assert that this (or any other statement) is false.

The logical axioms apply to truth in all meaningful senses of the word, not just the correspondence meaning of truth. More specifically, they define the coherence meaning of truth. We thus must rely on them in constructing systems of mathematical "truth". Similarly, we must use them when considering the metaphysical subjunctive. For instance, we may construct an imaginary universe, such as the "Life" universe (cf. Levy 1992) or other models using cellular automata, or the two-dimensional physical universe described in The Planiverse (Dewdney 1984). These are metaphysically subjunctive systems ("assumption contrary to fact"--the way the universe could be if it weren't the way it is) so that the correspondence meaning of truth cannot apply within them. (On the correspondence meaning of truth all we can say is that the entire system is false, that is, not the way the universe is.) Still, we may find it useful to require that such a subjunctive system be internally self-consistent. (Or as much so as possible; one may hypothesize that we live in the only completely self-consistent universe; see below).

2. Second, there are three metaphysical axioms. These are the axioms of existence, of identity, and of causality.

Rand takes it as axiomatic that "existence exists" (Rand 1961, 152). That is, there is something; the universe is not empty. This, again, is undeniable; if nothing exists, who is denying it, and to whom is he addressing his denial? Moreover, it is inescapable; one cannot assert anything to be correspondence-true without the assumption that a reality exists to which it corresponds.

The second metaphysical axiom, the axiom of identity, is Rand's "A is A." If something exists, then some thing, some specific thing with a specifiable identity, must exist. A thing is itself and cannot simultaneously be something else with a different identity. This axiom is equivalent to asserting that the laws of logic apply to the material universe. (Or: that everything which is correspondence-true is also coherence-true.) Again, this is undeniable and inescapable; if any statement about reality can be both true and false, how can anything be asserted of it?

Third, we have the axiom of causality. This may be taken to state that everything in the universe has a cause in the general Aristotelian (rather than the limited modern) sense. If some particular entity has certain characteristics at a given point in time, or some particular event occurs, there is a reason for it. It doesn't "just happen." This is equivalent to saying that the contents of the universe are related, that they in some way interact. Of course, if they do, they must do so in accord with logic, that is, there must be a reason for the behavior to occur as it does. Just as the axiom of identity asserts that logic applies to the properties of entities; so the axiom of causality asserts that the laws of logic apply to the properties of change. Again this is undeniable and inescapable; for if anything could become anything else without restriction, no entity could have an identity. (Cf. Rand 1961, 188.)

(We need not necessarily exclude the possibility of "metaphysical chance"; it is conceivable that causality may apply stochastically. For instance, there might be no specific cause for the decay of a particular radium atom, but a cause for the decay of radium atoms as a class which inclusively causes the decay of each one at some random time.)

These three metaphysical axioms are equivalent to an assertion of the correspondence meaning of truth. Taken together, they say that there are true statements about reality.

3. Finally, there are two epistemological axioms. These are the axioms of consciousness and volition. (Though I refer to them as "epistemological," they are just as much assertions about reality as the other axioms. They are not merely statements we are built to believe are true; they are true.)

The axiom of consciousness asserts that it is possible for consciousness (the perception of reality) to exist. This is undeniable and inescapable; he who denies it denies that he is conscious; since he cannot perceive reality, how can he make any assertions about what is possible or not possible? (Cf. Peikoff 1991, 5, 9-10.)

The axiom of volition asserts that free will is possible. Again, this is undeniable and inescapable. He who denies it is claiming that he is a deterministic mechanism; by what means does he establish that he is not merely programmed to deny volition, or indeed to make any other statement? (Cf. Branden 1963.)

The two epistemological axioms are equivalent to the statement (also of course axiomatic) that it is possible to know the truth.

Summing up, we may say that the three groups of axioms differ ultimately only in their scope:
# The logical axioms apply to everything that can be true.
# The metaphysical axioms apply to everything that is true.
# The epistemological axioms apply to everything that is known to be true.

The Interdependence of Axioms

It may be objected that the axioms as I have presented them are not independent. For instance, the axiom of volition bears a clear logical relationship to the axiom of consciousness. In fact, I suggest that it is a general characteristic of axioms (as opposed to arbitrary postulates) that they are non-independent. This is a necessary implication of the fact that axioms are inescapable.

Thus the postulates of Euclidean geometry are quite independent. Any one of them (not just the Parallel Postulate) may be replaced, while retaining the others, to create a new system. They have this independence precisely because they are mere postulates.

On the other hand, the three laws of logic, even though they are not logically equivalent, are completely interdependent; if any of them were false, neither of the others could be true. For if any one of them were false, there would be no such thing as truth. In the same way, it would not be possible for existence to exist if entities did not have identity. As Rand puts it, "Existence and identity are not attributes of existents, they are the existents." (Rand 1990, p. 56) Each and every axiom is inescapable--that is, it is assumed in every assertion made (within its context of applicability), including the assertion of another axiom.

Peikoff takes the position that some of the propositions I have characterized as axioms are actually "corollaries." (Peikoff 1991, 15-16) I put the term in quotes because he uses "corollary," not in the usual sense, but as: a statement which is "self-evident" once one has grasped the underlying axiom. This seems to express an over-anxious desire to defy Cartesian rationalism by carrying the "primacy of existence" to the ultimate extreme. The implied program is to demonstrate the literal truth of Rand's statement that her system is rooted in the single axiom that existence exists (Rand 1960, 152).

To account for all axiomatic knowledge as merely "the implications of 'existence exists'" is, to my mind, at best a tortuous exercise. But in the final analysis, my position and Peikoff's share the view that the various axioms must by their nature be taken together as an inseparable whole.

Do Axioms Matter?

There are opponents of the Aristotelian or Objectivist philosophies who say they will concede the validity of these (or equivalent) axioms, yet assert that they have no actual practical content. Hollinger, for instance, summarizes Rand's fundamental axioms (including "existence exists") then remarks:

This clearly will not do. Unless Rand's philosophy can be supported by more than the four axioms cited above, or until it can be shown that these axioms support only her philosophy, it will be difficult to take her views seriously. That is, she must show that her remarks are not merely the banalities they seem to be. (Hollinger 1984, 40.)

This sort of view seems to underestimate the potentialities of axiomatic systems. Consider, for example, the Peano Axioms which underlie the field of arithmetic and number theory. (These are, of course, not "axioms" in the strict sense used above, but postulates.) Here we have five propositions of stunning banality; for example: "'One' is a number." Yet these five statements, which seem to contain hardly any real information content, give rise to a literally infinite library of complex theorems. (Cf. Fulks 1961, 3-15.) We repeatedly encounter logical systems, not only in mathematics, but in other fields, in which a few trivial-seeming axioms generate subtle and intricate results with extraordinary fruitfulness. Among fields of knowledge which have been axiomatized with greater or lesser success are thermodynamics (cf. Thompson 1972, 32-52), quantum mechanics (cf. Jammer 1974, 384-399), and economics (von Mises 1966).

So we should not rule out the possibility that the axioms presented above, with the assistance of some minimal observational input, could generate significant knowledge. That would seem to be Rand's view. For instance, she appears to take the position that quantum-mechanical tunneling is impossible (Rand 1990, 293), and that a vacuum cannot exist (Rand 1990, 303). These facts can be known, she asserts, in advance of scientific experiment. Along the same lines, it is interesting to note that the axiom of consciousness appears to place very strict constraints on the laws of physics, and, in conjunction with a few basic physical principles, may even be able to circumscribe the values of physical constants such as the speed of light. (This assertion, the "Anthropic Principle," has generated heated debate. Cf. Leslie 1989; Carter 1990; Swinburne 1990.)

However, unlike mathematical postulates, philosophical axioms should not be expected to be "fertile," that is, capable of generating a body of knowledge by deduction. In the passages cited, Rand stresses that philosophy can delimit what is possible, but only science can determine what possibilities are actual. It is clear that Objectivism does not aim at developing philosophy as a system of deductive implications from its axioms, in the manner of the rationalists. For Rand, the purpose of axioms is to ground the knowledge gained by the senses, not to replace it.

So let us set aside the possibility of carrying out a rationalist program. We may still assert, contrary to Hollinger, that the Randian axioms (whether in her own words, or in the formulation I have presented) delineate a distinct philosophical position. Certainly Ayn Rand believed so; she said of Objectivism: "If one recognizes the supremacy of reason and applies it consistently, all the rest follows." (Cf. Binswanger 1986, 344.) This position is endorsed by other Objectivists, including those who vigorously disagree on certain specific issues (Peikoff 1991, 4; Kelley 1990, 63).

From the opposite perspective, one may ask: What modern philosophers would agree with the above eight axioms without reservation? Certainly no thinker in the tradition of Plato, of Hume, or of Kant could accept them all. Pragmatists, Logical Positivists, Linguistic Analysts--all these would have difficulty agreeing with these axioms. Can one imagine Rorty, Feyerabend, or Kuhn signing on to them?

Of course, philosophers who do agree to these axioms may yet differ because of disagreement about empirical facts. But in any case, surely we may conclude that only a relatively small subset of philosophical worldviews is compatible with acceptance of the eight axioms above. There is a sheaf of neo-Aristotelean philosophies, including Objectivism, that would claim to be consistent with them. Many other philosophical positions are excluded by them. So these axioms, whether "banal" or not, do carry some content; they are not trivial bromides that tell us nothing of use.

Axioms as Foundations of Knowledge

As Tibor Machan clearly explains, Rand's foundationalism is in a sense both rationalist and empiricist (Machan 1992). He brings out two key, and related, points. First, axioms ground not only deductive but inductive knowledge. Second, axioms themselves are known, not by pure introspection or ideation, but by abstraction from experience.

I think this position, though correct, is perhaps stated too weakly. In my view Objectivist epistemology, consistently applied, cannot even accept the traditional distinction between deductive and inductive reasoning as means of knowing reality. Denial of this dichotomy is at the conceptual root of the Rand- Peikoff argument against the analytic-synthetic dichotomy (Peikoff 1990). Strictly speaking, Rand's theory of concepts does not leave any room for pure a priori knowledge, for nothing can be known prior to or without reference to experience; all concepts are formed from percepts. Similarly, Objectivism cannot recognize pure a posteriori knowledge, for nothing can be understood without application (explicit or implicit) of axiomatic concepts; all percepts are integrated by means of these concepts.

Rand's crucial insight is that all knowledge is empirical-- that is, derived from perception of reality--and yet that we can be certain of empirical knowledge. Empirical knowledge is not merely "inductive," something only probably true because derived by statistical inference. Organization of our perceptual inputs by means of axioms, which are inescapably true, is what allows us to achieve certainty.

So the axiomatic method of reasoning is more than a technique of argument useful for refuting philosophical skeptics. It provides us with a method of grounding fundamental truths about reality. This makes the use of axiomatic reasoning a crucial resource in dealing with issues of causality and the problem of primaries, which lie at the root of the epistemology of science.

References

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