OPAR, Chapter Three
Concept-Formation
by James Leithhead

Date: 21 Nov 1992
Forum: Moderated Discussion of Objectivist Philosophy
Copyright: James Leithhead

OVERVIEW

Introduction

Peikoff opens the third chapter of his book by explaining that sensory material, discussed in the previous chapter, is only the first step on the road to knowledge. Human knowledge and action are not merely perceptual, but a *conceptual* phenomenon. Concepts are built on percepts, but represent a new scale of consciousness, one far more powerful than that open to animals, who function only on the perceptual level of consciousness. For man, the object of knowledge is not merely that which is before him, but "the universe in all its immensity".

Man ultimately relies on abstract principles to choose his values and his actions; therefore, "To understand man--and *any* human concern--one must understand concepts." Ayn Rand's theory of concepts, then, is of vital importance to the whole of her philosophy.

Peikoff indicates that this chapter is only meant to cover the essential aspects of Rand's concept-formation theory. It is merely an introduction to her Introduction. There is much more to say regarding the subject, but it is not to be found here. For further elaboration he refers the reader to the full text of Introduction to Objectivist Epistemology.

Section 1, "Differentiation and Integration as the Means to a Unit-Perspective"

The first section opens with an overview of the nature of a conceptual consciousness. Following Rand, Peikoff begins by tracing the development of the (implicit) concept "existent" in a child's mind.

This development progresses through three stages. The first stage is the child's awareness of things as objects. This represents the (implicit) concept "entity". The second stage is the child's ability to perceptually distinguish entities from one another and, further, to see the same object at different times and recognize that it is the same one. This represents the (implicit) concept "identity". The final stage is the grasping of differences and similarities among identities, that is, grasping entities as members of a group of similar members. This represents the development of the (implicit) concept "unit".

This stage, unlike the previous two, has no counterpart in the animal world. The ability to regard the world in terms of units constitutes "the great cognitive divide" between animals and man. As Ayn Rand puts it, "The ability to regard entities as units is man's distinctive method of cognition."

The unit-perspective represents a new scale of cognitive ability. It enables man to specialize intellectually and to learn inductively, gaining knowledge of all the units of a concept by studying a relative handful of them. Without the implicit concept "unit" we could not count, identify quantitative relationships, or do math. with this development we reach the conceptual level of consciousness and can perform all the advanced mental tasks it open up to us.

Peikoff proceeds to lay out a systematized account of the processes we perform in order to regard entities as units. The basic procedures involved are differentiation and integration.

"Differentiation" is "the process of grasping differences... distinguishing one or more objects of awareness from the others". "Integration" is the "process of uniting elements into an inseparable whole." On the perceptual level our sensations are automatically differentiated and integrated into percepts of the entities. The same two processes apply to the conceptual level, but in a different form and not automatically.

The process of concept-formation begins by isolating a specific group of concretes from all other concretes. This is done on the basis of *perceptually observed similarities* that distinguish the particular group of entities from the rest of one's perceptual field. In Peikoff's words, men "*abstract* such similarities from the differences in which they are imbedded". For example, one may study the similar shape of a number of tables while setting aside their other differences (size, color, etc). This power of abstraction "is the power to separate mentally and make cognitive use of an aspect of reality that cannot exist separately." Man makes such a recognition of similarities the basis of his cognitive organizational method.

The first step, then, is the mental isolation of a group of similars. Such an isolation, however, is not yet a concept and alone is not cognitively valuable. What is still required is an act of integration.

The integration of percepts is the "process of blending all relevant ones (eg. percepts of tables) into an inseparable whole." This whole is a new (mental) entity, a concept, which functions thereafter (in our consciousness) as a single, enduring unit unto itself. This new entity stands for an unlimited number of concretes; it subsumes all instances past, present, and future that belong to the group.

The tool that makes this integration possible is language. A word, a symbol that denotes a concept, is the only way we can retain such a large sum of concretes. Since only concretes exist, our concepts too must exist in some concrete form. This is the function performed by a word, which "transforms the sum of similars and the resolve to treat them together into a single (mental) concrete". This constitutes the completion of the integrative stage and is the form in which the concept exists. To use an analogy, the word is the body of the concept while the conscious perspective involved is the soul, and the two form an inseparable unity. "A concept without a word is at best an ephemeral resolve; a word without a concept is noise."

Peikoff finishes the section by posing the question: to what in reality do concepts refer? The question, he points out, has been a major problem in philosophy since the time of the ancient Greeks. Mysticism says they refer to supernatural entities grasped by man through revelation. Skepticism regards them as arbitrary construc- tions and a matter of social caprice. What is needed in philosophy is a solution to the problem of grounding concepts in reality.

Section 2, "Concept-formation as a Mathematical Process"

According to Ayn Rand, the solution to the problem lies in the connection between the concept-formation process and mathematics. To make this clear we must first consider measurement.

Measurement is defined by Ayn Rand as "the identification of a relationship established by means of a standard that serves as a unit." The process involves two concrete elements, the existent that is being measured and the existent that serves as the standard of measurement. An attribute is measured by a unit of that attribute, which must be in the range of human perception. This way we can grasp any size attribute, even one beyond our direct perceptual capacity, by relation to a concrete we can perceive directly. This expands the range of our consciousness beyond the perceptual level.

Conceptualization shares the same essential purpose and follows the same essential method as measurement. In both cases man identifies relationships among concretes, taking those directly perceived and relating everything else to them, including innumerable existents that are beyond his perceptual capacity. All of these relationships are made by *quantitative* means. Both processes involve the discovery of mathematical relationships among concretes.

Ayn Rand's fundamental observation is that similar concretes integrated into a concept differ only quantitatively, that is, in the measure of their characteristics. Therefore, when we form concepts our mental process is one of retaining the characteristics while omitting their measurements.

For example, take the formation of the concept "length". An attribute, length, is observed as common to several objects (a match, a pencil, a stick...), the difference in each case being only one of magnitude. The entities under scrutiny are all the same in regard to the attribute (they all possess it), but they differ in its measurement. In the abstraction the attribute is retained while the varying measurements are omitted with the understanding that they exist in some quantity, but it may be any quantity. By this method the subject integrates these first instances into a concept and is enabled to identify future instances, which will differ only in their specific measurements.

It's important to note that measurement omission does not imply that the measurements in question do not exist. As Rand explains, "It means that *measurements exist, but are not specified*...The principle is: the relevant measurements must exist in *some* quantity, but may exist in *any* quantity."

Grasping similarities is essential to the process of concept- ualization, as the above indicates. Peikoff cites Rand's formulation: "Similarity, in this context, is the relationship between two or more existents which possess the same characteristic(s), but in different measure or degree." When two things are similar it is their characteristic(s) that is the same. It's the measure of the characteristic(s) that differs.

In the process of concept-formation explicit knowledge of measurement is unnecessary. All that is needed is an awareness of perceptual similarity as described above, a form of "implicit" measurement. Since we must simply focus perceptually on attributes we need only discover commensurability, not specific quantitative information.

"Commensurable" means relatable to a common unit, differing only in specific measurements. To discover commensurability we need only observe variations in amount or degree such as longer/much longer/ shorter/much shorter, hotter/colder, rougher/smoother and so on. We experience this "continuum of more-or-less" directly, perceptually. By observing this continuum we can grasp, without explicit numerical measurement, that the relevant concretes can be related to a common unit. Such relations are the only form of measurement we need employ to perform concept-formation.

Measurement also plays a role in differentiating a group of existents from other existents. The process cannot be performed arbitrarily; there must be some basis. That basis lies in commen- surability. There is no relating table-shaped objects to red objects. What is needed is, in Rand's terms, a Conceptual Common Denominator (CCD for short). A CCD is "the characteristic(s) reducible to a common unit of measurement, by means of which man differentiates two or more existents from other existents possessing it."

Both integration and differentiation are essentially processes of measurement. We can differentiate groups only by reference to their commensurable characteristics. We integrate into units only concretes whose differences are of measurement. Concept-formation depends at root on the recognition of objective, mathematical relationships.

This brings us to Ayn Rand's definition of "concept", which captures in one sentence all of the key elements described above. "A concept is a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted."

To finish the section Peikoff discusses some practical questions regarding the Objectivist theory. He points out that "since the mind omits measurements whether it knows it or not, one may ask, what is the practical purpose of the Objectivist theory of concepts?"

Peikoff says the answer is in part that philosophers need to understand the mathematical aspects of concept-formation in order to define rules for the other, more consciously performed processes of thought. But deeper than that, the theory of measurement-omission is essential to the validation of conceptual knowledge and therefore of reason itself. "A proper theory of concepts is not sufficient to save to world. But it *is* necessary. The fact that concepts are valid tools of cognition whether we know it or not will not save us-- unless we *do* know it."

The practical accomplishment of the Objectivist theory is that it both defends man's mind at the fundamental level and disarms its worst enemies. The key is its demonstration that concepts are based upon and refer to reality.

Ayn Rand's answer to the "problem of universals" lies in the relationship she identifies between mathematics and universals, her discovery that the underlying method in both fields is the same. "Let those who attempt to invalidate concepts by declaring that they cannot find 'manness' in men," challenges Rand, "try to invalidate algebra by declaring that they cannot find 'a-ness' in 5 or 5,000,000."

Sections 3, "Concepts of Consciousness as Involving Measurement- Omission"

So far Peikoff's discussion has focused on first-level concepts, those that are formed directly from perceptual data. Starting from these, concept-formation continues to higher levels, forming more advanced concepts by abstracting from abstractions. This leads to wider and narrower concepts, representing more extensive and more intensive knowledge. For example, the first-level concepts "cat", "dog", and "horse" yield the wider concept "animal". Similarly, "man" is divided into narrower concepts based upon profession, such as "doctor", "lawyer", and "policeman". However, Peikoff does not wish to describe this process in any greater detain. Instead, he indicates that the material is covered in Rand's Introduction and moves on to the topic of concepts of consciousness.

Concepts of consciousness (such as "thought", "memory", and "love") are, in one sense, first-level abstractions and, in another, higher-level abstractions. They are first-level in that we experience them directly. They are higher-level in the sense that they can't be formed before we already have a great many previous concepts.

Concepts of consciousness possess two key attributes: content and action. The content is the object of consciousness, some aspect(s) of existence that we grasp, whether directly or indirectly. The action is the action of consciousness with regard to its content (remember that awareness is not a passive state but a continuous process of action).

As an illustration, Peikoff discusses a child forming his concept of "thought". The child experiences (performs) several acts of thought under different circumstances and with differing results. Eventually he grasps the similarity uniting the thinking processes he experienced and differentiates them from his other mental activities. He has isolated several instances of thought and these will serve as the first units of his future concept.

In abstracting thought processes, two key aspects are involved: content and intensity. Their characteristics are retained while their measurements are omitted. With regard to content, the concept "thought" omits all measurements distinguishing the content of one thought from another. As regards intensity, the concept omits all the measurable aspects there as well, such as the level of the abstraction(s) involved, the number involved, the degree of effort expended, and the clarity attained. By omitting the measurements and retaining only the characteristics of the process, which are the same for every unit, a universal concept is created that subsumes all of its instances.

Not all concepts of consciousness denote processes. Some are concepts of the products of processes, such as "knowledge", "science", and "concept". Others pertain to methods, such as "logic". Still others are integrations of concepts of consciousness with existential concepts, as is the case with "friendship". These last are especially prominent in philosophy since various of its branches are concerned with human action as directed by conscious choices and standards.

Section 4, "Definition as the Final Step in Concept-formation"

The final step in forming concepts is definition, which is essential to all concepts except axiomatic ones and those that denote sensations. The function of a definition, in Ayn Rand's words is "to distinguish a concept from all other concepts and thus keep its units differentiated from all other existents." The ostensive definitions a child employs become insufficient after a point. What becomes necessary are formal definitions that explicitly identify the nature of a concept's units.

A definition identifies a concept by specifying its essential characteristics. A concept's "essential" characteristics are those that make the units the kind of things they are and differentiates them from all other known existents.

There are two parts to a proper definition, both of which follow from the nature of concept-formation. In forming concepts we isolate units by grasping a distinguishing characteristic, what the medieval Aristotelians would call the differentia. We differentiate based on a wider characteristic, the CCD, which is shared by both the concretes being isolated and the concretes from which they are isolated. This yields the genus of the definition.

A definition in terms of genus and differentia is like a "logical x-ray" of the concept under investigation. It tells us what distin- guishes the units of the concept and from what they are distinguished. For example, if "man" is conceptualized by differentiating from dogs, cats, and horses, the "animal" would be the genus of the definition and "rational" its differentia.

Definitions, like concepts, are contextual. A context of knowledge is, in Rand's words, "the entire field of a mind's awareness or knowledge at any level of its cognitive development." The purpose of a definition is to differentiate a concept's units from all other existents *in a given context of knowledge*.

At an early age simple, obvious characteristics may suffice for a definition; but later, when one knows more aspects of reality, the same characteristic(s) may be inadequate to the task. At such a time the definition must be revised in light of the new context of knowledge.

For example, a child's definition of "man" early on might be "a thing that moves and makes sounds". After the child has discovered cats and dogs, however, his original definition is no longer sufficient to differentiate a man from other entities of which he is aware. He must then revise his definition to something like "a living thing that walks and has no fur", and so on. The process continues as the child's knowledge of the world increases until he eventually reaches the fully adult definition of "rational animal". All of these defin- itions formed by the child are valid within his (growing and changing) context of knowledge.

When a definition is contextually revised the new definition does not contradict the old. Facts remain facts and the earlier knowledge, out of which the definition was formed, is still knowledge. The old facts don't change or fade away; they are merely superseded by newly discovered, more fundamental ones.

Despite the separate context of each individual, objective, universal definitions valid for all men are possible. These must be determined "according to the widest context of human knowledge available to date." The context of knowledge is not that of any man, but of man. As Ayn Rand states, "An objective definition, valid for all men, is one that designates the *essential* distin- guishing characteristic(s) and genus of the existents according to all the relevant knowledge available at that stage of man's development."

Though contextual, definitions are not arbitrary constructions. Correct definitions are determined by the facts of reality (the nature of the entities subsumed by the concept). Peikoff stresses this fact and its relation to fundamentals. "Definitions are determined by the facts of reality--within the context of one's knowledge. Both aspects of this statement are crucial: reality *and* context of know- ledge; existence *and* consciousness."

Another important element here is the rule of *fundamentality*, which is often necessary to clarify what is the essential charac- teristic. This rule states that when a concept's units have more than one distinctive characteristic the definition must state the most significantly distinguishing feature--it must state the funda- mental. A feature is fundamental when it is responsible for all the units' distinctive characteristics, or at least a greater number than any other characteristic. The definitional principle is: "Wherever possible, an essential characteristic must be a fundamental."

A definition states only a few of a unit's characteristics, but it *implies* all the others one knows. After all, it is such know- ledge that determines what the definition is. Ayn Rand supplies a preamble that silently accompanies every definition: "Every definition begins with the implicit proposition: 'After full consideration of all the known facts pertaining to this group of existents, the following has been demonstrated to be their essential, therefor defining characteristic...'" Definition serves as a powerful tool of integration, reducing a complex sum to a few elements and expressing them in a brief, easily retainable statement.

Definition by nonessentials, in contrast, achieves just the opposite result. An arbitrary selection of features as definitional doesn't proceed from any process of cognition and cannot carry with it the units' other features. Instead it works to turn a concept into a floating abstraction, cutting it off from its units and thereby acting as a tool of disintegration, splintering and obliterating its data and rendering it any proposition that contains it useless.

The truth of a proposition depends on the validity of the concepts that constitute it, which in turn depend on the validity of their definitions. Rand emphasizes the importance of this fact: "The truth or falsehood of all of man's conclusions, inferences, thought, and knowledge rests on the truth or falsehood of his definitions."

A clear sign of the failure to formulate proper definitions is the claim that concepts are interchangeable with their definition. This indicates that concepts do not stand for existents, but only certain of their characteristics. This is not the case, even if the definition happens to be correct. A definition is not a shorthand tag that substitutes for other words. Man is not merely "animality" plus "rationality"; *all* of his characteristics are subsumed under the concept. The fallacious prescription above means dropping all the rest of one's knowledge of a concept's units, leaving only that chosen for the definition.

The selection of a few characteristics that differentiate and condense the content of a concept best is not a shrinking of that concept's content. In fact, it is only because a concept means its units, rather than its definition, that varying definitions in varying contexts of knowledge are possible.

Further, one's context of knowledge at any stage is not the limit of the concept's meaning. A concept is not restricted to its known characteristics. The units are what they are regardless of what anyone knows about them, and the concept refers to the units, not our (changing) knowledge of them. A concept is an "open-end" classification. It includes all of its referents' characteristics, the already discovered and the yet-to-be discovered.

Another important implication is that concepts, once formed, don't change. They are universal and stable. All men who use a concept use the *same* one--it is universal. When men use a concept it is the same as when they used it before--it is stable and unchanging. If these facts were not true no two people could ever be sure of having the same concept. Communication, education, and the division of cognitive labor would all be impossible. Rather than changing, a concept, if anything, is outright dropped and replaced.

To close this section Peikoff highlights an instructive metaphor developed by Rand. A concept is a file-folder, which is not inter- changeable with its label (definition), and is not limited to its current content. "The folder exists so that we can separate out as a single unit, and then study and interrelate, *all* the data ever to pertain to a given subject. This is precisely what the concept enables us to do."

Section 5, "Concepts as Devices to Achieve Unit-economy"

Peikoff concludes the chapter by discussing the fundamental cognitive role of concepts. They are devices to achieve unit-economy. He illustrates the issue with the "crow epistemology" example cited by Rand in her Introduction.

Like the crows in the example, man can only deal with a limited number of concretes. We can focus our awareness on about six or eight perceptual entities at a time. This is more than the crows, but still very limited. In order to deal with the vast totality of our experience we must have some way to compress the content of our mind, to *economize the units*. This is the function of concepts, to reduce a huge amount of information into a very small number of units.

A concept condenses a group or percepts into a single mental entity. This single unit takes the place of an endless series of concretes and clears the way for us to discover an unlimited amount of knowledge about the entity. They function not merely as time- savers (as many philosophers are apt to call them), but, more in- structively, as mental *space*-savers. Rand's theory of concepts teaches, in effect, that "A word is worth a thousand pictures."

Unit-economy is, predictably, essential to mathematics as well. Consider counting. Numbers serve the same essential purpose as concepts, allowing one to work with entities beyond direct perception. Algebraic equations, too, serve to achieve unit-economy, condensing long sequences of numerical calculations to single, brief formulas.

Unit-economy manifests itself many ways in concept-formation. A concept condenses its referents into a single mental unit. A definition condenses it concept's known characteristics to a single statement. Higher-level abstractions condense other concepts. By compressing data into fewer units the process allows us to deal with an ever-increasing scale of information, fulfilling our great cognitive need.

Comments and Questions

Peikoff (and Rand) refer to "implicit" concepts when discussing a child's development of "existent". However, Peikoff does not explain what constitutes an "implicit" concept in his book. The meaning is hardly obvious and could cause confusion for those less familiar with Rand's theory. A simple explanation would probably be helpful.

When discussing how we begin the process of concept-formation, Peikoff says we start on the basis of observed similarities among concretes and that the noting of these similarities makes possible our first differentiations. In a paper on abstraction, David Kelley seems to adopt the opposite perspective, giving primacy to differentiation. He argues that "given the perceptual basis of abstraction, differentiation has a certain primacy" in the first stage of concept-formation.

In the case of forming the concept "table" by contrasting with chairs, Kelley says that "Rand's theory is that the more radical difference between either one of the tables and a chair allows the subject to grasp the difference between the tables 'from the other side,' as it were: he sees that the two tables are not *as* different from each other as either one of them is from the chair. The awareness of the relation between tables as a less-than-comlpete difference is the enabling condition for the awareness of them as similar. The primary notion in her theory is therefore not comparative similarity, but comparative difference."

It seems that the two authors have contradicting views on this point. Do they? If so, who is right?

Peikoff refers to a concept as a "sum of concretes". Yet he also emphasizes that it is not merely a sum but the result of a blending that yields a new, separate unit. Is it wise then to continue to use the word "sum" when referring to concepts (Rand does also).

Peikoff cites Rand's formal definition of "concept", which refers to a concept as a "mental integration". A concept is the product of an integration, not the act of integration itself. This may or may not be obvious within context, but the phrasing could lead to confusion. Should Peikoff offer a slightly amended definition that makes this point more clear? (Note: I am suggesting this as a separate offering, not as a rewriting of what Rand herself supposedly said.)

On page 89 Peikoff states, rather eloquently, that a proper theory of concepts is so important that, though it alone is not sufficient to save to world, it is necessary if the world is to be saved. Is such a theory really so important? Could we not live just fine without it? How about the early United States? It did not arise from a good theory of concepts. Did it disintegrate because it did not have one?

When he reaches the issue of forming higher-level abstractions Peikoff prefers to say only a few words about them. He refers the reader to Rand's Introduction for an account of the process. This strikes me as a big omission on his part. It's not necessarily obvious how the process he described for first-level concepts applies to higher-level abstractions. The reader may get it or he may not. Is one to take Peikoff's word that the process works and is valid and move on to the next subject without wondering?

Peikoff and Rand speak of first-level and higher-level concepts. Is that all the distinction necessary? Does one need to know the ordinal progression of one's concepts as second-level, third-level, and so on as well?

Peikoff says that in one sense concepts of consciousness are first-level abstractions, in another sense higher-level abstractions. Is this duality tenable? Can one reasonably maintain that they are one thing in the one sense and another in a different sense? Are they at least more strongly one than the other?

Definition by nonessentials is claimed to "obliterate" a concept because it "does not carry with it the units' other features". However, Peikoff doesn't seem to formulate very strongly how and why this is so. Further, he doesn't seem to very strongly demon- strate how it is that a good definition *does* carry with it all the units' other features. How can his claims be strengthened? Can they be strengthened?

Do concepts, as Peikoff claims, really *never* change?

The discussion of the cognitive role of concepts (unit-economy) comes last. Would it have been better to bring up this issue earlier? He could have discussed first their function, then proceeded to show how they fulfill that function. Such a progression would allow the reader to see the direction of the process from the beginning. Would it make a difference, pedagogically, to have an end-in-sight?

Finally, a question of application that came to me the other day. How do we have a valid concept of "universe" by the Objectivist theory? There is only one, so we don't have entities to separate and unite into a unit. Can we contrast it with other, imaginary universes (unlikely), or is there simply no way to form the concept by the theory? Is it axiomatic (then there is no problem)? If so, how? Any insights?

References

Kelley, David "A Theory of Abstraction." Cognition and Brain Theory, 1984, 7(3&4), 329-357

Peikoff, Leonard Objectivism: The Philosophy of Ayn Rand New York: Dutton, 1991

Rand, Ayn Introduction to Objectivist Epistemology, 2nd ed. (H. Binswanger and L. Peikoff, ed.) New York: Meridian, 1990

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