An animal knows only a handful of perceptual concretes, and acts on them automatically. A man, by contrast, goes beyond his observations, he generalizes and identifies natural laws, he hypothesizes causal factors, and he projects alternative courses of action and long-range consequences. Man, in sum, is a conceptual being. That’s the theory, at least; in practice, we see animals all around.
In a more detailed view, the great cognitive divide is the ability to perceive units. Both animal and man have the implicit concept entity — they know something exist. They also have the implicit concept identity — they distinguish this from that. However, only man grasps similarities and differences among these entities. Only man understands a unit of a larger group of similar members, classified based not on arbitrary subjectivity, but on a criterion perceived in reality. The concept unit is thus a bridge between metaphysics and epistemology.
We begin the formation of a concept by isolating a group of concretes from the rest of our perceptions based on observed similarities. This is the process of differentiation. But we can do little with such a group. Man can deal with only a limited number of units, say, six or eight perceptual objects like tables; what to say of thousands or all tables in existence? For a consciousness to extend its grasp beyond a mere handful of concretes, it needs the capacity to condense its content, to economize its units. So we must proceed to integrate them. We blend the units into a new entity based on their essential characteristics, a mental entity that can now stand for an unlimited number of concretes — a concept. That is the power of abstraction so defining of man as such.
Ayn Rand says there is an essential connection between concept-formation and mathematics. Mathematics is the science of measurement. Measurement in turn is the process of bringing the universe to the scale of human knowledge. Both measurement and conceptualization, Ayn Rand states, involve the discovery of a mathematical relationship among concretes. When we form a concept, we retain the characteristics of the concretes, but omit their measurements. Such is the essence of abstraction.
In order to retain these concepts tied to reality (as against “floating abstractions”), however, man needs definitions. A definition identifies a concept’s units by specifying their essential characteristics, those which make the units what they are and differentiate them from the rest of existents in a given context of knowledge. But, the definition is not interchangeable with the concept itself; a concept includes all the characteristics of the existents it subsumes, whether definitional or not.
At last, a word is then just the “icing on the cake” (albeit an essential one), a linguistic shortcut to allow us to reference the whole sum of concretes at once, and to enable further thought. A concept is now a mental entity.
They say that “a picture is worth a thousand words”; Leonard Peikoff says that “a word is worth a thousand pictures”. I get it now.