Principle of Abstraction

a logical principle which forms the basis for definitions in terms of abstractions: any equality relation defined on some initial set of elements divides (or partitions, classifies) the initial set into paired nonintersecting classes of elements which are equal for the given relation. These classes are called abstraction classes of the relation, and the set of these classes is the factor set of the initial set with respect to the relation. Thus the principle of abstraction determines a process of abstraction: if we isolate a class of objects which are equal in some sense (that is, an abstraction or equivalence class), we have thus defined an “abstract” (arbitrary) object of this class, since, from the viewpoint of the goals defining the equality relation, each “concrete” object of the initial set is understood as an “abstract” object—that is, the carrier of a property which is common to all elements of the particular abstraction class. The principle of abstraction is used to introduce as abstract objects not only the “representatives” of abstraction classes obtained by partitioning the initial set Z by some relation R, but also the classes themselves. For example, if Z is the set of all straight lines (plane or space) and R is the relation of parallelness, then the abstraction class of an arbitrary line a, taken from Z with respect to R is the class of all lines taken from Z parallel to a₁, the abstraction class of a₂, taken from Z with respect to R is the class of lines parallel to a₂, and so on. But here we have introduced the new notion of direction as a new “object.” It is in this way that all abstract concepts are in effect formed. For example, the concept of a continuous function is one of the abstraction classes generated by partitioning the set of all (numerical) functions by an equivalence relation which relates those and only those functions satisfying the definition of continuity. In this typical case the factor set consists of a total of two elements—“continuous” (function) and “discontinuous”; here the principle of abstraction becomes an assertion that it is permissible to consider the class of continuous functions (or the concept of continuity) in a correct manner. The second abstraction class which appears in this example (leading to the negative concept of discontinuity) is a complement of the first and is not involved explicitly in the formulation of this particular case (moreover, the “negativeness” of the second concept is not material: in separating numbers into even and odd, human beings into male and female, vertebrates into warm-blooded and cold-blooded, etc., the second concept thus introduced is on an equal footing with the first). This form of the principle of abstraction (frequently called the principle of comprehension), which asserts that there “exists” an abstract class (or set) of all objects satisfying an arbitrary but intelligently stated property (or predicate), plays a fundamental role in set theory.