that which has a limit, boundary, or end. In philosophy the concept of the finite is used as a category characterizing every determinate, bounded object (thing, process, phenomenon, condition, property, and so forth). Every knowable object of reality appears finite in a certain respect. The boundary of the finite lends it determinateness. This boundary may be spatio-temporal, quantitative, or qualitative. A boundary separates a finite object from others, as well as connnecting it to them. Therefore, the finite, on the one hand, enjoys a relatively independent, detached existence and, on the other, is determined by something else and is dependent on it. This constitutes the contradiction in the finite. The most profound knowledge of the finite comes from a knowledge of the measure inherent in it. The presence of a boundary or measure necessarily implies the possibility of exceeding it, that is, the negation of the given finite entity, its transformation into another entity. The consideration of this leads to a dialectical conception of the finite, according to which it may be understood only as the unity of its own being and its own nonbeing, as the mutual transition of one into the other. In other words, the finite must be understood as moving, changing, transitory.
Consideration of the process of movement of a finite entity, in the course of which it continuously goes beyond its boundaries, leads to the idea of infinity. The connection between the finite and the infinite is twofold: first, every finite object is related to the infinite variety of other finite objects “external to itself (extensive infinity); second, it contains the infinite within itself as the expression of universal, invariant properties (intensive infinity). Consequently, in knowing any material object, we encounter a unity of the finite and the infinite. Every material object is inexhaustible (the principle of inexhaustibility of matter). Knowledge “consists . . . in seeking and establishing the infinite in the finite, the eternal in the transitory” (F. Engels, in K. Marx and F. Engels, Soch., 2nd ed., vol. 20, p. 548).
In mathematics the sense of the concept of the finite (like that of the infinite) depends on the specific character of mathematical objects. In the formulation of a particular mathematical theory, the concept is given various interpretations, which take into account only those means of defining and delimiting objects with which the theory operates. Objects which are finite in one respect and infinite in another are frequently said to be finite and unbounded, or infinite and bounded (for example, the set of points on a line segment is infinite but bounded; a closed elliptic Riemann space is finite and unbounded). In these cases, however, finiteness (infiniteness) also refers to the presence (absence) of some kind of bound, for example, the Riemann space is finite in the sense that there is a numerical bound characterizing the size of the largest distance in it. The most general mathematical definition of finiteness (finite set) occurs in mathematical logic and in set theory, for example, the Dedekind definition: a set M is finite if among its proper subsets there does not exist one equivalent to it. It has been proved that among the various definitions of finite set there can exist neither a “strongest” nor a “weakest” one, that is, for any of them there is both a definition which is logically deducible from it and one from which it itself may be deduced.
A. S. KARMIN