the Finite

Finite, the

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

References in classic literature ?
I may say that my first great lessons in true philosophy were obtained in these lectures, where I learned to distinguish between the finite and infinite, ceasing to envy any, while I inclined to worship one.
The 18 papers relate to the Monster group, character tables of finite simple groups, maximal subgroups of the listed groups, representation theory and general finite groups, Chevalley groups, constructing finite permutation representations of finitely presented groups, and the classification of the finite simple groups and their representations and subgroups.
The following three sections of the book present a more detailed development of the finite element method, then progress through the boundary element method, and end with meshless methods.
The mathematical root of the finite element method goes back to the history at least a half century.
Kant systematizes the contradictions inherent in the finite determinations of the understanding, and recognizes the necessity with which those contradictions arise.
Professors at City University London introduce the finite element method as a tool for simulating the behavior of electromagnetic fields in a photonic device, and explain the different forms of finite element-based methods that are widely used in the photonics field.
Because the analysed structure is of a complex geometrical shape, it is convenient to solve the problem of the temperature field by the finite element method.
The finite volume approach is used to solve a steady case problem of an epoxy disc with an inserted a crack and enclosed in a thin steel ring.
He offers basic knowledge of the finite element method, then sets out a procedure for performing an analysis, drawing on his graduate and professional lectures and on a public-private research project.
The positioning of the negative between finite and non-finite is ambivalent: depending on the finite, it may be either "premodifying" with respect to the non-finite or "post-modifying" with respect to the modal.
Some estimate the finite reinsurance market represents 5% to 10% of the total reinsurance premium, but Fuller said: "If you can't adequately define what finite reinsurance is, then it's difficult to capture what percentage of the market it is.
In a recent report on finite reinsurance, Fitch Ratings noted that "most arrangements that fall under the finite risk reinsurance umbrella are to lower reported losses or inflate reported surplus," but added that there is also "no simple definition for this complex product.

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