ОТНОШЕНИЕ

Relation

A necessary moment in the interconnection of all phenomena determined by the material unity of the world. The Relation of things is as objective as the things themselves. Things do not exist outside Relation and the latter is always the Relation of things. The existence of each thing, its specific features and properties, and its development depend on the sum total of its Relations to other things of the objective world. The properties themselves, necessarily inherent in one process or another or in a thing, are manifested only in their Relations to other things and processes.

Development of a phenomenon leads to a change in its Relations with other phenomena, the disappearance of some Relations and the appearance of other Relations. On the other hand, changes in the sum total of Relations in which the given object exists may lead to a change in the object itself. Relations are as diverse as things and their properties.

It is necessary to differentiate internal Relations of different, particularly opposite sides of an object and its external Relations with other objects. Account should be taken, first, of the relative nature of differences in internal and external Relations, second, the passage of one into another, and, third, the fact that external Relations depend on internal Relations, manifest and reveal them.

Social Relations are of a special nature. Man enters into Relations with the things he creates, the objective world, and other people. As a result, in the world he is mastering he contemplates himself and begins to treat himself as a man (gains self-consciousness) only by treating another man as his own likeness. This is what explains, on the one hand, the social nature of human consciousness, and, on the other, the necessity of studying social Relations in order to know history.

In dialectical logic, "the relations (= transitions = contradictions) of notions = the main content of logic, by which these concepts (and their relations, transitions, contradictions) are shown as reflections of the objective world". (Lenin, Vol. 38, p. 196.)

In mathematical logic, Relations are opposed to properties like multiple predicates to a singular predicate (see Predicate). "More", "equal", "cause" are the examples of dyadic Relations. "Among" and others are triadic Relations. In formal logic, the theory of Relations was developed by De Morgan, C. Peirce, and E. Schroder. The logical theory of Relations studies the general properties of Relations and the laws governing them. A calculus of Relations related to a calculus of classes forms an essential section of the theory of Relations. This studies the connections between Relations and operations with them and establishes the laws by which some Relations can be deduced from others.

Отношение

Двух чисел, частное от деления первого числа на второе. О. двух однородных величин называется число, получающееся в результате измерения первой величины, когда вторая выбрана за единицу меры. Если две величины измерены при помощи одной и той же единицы меры, то их О. равно О. измеряющих их чисел.

О. длин двух отрезков может выражаться рациональным или иррациональным числом. В первом случае отрезки называются соизмеримыми, а во втором - несоизмеримыми. Математики древнего мира не знали иррациональных чисел; для них понятие О. двух отрезков не сводилось к понятию числа; не зависимая от понятия числа геометрическая теория О. величин играла у них самостоятельную роль и заменяла в известном смысле теорию действительных чисел (см. Число). Действительно, по Евклиду, четыре отрезка а , b, а

’ b ’ составляют пропорцию а : b = а ’: b ’, если для любых натуральных чисел m и n выполняется одно из соотношений mа = nb , mа > nb , mа < nb всякий раз одновременно с соответствующим соотношением mа

’ = nb ’; mа ’ > nb ’ или mа ’ < nb ’. В случае несоизмеримости а и b это означает, что разбиение всех рациональных чисел ( х = m /n) на два класса по признаку а > xb или а < xb совпадает с разбиением по признаку а

’ > xb ’ или a ’ < xb ’ - в этом состоит идея современной теории дедекиндовых сечений. О двойном (иначе - сложном, ангармоническом) О. см. Двойное отношение.