Aggregate
Aggregate
1. In a general sense, a collection, a totality, a whole, a class, a group, a sum, an agglomerate, a cluster, a mass, an amount or a quantity of something, with certain definite characteristics in each case.
2. In Logic and Mathematics, a collection, a manifold, a multiplicity, a set, an ensemble, an assemblage, a totality of elements (usually numbers or points) satisfying a given condition or subjected to definite operational laws. According to Cantor, an aggregate is any collection of separate objects of thought gathered into a whole; or again, any multiplicity which can be thought as one; or better, any totality of definite elements bound up into a whole by means of a law. Aggregates have several propertiesfor example, they have the “same power” when their respective elements can be brought into one-to-one correspondence; and they are “enumerable” when they have the same power as the aggregate of natural numbers. Aggregates may be finite or infinite; and the laws applying to each type are different and often incompatible, thus raising difficult philosophical problems. See One-One; Cardinal Number; Enumerable. Hence the practice to isolate the mathematical notion of the aggregate from its metaphysical implications and to consider such collections as symbols of a certain kind which are to facilitate mathematical calculations in much the same way as numbers do. In spite of the controversial nature of infinite sets great progress has been made in mathematics by the introduction of the Theory of Aggregates in arithmetic, geometry and the theory of functions. (German, Mannigfaltigkeit, Menge; French, Ensemble).
3. In logic, an “aggregate meaning” is a form of common or universal opinion or thought held by more than one person.
4. In the philosophy of nature, aggregate has various meaningsit is a mass formed into clusters (anat.); a compound or an organized mass of individuals (zool.); an agglomerate (bot.) an agglomeration of distinct minerals separable by mechanical means (geol.); or, in general, a compound mass in which the elements retain their essential individuality. — T.G.
(in mathematics)The concept of an aggregate is now usually identified with that of a class (q.v.) — although as a historical matter this does not, perhaps, exactly represent Cantor’s notion. — A.C.