Stochastic-Conceptual Models Applied to Number Theory

Аннотация

Concepts are defined as couples (O, A) of sets O and A: the object O (a set of none, or one or more elements) is assigned to the set A of these elements’ (common) attributes. The objects change according to the sequence of attributes. Only couples of objects and attributes, that is concepts, are adequate for our world. The connections and linkswe need in databases and multimedia are expressed, naturally, by concepts, since concepts have been proved to dispose the order of a lattice (more complex and rich than linear and hierarchical ones). The lattice can be created by two algebraic perations: "intersection" as the multiplication and "symmetric-difference (!)" as the addition (!). There are, also, two other operations: the "union" and the "complement of a concept". Intersection and union (which cannot play the role neither of the addition nor of the multiplication) express similarities, while the other two operations express dissimilarities. The operation “complement of a concept” expresses the different, the uncommon, the variety. The symmetric-difference of two concepts has been proved to be a "distance" between them (in the mathematical sense!).