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Presented article is the first of two articles, the aim of which is to familiarize the uninitiated reader with the possibilities of application of mathematical methods developed in the scientific direction of “Conceptual analysis and design of systems of organizational management (CAD SOM)”, designed to solve various problems, both technical and in humanities. At the heart of this methodological approach is a process of conceptualization, which is understood as a theoretical study of qualitative aspects of a selected domain using mathematical forms (axiomatic) theory, the locking connection between the concepts of logical derivability characterizing this subject area. Designed axiomatic theory – conceptual scheme – is the basis for building database structures, decisionmaking processes, a variety of phenomena subject area, structure and genesis of domain analysis and other tasks. Given the non-mathematical preparation of the reader, the contents of the methods illustrated by conceptualizing a conceptually simple subject areas – areas related relations, as well as the choice of one of the most simple goals conceptualization – structuring the domain and build a variety of its phenomena. For a better understanding of an article in a lightweight form describes and explains the stages of conceptualizing technology, the main difficulties of postulating and typical solutions. The first article explains the issues positing (postulation) in constructing conceptual schemes. These include, in particular, the definition of the domain boundaries, the allocation of the basic concepts and the precise definition of the content. Considerable attention is paid to the impact of these decisions on the results of conceptualization.

The article substantiates the assumptions, axioms and theorems of the universal conflict model – the mathematical apparatus of the unified theory of conflict. The conflict is formalized as a loop of negative feedback, which, depending on the value of the feedback coefficient, can have both linear and nonlinear dynamics. In the first case attractors are of a constant or oscillating character. In the second case, the system tends to chaos and self-destruction. Rationality of axiomatization is caused by the absence in the modern scientific literature of a correct definition of conflict and, as a consequence, a correct understanding of its structure, functions and dynamics. The main advantage of the proposed axiomatization of the conflict is that it opens the way to uniting the unified theory of conflict being developed by the author with the mathematical theory of catastrophes, and the theory of complexity.