Metatheoretical semiotic modeling in science and technology
Mathematical models of the intertheory of sign systems are developed in terms of the vector fiber bundle (stratification) of feature spaces of differential geometry as part of the general complexes theory, using philosophical and system-semiotic representations of linguistic knowledge. The manifestation of metatheoretic (MT) laws from the equations of the individual signs formation to the analysis and synthesis of scientific texts and system theories is investigated. The universal equation proposed allows us to compare various objects and phenomena and use it as a device for quantitative and qualitative analysis. For each kind of sign and non-sign systems, specific conceptual and terminological base is created and a set of axioms of the connection of concepts is determined, i.e. a special system intertheory is formed. The semiotic intertheory studies the properties of linear order in the form of pure knowledge on laws abstracted from the contextual environment. The axioms of semiotics postulate the existence of different forms of sign systems of fiber bundles with a linearly ordered structure: the feature space for fibration, the base of fiber bundle, the vector space of bundle with fiber, and the connectedness of signs (symbols complex). Using the vectorization procedures and their generalization to a signs series, sign systems of different complexity are produced. An analogy of the text structure with cartographic and technical reality constructions is made. Sign changes occur with geometric transformations in various coordinate systems with the preservation of recognition capacity. The properties of simple signs extend to the properties of sign systems and MT-models of geographical and technical reality. The triadic form of the organization of MT-knowledge allows us to consider the signified object as the sum of the subject (concept) of research and signifier, which is logical connected with the philosophical model of transcendental argumentation. Order structures are modeled by a unit interval [0,1], which allows us to transfer the known properties of real and hyper-real numbers to complex sign systems (complexes) on complex sign systems (complexes) of semiotics, linguistics and other sciences of complex phenomena, and to present scientific texts of natural language in terms of the formal metalanguage of transformations and relations of algebraic systems. As a result, signs (schemes) are everything that reproduces and preserves linear order, that they are bounded, indexed, fractal and hierarchical organized, homotopical similar, etc.