Learning systems are in the forefront of analytical investigation in the sciences. In the social sciences they occupy the study of complexity and strongly interactive world-systems. Sometimes they are diversely referred to as symbiotics and semiotics when studied in conjunction with logical expressions. In the mathematical sciences the methodology underlying learning systems with complex behavior is based on formal logic or systems analysis. In this paper relationally learning systems are shown to transcend the space-time domain of scientific investigation into the knowledge dimension. Such a knowledge domain is explained by pervasive interaction leading to integration and followed by continuous evolution as complementary processes existing between entities and systemic domains in world-systems, thus the abbreviation IIE-processes. This paper establishes a mathematical characterization of the properties of knowledge-induced process-based world-systems in the light of the epistemology of unity of knowledge signified in this paper by extensive complementarities caused by the epistemic and ontological foundation of the text of unity of knowledge, the prime example of which is the realm of the divine laws. The result is formalism in mathematical generalization of the learning phenomenon by means of an operator. This operator summarizes the properties of interaction, integration and evolution (IIE) in the continuum domain of knowledge formation signified by universal complementarities across entities, systems and sub-systems in unifying world-systems. The opposite case of ‘de-knowledge’ and its operator is also briefly formalized.

Learning systems are in the forefront of analytical investigation in the sciences. In the social sciences they occupy the study of complexity and strongly interactive world-systems. Sometimes they are diversely referred to as symbiotics and semiotics when studied in conjunction with logical expressi...