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The doctrine of coincidentia oppositorum (the coincidence of opposites), the interpenetration, interdependence and unification of opposites has long been one of the defining characteristics of mystical (as opposed to philosophical) thought. Mystics of various persuasions have generally held that such paradoxes are the best means of expressing within language, truths about a whole that is sundered by the very operation of language itself. Any effort, it is said, to analyze these paradoxes and provide them with logical sense is doomed from the start because logic itself rests upon assumptions, such as the principles of non-contradiction and excluded middle, that are violated by the mystical ideas.

Mathematical development of a dialectical logic with truth-values in a complex field.

The coincidentia oppositorum is a common trope in many religious traditions, particularly those with a mystical or initiatory aspect, and fields of knowledge such as Psychology and Quantum Physics, with wave-particle duality. The aim of this paper is to present a mathematical theory of the coincidence of opposites, and where truth-values are expressed in a complex field. A propositional coincidentia oppositorum algebra is developed.

Although the literature of paraconsistent logics is abundant, the authors think that this is the first time that a dialectical logic has been developed with truth-values belonging to a complex field. The impossibility of reaching an absolute truth from opposite propositions is discovered, both being true, because of the existence of irrational numbers in the truth-values.

Mathematical models are constructed at the interface between practice, experience and theories. The function of models puts us on guard against the privilege granted to what is accepted as abstract and formal, and at the same time puts us on guard against a static and phenomenological conception of knowledge. The epistemology of models does not suppress in any way the objectives of science: only, a dogmatic conception concerning truth is removed, and dynamic and dialectical aspects of monitoring are stressed to establish the most viable model. The purpose of this paper is to examine hybrid methodologies (inductive-deductive) that may either propose hypothetical causal relations and seek support for them in field data or detect causal relations in field data and propose hypotheses for the relations detected.

The authors follow a dialectical analysis for a type of inductive-deductive model.

In this work, the authors present an inductive-deductive methodology whose practical result satisfies the Hegelian dialectic. The consequent implication of their mutual reciprocal integration produces abstractions from the concrete that enable thought. The real problem in this case is a given ontological system or reality.

The essential elements of the models – variables, equations, simulation and feedback – are studied using a dialectic Hegelian theory.

The distinction between essence and existence cannot be a distinction in God: in the actual infinite, essence and existence coincide and are one. In it, maximum and minimum coincide. Coincidentia oppositorum is a Latin phrase meaning coincidence of opposites. It is a neo-Platonic term, attributed to the fifteenth-century German scholar Nicholas of Cusa in his essay, Docta Ignorantia. God (coincidentia oppositorum) is the synthesis of opposites in a unique and absolutely infinite being. God transcends all distinctions and oppositions that are found in creatures. The purpose of this paper is to study Cusanus’s thought in respect to infinity (actual and potential), Spinoza’s relationship with Cusanus, and present a mathematical theory of coincidentia oppositorum based on complex numbers.

Mathematical development of a dialectical logic is carried out with truth values in a complex field.

The conclusion is the same as has been made by thinkers and mystics throughout time: the inability to know and understand the idea of God.

The history of the Infinite thus reveals in both mathematics and philosophy a development of increasingly subtle thought in the form of a dialectical dance around the ineffable and incomprehensible Infinite. First, the authors step toward it, reaching with their intuition beyond the limits of rationality and thought into the realm of the paradoxical. Then, they step back, struggling to express their insight within the limited scope of reason. But the Absolute Infinite remains, at the border of comprehensibility, inviting them with its paradoxes, to once again step forward and transcend the apparent division between finite and Infinite.

The purpose of this paper is to show that transmission of information and information storage or registration depends on structures. Structures emerge from coordinated sets of constraints. Complex systems depend on their structures to function. The temporal sequence of changes in the levels of the complex system determines its behavior. These three concepts are intimately linked with the environment. Environment, structure, function and behavior form a complex system–environment unit, which is the operational unit of existence for all open complex systems. Therefore, it becomes a point in the directional propagation of the cause, where stimulus environment becomes a Creaon, and then the Creaon becomes a Genon, becoming in turn the response to the experienced environment. The formation of structures is the main phenomenon of evolution. Evolution can also be accepted as free, in the sense that it does not cost additional deaths.

Mathematical and logical development of the structure and thermodynamics in complex systems.

Based on the above considerations, the authors are going to introduce two fundamental principles in Complex systems Theory: the Matthew Effect and the Principle of Sagan.

But as the authors’ purpose is to give a formal definition of a complex system from a totally theoretical point of view, they establish a relationship between concepts of General Systems Theory, Theory of the Environment, linguistics, Information Theory and thermodynamics.