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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.

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.

Organizational studies have been deeply influenced by three separate streams of research: the soft sciences; the hard sciences; and economics. This paper makes a case for an interdisciplinary approach, one that includes not only the social and physical and life sciences, but also methodologies that have a long history in mysticism. It illustrates how the similarities and relationships between depth psychology, in Jung’s theory of archetypes, and the “hard science” notion of complexity theory can reveal critical aspects of competition as expressed through capitalism. It also suggests that a methodology for accessing information about archetypes in general and capitalist competition in particular is through creative imagination.

Categories (particular (P) and general (V)) constitute a bipole with epistemological implications. The mutual categorical implication of this bipole is embodied in ordinary notions. It follows that a concept because it forms an element of concrete, sensible-rational, practical-theoretical activity has to unite the two inseparable poles, the general and the particular. If the concept of a physical quantity is abstract in relation to the physical object, it is concrete in comparison with mathematical quantity. This product of a secondary abstraction covers the background of physical qualities to extract the pure number, legitimately named abstract number. Both kinds of numbers are mutually exclusive: either the numbers are attached to a unit name and the number is concrete or nothing is attached and the number is abstract. However, in addition to their coordination in extension, they involve each other in comprehension: in fact, the pure number is the general pole V and concrete numbers form the particular pole of the dialectical concept of number K. The purpose of this paper is to provide a model for epistemological issues that arise in the context of meaning, concepts and use of words.

A dialectical theory of the binomial comprehension-extension of mathematical magnitudes.

The findings provide an objection to the traditional deductive order being also true in mathematics, and also that the reverse order cannot be considered as characteristic of mathematics, but show dialectic as universal. This opens the way to the special scientific deduction (mathematical, physical, biological, etc). going from the general to individual.

The structure of the mathematical concepts is elaborated.