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The purpose of this paper is to demonstrate mathematically the impossibility of achieving a utopian society. Demonstrate that any attempt to correct deviations from a hypothetical trajectory whose ultimate goal is the utopia, increasingly demands more work, including measures that lead to terror, which may even be absolute, leading to the horrible paradox that in seeking paradise hell is constructed.

Scientific tools that the authors have used are: the theory of the system linkage, alysidal algebra, kinematic theory and vector analysis.

Myths are the substrate of some complex systems of beliefs and utopia is its ultimate goal. The use of the combination of the theory of trajectories, belonging to the alysidal algebra, the theorem of unintended effects and kinematics theory provides an approximation to deviations suffering utopian ideological currents and their corrections.

This paper is a continuation of other previous papers developing the theory of complex societies.

The process of elaboration of the symbolic universe leads to important insights into the role of symbols in understanding human reasoning. Symbols become explanatory axes of universal global realities. Myths were constructed on these explanatory paths forming a superstructure of all belief systems with paraconsistent logic for the symbolism and a symbolic syntax. Myths and symbols are to be found in all cultures. Some of the most powerful and influential ones occur in popular culture since these often have the greatest immediate social impact.

Semiotic and logical development of the symbols is in mythical systems. The dissolution of the myth and the degradation of the myth's symbols constitute a long-drawn-out process in modern Western society and wherever s influence reaches. Myth is a story that may contain symbolic elements, but compared to the symbols or images of the exceptional, myth is characterized by a “story.”

Starting from a minimal definition to define myths and propose the following definition: Myth is a traditional tale that relates memorable and exemplary actions of extraordinary personages in prestigious and distant times, and myths have various forms and functions, perhaps some more clearly defined with a signifier than others, and different approaches can be combined for a better understanding of the myths. Dispensing with such simplistic assertions, and starting from a minimal definition to define myth, myth is a traditional tale that relates memorable and exemplary actions of extraordinary personages in prestigious and distant times.

Any symbol F originates in a unit that has two aspects and functions when the unit is restored. Thus, the symbol is rather “for something” than “of something” and the symbolic objects express the objects' correspondence in one unit or hendiadys. One semantic characteristic of symbols is “recognition”. The symbol F reveals a reality by means of the homogenous association of the signifier and significance in the symbol's constitution; although reality is separate, there is a homogeneous relation between the symbolizing and symbolized in symbolization.

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.

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.

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.

This paper aims to refer to a subjective approach to a type of complex system: human ecosystems, referred to as deontical impure systems (DIS) to capture a set of properties fundamental to the distinction between human and natural ecosystems. There are four main phenomenological components: directionality, intensity, connection energy and volume. The paper establishes thermodynamics of deontical systems based on the Law of Zipf and the temperature of information.

Mathematical and logical development of human society structure.

A fundamental question in this approach to DIS is the intensity or forces of a relation. Concepts are introduced as the system volume and propose a system thermodynamic theory. It hints at the possibility of adapting the fractal theory by introducing the fractal dimension of the system.

This paper is a continuation of other previous papers and developing the theory of DIS.

The purpose of this study is to examine the unintended consequences of actions as one of the central and constituent elements of sociological theory and long debated in the history of sociology. This question has been treated under varying sociological terminologies, including, providence, social forces, social paradoxes, heterogeneity of ends, immanent causality and the principle of emergency.

This paper is concerned with “adverse effects”. The thematic contexts of “unintended consequences of social action” the authors wish to focus attention on are specific types of consequences which may merit the adjective “adverse”.

The analysis of the intentions of our actions and their unwanted or foreseen consequences allows us to understand how societies work. Many historical facts are probably “unintentional.” But, most continuous or changing life forms must be interpreted as a mixture of intentional (social reproduction) and unintentional consequences (social change).

This paper focuses on four points of view: the object of sociology, the problems of order and social change, the methodological status of the discipline and the nature of social explanation, and mathematical theory. Four classifications of unintended consequences are formulated from the works of Boudon, Baert and Ramos, as well as the authors.

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 the study of the causal relationship. The concept called “naive” causality can be stated more generally as the belief (or knowledge) that results follow actions, and that these results are not random, but are consistently linked with causes. The authors have thus formed a very general and precarious concept of causality, but one that appropriately reflects the meaning of causality at the level of common sense.

Mathematical and logical development of the causality in complex systems.

There are three aspects of rationality that give the human mind a unique vision of reality: quantification: reduction of phenomena to quantitative terms; cause and effect: causal relationship, which allows predicting; and the necessary and valid use of (deterministic) mechanical models. This work is dedicated to the second aspect, that of causality, but at present leaves aside the discussion of possibility-necessity, proposing a modification to philosophical synthesis of causality specified by Bunge (1959), with contributions made by Patten et al. (1976) and LeShan and Margenau (1982).

Causality is an epistemological category, because it concerns the experience and knowledge of the human subject, without being necessarily a property of reality.