Abstract: The article focuses on the problems associated with the need to express in the language of science in a consistent way the movement and various kinds of changes taking place in the objective world by developing logics that would be tolerant of logical contradictions. The paper presents a brief history of the creation of paraconsistent (paraconsistent) logics, in which logical contradictions turned out to be quite permissible. It is noted that the priority in their development belongs to Russian scientists N. A. Vasiliev and I. E. Orlov, as well as Polish philosophers and logicians Y. Lukasevich and S. Yaskovsky; that since the seventies of the last century, the development of paraconsistent logics has assumed an international character; that interest in these logics is not abating at the present time. The possibilities of using paraconsistent logics presented in the works of foreign and domestic philosophers and logicians for the formalization of dialectics are discussed. The positive role of these logics is shown in solving many paradoxes in the foundations of classical propositional calculation and predicate logic, as well as in order to limit undesirable consequences when using IT technologies related to the processing of inconsistent or contradictory information. It is concluded that the use of paranoherent logics in the construction of models of individual fragments of dialectics contributes to its development as logic and as ontology, and indicates that the apparatus of non-classical logics as a whole is a very powerful means of studying and explaining many problems of theoretical cognition.

Abstract: Non-classical logicians have significantly expanded the traditional field of using logical methods. The first of them was the three-digit logic of Y. Lukasevich. Next came the three-digit logic of A. Bochvar, the "quantum logics" of G. Reichenbach and P. Detush-Fevrier, infinite-valued, probabilistic and other logics. The possibilities of non-classical logics have become widely used in various branches of scientific knowledge. Polysemantic, fuzzy, intuitionistic, modal, relevant and paranoherent, temporal and other non-classical logics are widely used today in physics, computational mathematics, computer science, linguistics, jurisprudence, ethics and other fields of natural science and socio-humanitarian knowledge. The recently increased interest in non-classical logics is explained, first of all, by the fact that various philosophical, syntactic, semantic and metalogical problems that were previously discussed in the scientific community are being replaced by practical interests. The main source of such interest is their wide application in computer science, artificial intelligence and programming. The logic of causality is used in the interpretation of the concepts of "law of nature", "ontological necessity" and "determinism"; temporal modal logics - for modeling, specification and verification of software systems of logical control; logics with vector semantics, combining the features of fuzzy and para-contradictory logics - in solving problems of dynamic verification of production knowledge bases and expert systems.

Abstract: The subject of this research is such philosophical and mathematical disciplines as Pythagoreanism and Platonism, which remain relevant at the present time. The author demonstrate the contribution of Pythagoreans to mathematics, their role in creation of geometric algebra, importance of the discovery of incommensurable segments that propelled the Pythagorean mathematics into crisis. The work examines the essence of the concept of mathematical Platonism, reveals its peculiarities, and demonstrates its dissimilarity from the concept of mathematical Pythagoreanism. The presently existing various forms of mathematical Platonism, as well as their peculiarities are explored. The article provides the main arguments of modern critics of Platonism in mathematics and their weaknesses. The author demonstrates the value of the concept of mathematical Platonism as a model visual thinking, and underlines that a large number of mathematicians remain its adherers. The scientific novelty is defined by the fact that the work actualized the ideas of Pythagoreanism and Platonism, as well as the consequence of a dispute that originated in ancient times and continues today between the supporters of Platonism and their opponents related to the fundamental grounds of mathematics. The author concludes that the results of modern mathematical science give valid arguments that confirm the performance and high efficiency of the concept of Platonism in comparison with other philosophical concepts of mathematics.

Abstract: The subject of this research is the problem of universality (singularity) and uniqueness (plurality) of the “first” or “experienced” mathematics and ordinary logic of reasoning. The article discusses the possibility of application of the sociocultural approach that efficiently functions in ethnomathematics, studying of the logical reasoning within the historical context. The author demonstrates that in modern history and philosophy of mathematics, there are two opposing points of view: supporters of the first one substantiate the singularity, universality of the “first” or “experienced” mathematics, while supporters of the second uphold the idea of plurality of mathematics, explained by the impact of sociocultural factors upon its development. It is underlined, that at present time, a similar situation can be noticed with regards to the problem of universality of the logic of reasoning. The scientific novelty consists on presentation of comparative analysis of the two opposing points of view pertaining to the universality/plurality of the logic of reasoning if compared with the analogous circumstances in mathematics. The following conclusions are made: 1) the question of universality, i.e. singularity for the entire population of the Earth, or uniqueness, plurality, close interconnection of the logic of ordinary reasoning (as the “first mathematics” with the culture that it emerges within and produces significant impact, remains relevant for the modern science and philosophy; 2) such question in many way correlates with the problem of occurrence of the abstract reasoning, as well as reasoning in general; 3) it would be appropriate at present stage to introduce the notion of “ethnologic” for assigning the area of research that examine the role of sociocultural factors in establishment of the ordinary logic in historical development.

Abstract: The subject of this research is constructivism, the interest to which, in the author’s opinion, is caused by the discontent with classical epistemology, awareness of its limitation, and also the fact that such variety as "epistemological constructivism" not only expresses a number of features of modern anthropological sciences and contemporary culture in general, but also gives answers to the questions, which encourage the understanding of opportunities and limits of human cognition and what is the role of the cognizing subject in cognitive activity. This article makes actual the problems of dispute between realists and antirealists, which have direct relation to the fundamental grounds of scientific knowledge. Based on the analysis and comparison of the ideas of certain representatives of epistemological, social, and radical constructivism, constructivism in mathematics, works associated with discussion of the problems of the effect of sociocultural factors upon the development of mathematics (particularly works in the area of ethnomathematics)  and science as a whole, as well as works of the Russian philosophers dedicated to the questions of constructivism in philosophy and science, the author makes the following conclusion: the results of modern science (including cognitivistics) provide fairly weighty arguments which confirm the realistic interpretation of cognitive process and its result – the knowledge. It is underlined that this interpretation contributes into more detailed understanding and more adequate explanation of the scientific factors, opens opportunities for development of the research programs, which in the context of antirealistic epistemology would be impossible.

Abstract: The subject of the research is the sociocultural (fundamentalist) approach to the philosophy of mathematics, in particular, ethnomathematics as the sphere of mathematical research. Particular attention is paid to the ideas of Oswald Spengler, the author of The Decline of the West, Neo-Kantian Hermann Cohen and Paul Natorp about the position and role of mathematics in the cognitive process as well as the relationship between mathematics and culture. The results of the research show that the human mind is able to offer many different ways to quantitative perception of the world, each of which arises from everyday practice, and that the paradigm adopted by contemporary mathematics is in fact only one of possibilities. The main research methods used by the author include generalization, logical and historical methods, as well as a comparative analysis. The novelty of the research is caused by the fact that it reveals the unity of ideas expressed in ethnomathematical researches that deal with the problems of occurrence and development of "experimental" mathematics as well as researches in the field of socio-cultural philosophy of mathematics. It is shown that these results support the idea of Spengler about the dependence of the forms of knowledge on human conditions of existence, about existence of not one but several mathematics, each of which is rooted in their own culture, as well as the ideas of the representatives of the Marburg Neo-Kantian School about mathematics being the "first principle" of thinking. The author of the article concludes that researches in the field of ethnomathematics, and in a broader context, as part of the socio-cultural approach to mathematical knowledge, are most effective when considering mathematics in terms of its historical development; that their results can serve as an additional argument in favor of mathematical empiricism in its confrontation with the mathematical apriorism. 

Abstract: AnnotationThe subject of investigation is the problem of rationality, attention to which today is associated with a certain distrust of classical rationality and the emergence of a new rationality that goes beyond scientific rationality and includes all kinds of cognitive practices. This new rationality goes beyond narrow for her scientific knowledge and incorporates all that "makes possible the existence of man in the modern world," it "more and more drawn into the myth and other traditional forms of knowledge, which equalized the rights of science"The main methods of research used by the author, is a logical method and the method of comparative analysis, you always get the tentative conclusions.The novelty of the work is connected with the idea of the author of that "new rationality" in which thinking manifests a kind of "flexibility" that extends to the rejection of the laws of traditional logic, to the assumptions in the thinking of contradictions, taking them as a fully legitimate, it is the result of the evolution of "pre-logical" or as it is called L. Levy-Bruhl "prelogicheskogo" thinking. In other words, modern rationality contains all the currently existing rationality, and thus permits and contradictory thinking, and others, in terms of traditional logic incongruities.The article tentative conclusion is that the development of rationality takes place by moving from the rationality of "primitive" thinking and rationality myth, then - to the rationality of religious, then - classical scientific thinking, and from there to the "new rationality", which is a synthesis of rationality existed in ancient times and exist today.

Abstract: The subject of the research is ethnomatematics as the field of knowledge that was created in the middle of the last century and is an interdisciplinary field of knowledge which includes both the actual math, as well as its history, and philosophy, ethnology, cultural studies, psychology, pedagogy and some other disciplines relating to mathematical knowledge in this way or another. One of the areas of ethnomathematics is pedagogy, within which researchers try to synthesize the achievements of philosophy, epistemology and history of natural science and mathematics focusing on educational activities, in particular, on teaching mathematics in schools and universities. The question whether it would be reasonable to appeal to personal experience of students and their ethnonational and socio-cultural background are actively discussed in Russia within the framework of such branches of knowledge as ethnodidactics and pedagogy. The author of the article shows common problems in ethnomathematics and ethnodidactics and views arguments of the followers and critics of the ethnonational approach to education. The main research methods used by the author include analysis of literature as well as comparative analysis and summarization. The novelty of the research is caused by the fact that the author analyzes works of foreign authors in the field ethnomathematics, these works are not so well-known by Russian philosophers and scientists dealing with the same problems. The novelty is also caused by the author's focusing on the point of contact between ethnomathematics as a rather young branch of knowledge and ethnodidactics which is successfully used at national schools of the Russian Federation.