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One of the main requirements for mathematical proofs is that of rigor. Rigor guarantees that there is a logical connection between the proposition being proved and the assumptions or axioms being accepted. Traditionally, the rigor of a mathematical proof is thought to be ensured by appealing to formalization. However, this understanding does not correspond to mathematical practice. The vast majority of proofs (both in the past and at present) are given informally and accepted by the community of expert mathematicians without any formalization. The article presents an alternative position regarding the rigor of mathematical proof proposed by S. De Toffoli, which is supposed to be consistent with practice. The article is divided into four parts. The first part considers the traditional understanding of proof as a surveyable and formalizable construct used for persuasion; it discusses the structure of proof and the standard understanding of rigor provided by formalization. The second part of the article discusses the practice of referring to “manipulative imagination” with respect to topology and shows the special status of this ability in “filling in” gaps in informal proofs. It is argued that this ability is developed through an appropriate training and involves only such manipulations that are justified from a strictly formal point of view. The third part discusses an alternative to the traditional notion of proof as a shareable, transferable and formalizable construct, components (“steps”) of which can be enumerated. Also, it introduces the notion of simil-proof which is a means of explaining the mathematical practice of accepting incomplete or error-containing proofs. In the fourth part of the article, the final points are formulated concerning the commitments of accepting relativism and anti-realist ontology when explaining the notions of proof and rigor.

The article provides a philosophical analysis of the methods for describing and explaining the properties of light in geometric optics. The causes and factors that have had a defining effect on the methodology of description and explanation in geometric optics are revealed.

The article is concerned with analyzing the main stages in the development of the physiology of the organ of hearing. In the era of the Ancient World and the Middle Ages, the preconditions for the further development of this scientific area were formed. In modern times, the accumulation of knowledge, acquired through the use of new theoretical and experimental approaches to understanding the processes in the body, continues and important discoveries are made. At present, the physiology of the hearing organ is one of the important areas of physiology. The article is aimed at identifying and describing the basic periods of formation and development of the physiology of the hearing organ. The materials used for writing this work were domestic and foreign studies on the issue. The main results consist in presenting the stages of development of the physiology of the hearing organ along with providing the description of the contribution of leading researchers in this field. The conclusion is made about the process of formation and development of the physiology of the organ of hearing as a branch of biomedical knowledge.

There are not many examples in history when the world scientific thought was able to respond so quickly to a common challenge, which became a coronavirus infection in 2020-2022. During this period, domestic economic researchers published 1615 Russian-language articles in journals, so that a wide range of their views was reflected. The semantic and cluster analysis of keywords and frequencies of references to concepts in the titles of these articles made it possible to identify a number of patterns. The collection of article titles can be considered as a sublimated text reflecting the researchers’ cumulative understanding of the economic processes that took place during the examined period. It was possible to determine a strong correspondence of the frequencies of the used concepts to the distribution according to Zipf's law. This should be interpreted as a reflection of a certain unity of the process of public understanding of both the circumstances of the pandemic and its impact on the economy in various aspects

Introduction. Dramatic changes in the social structure and political regulation of the public system necessitated a rethinking of the characteristics of the educational system and its resources. The position is represented by a socio-philosophical analysis of the characteristics of the modern Russian Institute of Education, dynamics trends due to the current types of postmodern rationality. Methodology. The theoretical and methodological basis of the author’s research was the dialectical method, historical, systemic, comparative analysis. Explored the signs of the institute within the postmodern paradigm of social structure. The problem is investigated from the standpoint of socio-philosophical analysis, which allows exploring the multivariant discursive space of the institute of education, its current state, characteristic features and projections of the social future. Discussion. From the standpoint of social significance, social processes and characteristics of the education system are characterized. Among them, the author identifies the most significant: the introduction of the principles of consumption culture; the primacy of simulacres and simulations in the system reform program; management in the implementation of personnel management of the educational organization; the establishment of the principles of macturing (efficiency, calculability, predictability); substituting qualitative and quantitative approaches; the breakdown of normative images of education; an irrational imaginery that determined the vector of dehumanization of the educational system. The destructive impact of these characteristics on the functioning of the institute, destroying the rational and humanitarian principles of the education system itself, is justified. Destruction of modern strategies in education based on analysis of virulence and contagiousness of social consequences of devoid of national specificity and state goal setting of educational policy are disclosed and analyzed. The conclusion fixed position on the significant updating of the research topic and the growing need for reflection and socio-philosophical analysis of the development trends of the Institute of Education from the standpoint of public inquiry and solving national problems.

Introduction. The problem of the traditional and innovative scientific and educational space of the university is actualized in the aspect of the uncertainty of innovative changes, the ambiguity of socio-educational priorities, the lack of a holistic methodology of philosophical understanding of certainty and uncertainty in ontological, epistemological, epistemological, axiological research programs, which affects the possibility of forecasting in education. Methodology. The methodological basis is the theories of pedagogical innovation by V. I. Zagvyazinsky, M. M. Potashnik, V. A. Slastenin, and the conceptual foundations of the philosophy of education (forecasting, modeling, design) by V. P. Bespalko and B. S. Gershunsky. To determine the traditional and innovative component of the evaluation system of the scientific and educational space of the university, materials are used that can show the state of the object (university), the results of statistical monitoring, monitoring of graduates’ employment, etc. Discussion. The directions of research in the issues of continuity of the organization and development of the scientific and educational space of the university are highlighted. A working definition of the concept of “innovative changes” is given, which demonstrates the combination of traditions and the implementation of innovative changes taking into account continuity, which will contribute to achieving a new quality of education. The interrelation and subordination of the categories “traditions” and “innovations” is established, which allows us to more fully characterize the processes of succession. It is determined that in the future, the coordinated development of the categories “traditions” and “innovations” in the field of educational practice can be fixed in the following directions: multilevel system modeling (including characteristics of federal, regional and other levels), a combination of internal and external assessments; development of criteria for educational achievements related to the use of innovative changes; evaluation of innovative changes in in the context of the organization of international research (participation in international rankings). Conclusion. The evaluation system of the scientific and educational space of the university includes an assessment of traditional and innovative changes in internal and external conditions and mechanisms, their speed of response to changes in the external environment, the digital transformation of the university, and the possibility of institutionalization of innovative changes.

Introduction. The presented article is the fourth and final part of the series of articles “On the way to the health of society” published in the “Philosophy of Education” journal. It reveals the following fundamental provisions of knowledge studies in education: its ontology, epistemology and axiology. These are the social ontology of educational processes; the main issue of education; epistemology of education; basic intelligence of students; polar constructive and destructive transformations of education; culture of education in the analysis of polar social processes. The methodology is represented by the concepts of cyclism, dialectics of social processes; system philosophy; comparativism; axiology of society and education. Discussion. 1. The attributive structure of education in society is shown. 2. The content of the main question of education, including the subsections: how to teach; what to teach; whom to teach? 3. Epistemology and knowledge studies in education, the main learning strategy, the main blocks of vital knowledge are analyzed. 4. The concept of basic intelligence in education is discussed. 5. The polar transformations in education are traced: along the path of normogenesis (normosa) with the formation of a comprehensively developed personality, along the path of pathogenesis (pathos), with the appearance of “human marriage” in education. 6. The culture of distinction is analyzed as an important aspect of the axiology of education in identifying polar transformations of social processes. Conclusion. A number of materials on the philosophy of education and knowledge studies related to the problems of education and training that determine the healthy existence of modern society, normos and normogenesis in the field of education are summarized.

Introduction. The analysis of the development of Internet technologies and their adaptation to the education system is an important direction of modern philosophy. The purpose of the research is to find answers to the questions: how does the student community use electronic technologies; who and in what types of educational practices do students trust more? And also on the basis of existing and conducted new research to identify changes and track the transformational process in the problem of trust. Methodology. A sociological survey combined with an analytical approach allowed not only to present the problem of the phenomenon of trust itself, but also to specify it in application to the interaction of three compositional elements of educational interactions: the pedagogical community, the student community and Internet technologies. Discussion. The social attitude is that with the development of digital means in educational interaction, the need for direct communication between students and a teacher is reduced if their digital literacy is coordinated (according to the level of development) in both subjects. It is established that during the sessions (exams, tests, defense and preparation of term papers and theses), students are more willing to resort to personal communication with the teacher. Conclusions. The results of the study indicated that: trust in the book as a source of information decreases, which determines new value orientations and requires the formation of appropriate skills, including psychological readiness for such a “turning point”; school preparation does not provide a sufficient level of preparedness, therefore mastering the function of a personifier of educational interactions falls a serious responsibility on a university teacher, which is reflected in the “critical points” of the educational process.

Introduction. The problem of humanitarian content human capital lies in the technocratization of education and its consideration through an economic prism. An illustrative example of it is the Bologna process, which main reason was not the cultural transmission of national values within European states, but the inability to meet economic segment needs. A comparative analysis of educational systems reforming in European countries clearly demonstrated that the main trends in educational policy are leading of market model sample, where instead of pedagogical tasks providing educational services is imposed. Due to this problem solution is in rethinking human capital and considering it through the pedagogical science, defining its humanitarian component as a key one. Methodology. As a methodological basis, a comparative analysis of educational programs contents in the countries of Western Europe, Asia and Russia is used. Discussion. The main characteristics of educational systems of countries-initiators Bologna process are presented. It is noted that the Bologna process (as an economic concept of human capital formation), assuming a two-level education, contributed to economizing resources with the possibility of expanding higher education, profession choice, educational mobility, but, could not prevent a radical reconstruction of existing educational institutions, almost complete loss of national education systems` identity and originality. Conclusion. The main problem of modern education system is the lack of definitions of its goals and means that correspond to modern realities, requiring awareness of who and what is the subject of educational process, and such things as moral, spiritual and psychological values, moral beliefs, internal culture personality and. etc. That is why there is an urgent need for a new look and new interpretation of human capital phenomenon (education) in pedagogical science context.

Introduction. The problems of protecting human rights and freedoms in the field of education and upbringing are at the intersection of philosophy, jurisprudence, and pedagogy. This is inherent in the idea of a social and legal state. The relevance of the topic is due to the need to rethink the idea of mercy in the system of legislative projects at the federal and regional levels of the Russian Federation in order to harmonize the relationship between the individual and the social in a person. The article is devoted to the actual processes taking place in the educational, educational and preventive activities of educational organizations in the context of a deep transformation of public relations. The purpose of the study is to identify the systemic factors underlying the mechanisms of modernization of the educational and educational processes of modern educational organizations. The novelty consists in the desire to highlight charity as an amplifying contour of the main activities of educational organizations, representation and practical application of the current laws of the Russian Federation. Methodology. The study was conducted as part of the collection and interpretation of empirical information on the formation of Russian educational policy based on the methodology of the axiology of education. The axiological methodology allowed the authors to investigate: the institutional approach of the place and role of educational organizations in the prevention of deviant behaviors of children and youth; the need to model the position and role of a teacher (teacher, mentor) as a professional and competent specialist capable of translating the soul’s abilities to goodness, justice and mercy; the regional component of the award policy preserved in the legislative acts of education, upbringing, prevention of offenses on the principles of mercy. Discussion. The purpose of an educational organization in its current state is rethought, the factors preventing the disclosure of its potential are analyzed and it is determined that the basis for adapting educational activities to socio-political realities is a developed spiritual, moral and legal culture, traditional values of Russian society. Based on the understanding of humanistic problems that inevitably accompany new trends in education, two reinforcing contours of mercy are discussed (the potential role of an educational institution in the prevention of deviant behaviors and juvenile delinquency; the role of the representation of ideas of mercy in the system of evaluating the activities of educational workers). Conclusion. The proposed recommendations can be used to increase the range of scientific research in the field of spiritual and moral values, the key of which is charity. The review and interpretation of federal and regional laws presented in the article allow us to rationally explain the phenomenon of mercy as an unconditional value of public and individual consciousness, which allows us to gain confidence and certainty in the conditions of the collapse of normative patterns of education and the uncertainty of being. Such a socio-philosophical understanding of the relationship between educational policy and individual-social consciousness is the basis for reforming and updating theoretical and applied research in the field of philosophy of education.