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In this article, we are interested in the complex version of Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation for grayscale image inpainting as well as the multi-component Cahn-Hilliard systems for image inpainting, that is an extension approach for color image inpainting. We have studied well-posedness of the steady state problem associated to the complex Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation as well as to Bertozzi-Esedoglu-Gillete-Cahn Hilliard systems. Then, Backward Euler discretization on time has been considered in both models mentioned above. We were able to prove the stability of the backward Euler scheme. Finally, we do some numerical simulations that confirm the theoretical results and show the efficiency of the scheme. The simulations were done using FreeFem++.

A method is proposed for restoring two coefficients in a mixed boundary value problem for an inhomogeneous hyperbolic partial differential equation of the second order based on additional information about the solution of the boundary value problem for a given fixed value of the spatial argument of the solution. The problem simulates the propagation of small transverse vibrations of a finite string with an end affected by the gravity of a body with varying mass. The proposed algorithm provides a sequential restoration of the multiplier in the heterogeneity of the oscillation equation and the coefficient in the nonclassical boundary condition based on the values of one additional function of one argument.

The article considers the possibility of applying Karl Popper’s principle of falsification to mathematics. A common position is that the principle of falsification can only be applied to the empirical sciences, since statements not about the surrounding world are inherently unfalsifiable. The purpose of this article is to demonstrate that applying the principle of falsification to mathematics can be pragmatically interesting and feasible. The first part of the article discusses the principle of falsification itself and why the question of the falsifiability of mathematics is basically worthy of study. The second part presents an argument that the main cause for the apparent unfalsifiability and the lack of scientific status of mathematics is the implicit acceptance of traditional mathematical Platonism by researchers. The third part examines the possibility of employing abstrаction principles to transform traditional Platonism into a version of naturalism. By adopting the metaphysics and epistemology of such position, mathematics can be considered falsifiable on par with other sciences.

Throughout the human history, innumerable religious, philosophical, and scientific models of our world have been developed. But, to date, no model of the Universe has been proposed that would be supported by all the confessions of modern society. However, recently, the advent of such a tool as the computer has enabled the rapid development of artificial intelligence. In this paper, the new model of the Universe is based on advances in computer technology, information processing, and AI development, as well as new astrophysical data, and also previously established philosophical and theological approaches were involved.

The article considers a technotropic approach to the development of artificial intelligence (AI) that is focused on creating systems capable of conscious activity. In the context of the dominance of traditional methods and optimization research, the author proposes a transdisciplinary paradigm comprising four levels: philosophical (an axiomatic system), methodological (integration of principles into the phenomenon of jimination generalizing and expanding the recursion mechanism), formal (conceptualization and formalization of jimination with an example of application to Gоdel’s theorem), and applied (comparison of jimination with some modern AI technologies). The concept of technotropy is analogous to the concept of anthropy and emphasizes the possibility of realizing the phenomenon of self-organization by technology with the correct approach. This work lays the theoretical and practical foundation for the development of a new generation of AI that goes beyond existing models and offers a methodological basis for further research into conscious systems in artificial intelligence.

The article analyzes the distribution of responsibility between humans and artificial intelligence as the autonomy of digital systems increases. The purpose of the study is to determine whether an algorithm can act as a carrier of moral responsibility and how human subjectivity changes when decision-making functions are delegated to artificial intelligence. The study uses an interdisciplinary approach based on the analysis of philosophical concepts of responsibility (deontology, utilitarianism, existentialism), as well as modern trends in the philosophy of technology and digital ethics. The authors show that, despite the functional autonomy and ability of algorithms to make decisions without human intervention, artificial intelligence lacks consciousness, intention, and the ability to realize the consequences of its actions, which means it cannot act as a moral agent. Responsibility inevitably remains with the human developer, user, or owner of the system. The study’s peculiarity is the development of a philosophical model of the distribution of responsibility among participants in interactions with artificial intelligence which demonstrates that the primary criterion is not the complexity of the algorithm, but the preservation of human control. The importance of the work is in substantiating the need for digital ethics and the preservation of human subjectivity in the context of the growing autonomy of artificial intelligence.

The article analyzes the concept of basal (basic) cognition, which, through the study of organisms functioning on a non-neural basis, concludes that they have rather complex information processing mechanisms necessary to learn about and assess the features of their internal state and the environment and interact with them productively in order to find ways to meet existential needs, the main of which are survival/sustainability, growth/prosperity and reproduction. Researchers call the mechanisms discovered at this level of biological organization basic, since they persist at higher levels of development. Basal cognition itself is considered as a method for studying cognitive systems on a non-neural basis, which can also be applied to the study of artificial intelligence. All this, according to the developers of the concept, makes it possible to build a synthetic theory of cognition that covers all levels of cognitive complexity. However, the authors of the article express doubts about the possibility of building such a theory, given that the analyzed concept describes cognitive processes in the language of physical-chemical-electrical processes, but this is not enough to reveal cognition at a higher level of cognitive complexity. The concept of basal cognition also fails to answer a number of questions: what is the demarcation criterion for drawing a line between the cognitive and non-cognitive, how does intentionality emerge, etc. At the same time, the concept has promising prospects not only for studying the development of the biological foundations of cognition but also for solving problems of artificial intelligence.

Science has yet to build a synthetic concept from working and additional blocks, which should result in a product that combines a variety of approaches, teachings, etc., but the main thing is that it more fully explains the structure of physical systems of organic and inorganic nature and can overcome the revealed dead ends of modern scientific ideas. The paper proposes a variant of such a synthetic concept, based on the development of an information-logic approach and allowing for a shorter research path compared to the thermodynamic approach. Within the former, information is understood as the leading component in an abstrаct model of the physical system, within the latter, matter and energy play a primary role. As a result of the research, the author concludes that the arsenal of science can include a first version of the origin of living nature (and the properties of subjectivity in it) relying on a reductionist way of explanation and, accordingly, an alleged “third” principle connecting physics and biology with a “seamless bond”.

The paper examines the opposition between Newton’s corpuscular theory of light and Huygens’ wave theory that arose in the mid-seventeenth century. From the height of modern knowledge, based on the concept of a light beam as a stream of photon corpuscles and using well-known laws of mechanics, basic phenomena are explained and important optical formulas are derived. This speaks in favor of the corpuscular theory as a physical model of light. The wave theory plays an additional role - it is an averaged mathematical tool of optics that serves as an abstrаct (conditional) model.

The article examines the phenomenon of citizen science and its role in transforming the status of scientific knowledge in modern society. The author notes that science has evolved from a disinterested search for truth to a socially significant institution engaging non-professionals. By analyzing T. Kuhn’s concept of normal science, and also concepts of post-normal science and Mode 2.0 science, the study demonstrates that citizen science can manifest in several forms. It often serves as an auxiliary tool for professional scientists, yet civil researchers may also exhibit considerable autonomy within various activist projects. The article also notes that denialism can be considered a form of citizen science skeptical of the scientific community. All this indicates that, in an era of interdisciplinarity and social engagement, science is becoming a boundary object requiring new forms of dialogue between scientists and society.