The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of the community, as (...) well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games. (shrink)

The article is devoted to the comparison of two types of proofs in mathematical practice, the methodological differences of which go back to the difference in the understanding of the nature of mathematics by Descartes and Leibniz. In modern philosophy of mathematics, we talk about conceptual and formal proofs in connection with the so-called Hilbert Thesis, according to which every proof can be transformed into a logical conclusion in a suitable formal system. The analysis of the arguments of the proponents (...) and opponents of the Thesis, “conceptualists” and “formalists”, is presented respectively by the two main antagonists – Y. Rav and J. Azzouni. The focus is on the possibility of reproducing the proof of “interesting” mathematical theorems in the form of a strict logical conclusion, in principle feasible by a mechanical procedure. The argument of conceptualists is based on pointing out the importance of other aspects of the proof besides the logical conclusion, namely, in introducing new concepts, methods, and establishing connections between different sections of meaningful mathematics, which is often illustrated by the case of proving Fermat’s Last Theorem (Y. Rav). Formalists say that a conceptual proof “points” to the formal logical structure of the proof (J. Azzouni). The article shows that the disagreement is based on the assumption of asymmetry of mutual translation of syntactic and semantic structures of the language, as a result of which the formal proof loses important semantic factors of proof. In favor of a formal proof, the program of univalent foundations of mathematics In. Vojevodski, according to which the future of mathematical proofs is associated with the availability of computer verification programs. In favor of conceptual proofs, it is stated (A. Pelc) that the number of steps in the supposed formal logical conclusion when proving an “interesting” theorem exceeds the cognitive abilities of a person. The latter circumstance leads the controversy beyond the actual topic of mathematical proof into the epistemological sphere of discussions of “mentalists” and “mechanists” on the question of the supposed superiority of human intelligence over the machine, initiated by R. Penrose in his interpretation of the Second Theorem of Goedel, among whose supporters, as it turned out, was Goedel himself. (shrink)

The article discusses the V.V. Tselishchev’s original and unique systematic study of the specific and extremely complicated problems of Gödel results regarding the question of artificial intelligence essence. Tselishchev argues that the reflexive property should be considered not only as an advantage of human reasoning, but also as an objective internal limitation that appears in case of adding Gödel sentence to a theory to build a new theory. The article analyzes so-called mentalistic Gödel’s argument for fundamental superiority of (...) human intelligence over machine one and the non-algorithmic nature of natural thinking. The discussion about the Gödel argument is not entirely speculative, but contains new knowledge. An example of such knowledge are the results of R. Smullyan levels of computers “awareness,” which are may be interpreted in a psychophysical sense. The concept of “zero level of intelligence” is proposed for such a reflexive property as “awareness of selfconsciousness.” Reflexive ranks below the awareness of self-consciousness can be considered negative levels of thinking in the sense that the intelligence, being reduced to them, significantly loses its completeness. Even self-consciousness turns out to be a negative level of thinking, since, according to Smullyan, the subject of self-consciousness is unaware of the type of thought to which he belongs. A thought experiment is proposed that allows us to establish the distribution of the properties of Smullyan stability and normality and to answer the question “Does an intuitive belief in the truth of a formal proof affect the truth of a proposition being proved?” According to intuitionism, the most unpleasant epistemic property is instability: beliefs that are not based on deep intuitions have no value. According to the constructivist philosophy of mathematics, instability is a less negative property than abnormality: the fact that high-ranking beliefs cannot be immersed to the very foundations is not significant because violation of truth due to lowering the rank of reflection is not critical. (shrink)

Russkai︠a︡ t︠erkovʹ i drugie statʹi -- V temnykh religioznykh luchakh.

В книге представлено понимание границ в психотерапии с позиций аналитической психологии. Для практикующих психологов и психотерапевтов, а также для студентов психологических вузов.

Исследование иллюстрируется примерами анализа самого современного искусства в сферах живописи, театра, кино, литературы.

The attention of scientists is now being drawn to a new branch of knowledge known as the “philosophy of science.” It is true, however, that philosophers of this country are not very happy about this word combination and often identify it with logical, positivism. Indeed, it would seem better to speak not of the philosophy, but of the logic of scientific development. Science has become an object of study, and there has emerged metascience, i.e., a science studying the logic of (...) scientific structures. This field of knowledge cannot so far boast of generally accepted results. But it has done something else: new acute questions have been formulated and discussed fruitfully and originally. With this paper we are making an attempt to enter the discussion. The philosophy of science has not certainly taught scientists to discover the truth but it has indubitably increased their criticism towards their own activities, and this is really a very important accomplishment. (shrink)

В сборнике представлены статьи, посвященные проблеме построения социальной онтологии общества различия. Для студентов, аспирантов, преподавателей философии и всех интересующихся проблемами современной философии.

Анализируется круг проблем, к которым было приковано преимущественное внимание русских философов указанного периода. Это - идущее от А. С. Хомякова философское осмысление Божественного Триединства.

Для студентов, аспирантов, преподавателей и исследователей, специализирующихся в области философии, социологии, политологии, психологии и педагогики.

This volume brings together the primary challenges for 21st century cognitive sciences and cultural neuroscience in responding to the nature of human identity, self, and evolution of life itself. Through chapters devoted to intricate but focused models, empirical findings, theories, and experiential data, the contributors reflect upon the most exciting possibilities, and debate upon the fundamental aspects of consciousness and self in the context of cultural, philosophical, and multidisciplinary divergences and convergences. Such an understanding and the ensuing insights lie in (...) the cusp of philosophy, neurosciences, psychiatry, and medical humanities. In this volume, the editors and contributors explore the foundations of human thinking and being and discuss both evolutionary/cultural embeddedness, and the self-orientation, of consciousness, keeping in mind questions that bring in the interdisciplinary complexity of issues such as the emergence of consciousness, relation between healing and agency, models of altered self, how cognition impacts the social self, experiential primacy as the hallmark of consciousness, and alternate epistemologies to understand these interdisciplinary puzzles. (shrink)