Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)

A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...) by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG , it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter). (shrink)

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.Theorem 2.Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.This result has been proved in [1] with the additional assumption thatGis connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in (...) the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.The second author has shown in [11] that a nilpotentω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus. (shrink)

The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement:Fact1.1 (Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G.The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. (...) The original statement stipulated one ofNorG/Nto be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces eitherNorG/Nto be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem).The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a groupGof finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a groupHis called aπ-group, where π is a set of prime numbers, if elements ofHhave finite orders whose prime divisors are from π. Maximal π-subgroups of a groupGare called π-Hall subgroups. They exist by Zorn's lemma. Since a normal π-subgroup ofGis in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.)The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider thep-Sylow for any primep, or even the torsion part of ℂ*. LetHbe this subgroup.Hhas a complement, but this complement is found by Zorn's Lemma (consider a maximal subgroup that intersectsHtrivially) and the use of Zorn's Lemma is essential. In fact, by Zorn's Lemma, any subgroup that has a trivial intersection withHcan be extended to a complement ofH. Since ℂ* is abelian, these complements cannot be conjugated to each other. (shrink)

In the course of his collaboration with GAKhN, whose task was to create a systemic 'scientific' theory of art, Losev undertook a systematic interpretation of German classical aesthetics as the historical presupposition for his own Christian, Platonist doctrine of art conceived as a dialectical universe comprising totalizing connections at all levels. This interpretation was concealed in a masterful way within the ' Commentaries' to Dialektika khudozestvennoj formy. Independently of the significant results achieved by this revival of the classical tradition, Losev (...) 's mythologized theory of art called forth a critical reaction on the part of his GAKhN colleagues and brought to light some of the broader theoretical attitudes present among GAKhN 's collaborators. (shrink)

The author shows how mathematics in ancient and early medieval Europe was constrained by deeply rooted metaphysical conceptions and how these constraints have been overcome since the late medieval period. As examples, he focuses on changing conceptions of chance, motion, and infinity.

Recently, we have faced the outbreak of the coronavirus disease 2019 in the world, which has attracted the attention of all people. Stress has become a word familiar to all people. The stressors of life are relatively clear and some of them cannot be eliminated by humans. One of the stressors in the life of humans is the COVID-19 pandemic. Doctors believe that the virus is controllable but its prevalence is quicker and deadlier than other viruses. In addition, the virus (...) puts the elderly and those with underlying diseases at extreme risk. Therefore, more care is seriously required to protect members of society. However, this psychological imbalance caused by the virus is associated with stress and anxiety. Lack of proper management of this stress will be associated with emotional impacts, physiological impacts, cognitive impacts and behavioural impact.Contribution: Given the importance of this topic, this study aimed to evaluate the relationship between religious beliefs and coping with COVID-19-related stress amongst Muslim students in three cities of Russia – Ingushetia, Chechnya and the Republic of Dagestan – in 2020. Data were collected using questionnaires, the validity and reliability of which were previously confirmed. According to the results, religious beliefs had an impact on stress coping by p = 0.657. In other words, students’ ability to cope with stress improved by 0.657 units, with each unit of enhancement in their religious beliefs. (shrink)

We classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3.

One of the main trends of modern society is a marked increase of the theoretical interest to the sphere of mythology and the theory of myth. And in light of globalization approaches explicitly focus on the impact study of the regional aspects of social phenomena. In this regard, in regional studies of mythology appear attempts seen the myth city's as a socially formalized knowledge about the city or region, a strong regulatory and mobilization element in society, which is capable of (...) forming a regional entity and identity. A problem is in that every city has the own myths and specific municipal culture. It can be one of reasons of conflicts at co-operation of townspeople of different regions because all of us are inclined to identify itself with a sub-group on the sign of place of inhabitation, line of business or the world view. Thus, at all cities have different myths and symbols. We are inclined to see our city as city of the chosen townspeople, while other cities and regions are inhabited by demons and monsters. The article suggests to consider the city like a self-valuable phenomenon. The article submitted to scientific use of the term "the myth city's" which, the author's perspective is not identical to the concept of "myths of the city" and "Mythology of the city." also authors conducted analysis the history of the formation of the Donetsk's myth, identified the main stages of its formation, and conducted explication of its meaningful and functional characteristics. (shrink)

У статті оцінюється значення онтологічної та метафізичної проблематики у сучасному філософському дискурсі, критично аналізується концепція монографії Андрія Баумейстера «Буття і благо» та пропонується інтерпретація буття як ствердження реальності в соціально детермінованій дії.

У статті досліджується ситуація екзистенційної загрози, пов'язана з руйнуванням со­ціальної реальності. Зроблено припущення, що саме загроза існуванню тієї структурної частини реальності, яка походить від релігійно-міфологічних форм соціального буття, сприй-мається людиною як негайний привід до самозахисту. Навмисні зовнішні впливи на людську свідомість призводять до формування ерзац-реальностей, у яких "боротьба за цінності" є лише раціоналізованою формою дораціонального намагання зберегти ідентичність. Але саме пов'язаний з цим самозахистом стан розуму й дозволяє розпалювати конфлікти, подібні до східноукраїнського.

I. "Filosofskiĭ paroxod": k istorii voprosa -- II. Respressii i permanentnai︠a︡ ėmigrat︠s︡ii︠a︡ -- III. Overkili russkoĭ filosofii -- IV. Postsovetskiĭ period: v poiskakh identichnosti -- V. Vozvrashchenie "filosofskogo paroxoda" (2003) -- VI. "Filosofskie avtobusy" -- VII. Samolet-paroxod-"filosofskiĭ poezd" -- VIII. Afinskiĭ kongress: prikosnovenie k istorii (2013) -- IX. "Uchitʹsi︠a︡ bytʹ chelovekom": XXIV vsemirnyĭ filosofskiĭ kongress -- X. Mezhdu proshlym i budushchim.

Frankfurt-type cases with covered manipulation received a great attention in the debates about freedom of will and moral responsibility. They pretend to give the refutation of the Principle of Alternative Possibilities and to show that we can intuitively blame or praise an agent who was not able to do otherwise. In this paper, I will try to make explicit some basic intuitions underlying the agent’s responsibility in Frankfurt-type cases, which were surprisingly ignored in the contemporary debates. The key intuition is (...) that the responsibility of the agent in Frankfurt-type cases is always grounded at the point of overcoming the uncertainty preceding action. This overcoming is crucially important for agent’s responsibility and immune to any manipulation of counterfactual intervener. (shrink)

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

This article discusses the foundation of ideas for Vyacheslav Ivanov’s interpretation of Pushkin’s poem. In The Gypsies, Ivanov sees a conflict between personal freedom and sobornost’ as revealed b...

About 130 languages are currently spoken in the USSR. These languages differ considerably in their numbers of speakers, social status, scope and viability. Our primary interest in this paper will be with those languages that are in extreme danger of extinction in the near future.

У статті проаналізовано вчення П'єра Абеляра в частині його впливу на виокремлення світської сфери знання з тотально релігійного середньовічного дискурсу. Схоластична теологія, творцем якої є Абеляр, розглядається автором статті як специфічний результат філософського мислення в граничних екзистенційних обставинах та маркер секулярних зрушень у середньовічній християнській культурі.

The article analyzes the problem of the correlation of morality and ideology. The author believes that the distinction between morality and ideology can be made only at the level of their genealogy: if ideology is a socio-cultural product, then morality is rooted in human nature, in "moral intuitions", which medieval scholastic philosophy called synderesis. In modern ethics, synderesis can be identified with the neurophysiological prerequisites of morality. Opponents of this approach argue that, firstly, it excludes free will, and secondly, it (...) does not explain the content of moral values and norms in any way. The author of the article criticizes these statements and offers his original theory of synderesis. The article draws an analogy between ethics and intuitionistic mathematics, which considers mathematical objects as the results of intellectual construction based on initial intuitions. Ethical objects should also be considered as constructions based on innate intuitions. Moral intuitions do not exclude free will, since they rely on rationality inherent in our nature, the main property of which is the ability to make decisions regardless of external stimuli and internal states. In addition, these intuitions influence the content of our norms, as they are the "operators" of their construction and are included in the "final product". (shrink)

Boston Studies in the Philosophy of Science are devoted to symposia, con gresses, colloquia, monographs and collected papers on the philosophical foundations of the sciences. It is now our pleasure to include A. A. Zi nov'ev's treatise on complex logic among these volumes. Zinov'ev is one of the most creative of modern Soviet logicians, and at the same time an innovative worker on the methodological foundations of science. More over, Zinov'ev, although still a developing scholar, has exerted a sub stantial (...) and stimulating influence upon his colleagues and students in Moscow and within other philosophical and logical circles of the Soviet Union. Hence it may be helpful, in bringing this present work to an English-reading audience, to review briefly some contemporary Soviet investigations into scientific methodology. During the 1950's, a vigorous new research program in logic was under taken, and the initial published work -characteristic of most Soviet pub lications in the logic and methodology of the sciences - was a collection of essays, Logical Investigations (Moscow, 1959). Among the authors, in addition to Zinov'ev himself, were the philosophers A. Kol'man and P. V. Tavanec, and the mathematicians and linguists, S. A. Janovskaja, A. S. Esenin-Vol'pin, S. K. Saumjan, G. N. Povarov. (shrink)

In the course of his collaboration with GAKhN, whose task was to create a systemic ‘scientific’ theory of art, Losev undertook a systematic interpretation of German classical aesthetics as the historical presupposition for his own Christian, Platonist doctrine of art conceived as a dialectical universe comprising totalizing connections at all levels. This interpretation was concealed in a masterful way within the ‘Commentaries’ to Dialektika khudožestvennoj formy. Independently of the significant results achieved by this revival of the classical tradition, Losev’s mythologized (...) theory of art called forth a critical reaction on the part of his GAKhN colleagues and brought to light some of the broader theoretical attitudes present among GAKhN’s collaborators. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.