We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.

In this paper I reconstruct Schiller's theory of virtue from his essay “On grace and dignity” and discuss how it fits into Kant's moral philosophy. According to Schiller, a virtuous person – a beautiful soul – is characterised by harmony between reason and sensibility. I show that despite some misleading textual clues and interpretations based on them, Schiller does not regard the harmony between reason and sensibility as an equal partnership. On the contrary, just like Kant, Schiller fully recognises the (...) priority of reason. The difference between their positions is more subtle. Whereas Kant holds that sensibility must be overtly subordinated to reason, Schiller believes that there should be a semblance of an equal partnership between reason and sensibility, provided that the latter is sufficiently ennobled. (shrink)

This paper is devoted to Leonardo da Vinci’s ‘The Arno Valley Landscape‘. The historical aspect of the subject under study has been considered, as well as the methodological issues in learning the professional drawing techniques in the framework of Leonardo’s ten attributes of sight and the Thinking through Drawing methodology. Our approach allows formulating the figurative and artistic component of the masterpiece and find out Leonardo’s hidden messages, full of mystics and symbolism.

We study punctual categoricity on a cone and intrinsically punctual functions and obtain complete structural characterizations in terms of model-theoretic notions. As a corollary, we answer a question of Bazhenov, Downey, Kalimullin, and Melnikov by showing that relational structures are not punctually universal. We will also apply this characterisation to derive an algebraic characterisation of relatively punctually categorical mono-unary structures.

We show that for every computable ordinal of the form β=δ+2n+1>1, where δ is zero or a limit ordinal and n∈ω, there exists a torsion-free abelian group having an X-computable copy if and only if X is nonlowβ.

We prove that for any computable successor ordinal of the form α = δ + 2k there exists computable torsion-free abelian group that is relatively Δα0 -categorical and not Δα−10 -categorical. Equivalently, for any such α there exists a computable TFAG whose initial segments are uniformly described by Σαc infinitary computable formulae up to automorphism, and there is no syntactically simpler family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples (...) of Δα0-categorical TFAGs for arbitrarily large α was first raised by Goncharov at least 10 years ago, but it has resisted solution 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a 0″-computable worthy labeled tree into a computable TFAG. We expect that this techni... (shrink)

We review publications in automated scientific discovery using deep learning, with the aim of shedding light on problems with strong connections to philosophy of science, of physics in particular. We show that core issues of philosophy of science, related, notably, to the nature of scientific theories; the nature of unification; and of causation loom large in scientific deep learning. Therefore, advances in deep learning could, and ideally should, have impact on philosophy of science, and vice versa. We suggest lines of (...) further research, and highlight the role ‘theory-based’ AI could have in future developments of the field. (shrink)

We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{}$-computable Polish space; this answers a question of Nies. (...) We also prove analogous results for compact Polish groups and locally path-connected spaces. (shrink)

We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are (...) not necessarily direct or expected. (shrink)

The following text is a translation of Semyon Frank’s “L’intuition fondamentale de Bergson” published in Henri Bergson: Essais et témoignages inédits, edited by Albert Béguin and Pierre Thévenaz, Neuchâtel: Éditions de la Baconnière, 1941. In this article, Frank addresses Bergson’s notion of intuition, his anti-intellectualism, his mysticism, his closeness to Lebensphilosophie, the notion of lived experience, the distinction between intuition as pure contemplation and intuition as living knowledge, the distinction between cognition of the atemporal essence of reality and cognition (...) of the world of becoming, pragmatism, the disinterested and open intuitive spiritual attitude vs the utilitarian attitude of social groups closed in on themselves, the transrational vs the irrational, spiritual life, psychic life, the Absolute, and the temporal flux. The article contains criticisms of Bergson on the issues of time and intuition: the durée, the intuitive time, is incomprehensible without an atemporal foundation. Therefore, according to Frank, Plato was correct—contra Bergson—to define time as the moving image of eternity. And, if there is such an atemporal foundation, then, contrary to what Bergson seems to think, intuition as living knowledge cannot be the sole mode of intuition. Moreover, unlike what Bergson appears to think, an intuition of the Absolute would be an intuition of the transrational rather than of the irrational. The translation is preceded by an introduction tracing the genesis of the article, which was commissioned by the Swiss philosopher Pierre Thévenaz. (shrink)

We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is$0''$. We also prove that, with a bit of work, the latter result can be pushed beyond$\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex (...) automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability. (shrink)

We solve a problem posed by Goncharov and Knight 639–681, 757]). More specifically, we produce an effective Friedberg enumeration of computable equivalence structures, up to isomorphism. We also prove that there exists an effective Friedberg enumeration of all isomorphism types of infinite computable equivalence structures.

Review by: Alexander G. Melnikov The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 400-401, September 2013.

The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma (...) }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel. (shrink)

The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelian p-groups, the class of Abelian torsion groups, and the special class of "rank-homogeneous" trees. We consider these conditions as a possible definition of what it means for a class of structures to have "Ulm type". The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply (...) this result to show that there is no Turing computable embedding of the class of graphs into the class of "rank-homogeneous" trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a "rank-preserving" Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank ${\mathrm{\omega }}_{1}^{\mathrm{C}\mathrm{K}}$. (shrink)

We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.

We show, for each computable ordinal α and degree $\alpha > {0^{\left( \alpha \right)}}$, the existence of a torsion-free abelian group with proper α th jump degree α.

We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.

Russian philosophy is “a sphere of conversation” in which thought is “divined”. It is a realm of search for “universal meaning” and “cultivation” of historical reality. Such a “conversation” around the work of Dostoyevsky took place in the 1920s among philosophers (including members of the Free Philosophical Association or Volfila in its abbreviated form). The theme takes on added significance at the hands of Ya. E. Golosovker and S. L. Frank whose intellectual affinity manifests itself today in the way they (...) interpret Kantian motives in the work of Dostoyevsky. Reflecting on “the logic of imagination” Golosovker draws attention to the fact that the concrete-metaphysical method of seeing the world is not identical to the amphiboly of Kant’s reflexive concepts. He made it the central theme of his book, Dostoyevsky and Kant, immersing the philosophical component of the novel The Brothers Karamazov in the context of Kant’s antinomies. The later Frank took up the subject of Dostoyevsky’s worldview in response, as it were, to the discussion of the problem of cultural crisis at Volfila (including at meetings “In Memory of Dostoyevsky”). Following Yakov Golosovker, Andrey Bely, Aron Steinberg, Semyon Lurie and others, he continued the discussion of the crisis in its organised forms. In this context, too, Kant is a significant presence. For Frank, as for Golosovker, Dostoyevsky’s world view has a “concrete-metaphysical character”. They come to this conclusion reflecting on the problem of freedom as seen by Dostoyevsky who sought to overcome Kant’s antinomies. (shrink)

This article addresses the famous 1910 debate between Vladimir F. Ern and Semyon L. Frank centered around the problem of identifying the distinguishing features of Russian philosophy. The debate was a continuation of Ern’s debates with Russian philosophers associated with the international journal Logos (Sergei I. Hessen, Fyodor A. Stepun, Boris V. Yakovenko, and others). The author shows that Ern’s understanding of an original Russian philosophy is organically related to his overall philosophical doctrine. As for Frank, his views during (...) the 1910s on the distinguishing features of Russian philosophy were different in nature. It is worth noting that in the 1920s articles written during his émigré period, Frank’s views on the distinguishing features of Russian philosophy undergo a decisive change. Frank’s new ideas about Russian philosophy turn out to be extremely close to those of his late opponent, both on the whole and in the details. At the same time, Frank’s overall philosophical views do not undergo as dramatic a change and develop in a fully evolutionary mode. The author attempts to identify the reason for these changes. (shrink)

This essay introduces Semyon Lyudvigovich Frank as a philosopher who deservedly may be called a revolutionary thinker: he introduced a remarkable social ontology that foregrounds service. His oeuvre presents service as the supreme principle of personal and hence social life. The singular personality is seen as being there to creatively serve itself: his view of man focuses on the human soul as being there to bring forth creative action—to serve those who will come after, the community, society, and the (...) Christian Churches. Service, then, is the source for freedom as a derivative principle. Consequently, and in opposition to the fundamental idea of the “Charter of Human Rights,” freedom in Frank has no absolute value, but only a functional one. It is justified by the ontological principle of service. All governmental organization is, ideally, the organization of freedom, the planned, systematic formation of free, spontaneous cooperation. Spontaneous cooperation makes up part of his concept of sobornost’, the empirical substrate of social culture. Frank would have agreed with Karl Popper’s notion of the “open society,” yet he would have certainly added that accessibility and transparency, be they spiritual or social, emanate from the principle of the universality of service. The true ontological meaning and the true source of democracy is, in his eyes, not the rule of all, but the service of all. (shrink)

This essay introduces Semyon Lyudvigovich Frank as a philosopher who deservedly may be called a revolutionary thinker: he introduced a remarkable social ontology that foregrounds service. His oeuvre presents service as the supreme principle of personal and hence social life. The singular personality is seen as being there to creatively serve itself: his view of man focuses on the human soul as being there to bring forth creative action—to serve those who will come after, the community, society, and the (...) Christian Churches. Service, then, is the source for freedom as a derivative principle. Consequently, and in opposition to the fundamental idea of the “Charter of Human Rights,” freedom in Frank has no absolute value, but only a functional one. It is justified by the ontological principle of service. All governmental organization is, ideally, the organization of freedom, the planned, systematic formation of free, spontaneous cooperation. Spontaneous cooperation makes up part of his concept of sobornost’, the empirical substrate of social culture. Frank would have agreed with Karl Popper’s notion of the “open society,” yet he would have certainly added that accessibility and transparency, be they spiritual or social, emanate from the principle of the universality of service. The true ontological meaning and the true source of democracy is, in his eyes, not the rule of all, but the service of all. (shrink)

The widespread assessment of the early period of Semyon L. Frank’s work as being influenced by German Neo-Kantianism is in need of a critical scrutiny. There are several reasons why the Russian philosopher’s interest in Neo-Kantianism merits a closer look. First, two systemic theories belonging to different trends exerted a decisive influence on Russian philosophy in the late nineteenth and early twentieth centuries: German Neo-Kantianism and Vladimir Solovyov’s school of all-unity. Second, Frank himself and the German Neo-Kantians considered Nicholas (...) of Cusa to be one of their forerunners and pointed out the importance of his mathematical ideas for their mathematical studies. Thirdly and lastly, Frank paid particular attention, especially in his formative period as a philosopher, to the leading trend in the early twentieth century, namely German Neo-Kantianism, which led some students of his work to believe that German Neo-Kantianism played the decisive role in Frank’s abandonment of Marxism in favour of religious ontology. Frank’s fundamental disagreement with German Neo-Kantianism was expressed in his work The Object of Knowledge in which he criticised the Neo-Kantian concepts of number and time. Although Frank rightly points out the one-sidedness of the Neo-Kantian definition of number, most notably in the works of the Marburg philosopher Paul Natorp, on the whole his criticism of the Neo-Kantian concept of number and time as being different from his own is not entirely convincing. In my opinion, Frank’s attempt to explain the abstract concept of number through a still more abstract concept of all-unity was not crowned with success because he ignored the experience of Christian theology and the Trinity dogma as well as the profound thoughts of the Rev. Pavel Florensky on this topic. (shrink)

Semyon Frank opposed the Neo-Kantian School and admitted the real existence of the objects of cognition. He treated ontologism as essential to the entire movement of Russian religious philosophy. For Frank, one can only know about something thanks to the absolute, which exists prior to the knowing subject. Ontologism, affirming the priority of being over cognition, has a great significance not only for metaphysics and epistemology, but also for the philosophy of religion. In particular, Frank taught that the most (...) privileged mode of cognition of God is intuition, an immediate experience of God or faith. Intuition is at the heart of the ontological proof, which can be found in St. Anselm, Descartes, and Frank himself. Frank dedicated a number of articles to this topic: “K istorii ontologicheskogo dokazatel’stva”, “Ontologicheskoe dokazatel’stvo bytiya Boga”, as well as texts recently discovered at the Bakhmeteff Archive, including “Dokazatel’stvo bytiya Boga”. In this way, Russian ontologism leads to a new interpretation of the traditional ontological proof, one which acknowledges the existence of God, not on the basis of arguments resting on His definition, but on the basis of the intuitive recognition of His being. (shrink)

This article is devoted to Russian religious thinker Semyon L. Frank’s philosophical interpretation of Alexander S. Pushkin’s work. The article identifies the place and significance of the Pushkin...

The paper focuses on the mutual interaction as well as the impact of the Scottish Enlightenment on the formation of the Enlightenment in Russia during the reign of Catherine the Great. It focuses on the relationship between the work of Adam Smith and Semyon Efimovich Desnitskii, who, thanks to Desnitskii’s studies at the University of Glasgow, got to know each other as teacher and student. The central point of their interaction is the issues of the philosophy of history based (...) on natural-law assumptions and focus on understanding the formation of history, culture, citizenship, and social transformations through morality and law. The paper examines Desnitskii’s ideas on the stages of human coexistence in which Smith’s concept is projected. The paper is not only an attempt to compare these concepts, but it also characterizes Desnitskii’s interest in adapting the ideas of the Scottish Enlightenment to Russian reality and needs and using them in the environment of the advancing feudal empire, to which Desnitskii wants to contribute with his work. (shrink)

The article analyzes the problems of unknowability and total unity in the light of philosophy of Semyon L. Frank, set forth by him in the work The Unknowable. The author of the paper considers all the problems that arise as “icebergs” and tries to find the reasons for the distortion of the vision of the “top of the iceberg” and the “underwater part of the iceberg,” which is often unaware. The author examines the problems of the inadequate perception of (...) reality: a narrow “one’s own little world” of the world perception, passed off as the truth in the final instance; absolutization of the mind, which considers itself able to know everything; the cultivation of negative information and cruelty in society; the role of media forming world perception; antagonistic dichotomy of the world perception; lack of a sense of connection of everything that exists, in particular of a sense of unity of “I” and “Thou;” loss of the “culture of heart” and the ability to love. As a result of the analysis undertaken, the author concludes that only by changing the causes that give rise to the “world of evil and separation” can the lost integrity and harmony of man, society and world civilization be restored. The revival of the earth’s civilization is possible only in total unity. (shrink)

The subject of the article is the problem of the crisis of ideals generated by the European Enlightenment and the Modern era, as interpreted by Russian thinkers of the first third of the 20th century. Building on the conception of Semyon Frank, the author analyzes the relationship between the socio-political ideals of the Enlightenment in the Russian version and false “idols” that led Russia and Europe to an unprecedented escalation of evil in the early 20th century, is analyzed. Particular (...) attention is paid to the assumption that the ideals of democracy as a true power of the people, as a result of uncritical imple­mentation, lead to revolutionary violence and further reproduction of tyranny. Two ver­sions of democracy are considered: according to Frank, one of them is genuine, based on the idea of disinterested service to the truth, and the second is “false”, oriented to­wards a totalitarian utopia, fundamentally hostile to freedom. The author also analyzes Frank’s views on the phenomenon of the “idol of culture”, which is based on the idea of linear progress. According to Frank, linear progress cannot prevent Russia and Europe from falling into barbarism. Moreover, it even significantly contributed to that break­down. It is concluded that Frank as a Christian philosopher and political thinker became one of the relatively few (along with, for example, Fyodor Stepun and Vladimir Weidle), but all the more significant conductors of the Russian cultural tradition in Europe and the European cultural tradition in Russia: that is, a genuine personification of the phe­nomenon of a “Russian European”. (shrink)

This article examines the philosophy of Semyon Frank and the theology of Karl Rahner in the light of transcendental themes. It demonstrates that the appeal to transcendental an­thropology and to anthropology in general is perceived by Franck and Rahner as a way of renewal of religious metaphysics, avoiding accusations of both «dogmatism» and re­moteness from living religious experience. Both thinkers, appealing to the Christian tradi­tion and Western philosophy, interpret the «non-thematic» and «non-conceptual» experience of being as a primordial existential (...) state. In the author’s opinion, Frank and Rahner offer original approaches to the problem of the foundations of religious experience by de­veloping a philosophical theology «from below» with a fundamental focus on phenomeno­logical and transcendental consideration of subjectivity. (shrink)