This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U (n) and symmetric groups $S_{n},\ n\in {\Bbb N}$ . Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular (...) it is still unknown if every group is hyperlinear and/or sofic. (shrink)

The problem of commensurability/incommensurability of different cultural codes is a key problem of modern civilizational development. This is the problem of the search for communicative unity in the world of cultural and biological diversity, which has to be protected, and the search for the cohesion of different Umwelten, of semiotically-defined artificial and natural environments, of ecological and cognitive niches, taking into account that each of them has their own identity and uniqueness. The purpose of the article is to draw attention (...) to the fact that the question of the so-called incommensurability of different conceptual schemes, paradigms, language consciousnesses is widely discussed not only in cross-cultural studies and philosophical problems of translation but also in connection with the problems of incommensurability between the language of classical physics and the language of relativistic quantum physics. Attention is drawn to the problem of the incommensurability and correlation of different languages that are used in debates about the foundations of quantum mechanics, its interpretation, comprehension and ontology. Two approaches stand out in this debate. The first approach is based on the language of the formed being, on the language of things localized in time and on the logic of Aristotle. The second approach is based on the language of the becoming, process and nonlocality, on the search for various processual-oriented temporal logics. In this regard, we discuss the processual approach to understanding quantum mechanics, proposed in the philosophical and physical works of D. Bohm. The authors argue that the experience of constructive understanding of the metaproblems of the interpretation of quantum mechanics, the critical reception of the legacy of such philosophers of the process as Peirce, Bergson and Whitehead, the deep reflection on the problems of commensurability/ incommensurability of linguistic consciousnesses of different cultures – will eventually create a common synergetic-interdisciplinary space of cooperation for the solutions of the above-mentioned issues. (shrink)

The main problem discussed in this paper is: Why and how did animal cognition abilities arise? It is argued that investigations of the evolution of animal cognition abilities are very important from an epistemological point of view. A new direction for interdisciplinary researches – the creation and development of the theory of human logic origin – is proposed. The approaches to the origination of such a theory (mathematical models of ``intelligent invention'' of biological evolution, the cybernetic schemes of evolutionary progress (...) and purposeful adaptive behavior) as well as potential interdisciplinary links of the theory are described and analyzed. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...) the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy’s mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno’s flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

We have synthesized a 582,970-base pair Mycoplasma genitalium genome. This synthetic genome, named M. genitalium JCVI-1.0, contains all the genes of wild-type M. genitalium G37 except MG408, which was disrupted by an antibiotic marker to block pathogenicity and to allow for selection. To identify the genome as synthetic, we inserted "watermarks" at intergenic sites known to tolerate transposon insertions. Overlapping "cassettes" of 5 to 7 kilobases (kb), assembled from chemically synthesized oligonucleotides, were joined by in vitro recombination to produce intermediate (...) assemblies of approximately 24 kb, 72 kb ("1/8 genome"), and 144 kb ("1/4 genome"), which were all cloned as bacterial artificial chromosomes in Escherichia coli. Most of these intermediate clones were sequenced, and clones of all four 1/4 genomes with the correct sequence were identified. The complete synthetic genome was assembled by transformation-associated recombination cloning in the yeast Saccharomyces cerevisiae, then isolated and sequenced. A clone with the correct sequence was identified. The methods described here will be generally useful for constructing large DNA molecules from chemically synthesized pieces and also from combinations of natural and synthetic DNA segments. 10.1126/science.1151721. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...) Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...) interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (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)

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.

This volume explores how Catholicism began and continues to open its doors to the wider world and to other confessions in embracing ecumenism, thanks to the vision and legacy of the Second Vatican Council. It explores such themes as the twentieth century context preceding the council; parallels between Vatican II and previous councils; its distinctively pastoral character; the legacy of the council in relation to issues such as church-world dynamics, as well as to ethics, social justice, economic activity. Several chapters (...) discuss the role of women in the church before, during, and since the council. Others discern inculturation in relation to Vatican II. The book also contains a wide and original range of ecumenical considerations of the council, including by and in relation to Free Church, Reformed, Orthodox, and Anglican perspectives. Finally, it considers the Council’s ongoing promise and remaining challenges with regard to ecumenical issues, including a groundbreaking essay on the future of ecumenical dialogue by Cardinal Walter Kasper. (shrink)

The article discusses the attitude of Georg Lukács and his adherents who formed a circle “Techeniye” toward the phenomenon of Stalinism. Despite the political nature of the topic, the authors are aspired to provide an unbiased research. G. Lukács’ views on the theory and practice of Stalinism evolved over time. In the 1920s Lukács welcomes the idea of creation of socialism in one country and abandons the former revolutionary ideas expressed in his bookHistory and Class Consciousness. This turn is grounded (...) by new interpretation of Hegel as “realistic” thinker whose “realism” was shown in the aspiration to find “reconciliation” with reality and in denial of any utopias. The philosophical evolution leading to “realism” assumes integration of revolutionaries into the hierarchy of existing society. The article “Hölderlin’s Hyperion” represents attempt to justify Stalinism as a necessary and “progressive” phase of revolutionary development of the proletariat. Nevertheless, events of the second half of the 1930s force Lukács to realize the catastrophic nature of political strategy of Stalinism. In his works, Lukács ceases to analyze political topics and concentrates on problems of aesthetics and literary criticism. However, his aesthetic position allows to reconstruct the changed political views and to understand why he had earned the reputation of the “internal opponent” to Stalinism. After 1956, Lukács turns to political criticism of Stalinism, which nevertheless remains unilateral. He sees in Stalinism a kind of the left sectarianism, the theory and practice of the implementation of civil war measures in the era of peaceful co-existence of two systems. (shrink)

We conducted two experiments to determine whether common folk think that so-called _tell-tale sign_ bald-faced lies are intended to deceive—since they have not been tested before. These lies involve tell-tale signs (e.g. blushing) that show that the speaker is lying. Our study was designed to avoid problems earlier studies raise (these studies focus on a kind of bald-faced lie in which supposedly everyone knows that what the speaker says is false). Our main hypothesis was that the participants will think that (...) the protagonists from our examples lied without intending to deceive, and the results of our surveys confirmed this hypothesis: most of our participants rated tell-tale sign lies as lies not intended to deceive. Therefore, our analysis suggests that common folk think that some lies are not intended to deceive. (shrink)

Arguably, the existence of bald-faced (i.e. knowingly undisguised) lies entails that not all lies are intended to deceive. Two kinds of bald-faced lies exist in the literature: those based on some common knowledge that implies that you are lying and those that involve tell-tale signs (e.g. blushing) that show that you are lying. I designed the tell-tale sign bald-faced lies to avoid objections raised against the common knowledge bald-faced lies but I now see that they are more problematic than what (...) I initially thought. Therefore, I will discuss these lies in more detail, refine the existing cases, and resolve some anticipated objections. I conclude that tell-tale sign bald-faced lies are genuine lies not intended to deceive. (shrink)

According to one influential argument put forward by, e.g. Chisholm and Feehan, Pfister, Meibauer, Dynel, Keiser, and Harris, asserting requires intending to give your hearer a reason to believe what you say (first premise) and, because liars must assert what they believe is false (second premise), liars necessarily intend to cause their hearer to believe as true what the liars believe is false (conclusion). According to this argument, that is, all genuine lies are intended to deceive. ‘Lies’ not intended to (...) deceive are not genuine lies because they do not involve assertions and you need to assert in order to lie. In this paper, I reject this argument by arguing that the first premise is false: intending to give your hearer a reason to believe what you say is not necessary for asserting. (shrink)

In this paper we prove the following theorem: Any one-dimensional definably connected group G over an o-minimal structure is, as an abstract group, isomorphic to either pPp∞δ or δ.

Although it is well documented that language plays an important role in cognitive development, there are different views concerning the mechanisms underlying these effects. Some argue that even early in development, effects of words stem from top‐down knowledge, whereas others argue that these effects stem from auditory input affecting attention allocated to visual input. Previous research (e.g., Robinson & Sloutsky, 2004a) demonstrated that non‐speech sounds attenuate processing of corresponding visual input at 8, 12, and 16 months of age, whereas the (...) current study demonstrates that words attenuate visual processing at 10 months but not at 16 months (Experiment 1). Furthermore, prefamiliarization with non‐speech sounds (Experiment 2) resulted in able processing of visual input by 16‐month‐olds. These findings suggest that some effects of labels found early in development may stem from familiarity with human speech. The possibility of general‐auditory factors underlying the effects of words on cognitive development is discussed. (shrink)

In 1995 N. C. A. da Costa and F. Doria proposed the modaltype elegant axiomatization of Jaśkowski’s discussive logic D2. Yet his ownproblem which was formulated in 1975 in a following way: Is it possible toformulate natural and simple axiomatization for D2, employing classical disjunction and conjunction along with discussive implication and conjunctionas the only primitive connectives? — still seems left open. The matter of factis there are some axiomatizations of D2 proposed, e.g., by T. Furmanowski, J. Kotas and N. (...) C. A. da Costa , G. Achtelik, L. Dubikajtus,E. Dudek and J. Konior , satisfying da Costa’s conditions, but they arerather looking very complicated and unnatural. An attempt is made to solveda Costa’s problem. The new axiomatization of D2 is proposed essentiallybased on da Costa’s-Doria axiomatization from 1995. (shrink)

ABSTRACT Two models of the Aristotelian syllogistic in arithmetic of natural numbers are built as realizations of an old Leibniz idea. In the interpretation, called Scholastic, terms are replaced by integers greater than 1, and s.Ap is translated as “s is a divisor of p”, sIp as “g.c.d. > 1”. In the interpretation, called Leibnizian, terms are replaced by proper divisors of a special “Universe number” u < 1, and sAp is translated as “s is divisible by p”, sIp as (...) ‘l.c.m. < u”. Both interpretations are proved to be adequate to the Aristotelian syllogistic. They are extended to syllogistic including term negation and term conjunction as well. (shrink)

Philosophers like G.H. von Wright and D. Makinson have pointed to serious challenges regarding the foundations of deontic logic. In this paper, I suggest that to deal successfully with these challenges a reconsideration of the research program of the discipline is useful. Some problems that have troubled this particular field of logical study for decades may disappear or appear more tractable if we view them from the perspective of a language game introduced by D. Lewis involving three characters: the Master, (...) the Slave and the Kibitzer. The adoption of this perspective opens a natural approach to a new layout of the domain of deontic studies. I propose dividing deontic logic into six sub-areas which are distinguished by their focus on the different idioms typical of the individual players, by conceiving the language game as either being static or as dynamic and by the aims of the logical inquiry. What kind of insights the proposed perspective provides is illustrated by an analysis of the so-called Ross paradox—a problem that has troubled deontic logic since its origins and, though it was many times pronounced solved, still keeps coming back ‘alive and kicking’. (shrink)

In “Elementa Juris Naturalis” (1671) G.W. Leibniz formulated the genius intuition about the fundamental unity of juridical and Aristotelian modalities. Interpreting and explicating this intuition by virtue of the deontic logic G.W. Wright arrived to the conclusion that the unity of modalities in question is not an equivalence relation but an analogy one. However a complement for G.W. Wright’s explication of the intuition of G.W. Leibniz was submitted. This complement was a two-valued algebra of formal-natural-law philosophy of juridical and Aristotelian (...) modalities as moral-legal evaluation-functions determined by one variable. In thepresent paper I submit a binary generalization of the mentioned unary complement. By virtue of this generalization some significant shortcomings of the complement in question are eliminated. The generalization deals with the modalities as moral-legal evaluation-functions determined by two variables. (shrink)

Despite continuous efforts to reduce poverty on the global, national and communal levels the problem of closing the income gap remains serious. In this work, we analyze why the traditional infrastructure-based approach to development is becoming less effective. We have identified some features of the promising new approach, described in the collection of works "Self-Transformation of the Forgotten Four-Fifths" edited by Robert G. Dyck and Matjaz Mulej. The new approach could be applied in the information age when knowledge becomes a (...) key variable of development. (shrink)

This article considers the contribution of Ernst Mally to the theory of objects, namely he introduced a distinction of two types of properties (determinations) — formal and extraformal, that allowed Meinongian theory of objects to provide a satisfactory response to the criticism of Bertrand Russell, related to the question of the existence of the golden mountain. According to Meinong, object's so-being (Sosein) is logically independent of its ontic status (Sein). In this case Russell asks, how the existent golden mountain can (...) be existent, golden and a mountain, because golden mountains do not exit (as well as round squares). So it was obvious that the independence of Sosein from Sein thesis should be revised. That's why Mally made a remark that the properties that make object's Sosein are extraformal. Later these ideas have formed the basis of the contemporary study of nonexistent objects, the basis of the theories of Terens Parsons, Richard Routley etc. In particular Parsons's distinction of nuclear and extranuclear properties is a revised version of Mally's distinction.In addition, the article analyzes Ernst Mally's critics of epistemological idealism and its connection with the famous psychologism dispute in logic and semantics in the late 19th and early 20th century. This dispute is more associated with the works of G. Frege and E. Husserl, which accused of psychologism many representatives of Brentano's School. Particularly Alexius Meinong was also called a psychologist. That's it's important to find out, which side was Mally on in this dispute. (shrink)

Based on some materials from the A.V. Florovsky's Foundation in the Archive of the Russian Academy of Sciences, the author of the article reconstructs a little-known page from the history of academic and philosophical life of the Russian migr, that is the history of the Philosophical Society at the Russian Free University in the 1930s-1940s, including during the Nazi occupation of Czechoslovakia. It is justified that the reconstruction of the history of Russian philosophical institutions can set a new research perspective (...) that allows us to consider the history of Russian philosophy as a process of formation and development of institutionalized forms of production, reproduction and translation of philosophical knowledge, which would allow to create a new historical and philosophical cross-section of Russian thought, complementing traditional approaches to the historiography of Russian philosophy. The article reveals new aspects of the scientific activity of N.O. Lossky and I.I. Lapshin, who played a key role in the work of the Philosophical Society. In addition, the analysis of archival materials made it possible to identify and reconstruct new facts from the creative biography of such thinkers of Russian emigration as S.L. Frank, E.V. Spektorsky, B.P. Vysheslavtsev, G.V. Florovsky. Russian Free University's Philosophical Society report for 1941 is given as an appendix to the article, as well as the reconstruction of the problems of the circle for the study of the Russian national idea. It is concluded that scientific and philosophical activity at the RSU was carried out through an extensive network of scientific societies, associations, circles and seminars, which allowed developing a wide variety of issues - aesthetics, the history of Russian and Western philosophy, philosophical problems of Russian literature and history, the Russian national idea. (shrink)

Objects of consideration are various non-classical connectives “hidden” in the classical logic in the form of G˛s with ˛ —a classical connective, and s—a propositional variable. One of them is negation, which is defined as G ⇒ s; another is necessity, which is defined as G ∧ s. The new operations are axiomatized and it is shown that they belong to the 4-valued logic of Lukasiewicz. A 2-point Kripke semantics is built leading directly to the 4-valued logical tables.

In the field of geographical limology, the new science of international borders, it is possible to distinguish several consecutive theoretical approaches that can be seen as either traditional or postmodern. At each consecutive stage new approaches have been applied together with, not in place of, the traditional, in order to continue to perfect the discipline (see, for instance, Minghi, 1963; Rumley and Minghi, 1991). Traditional approaches, such as historical mapping, the typological, the functional and the political, have been described in (...) previous theoretical works (e.g. Kolossov and O'Loughlin, 1998; Kolossov and Turovsky, 1998; Kolossov and Mironenko, 2001). Here we are concerned with the postmodern approaches, which have been emerging mostly during the 1990s and the present decade. (shrink)

The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms.

The article refutes the widespread view that Dostoevsky's Christian beliefs were strictly Orthodox. It is proved that Dostoevsky's religious and philosophical searches' central tendency is the criticism of historical, ecclesiastical Christianity as a false, distorted form of the teaching of Jesus Christ and the desire to restore this teaching in its original purity. Modern researchers of the history of early Christianity find more and more arguments in favor of the fact that the actual teaching of Jesus Christ is contained in (...) that religious movement, which the church called the Gnostic heresy. The exact philosophical expression of the teaching of Christ was received in the later works of J.G. Fichte, whose ideas had a strong influence on the Russian writer. Like Fichte, Dostoevsky understands Christ as the first person who showed the possibility of revealing God in himself and gaining divine omnipotence and eternal life directly in earthly reality. In this sense, every person can become like Christ. Dostoevsky's main characters walk the path of Christ and show how difficult this path is. The article shows that Dostoevsky used in his work not only the philosophical version of true Christianity developed by German philosophy, but also the key motives of the Gnostic myth, primarily the idea that our world, filled with evil and suffering, is created not by the supreme, good God-Father, but by the evil Demiurge, the Devil. (shrink)

In the history of Russian philosophy, V. S. Solovyov is known as one of the prominent spokesmen of theological idealism. Lenin more than once pointed to the need to combat the ideas of this philosopher . He quite properly regarded them as the theoretical foundation of the ideology of "liberal renegacy.".

The title of this article will perhaps seem excessively contentious to the reader. What solitude of Vladimir Ivanovich Vernadskii, he might ask, if he has even the slightest familiarity with the scientist's biography?