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 paper focuses on the review of current literature on formal philosophy. Special attention is paid to the review of the book «Introduction to Formal Philosophy» [Hansson, Hendricks, 2018]. The book is a consistent introduction to the problems of formal philosophy, a research tradition that relies on the precise mathematical tools in order to study traditional philosophical problems. The methods of formal philosophy are successfully applied not only to the problems of ontology, epistemology and philosophy of language but also relevant (...) for the problems of ethics, axiology and social philosophy. The book demonstrates that it is not correct to identify formal philosophy with another area of study – philosophical logic, since formal philosophy uses not only logical methods of analysis, but also uses the tools of game theory, decision theory, probability theory, Bayesian statistics, and other theories. Although the book has a propaedeutic character, it also contains some open problems. These problems include the aggregation of the opinions of the group under the condition of a conflicting base of premises in the theory of public choice, there are still open problems in the interpretation of Arrow’s impossibility theorem and others. Certainly, formalization in itself is not a general solution to the particular philosophical problem, but only a tool that allows to formulate a problem in a more rigorous and precise way, which sometimes allows to reveal some unexpected consequences, some implicit contradictions and new solutions. Despite the importance of the concept of coherence in ethics, decision theory, philosophy of law, Bayesian epistemology, philosophy of science, the existing formalizations of the concept of coherence are highly specialized for epistemology, researchers recognize the lack of the relevant explanatory models. Overall, the book is an excellent introduction in to the field of formal philosophy, which provides a general overview of different aspects of formal philosophy and the opportunity to study particular research topics by means of an extensive bibliography accompanying each of the chapters. (shrink)

The article presents a careful analysis of the idea of the “open texture” of empirical concepts and the problems of verification in the way that they were formulated by Friedrich Waismann. The idea of the “open texture” means for Waismann a certain type of a linguistic indeterminacy or a sort of lack of definition, which must be distinguished from, and linked to, another types like vagueness or ambiguity. It is shown that empirical statements are not conclusively verifiable for two different (...) reasons: the incompleteness of description of the material object and the open texture of the terms involved. We cannot conclusively verify statements in which the empirical concepts are used, because we cannot define these concepts in an exhaustive way because of their open texture. Thus, the definition of the concept will be incomplete. Waismann’s approach to definition plays here a key role, and it is directly related to the open texture of concepts. The author proposes interpreting the open texture as an immanent property of the concept, as something that is embedded in it a priori, and which can cause a vagueness. Nevertheless, an open texture must be distinguished form a vagueness. This leads to the conclusion that an open texture is a possibility of vagueness; vagueness can be remedied by giving more accurate rules, open texture cannot. In this sense, the “open texture” of a language allows for a more precise definition of concepts (by adjusting the definition) if appropriate circumstances arise. This justifies the thesis that the argument of the open texture is the ontological basis of the linguistic anti-reductionism of Friedrich Waismann. (shrink)

The article presents a detailed consideration of the arguments from the symposium “Verifiability”, which was held on July 14, 1945 in London, proposed by Scottish philosopher and theologian Donald MacKinnon, Austrian logician and mathematician Friedrich Waismann and English logician and philosopher of science William Kneale. MacKinnon’s approach to verifiability was based on the metaphysics of fact, while Waismann and Kneale’s approach was based on the semantic specificity of empirical concepts (“open texture” and context of use) and on the truth-values of (...) empirical propositions. The symposium in question is interesting primarily because it is the last meaningful discussion on the concept of verifiability in the form in which it was understood by the Vienna Circle. Despite the fact that, in the coming years, this concept received a completely different interpretation, which stands away from the original source, the 1945 symposium could be rightfully treated as a canonical discussion on verifiability in the sense that nothing like this has ever happened in the philosophy after. (shrink)

The article presents the philosophical and linguistic conception of Friedrich Waismann – the theory of many-level-structure of language. The key point of this theory is that each language stratum has its own logic: different concepts of truth, the methods of verifiability and the completeness of description are used in different strata. All this has an influence on the structure of logic itself. This approach suggests that all homogeneous statements (identical in a logical sense) are grouped in one stratum. The relations (...) between the different strata are of a most complicated, peculiar and elusive nature and that does not allow to reduce statement of one stratum to another. According to Waismann, these considerations lead to a new picture of the language, naturally divided into strata. (shrink)

The article examines K.A. Rodin’s thesis on the possibility of including Wittgenstein’s remarks on intention and action in the context of legal philosophy research. It is shown that although the concepts of intention and intentional action are relevant to the philosophy of law, Wittgenstein’s own ideas did not have a significant impact on their relevance (and some of them did not have it at all). This influence is confined to the fact that, like Wittgenstein, many jurists and legal theorists, mainly (...) those who were influenced by the ideas of H.L.A. Hart, also objected to the causal theory of action. However, the influence is limited to this approach. (shrink)

During its hundred-year history, Ludwig Wittgenstein’s Tractatus Logico-Philosophicus has undergone a variety of interpretations and explanations. But the significance of this work cannot be limited to an assessment of whether it had an impact on the development of logical positivism or not. Similarly, the reading of Tractatus cannot be reduced to just an ethical or some other readings. This article proposes to study a possible reading of “Tractatus” in terms of legal philosophy, which is based on the relation between facts, (...) values and norms, as well as on how interesting it could be (or would be) for legal philosophy. (shrink)

The article presents an analysis of K.A. Rodin’s argument that after publishing of Peter Winch’s book “The Idea of Social Science” (1958) the discussions of rule-following problem concerning to social epistemology and the methodology of social studies have not had tangible results. It is shown by the example of modern legal studies that this conclusion is not valid. On the contrary, Wittgenstein’s problem of rule-following and the very idea of rule-shaped activity have proved to have a great importance for an (...) analytical legal philosophy and turned into an independent subject of study. (shrink)

The progress in computer programming leads to the shift in traditional correlation between intuitive and formal components of mathematical knowledge. From epistemological point of view the role of intuition decreases in compare with formal representation of mathematical structures. The relevant explanation is to be found in D. Hilbert’s formalism and corresponding Kantian’s motives in it. The notion of sign belongs to both areas under consideration: on the one hand it is object of intuition in Kantian de re sense, on the (...) other hand, it is part of formal structure. Intuitive mathematical knowledge is expressed by primitive recursive reasoning. The W. Tait’s thesis, namely, that finitism as methodology of mathematics is equivalent to primitive recursive reasoning is discussed in connection with some explications of Kantian notion of intuition. The requirements of finitism are compared with normative role of logic. (shrink)

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)

У статті аналізуються свідчення вірменських істориків ХІІІ - початку XIV ст. відносно подій хрестових походів 1147 - 1149 рр. і 1189 - 1192 рр.; визначено специфіку традиції історіописання, яка була притаманна середньовічним вірменським хроністам. На основі джерелознавчого дослідження з'ясовано рівень достовірності повідомлень вірменських наративів, ступінь їхньої об'єктивності та інформативності.

Betrifft die Handschrift Cod. 555 der Burgerbibliothek Bern.

Verification and validation of computer codes and models used in simulations are two aspects of the scientific practice of high importance that recently have been discussed widely by philosophers of science. While verification is predominantly associated with the correctness of the way a model is represented by a computer code or algorithm, validation more often refers to the model’s relation to the real world and its intended use. Because complex simulations are generally opaque to a practitioner, the Duhem problem can (...) arise with verification and validation due to their entanglement; such an entanglement makes it impossible to distinguish whether a coding error or the model’s general inadequacy to its target should be blamed in the case of a failure. I argue that a clear distinction between computer modeling and simulation has to be made to disentangle verification and validation. Drawing on that distinction, I suggest to associate modeling with verification and simulation, which shares common epistemic strategies with experimentation, with validation. To explain the reasons for their entanglement in practice, I propose a Weberian ideal–typical model of modeling and simulation as roles in practice. I examine an approach to mitigate the Duhem problem for verification and validation that is generally applicable in practice and is based on differences in epistemic strategies and scopes. Based on this analysis, I suggest two strategies to increase the reliability of simulation results, namely, avoiding alterations of verified models at the validation stage as well as performing simulations of the same target system using two or more different models. In response to Winsberg’s claim that verification and validation are entangled I argue that deploying the methodology proposed in this work it is possible to mitigate inseparability of V&V in many if not all domains where modeling and simulation are used. (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)

The Gesta Normannorum Ducum is one of the most important sources for the history of Normandy and England in the eleventh and twelfth centuries, and contains the earliest prose account of the Norman Conquest. It was written by a succession of authors, the first of whom was William of Jumieges, who wrote for William the Conqueror. Later historians, such as Orderic Vitalis and Robert of Torigni, interpolated and extended the chronicle as far as King Henry I. The later accretions reveal (...) much not only about changing attitudes towards the Norman invasion of England, but also about views of the early Viking foundations of Normandy.Elisabeth van Houts's two-volume edition is based on a study of all forty-seven extant manuscripts of the Gesta, including the earliest surviving copy of c. 1100, unknown until very recently. The full original text of William of Jumieges is supplied, as well as the integral text of the subsequent revisions and additions. Volume I contains Dr van Houts's introduction to the whole work, together with the text and translation of books i-iv. Volume II contains books v-viii. The edition forms an important contribution to our understanding of Anglo-Norman politics. (shrink)

The Gesta Normannorum Ducum is one of the most important sources for the history of Normandy and England in the eleventh and twelfth centuries, and contains the earliest prose account of the Norman Conquest. It was written by a succession of authors, the first of whom was William of Jumièges, who wrote for William the Conqueror. Later writers, such as Orderic Vitalis and Robert of Torigni, interpolated and extended the chronicle as far as King Henry I. The later accretions reveal (...) much not only about changing attitudes towards the Norman invasion of England, but also about views of the early Viking foundation of Normandy. Elisabeth van Houts's two-volume edition is based on a study of all forty-seven extant manuscripts of the Gesta, including the earliest surviving copy of c. 1100, hitherto unknown. The full original text of William of Jumièges is supplied, as well as the integral text of the subsequent revisions and additions. Volume I contains Dr van Houts's introduction to the whole work, together with the text and translation of books i-iv. Books v-viii will appear in Volume II. The edition forms an important contribution to our understanding of Anglo-Norman politics. (shrink)

t. 1. Dialog s V. V. Davydovym -- t. 2. Nereshennye problemy razvivai︠u︡shchego obrazovanii︠a︡.

In spite of all the industrious efforts Peirce scholars have made so far, Peirce’s biography still retains a number of gaps, among which the problem of identity of Peirce’s second wife, Juliette Froissy, stands out most significantly. It is all the more important that, as some scholars suggest, the discovery of any reliable facts about Juliette could provide an explanation to some of the decisions Peirce had made, which irrevocably changed the course of his life, as well as his semiotic (...) theory. By courtesy of Professor Dr. Nathan Houser and the Peirce Edition Project, the writer of the present paper was granted access to the archive materials containing the Max H. Fisch — Maurice Auger correspondence and Victor Lenzen’s notes on Juliette. The paper aims at arranging the dispersed data obtained from these and other sources into a set of several distinct versions, which curiously refer to each other and collectively impose a certain order on some major abductions concerning Juliette’s identity. (shrink)

The relationship between logic and ethics is one of the basic and most essential questions of classical philosophical analysis. Since the time of the Pythagoreans, the fundamental unity of the two – whether by means of vague intuition, an elaborate conceptual scheme, or even a carefully crafted lifestyle – has led philosophers to identify truth and virtue. In his critical philosophy Kant put this unity of truth and virtue to extensive and rigorous trial to determine what conditions, if any, a...

In The Sources of Normativity Korsgaard introduces her conception of practical identities understood as the source of moral obligations. This conception forms a point of transition from Korsgaard’s theory of action to her solution to the problem of the authority of moral norms. In order to describe how universal categorical reasoning is compatible with the moral content of particular practical decisions, Korsgaard needs to show how our contingent practical identities can be reconciled with what she defines as the universally shared (...) identity that expresses our humanity. To make this reconciliation work, she devises an argument for the public shareability of reasons amended by her reinterpretation of the Kantian as if principle. I suggest that, in doing so, Korsgaard steps too far away from Kant’s architectonic approach to the question of why moral norms bind us, and that, consequently, the Korsgaardian reconciliation between the two kinds of identities, as it stands, cannot be accomplished. (shrink)

This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of (...) the very process of the reasoning itself. Thus conceived, one's capacity to diagram their thought reveals a set of characteristics common to ordinary language, visual perception, and necessary mathematical reasoning. The book offers an original synthetic approach that allows tracing the roots of Peirce’s conception of a diagram in certain patterns of interrelation between his semiotics, his pragmaticist philosophy, his logical and mathematical ideas, bits and pieces of his biography, his personal intellectual predispositions, and his scientific practice as an applied mathematician. (shrink)

The conception of two kinds of practical identities, which Korsgaard introduces in the Sources of Normativity, helps her explain how universal categorical reasoning is compatible with the moral content of individual practical decisions. Based on this conception, she devises an interpretation of the Kantian as if principle amended by her argument for the public shareability of reasons. I suggest that, in doing so, Korsgaard steps too far away from Kant’s architectonic approach to the question of why moral norms bind us, (...) and that, as a result, the Korsgaardian explanation, as it stands, cannot be accomplished. (shrink)

InMind and World,John McDowell claims that we need to steer our way between bald naturalism and rampant platonism as two ways to explain our capacity to use concepts. Performing this task requires an explanation of how concepts can be both socially charged and, at the same time, genuinely involving the world as it really is. I suggest that McDowell’s explanation is insufficient and that Wilfrid Sellars’s idea of sense impressions might be used to clarify the relationship between social practices and (...) conceptual knowledge without incurring too much damage to the overall architectonics of McDowell’s theory.DansMind and World, John McDowell affirme que nous devons nous éloigner du naturalisme dépouillé et du platonisme exubérant comme étant deux manières d’expliquer notre capacité à utiliser des concepts. Pour accomplir cette tâche, il est nécessaire d’expliquer comment les concepts peuvent être à la fois chargés socialement et réellement engagés avec le monde tel qu’il est. Je suggère que l’explication de McDowell est insuffisante et que l’idée de Wilfrid Sellars des impressions sensorielles peut être utilisée pour clarifier la relation entre les pratiques sociales et la connaissance conceptuelle, et ceci sans faire encourir trop de dégâts à l’architectonique générale de la théorie de McDowell. (shrink)

This paper compares Charles Peirce’s and Robert Brandom’s conceptions of normative objectivity. According to Brandom, discursive norms are instituted by practical attitudes of the members of a community, and yet the objectivity of these norms is not reducible to social consensus. Peirce’s conception of normative objectivity, on the contrary, is rooted in his idea of a community of inquiry, which presupposes a consensus achievable in the long run. The central challenge in both cases is to explain how the norms that (...) all members of a community take to be correct differ from those that are correct objectively. I argue that, in meeting the challenge of reconciling the social character of knowledge and the objectivity of norms shared by a community of knowers, Brandom’s approach might benefit from the Peircean idea of the ultimate agreement. (shrink)

The present paper is aimed at showing some correlations between Charles Peirce’s life, his intellectual habits as a logician and mathematician, his semiotic theory and his practice as a geodesist. For this purpose, it makes use of Peirce’s ideas about the nature of visual experience, some facts of his intellectual biography, and his definitions of sign and the term “virtual.” It appears that Peirce’s mature pragmatist and semiotic ideas find some support in his early practice as a scientist and a (...) mathematician, thus providing an interesting example of the intersection of scientific practice and philosophical speculation. (shrink)