Dans cet article je présente une analyse critique de l’approche habituelle de l’identité mathématique qui a son origine dans les travaux de Frege et Russell, en faisant un contraste avec les approches alternatives de Platon et Geach. Je pose ensuite ce problème dans un cadre de la théorie des catégories et montre que la notion d’identité ne peut pas être « internalisée » par les moyens catégoriques standards. Enfin, je présente deux approches de l’identité mathématique plus spécifiques: une avec la (...) fibration catégorielle et l’autre avec des catégories supérieures faibles. (shrink)

Dans cet article je présente une analyse critique de l’approche habituelle de l’identité mathématique qui a son origine dans les travaux de Frege et Russell, en faisant un contraste avec les approches alternatives de Platon et Geach. Je pose ensuite ce problème dans un cadre de la théorie des catégories et montre que la notion d’identité ne peut pas être « internalisée » par les moyens catégoriques standards. Enfin, je présente deux approches de l’identité mathématique plus spécifiques: une avec la (...) fibration catégorielle et l’autre avec des catégories supérieures faibles. (shrink)

The popular view according to which category theory provides a support for mathematical structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies ‘invariant form’ (Awodey) categorical mathematics studies covariant and contravariant transformations which, generally, have no invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics.

When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised intuitions are (...) further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified. (shrink)

This article asks whether a country that suffers from serious environmental problems caused by another country could have a just cause for a defensive war? Danish philosopher Kasper Lippert-Rasmussen has argued that under certain conditions extreme poverty may give a just cause for a country to defensive war, if that poverty is caused by other countries. This raises the question whether the victims of environmental damages could also have a similar right to self-defense. Although the article concerns justice of war, (...) we will concentrate only on the issue of what can be just causes of war, instead of evaluating the entire justification of war. This is to say that we will limit our discussion to the question concerning just cause and leave aside more general questions concerning justness and moral permissibility of war. Our aim is to list the questions that must be made and settled if defensive war in the case of serious environmental problems is said to have (or not to have) a just cause. We will argue that there are three questions that are most important in this context. They are the question concerning liability, the question of collective responsibility, and the question whether environmental harms may create a “sufficient reason” for raising a war. (shrink)

There are various ways to describe the type of society that developed, at least in America and Europe, over the last two centuries. One of the better known ones is the civilization of money. Different people, depending on their world view, can judge this fact differently: to deny it, however, is impossible. This is especially obvious now, when the most grandiose and stubborn attempt at liberating society from the power of money – allegedly in order to subordinate money to higher (...) ideals – has ended in total failure. (shrink)

Aleksandr Pushkin (1799-1837) occupies a special place in the development of Russian culture. He was at the same time a great poet, the reformer of Russian literary language, a historian and a political thinker. In the enormous mass of work devoted to Pushkin, a certain number of articles are concerned with his ideas on economics and the reflection of socio-economic problems in his writing. Until now, however, this theme has been studied in only a fragmentary way and less from the (...) point of view of the professional economist than from that of a literary historian. In an attempt to enlarge and complete the idea we may have of Pushkin's economic views, I propose also to show the importance the politico-economic thought of Western Europe had for his work. One person comes to the fore when we study the subject from this angle: Nikolai Turgenev, a Decembrist, one of the “masters” of the poet's youth, who played an important role in the rapprochement of Russian and Western European cultures. (shrink)

Category theory doesn't support Mathematical Structuralism but suggests a new philosophical view on mathematics, which differs both from Structuralism and from traditional Substantialism about mathematical objects. While Structuralism implies thinking of mathematical objects up to isomorphism the new categorical view implies thinking up to general morphism.

In addition to the “universal glue,” which is the local mechanical causation, the standard explanatory scheme of classical science presumes two “universal vessels,” which are global space and time. I call this outdated metaphysical setting “black-and-white” because it allows for only two principal scales. A prospective metaphysics able to bind existing sciences together needs to be “colored,” that is, allow for scale relativity and diversification by domain.

J. К. Galbraith begins his book on the history and current problems of money with the following statement: “There is much that is diverting in the history of money, but even more that provides a rich reflection of human behavior and human folly.”.

In 1900 David Hilbert announced his famous list of then-opened mathematical problems; the problem number 6 in this list is axiomatization of physical theories. Since then a lot of systematic efforts have been invested into solving this problem. However the results of these efforts turned to be less successful than the early enthusiasts of axiomatic method expected. The existing axiomatizations of physical and biological theories provide a valuable logical analysis of these theories but they do not constitute anything like their (...) standard presentation,which can be used for transmission, evaluation, and justification of physical and biological knowledge. This state of theart in the axiomatization of physics is strong evidence that the standard notion of axiomatic theory stemming from Hilbert and Tarski is not appropriate for thet ask. However in the recent years in mathematics there emerged a new axiomatic approach best represented by the Homotopy Type theory (HoTT). We argue that the constructive axiomatic architecture used in HoTT has better chances to be successfully applied in physics as well as in computer science and engineering. (shrink)

The author attributes the social evils of present-day Russia to a distorted "pseudoliberal" understanding of freedom as a condition unlimited by internal or external constraints.

In 1993, the American logic S. Yablo was proposed an original infinitive formulation of the classical ≪Liar≫ paradox. It questioned the traditional notion of self-reference as the basic structure for semantic paradoxes. The article considers the arguments underlying two different approaches to analysis of proposals of the ≪Infinite Liar≫ and understanding of the genuine sources for semantic paradoxes. The first approach (V. Valpola, G.-H. von Wright, T. Bolander, etc.) imposes responsibility for the emergence of semantic paradoxes on the negation of (...) the truth predicate. It deprives the ≪Infinite Liar≫ sentences of consistent truth values. The second approach is based on a modified version of anaphoric prosententialism (D. Grover, R. Brandom, etc.). The concepts of truth and falsehood are treated as special anaphoric operators. Logical constructs similar to the ≪Infinite Liar≫ do not attribute any definite truth values to sentences from which they are composed, but only state certain types of relations between the semantic content of such sentences. (shrink)

This article discusses the theoretical core and evolution of Ruben Apressyan’s moral philosophy. The starting point in the development of his views was his clarification of the “idea of morality,”...

Drawing from the complete normative resemblance between the questions of moral justification of the use of force and the moral justification of deception, the author finds in Kant's ethics an example of theoretical inconsistency. He shows that Kant's conclusion about the situational moral permissibility of forcible compulsion necessarily entails a decision in favor of the moral permissibility of lying for philanthropic reasons, which Kant denies.

The development of the science of economics is closely linked to the structure of capitalism. Even though ancient and medieval thinkers had already stated a certain number of ideas in this domain, the science of economics, in the modern sense of the word, did not truly begin until the 17th Century and the early 18th Century. At that time the methodology for research in the natural sciences was developed, and the first scientific academies and societies were founded (England, France, Prussia, (...) Russia). The initial step in the development of the science of economics occurred in conjunction with this process. Moreover, the ideas of the first economists were, as a general rule, centered on the practical problems and needs of the State. And in that period the prevalent state structure in Europe was the absolute monarchy. Consequently, the first economists were, for the most part, counsellors to the sovereign, authors of studies and reports containing a variety of proposals regarding economic policy. With regard to a great many distinguished economists of the 17th and 18th Centuries, we can speak of them being “men of proposals”. (shrink)

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. (...) Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method. (shrink)

In [5] it has been shown that for first-order definability over the reals there exists an effective procedure which by a finite formula with equality defining an open set produces a finite formula without equality that defines the same set. In this paper we prove that there exists no such procedure for Σ-definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of Σ-definable sets into new definitions of Σ-definable sets in such (...) a way that the results will define open sets, and if a definition defines an open set, then the result of this transformation will define the same set. These results highlight the important differences between Σ-definability with equality and Σ-definability without equality. (shrink)

Russian Marxism was fairly late to address building its own understandings of the Russian historical process. Moreover, the Bolsheviks did not have their own historiography of “Russian history” despite the fact that, beginning in 1918, they began more and more vehemently claiming not just total ideological control but also intellectual hegemony. A confrontation between “Marxist” and “non-Marxist” understandings arose. At the same time, the real disputes within the camp of Marxist historians came down to a confrontation between the versions of (...) the historical process proposed by Georgi V. Plekhanov and Mikhail N. Pokrovskii back in the 1910s. This article broadly analyzes the disputes in the Marxist camp, from pressing political implications such as attitudes toward the state to the definition of the place of “historical facts” in theory and interpretation. We also demonstrate that it was, in fact, the understandings of Plekhanov, Leon D. Trotsky, and Pokrovskii that continue, both explicitly and implicitly, the legacy of Vasily O. Klyuchevsky’s historical schema and his understanding of the “state school,” a legacy that has remained unstudied until now. (shrink)