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)

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and (...) clearer system. Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals. (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)

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)

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.

В книге собраны статьи и эссе последнего десятилетия о науке и ее месте в современном обществе. Для читателей, интересующихся наукой и ее людьми.

The article examines the role and place of the Mystery within the structure of Mamleev’s metaphysics. The author of the article implicates the concept of “chaosmos’’, introduced by James Joyce in the experimental novel “Finnegans Wake”, to the Mamleev’s Universe. This leads to the transformation of the formula “chaos – osmosis – cosmos”, actualized by postmodern discourse, into the formula “Chaos – Osmosis – Cosmos”. Chaos here is Sacred Chaos, being one of the metaphysical manifestations of Eternal Russia. Osmosis appears (...) to be an antinomic Russian life, which combines the incompatible, and sets itself fundamentally impossible tasks. The Сosmos brings us to the tradition of Russian cosmism with its problem of microcosm and macrocosm conformity. The author of the article points out that the all-pervading and unifying element of Mamleev’s chaos is the Mystery. It is one of the five key metathemes of Mamleev’s oeuvre. The particularity of the Mystery is that it provides a link between other metathemes: utrism “I”, Death, the Abyss and Russia. In the plane of utrism “I” the Mystery is seen as the mystery of man. The Mystery is related to Death by the person’s interest of his own destiny and finiteness. The problem of overcoming death and the passion for eternal life are the central themes that concern almost all the heroes of Mamleev’s novels and short stories. For Mamleev they are the source of values aimed at the human need to achieve a higher “I”. The connection between the Mystery and the Abyss, as well as the connection between the Mystery and Russia are the strongest and the most important for Mamleev. Consideration of these connections prompts the author to conclude that Yuri Mamleev describes such dimensions of the Mystery, which go beyond the categories “unfathomable-to-us” and “unfathomable-in-itself”, introduced by S. Frank. This requires the introduction of new concepts that would bring out the essence of Mamleev’s innovations. The author of the article proposes to consider the Abyss as a “Last Mystery” by the fact it is transcendent to the transcendents themselves, that is it goes beyond the Absolute, as well as to consider Eternal Russia as a Russia-Mystery on the grounds that Mamleev gives it the status of the third metaphysical Beginning, along with the Absolute and the Abyss. (shrink)

The article dealt with the question of the ratio of Yuri Mamleev’s I-utrism to Tradition. The problem is in the contradiction between two positions, which Mamleev notes in different places of “The Destiny of Being” – a view of the I-utrism as a new approach within the framework of Tradition, and as going beyond it with the introduction of extraneous elements. Based on the connection between the I-utrism and the “Last Doctrine”, which clearly lies outside the Tradition, the author of (...) the article concludes that the I-utrism should not be considered as a new approach within the framework of the Tradition, but as an autonomous construction to it. According to the author, the I-utrism due to its own radicalism contains two key problems that lead not only to a break with Tradition but also to a violation of the inner integrity of Mamleev’s philosophy itself. These two problems are defined as the radicalization of the Vedanta formula “Atman = Brahman” and the absence of the figure of the Other. I-utrism forms unified I, which the author of the article proposes to call “I-total”, because it displaces everything but itself, including the classical concept of the Absolute. Hence the lack of a topos for Mamleev’s concepts such as “Abyss” and “Eternal Russia”. It is compounded by the prevention of the metaphysical event of the Other, which is especially important for the idea of a metaphysical and mysterious Russia that requires intersubjectivity. In the process of analyzing these problems and issues, the article compares the philosophical positions of Stirner and Mamleev, as well as addresses the topic of demonic I-utrism. After all, the author concludes that the I-utrism needs to be revised and softened. This would allow it to exist organically in the overall picture of Mamleev’s worldview and not lose touch with Tradition despite the inevitable exit from it. (shrink)

The author of the article studies the creative legacy of Yuriy Mamleev, trying to establish the correlation between of his philosophical and fiction writings. The conventional editorial distinction between philosophical and literary texts as different and autonomous categories is rejected. Instead of the idea, that philosophical and fictional aspects of texts by Mamleev are inextricably linked, it is claimed that they should be considered together. In the author’s opinion, these texts present different perspectives and implementations of his oeuvre that contains (...) the rational dimension and on the other hand – irrational dimension of artistic images and obscure allusions. Mamleev’s writings considered by means of 5 crucial “metathemes”: Аbyss and the “Ultimate relationships” concerned with it, the Mystery which has two levels – one cognizable and the another – uncoognizable in principal, the “root of self-being” and metaphysic of “I”, death and immortality, Russia and the Russian soul which are existed between West and East with special historical and cosmological mission. It is concluded that these 5 metathemes pervade all Mamleev’s texts and comprise his spiritual search. The characters of Mamleev’s fiction are mostly people of post-Soviet world, who ask questions with no set answers to become acquainted with the secrets of being. At the same time they are concerned with their own existence, their self destiny. Their actions are extraordinary and illogical. Supposedly Mamleev’s artistic images not only correspond with his philosophical ideas, for instance – utrism or the “Ultimate relationships”, but also reveal them from unexpected perspective. The author determines that distinction between philosophy and literature of Mamleev is incorrect, as these areas are conflated. Any distinction of this kind will ruin the logic of Mamleev’s thinking and outlook. Thus, metaphysical realism can be viewed as the crossing, distinction blurring area, productive and creative transgression. Or as the situation when philosophy is more than it is itself. (shrink)

The article attempts to analyze the concept of gender, study philosophical preconditions of its emergence and trace the main postmodern aspects of the gender category. It proves that gender research in the postmodern era is not identical to the theories of feminism. It deals with social life of both sexes, their behavior, roles, characteristics, common and different between them, the social relationships of the sexes, considering the world from the standpoint of both socio-gender groups. The article shows that an urgent (...) need for more purposeful development of independent women's research in the developing countries. Such research should holistically reflect and study the lives of women on the basis of the so-called women's rather than universalized man experience. The main idea of this scientific research was to emphasize that the world can be explored not only from a man perspective, but also from the standpoint of woman experience. The level of scientific study of the gender conceptual sphere is clarified, in particular, the gender concept is a multidimensional complex represented in the language, which has a certain ethnocultural specificity. The concept has an unstable structure, which is reflected in the model of the concept developed by us, in which we highlight the root and applications of the concept, the possible movement of features in the relevant semantic directions from and to the root. The unstable structure of the concept is also characteristic of its root: during historical development, the root may change, but the semantic meaning is not lost, but only replaced by synonyms or verbal innovations relevant to today. (shrink)

The following article by Semen Liudvigovich Frank was published in the German monthly literary periodical Gral, which came out between 1906 and 1937 in Ravensberg. The editor of the periodical was Franz Eichert. For the most part, it presents the contents of one of the lectures Frank gave to a West European audience, familiarizing them with the philosophical legacy of "enigmatic" Russia, which had been through an unprecedented historical cataclysm. Interest in Russian philosophy was no accident either for the (...) audience or for the lecturer, an eyewitness and even participant—as essayist and ideologist of the liberal bourgeoisie—of the violent political events in Russia. More and more new historical, cultural, and philosophical materials were being drawn, as into a whirlpool, into the process of interpreting the Russian Revolution. (shrink)

This brief essay discusses some of the medical and social uses of banking human semen as well as many of the ethical issues which emerge from the application of this technology.

Using a sample of 652 college students, we examined several implications of the hypothesis that the shape of the human penis evolved to enable males to substitute their semen for those of their rivals. The incidence of double mating by females appears sufficient to make semen displacement adaptive (e.g., one in four females acknowledge infidelity, one in eight admit having sex with two or more males in a 24-hour period, and one in 12 report involvement in one or (...) more sexual threesomes with two males). We also document several changes in post-ejaculatory behavior (e.g., reduced thrusting, penis withdrawal, loss of an erection) which may have evolved to minimize displacement of the male’s own semen. Consistent with predictions derived from a theoretical model (Gallup and Burch 2006), we discovered that most females report waiting at least 48 hours following an instance of infidelity before resuming sex with their in-pair partners. (shrink)

ABSTRACTBrazil has not yet approved legislation on assisted reproduction. For this reason, clinics, hospitals and semen banks active in the area follow Resolution 1358/92 of the Conselho Federal de Medicina, dated 30 September 1992. In respect to semen donation, the object of this article, the Resolution sets out that gamete donation shall be anonymous, that is, that the donor and recipients shall not be informed of each other's identity. Thus, since recipients are unaware of the donor's identity, (...) class='Hi'>semen banks and the medical teams involved in assisted reproduction become the intermediaries in the process. The objective of this article is to show that, in practice, this represents disrespect for the ethical principles of autonomy, privacy and equality. The article also stresses that the problem is compounded by the racial question. In a country like Brazil, where racial classification is so flexible and goes side by side with racist attitudes, the intermediary role played by semen banks and medical teams is conditioned by their own criteria of racial classification, which are not always the same as those of donors and semen recipients. The data presented in this paper were taken from two semen banks located in the city of São Paulo . At the time of my research, they were the only semen banks in the state of São Paulo and supplied semen to the capital , the state of São Paulo, and to cities in other Brazilian states where semen banks were not available. (shrink)

Brazil has not yet approved legislation on assisted reproduction. For this reason, clinics, hospitals and semen banks active in the area follow Resolution 1358/92 of the Conselho Federal de Medicina, dated 30 September 1992. In respect to semen donation, the object of this article, the Resolution sets out that gamete donation shall be anonymous, that is, that the donor and recipients (and the children who might subsequently be born) shall not be informed of each other's identity. Thus, since (...) recipients are unaware of the donor's identity, semen banks and the medical teams involved in assisted reproduction become the intermediaries in the process. The objective of this article is to show that, in practice, this represents disrespect for the ethical principles of autonomy, privacy and equality. The article also stresses that the problem is compounded by the racial question. In a country like Brazil, where racial classification is so flexible and goes side by side with racist attitudes, the intermediary role played by semen banks and medical teams is conditioned by their own criteria of racial classification, which are not always the same as those of donors and semen recipients. The data presented in this paper were taken from two semen banks located in the city of São Paulo (Brazil). At the time of my research, they were the only semen banks in the state of São Paulo and supplied semen to the capital (São Paulo city), the state of São Paulo, and to cities in other Brazilian states where semen banks were not available. (shrink)

The article contains a comparative analysis of the thought of Russian émigré philosopher Semen Frank and one of the most prominent representatives of process philosophy and theology Charles Hartshorne. Among the points of convergence, their integral vision of reality was pointed out. Frank’s and Hartshorne’s approaches to the question of cognition of God were considered, with special attention paid to their interpretation of the ontological proof. Hartshorne was familiar with Russian thought and even wrote reviews on Zenkovsky and Lossky’s (...) classic books on the history of Russian philosophy, where he mentioned Frank more than once. One cannot speak of the two thinkers’ influence on each other, but rather of a common philosophical heritage going back to Plato. (shrink)

Semen Frank is one of the most interesting and exciting pre-revolutionary Russian religious philosophers to be “rediscovered” after the fall of the Soviet Union. His involvement in Russian pre-revolutionary political and academic life brought Frank into contact with an imaginative, progressive and idealistic group of thinkers whose ranks he then joined. Like Nicholas Berdyaev and Fr. Sergei Bulgakov, Frank put forward his own philosophical views about their world, which was in upheaval, and about human nature. After emigration from the (...) then young Soviet Union, Frank continued to produce his impressive body of work. His ideas are remarkable for their considered, moderate and tolerant nature.This book is a contribution to the ever-growing body of scholarship, which strives to connect Russian thought to its contemporary Western context and continues to facilitate the revival of these fascinating thinkers. In addition to situating Frank’s thought in a Western context, this work seeks to provide an overview of his thought, and introduce his key arguments to those unfamiliar with Russian religious philosophy. (shrink)

Semen Frank is one of the most interesting and exciting pre-revolutionary Russian religious philosophers to be “rediscovered” after the fall of the Soviet Union. His involvement in Russian pre-revolutionary political and academic life brought Frank into contact with an imaginative, progressive and idealistic group of thinkers whose ranks he then joined. Like Nicholas Berdyaev and Fr. Sergei Bulgakov, Frank put forward his own philosophical views about their world, which was in upheaval, and about human nature. After emigration from the (...) then young Soviet Union, Frank continued to produce his impressive body of work. His ideas are remarkable for their considered, moderate and tolerant nature.This book is a contribution to the ever-growing body of scholarship, which strives to connect Russian thought to its contemporary Western context and continues to facilitate the revival of these fascinating thinkers. In addition to situating Frank’s thought in a Western context, this work seeks to provide an overview of his thought, and introduce his key arguments to those unfamiliar with Russian religious philosophy. (shrink)