The philosophy of mathematical practice sometimes investigates the social constitution of mathematics but does not always make explicit the philosophical-normative framework that guides the discussion. This chapter investigates some recent proposals in the philosophy of mathematical practice that compare social facts and mathematical objects, discussing similarities and differences. An attempt will be made to identify, through a comparison with three different perspectives in social ontology, the kind of objectivity attributed to mathematical knowledge, the type of representational or non-representational semantics adopted, (...) and the justificatory or coordinative role entrusted to axiomatics. After a brief introduction to key issues in social ontology, Sect. 3 of the chapter offers a survey of contributions by Feferman, Ferreirós, and Cole, highlighting a difference between approaches based on rules, practices, and intentional states, respectively. Section 4.1 also discusses results by Carter, Collin, Giardino, and Pantsar, focusing on differences between the objectivity of knowledge and objectivity of objects and showing that many authors combine a realist, structuralist, and pragmatist perspective, as well as the idea that objectivity comes in degrees. Section 4.2 focuses on the kind of semantics that is adopted in socially oriented approaches. A representational semantics is preferred by authors grounding their views on mental states, whereas a non-representational semantics best fits with views based on practices. Section 5 considers how axiomatics can be understood in a social perspective. Axiomatics generally plays a justificatory role also in theories that aim to explain the social constitution of mathematical knowledge. Yet, if one shifts from approaches based on mental states to approaches anchored on rules or habits, axiomatics can be viewed as an institution based on obligations, functions, coordination problems, and agent’s actions and roles, thus playing an organizational and coordination role. (shrink)

The paper introduces Vailati’s life and works, investigating Vailati’s education, the relation to Peano and his school, and the interest for pragmatism and modernism. A detailed analysis of Vailati’s scientific and didactic activities, shows that he held, like Peano, a a strong interest for the history of science and a pluralist, anti-dogmatic and anti-foundationalist conception of definitions in mathematics, logic and philosophy of language. Vailati’s understanding of mathematical logic as a form of pragmatism is not a faithful interpretation of Peano’s (...) conception, but it is essential to understand the relations of Peano’s logic with other philosophical traditions and some epistemological aspects of Peano’s perspective, such as the search for a universal language. (shrink)

This high-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether. Additional topics include Einstein's electrodynamics of moving bodies, Minkowski spacetime, gravitational geometry, time and causality, and other subjects. Highlights include a rich exposition of the elements of the special and general theories of relativity.

Recordando a Bachelard a medio siglo de su muerte (2011) -- Filosofia de la relatividad (2011) -- La proliferación de los conceptos de especie en la biología evolucionista (2010).

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...) analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers. (shrink)

This paper is concerned with certain ontological issues in the foundations of geographic representation. It sets out what these basic issues are, describes the tools needed to deal with them, and draws some implications for a general theory of spatial representation. Our approach has ramifications in the domains of mereology, topology, and the theory of location, and the question of the interaction of these three domains within a unified spatial representation theory is addressed. In the final part we also consider (...) the idea of non-standard geographies, which may be associated with geography under a classical conception in the same sense in which non-standard logics are associated with classical logic. (shrink)

Volume 19, Issue 9, September 2019, Page 37-39.

Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, and (...) hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

A critical survey of the main philosophical theories about events and event talk, organized in three main sections: (i) Events and Other Categories (Events vs. Objects; Events vs. Facts; Events vs. Properties; Events vs. Times); (ii) Types of Events (Activities, Accomplishments, Achievements, and States; Static and Dynamic Events; Actions and Bodily Movements; Mental and Physical Events; Negative Events); (iii) Existence, Identity, and Indeterminacy.

This paper tackles the question of whether the order of concepts was still a relevant aspect of scientific rigour in the 19th and 20th centuries, especially in the case of authors who were deeply influenced by the Leibnizian project of a universal characteristic. Three case studies will be taken into account: Hermann Graßmann, Giuseppe Peano and Kurt Gödel. The main claim will be that the choice of primitive concepts was not only a question of convenience in modern hypothetico-deductive investigations, but (...) sometimes also the result of philosophical investigations onto the foundation of scienti­fic disciplines. The question of the "right" order of concepts is an ideal to be followed rather than a task that can be fulfilled, but remains nonetheless an essential part of the axiomatic enterprise. This paper aims to question whether there is in fact such a stark contrast, as there is often claimed to be in the literature, between the debates relating to the right order of concepts and the foundational questions concerning modern axiomatics. The scientific rupture determined by the appearance of hypothetico-deductive systems in mathema­tics and logic should thus not be dissociated from some relevant continuities concerning the ideal of knowledge as the search for a general theory of concepts deriving from some fundamental elements. (shrink)

This paper tackles the question of whether the order of concepts was still a relevant aspect of scientific rigour in the 19th and 20th centuries, especially in the case of authors who were deeply influenced by the Leibnizian project of a universal characteristic. Three case studies will be taken into account: Hermann Graßmann, Giuseppe Peano and Kurt Gödel. The main claim will be that the choice of primitive concepts was not only a question of convenience in modern hypothetico-deductive investigations, but (...) sometimes also the result of philosophical investigations onto the foundation of scienti­fic disciplines. The question of the "right" order of concepts is an ideal to be followed rather than a task that can be fulfilled, but remains nonetheless an essential part of the axiomatic enterprise. This paper aims to question whether there is in fact such a stark contrast, as there is often claimed to be in the literature, between the debates relating to the right order of concepts and the foundational questions concerning modern axiomatics. The scientific rupture determined by the appearance of hypothetico-deductive systems in mathema­tics and logic should thus not be dissociated from some relevant continuities concerning the ideal of knowledge as the search for a general theory of concepts deriving from some fundamental elements. (shrink)

The question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: indispensability arguments that are aimed at justifying mathematics itself; philosophical justifications of the successful application of mathematics to scientific theories; and discussions on the application of real (...) numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant’s position on the problem, several neo-Kantian perspectives, early analytic philosophy, and late 19th century mathematicians. Finally, the debate on the cogency of an application constraint in the definition of real numbers is discussed in relation to a contemporary debate in neo-logicism, in order to suggest a comparison not only with Frege’s original positions, but also with the ideas of several neo-Kantian scholars, including Hölder, Cassirer, and Helmholtz. (shrink)

Considering topology as an extension of mereology, this paper analyses topological variants of mereological essentialism (the thesis that an object could not have different parts than the ones it has). In particular, we examine de dicto and de re versions of two theses: (i) that an object cannot change its external connections (e.g., adjacent objects cannot be separated), and (ii) that an object cannot change its topological genus (e.g., a doughnut cannot turn into a sphere). Stronger forms of structural essentialism, (...) such as morphological essentialism (an object cannot change shape) and locative essentialism (an object cannot change position) are also examined. (shrink)

El autor analiza algunos aspectos de los debates sobre la democracia en la llamada transición de los años ̓80, para advertir sobre la preeminencia que adquirieron las disposiciones procedimentales y representativas —elevadas a condición metanormativa— y la conceptualización de la política como producción de consensos. Esas orientaciones impidieron una recuperación de las experiencias democráticas que la dictadura intentó cancelar, y fueron incapaces de poner en foco las figuras de subjetividad que el terror estatal y el neoliberalismo produjeron en tanto intervenciones (...) sobre las tramas sociales. Finalmente, el autor se detiene brevemente en algunas experiencias políticas democratizadoras, en tanto formas de aparición de los pueblos en estos últimos 40 años. (shrink)

Background The COVID-19 pandemic has influenced the approach to the health-disease system, raising the question about the principles of bioethics present in physician–patient relations. The principles while widely accepted may not be sufficient for a comprehensive ethical analysis. Therefore, the aim of this study was to explore the perception of these principles and the physician–patient relationship during a hospital stay through a qualitative approach. Method Sixteen semi-structured interviews took place to know the patients’ perception during their 2020 hospitalization for COVID-19. (...) The data was analyzed through the constant comparison method, creating categories and comparing them. In the end, seven categories were established and were grouped in three: bioethical principles (dignity, charity, vulnerability, autonomy), doctor-patient relationship (participant commitment, informed consent, health staff-patient relationship) and the experience of the disease (illness, the role of the family). Results The research found that most patients described a positive experience, with the feeling of having been well cared for with no sense of discrimination or injustice done. The majority also reported that their autonomy was respected in the treatment decisions. The evaluation of these attitudes is an area of opportunity, especially when the patients' vulnerability is at risk. Conclusions The ethics of virtue offers a better reflection of how human beings manifest themselves by emphasizing the development of virtuous character and behaviors that allow them to realize their values in life. Authorized by the Research Ethics Committee with registration: DI/18/105-B/3/308. (shrink)

In this paper, we extend the expressive power of the logics K3, LP and FDE with anormality operator, which is able to express whether a for-mula is assigned a classical truth value or not. We then establish classical recapture theorems for the resulting logics. Finally, we compare the approach via normality operator with the classical collapse approach devisedby Jc Beall.

Reasons-first theorists claim that facts about reasons for attitudes are normatively primitive, and that all other normative facts ultimately reduce to facts about reasons. According to their view, for example, the fact that something is good ultimately reduces to facts about reasons to favour it. I argue that these theories face a challenging dilemma due to the normativity of arational lifeforms, for instance the fact that water is good for plants. If all normative facts are, ultimately, facts about reasons for (...) attitudes, then reasons-first theorists must either (a) show that these facts do reduce to facts about reasons, or (b) concede that they do not and, instead, show that this is not a problem for their view. Both options, however, are riddled with difficulties—or so I will try to argue. (shrink)

This book describes the mathematical aspects of the semantics of programming languages. The main goals are to provide formal tools to assess the meaning of programming constructs in both a language-independent and a machine-independent way, and to prove properties about programs, such as whether they terminate, or whether their result is a solution of the problem they are supposed to solve. In order to achieve this the authors first present, in an elementary and unified way, the theory of certain topological (...) spaces that have proved of use in the modelling of various families of typed lambda calculi considered as core programming languages and as meta-languages for denotational semantics. This theory is now known as Domain Theory, and was founded as a subject by Scott and Plotkin. One of the main concerns is to establish links between mathematical structures and more syntactic approaches to semantics, often referred to as operational semantics, which is also described. This dual approach has the double advantage of motivating computer scientists to do some mathematics and of interesting mathematicians in unfamiliar application areas from computer science. (shrink)

Neste artigo, temos como objetivo perscrutar o funcionamento da verbo-visualidade de uma perspectiva dialógica do discurso, utilizando, como material de análise, uma videomontagem humorística acerca do ator político Dilma Rousseff, durante a campanha presidencial brasileira do ano de 2010. Primeiramente, procuramos refletir acerca de alguns pressupostos teórico-metodológicos de Bakhtin e de Brait, qual seja, o de verbo-visualidade, alicerçado na concepção bakhtiniana de linguagem. Em um segundo momento, detivemo-nos na análise do texto objeto do artigo, que se pautou na seguinte indagação: (...) como contribuem e funcionam os elementos verbais e visuais para produção e direcionamento de sentidos na videomontagem? Nossa hipótese é a de que os discursos carregados pela montagem criam uma trilha ideológica de sentido que leva os interlocutores a interpretar a então candidata como uma pessoa sem discurso, manipulável. Diante do exposto, tentaremos demonstrar a hipótese que sustentamos. (shrink)

In 1 Corinthians 15:35–58, Paul combats disagreement in the Corinthian church over the nature of the resurrection. Paul’s argument for the physicality of the resurrection, and his location of the individual’s resurrection within a larger transformation and restoration of all creation, changes how the contemporary church understands and teaches Christian hope, sanctification, and mission.

Recent years have featured the existence of a variety of structuralisms, with an important partition between methodological versus philosophical structuralism. Inside philosophical structuralism, many trends can be identified, corresponding to various ontological stances. We argue here that another main partition has contributed to organize structuralism in the twentieth century, rooted in different technical and theoretical interests. This partition is largely transversal to the ones classically identified. Concretely, the paper will focus on possible differences between an arithmetical and logical notion of (...) structure that can be traced back to the writings of Bertrand Russell and Rudolf Carnap, and a mathematical notion of structure, exemplified in the works by Bourbaki. This coexistence gives rise to a fundamental ambiguity that affects contemporary structuralism. Philosophically, in one case the attention is rather centered on a foundational and reductionist perspective, as featured by the Whitehead-Russell Principia and the Carnapian project of the Aufbau: the scientific construction of the world around the idea of structure. In the other, the focus is on epistemological and dynamical issues, as exemplified by two key issues in Bourbaki’s treatise: understanding the architecture of mathematics, offering a tool-kit to mathematicians. These two distinct meanings still coexist inside contemporary scientific practices and lead to different theoretical interests, as we will show thanks to various recent examples. (shrink)

Giuseppe Veronese (1854-1917) est connu pour ses études sur les espaces à plusieurs dimensions ; moins connus sont les écrits « philosophiques », qui concernent les fondements de la géométrie et des mathématiques et qui expliquent les raisons pour la construction d’une géométrie non-archimédienne (une dizaine d’années avant David Hilbert) et la formulation d’un concept de continu, qui contient des éléments infinis et infiniment petits. L’article esquissera quelques traits saillants de son épistémologie et analysera le rapport entre géométrie et intuition (...) spatiale, en comparant les conceptions de l’espace géométrique dans les œuvres de Veronese, Helmholtz et Poincaré. Selon Veronese l’espace intuitif, de même que l’espace géométrique, n’est pas euclidien et il n’a pas un nombre déterminé de dimensions : on peut donc avoir l’intuition aussi bien d’un espace à quatre dimensions que d’un continu non-archimédien. (shrink)

The turn of the last century was a key transitional period for the development of symbolic logic and scientific philosophy. The Peano school, the editorial board of the Revue de Métaphysique et de Morale, and the members of the Vienna Circle are generally mentioned as champions of this transformation of the role of logic in mathematics and in the sciences. The articles contained in this volume aim to contribute to a richer historical and philosophical understanding of these groups and research (...) areas in Italy, France and Austria. Specifically, the contributions focus on the following topics: a detailed investigation of the relation between structuralism and modern mathematics; different notions of definition and interpretation at the turn of last century; a closer understanding of the relation between the Vienna Circle, the Peano School and French philosophy in the first half of the twentieth century. (shrink)

ABSTRACT The philosophical tradition approaches to morals have their grounds predominantly on metaphysical and theological concepts and theories. Among the traditional ethics concepts, the most prominent is the Divine Command Theory (DCT). As per the DCT, God gives moral foundations to the humankind by its creation and through Revelation. Morality and Divinity are inseparable since the most remote civilization. These concepts submerge in a theological framework and are largely accepted by most followers of the three Abrahamic traditions: Judaism, Christianity, and (...) Islam: the greatest part of the human population. Holding faith and Revelation for its grounds, the Divine Command Theories are not strictly subject to the demonstration. The opponents to the Divine Command conception of morals, grounded in the impossibility of demonstration of its metaphysical and religious assumptions, have tried for many centuries (albeit unsuccessfully) to devalue its importance. They held the argument that it does not show material evidence and logical coherence and, for this reason, cannot be taken into account for scientific nor philosophical purposes. It is just a belief and, as so, should be understood. Besides these extreme oppositions, many other concepts contravene the Divine Command theories, in one or another way, in part or in full. Many philosophers and social scientists, from the classic Greek philosophy up to the present date, for instance, sustain that morality is only a construct, and thus culturally relative and culturally determined. However, this brings many other discussions and imposes the challenge to determine what is the meaning of culture, which elements of culture are morally determinant, and finally, what are the boundaries of such relativity. Moral determinists claim that everything related to human behavior, including morality, is determined, once free will does not exist. More recently, modern thinkers argued that there is a strict science of morality. However, the scientific method alone, despite explaining several facts and evidence, cannot enlighten the entire content and full meaning of ethics. Morals’ understanding requires a broader perception, and an agreement among philosophers, which they have never achieved. All of these questions have many different configurations depending on each philosophical strand, and start complex analysis and endless debates, as long as many of them are reciprocally conflictive. The universe and the atmosphere involving this thesis are the dominions of all these conceptual conflicts, observed from an objective and evolutionary standpoint. Irrespective of this circumstance and its intrinsic importance, however, these questions are far distant from the methodological approach of an analytical discussion on objective morals, what is, indeed, the aim and scope of this work. We should briefly revisit these prominent traditional theories because this thesis shelters a comparative study, and its assumptions at least differ profoundly from all traditional theories. Therefore, it becomes necessary offering direct and specific elements of comparison to the reader, for the right criticism, dispensing interruptive researches. However, even revisiting the traditional theories, for this comparative and critical exposure purpose, they will be kept by the side of our main concerns, as “aliena materia.” Irrespective of the validity of any or all of the elements of this discussion, and their meaning as the philosophical universe of this thesis, the purpose of this work is demonstrating and justifying the existence and meaning of prehistoric moral archetypes arisen directly from the very first social needs and efforts for survival. These archetypes are the definition of the essential foundation of ethics, its aggregation to the collective unconscious and corresponding logic organization and transmission to evolutionary stages of the human genome and different relations space-time, irrespective of any contemporary experience of the individuals. The system defined by these archetypes composes an evolutionary human social model. Is this a metaethical position? Yes, it is. Moreover, as in any metaethical reasoning, we should look carefully for the best and coherent routes, as the Analytical Philosophy offers them. Thus, this work should reasonably demonstrate that morals are not a cultural product of the civilized men or modern societies and that despite being subject to several cultural relative aggregations and subtractions, its essential foundations are archetypal and have never structurally changed. This reasoning induces that morality is an original attribute of the “homo sapiens”; it is not a property and nor an accident: it integrates the human essence and belongs to the realm of the ontological human identity. The human phenomena is a continuing process, playing its role between random determination and free will, and we need to question how morality began and how did it come to us in the present. (shrink)

The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required (...) by the modern version of the theory of proportions. Secondly, Grassmann’s conception of mathematical knowledge will be investigated. Parting from the traditional definition of mathematics as a science of magnitudes, Grassmann considered mathematical forms as particulars rather than universals: the classification of the branches of mathematics was thus based on different operational rules, rather than on empirical criteria of abstraction or on the distinction of different species belonging to a common genus. It will be argued that a different notion of generalization is thus involved, and that the knowledge of mathematical forms relies on the understanding of the rules of generation of the forms themselves. Finally, the paper will analyse if Grassmann’s approach in the first edition of the Ausdehnungslehre should be explained in terms of the notion of purity of method, and if it clashes with Grassmann’s later conventionalism. Although in the second edition the features of the operations are chosen by convention, as it is the case for the anti-commutative property of the multiplication, the choice is oriented by our understanding of the resulting forms: a simplification in the algebraic calculus need not correspond to a simplification in the ‘dimensional’ interpretation of the result of the multiplicative operation. (shrink)

"Perhaps not all the stories that follow are true. They could, however, be true, and the Reader is invited to ponder this." So begins _Insurmountable Simplicities_, Roberto Casati and Achille Varzi's colorful incarnation of the many philosophical conundrums that hide in the wrinkles of everyday life. Why do mirrors seem to invert left and right but not up and down? How do we know whether strawberries taste the same for everyone? Where is it written that we must observe the (...) law, and if it is not written, why should we observe it? What if we could swap brains-or the rest of our bodies? _Insurmountable Simplicities_ is filled with stories, dialogues, and epistolary exchanges that cover a range of themes-such as personal identity, causality and responsibility, fortune, the nature of things, the paradoxes of time and space, the interface between logic and language-in captivating and inventive ways. Clear, concise, and intellectually engaging, this internationally acclaimed book brilliantly demonstrates that the beauty of philosophy resides in its thorough engagement with the simplicities of the world, insurmountable as they might initially appear. (shrink)

The paper advocates an epistemological interpretation of the Peano School axiomatics. The construction of axiom systems is presented as a cognitive enterprise unveiling the internal dynamics, evolution, and architecture of axiomatic systems as well as connections to applications. This approach reveals that the study of the relation between axioms and theorems not only serves to reduce a theory to a minimum number of principles and increase the certainty or justification of the latter, but also to study alternative settings of a (...) given mathematical theory. The paper explains why the Peanian conception of axiomatics differs from axiomatic deductivism, if-thenism and scientific deductivism, and analyzes Peano’s conception of rigor. The paper also postulates that Peano had a set-theoretic but also an algorithmic conception of function, and prioritized functions over relations. Two seemingly contradictory aspects of Peano’s axiomatics, global and local reasoning, are illustrated through the examples of rigor and simplicity. Finally, the paper claims that Peano’s axiomatics was a dynamic mathematical enterprise rather than an a priori axiomatization grounded on logical principles, which explains the difference with respect to Frege’s project. (shrink)