It is often assumed that concepts from the formal sciences, such as mathematics and logic, have to be treated differently from concepts from non-formal sciences. This is especially relevant in cases of concept defectiveness, as in the empirical sciences defectiveness is an essential component of lager disruptive or transformative processes such as concept change or concept fragmentation. However, it is still unclear what role defectiveness plays for concepts in the formal sciences. On the one hand, a common view sees formal (...) concepts to be protected against defects because of their precise and stable nature. On the other, studies going back as far as Lakatos (1963) showcase the changeability of such concepts. -/- In this paper, I will investigate if and how defectiveness based on the occurrence of inconsistencies can appear with formal concepts. To make the case as strong as possible, I employ a strict notion of formal concept that assumes the concept to have a fixed and definite extension. I will show that there are indeed certain types of defectiveness that cannot occur with such concepts; but that there are other types of defectiveness that do occur. This means that while formal concepts have to be treated differently than non-formal concepts, questions about defectiveness---as raised in the conceptual engineering debate and philosophy of science---can still be applied to them. I will highlight this point by showing how formal sciences have special strategies available to them that allow them to resolve defectiveness of their concepts in a flexible and informative manner. (shrink)

This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins’ claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy’s naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how set-theoretic practice can be said to introduce models as (...) new entities. In conclusion, I argue for a view that takes both sets and models to be fundamental entities in set theory. (shrink)

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...) seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist. (shrink)

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni-Friedman :77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

In this article we show that Morse-Kelley class theory provides us with an adequate framework for class forcing. We give a rigorous definition of class forcing in a model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$$$ \end{document} of MK, the main result being that the Definability Lemma can be proven without restricting the notion of forcing. Furthermore we show under which conditions the axioms are preserved. We conclude by proving that Laver’s Theorem does not hold for (...) class forcings. (shrink)

We survey recent developments in the theory of class forcing for- malized in the second-order set-theoretic setting.

This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John (...) Templeton Foundation from January 2013 until September 2015. (shrink)

In this article we introduce and study hyperclass-forcing in the context of an extension of Morse-Kelley class theory, called MK∗∗. We define this forcing by using a symmetry between MK∗∗ models and models of ZFC− plus there exists a strongly inaccessible cardinal. We develop a coding between β-models ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} of MK∗∗ and transitive models M+ of SetMK∗∗ which will allow us to go from ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} to M+ and vice versa. So instead of forcing with a hyperclass in MK∗∗ we can force over the corresponding SetMK∗∗ model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK∗∗ model M+ can be forced to look like Lκ∗[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L_{\kappa ^*}[X]$$ \end{document}, where κ∗ is the height of M+, κ strongly inaccessible in M+ and X ⊆ κ. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK∗∗, which will in turn be an extension of the original ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document}. Our main result combines hyperclass forcing with coding methods of Beller et al. and Friedman to show that every β-model of MK∗∗ can be extended to a minimal such model of MK∗∗ with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic. (shrink)

Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure and (...) aims of the Second Philosopher’s investigation into set-theoretic methodology; pro- vides a platform to analyze the Second Philosopher’s methods themselves; and can be applied to further questions in the philosophy of set theory. (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

In the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the better-known dialogical logic, and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s well-known rejection of the actual infinite, as stated in Lorenzen, was not a major motivation for operative (...) logic and mathematics. Rather, we argue that a shift happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operationism is concerned with a critique of the Cantorian notion of set and with related questions about the notions of countability and uncountability; it is only later that his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

The book deepens our understanding of Arendt's conception of the role of violence in her political theory. But it also uses her work as a provocation to think about how we might engage with, build on, or criticize contemporary ideas of the political that have drawn on Arendtian themes-notably via the notion of "agonal" or "agonistic" politics as theorized in recent years by thinkers such as Chantal Mouffe and Bonnie Honig-and how we can read Arendt in different ways to challenge (...) or further our understanding of the political. (shrink)

How can dispute records shed light on the study of dispute settlement processes and their social and political underpinnings? This volume addresses this question by investigating the interplay between record-making, disputing process, and the social and political contexts of conflicts. The authors make use of exceptionally rich charter materials from the Iberian Peninsula, Italy, and Scandinavia, including different types of texts directly and indirectly related to conflicts, in order to contribute to a comparative survey of early medieval dispute records and (...) to a better understanding of the interplay between judicial and other less formal modes of conflict resolution. Contributors are Isabel Alfonso, José M. Andrade, François Bougard, Warren C. Brown, Wendy Davies, Julio Escalona, Kim Esmark, Adam J. Kosto, Juan José Larrea, André Evangelista Marques, Josep M. Salrach, Igor Santos Salazar, and Francesca Tinti. (shrink)

In the first edition of The Gay Science (GS), Nietzsche proposes that we treat knowledge as unconditionally valuable and life as a tragic quest for truth. In the second edition of GS, he seems to retract this proposal, suggesting that we substitute “incipit parodia” for “incipit tragœdia.” But Nietzsche does not say what he means by “parody,” or what role he believes it should play in our evaluative lives. This article proposes that by introducing parody into GS, Nietzsche intends not (...) only to correct the original text by mocking his earlier treatment of science as tragic, but also to designate parody as a basis for new tasks, duties, and values beyond what science can offer. This article speculates that parodic value creation might consist in the imitation of behaviors and practices associated with preexisting values without the reproduction of the beliefs and feelings corresponding to those practices. (shrink)

Le « Bulletin bibliographique de philosophie politique et sociale » est réalisé pour la revue Philosophique par une équipe de rédactrices et de rédacteurs. Il est coordonné par Caroline Guibet Lafaye (CNRS) et Fabien Ferri (Université de Franche-Comté) au sein de Centre de Documentation et de Bibliographie Philosophiques de l’Université de Franche-Comté (UR 2274 CDBP–Logiques de l’Agir), et propose de brèves recensions d’ouvrages récemment parus, réparties dans diverses rubriques thématiques....

Énoncer que la civilisation traverse, en ce moment, une crise extrêmement grave est devenu un lieu commun ; cependant, cela demeure une source d'interrogations pour prétendre à en sortir. Cette situation de crise laisse craindre ou espérer un changement profond. Elle est aussi celle des mots et celle des images, aucun domaine n'échappe à la crise. Et si les frontières de l'oeuvre d'art ne sont pas clairement définies, celles du sujet ne le sont pas davantage. Les frontières mêmes du sujet (...) sont malléables, dépendantes des contrées extérieures auxquelles il se frotte sans cesse. Aussi, le sujet lui-même est un sujet en mouvement dans une société toujours plus rapide, plus performante, plus financiarisée. Cependant, l'agitation ou le frémissement qui habite le sujet n'a pas pour seule origine un univers sociétal économique, les espaces ou objets esthétiques, naturels ou méditatifs influent également. Ainsi, en tant que créateur ou spectateur, le sujet se confronte à l'oeuvre d'art. S'il peut être touché, provoqué parfois, par l'extérieur, il est aussi travaillé de l'intérieur. Les arts et la littérature sont ici convoqués pour penser le rapport du sujet à son identité et au monde ; des études de cas porteront donc sur les démarches des artistes. En effet, en quoi les arts visuels et les récits contribuent-ils à la construction du sujet? En quoi la rencontre avec l'oeuvre agit-elle sur l'identité du sujet? Comment le sujet est-il « à l'oeuvre » lorsqu'il entreprend de rencontrer ou de créer des oeuvres? (shrink)

Experience offers a reading experience like no other. A heat-sensitive cover by Olafur Eliasson reveals words, colors, and a drawing when touched by human hands. Endpapers designed by Carsten Holler are printed in ink containing carefully calibrated quantities of the synthesized human pheromones estratetraenol and androstadienone, evoking the suggestibility of human desire. The margins and edges of the book are designed by Tauba Auerbach in complementary colors that create a dynamically shifting effect when the book is shifted or closed. When (...) the book is opened, bookmarks cascade from the center, emerging from spider web prints by Tomas Saraceno. Experience produces experience while bringing the concept itself into relief as an object of contemplation. The sensory experience of the book as a physical object resonates with the intellectual experience of the book as a container of ideas. Experience convenes a conversation with artists, musicians, philosophers, anthropologists, historians, and neuroscientists, each of whom explores aspects of sensorial and cultural realms of experience. The texts include new essays written for this volume and classic texts by such figures as William James and Michel Foucault -- Provided by the publisher. (shrink)

Inspired by Voltaire's advice that a text needs to be concise to have real influence, this anthology contains fiery extracts by forty eighteenth-century authors, from the most famous philosophers of the age to those whose brilliant writings are less well-known. These passages are immensely diverse in style and topic, but all have in common a passionate commitment to equality, freedom, and tolerance. Each text resonates powerfully with the issues our world faces today. Tolerance was first published by the Société française (...) d'étude du dix-huitième siècle (the French Society for Eighteenth-Century Studies) in the wake of the Charlie Hebdo assassinations in January 2015 as an act of solidarity and as a response to the surge of interest in Enlightenment values. With the support of the British Society for Eighteenth-Century Studies, it has now been translated by over 100 students and tutors of French at Oxford University."--Publisher's website. (shrink)

Moral injury emerged in the healthcare discussion quite recently because of the difficulties and challenges healthcare workers and healthcare systems face in the context of the COVID-19 pandemic. Moral injury involves a deep emotional wound and is unique to those who bear witness to intense human suffering and cruelty. This article aims to synthesise the very limited evidence from empirical studies on moral injury and to discuss a better understanding of the concept of moral injury, its importance in the healthcare (...) context and its relation to the well-known concept of moral distress. A scoping literature review design was used to support the discussion. Systematic literature searches conducted in April 2020 in two electronic databases, PubMed/Medline and PsychInfo, produced 2044 hits but only a handful of empirical papers, from which seven well-focused articles were identified. The concept of moral injury was considered under other concepts as well such as stress of conscience, regrets for ethical situation, moral distress and ethical suffering, guilt without fault, and existential suffering with inflicting pain. Nurses had witnessed these difficult ethical situations when faced with unnecessary patient suffering and a feeling of not doing enough. Some cases of moral distress may turn into moral residue and end in moral injury with time, and in certain circumstances and contexts. The association between these concepts needs further investigation and confirmation through empirical studies; in particular, where to draw the line as to when moral distress turns into moral injury, leading to severe consequences. Given the very limited research on moral injury, discussion of moral injury in the context of the duty to care, for example, in this pandemic settings and similar situations warrants some consideration. (shrink)

The practice of relationship itself is seen to be the central vehicle through which human beings learn and understand themselves, others, and the world around them. The politics of sexual difference insists that the flourishing of sexual difference in both women and men, girls and boys, relies, ultimately, on both sexes taking up the challenge to rethink themselves and the world. This chapter explores the emergence of a whole new philosophical idea, brought into being by Luce Irigaray in the context (...) of the women's movements of the 1960s and 1970s in Europe. It then outlines how this idea was developed in Italy and Spain to restore the work of the mother to the origins of what we understand as education and philosophy, emphasising the need to bring relationship itself as the vehicle for this process onto centre stage and keep it there. (shrink)