The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...) type theory. It also gives the new system of foundations a distinctly structural character. (shrink)

An affirmative answer is given to the question quoted in the title.

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired (...) by the groupoid model of (intensional) Martin–Löf type theory [14] due to Hofmann and Streicher [9]. In particular, we show that a form of Martin–Löf type theory can be soundly modelled in any model category. This result indicates moreover that any model category has an associated “internal language” which is itself a form of Martin-Löf type theory. This suggests applications both to type theory and to homotopy theory. Because Martin–Löf type theory is, in one form or another, the theoretical basis for many of the computer proof assistants currently in use, such as Coq and Agda (cf. [3] and [5]), this promise of applications is of a practical, as well as theoretical, nature. This paper provides a precise indication of this connection between homotopy theory and logic; a more detailed discussion of these and further results will be given in [20]. (shrink)

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this (...) case 2.In the present work, we generalize the entire arrangement from propositional to first-order logic, using a representation result of Butz and Moerdijk. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed primarily in the form of a contravariant adjunction, is established by homming into a common dualizing object, now Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology.The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we give an elementary proof of Butz and Moerdijkʼs result in logical terms. (shrink)

Wittgenstein's philosophy of mathematics has long been notorious. Part of the problem is that it has not been recognized that Wittgenstein, in fact, had two chief post-Tractatus conceptions of mathematics. I have labelled these the calculus conception and the language-game conception. The calculus conception forms a distinct middle period. The goal of my article is to provide a new framework for examining Wittgenstein's philosophies of mathematics and the evolution of his career as a whole. I posit the Hardyian Picture, modelled (...) on the Augustinian Picture, to provide a structure for Wittgenstein's work on the philosophy of mathematics. Wittgenstein's calculus period has not been properly recognized, so I give a detailed account of the tenets of that stage in Wittgenstein's career. Wittgenstein's notorious remarks on contradiction are the test case for my theory of his transition. I show that the bizarreness of those remarks is largely due to the calculus conception, but that Wittgenstein's later language-game account of mathematics keeps the rejection of the Hardyian Picture while correcting the calculus conception's mistakes. (shrink)

This brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of (...) ways, including topologically, type-theoretically, and through variation. Many previous results from topos theory involving realizability, permutation, and sheaf models of set theory are subsumed, and the prospects for further such unification seem bright. (shrink)

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

Oliver Cromwells historical reputation underwent significant change during the nineteenth century. Writers such as Thomas Carlyle were prominent in this reassessment, creating a Cromwell that found particular support among Nonconformists in the north of England. Projects to memorialize Cromwell included the raising of public statues. This article traces the history of the Manchester statue, the first major outdoor statue of Cromwell to be unveiled in the country. The project originated among Manchester radical Liberal Nonconformists in the early 1860s but was (...) not realized until 1875. It was the gift of Elizabeth Heywood; the sculptor was Matthew Noble. The project, including its intended site in Manchesters new Town Hall, was contentious, exposing political and religious divisions within the community, reinforcing the view that the reassessment of Cromwells place in the making of modern Britain was far from settled. (shrink)

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E. In contrast to the well-known interpretation of higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE, but rather by a suitable complete Heyting algebra H. The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from surjective geometric morphisms (...) f : F → E, where H = f∗ΩF. The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are not longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion. (shrink)

The purpose of the Cambridge Edition is to offer translations of the best modern German edition of Kant's work in a uniform format suitable for Kant scholars. When complete the edition will include all of Kant's published writings and a generous selection from the unpublished writings such as the Opus postumum, handschriftliche Nachlass, lectures, and correspondence. This volume contains the first translation into English of notes from Kant's lectures on metaphysics. These lectures, dating from the 1760s to the 1790s, touch (...) on all the major topics and phases of Kant's philosophy. Most of these notes have appeared only recently in the German Academy Edition; this translation offers many corrections of that edition. As is standard with the volumes in the Cambridge Edition there is an extensive editorial apparatus, including extensive linguistic and explanatory notes, a detailed subject index, and glossaries of key terms. (shrink)

This paper continues a conversation about Wittgenstein’s picture of language and meaning and its potential applications for educational theorising. It takes the form of a response to Wolff-Michael Roth’s earlier paper “Heeding Wittgenstein on “understanding” and “meaning”: A pragmatist and concrete human psychological approach in/for education,” in which Roth problematizes the use of the terms “understanding” and “meaning” in education discourse and proposes their abandonment. Whilst we agree with Roth about a series of central points, at the same time we (...) maintain that he has taken his argument in directions antithetical to our reading of Wittgenstein’s work. We offer four points of departure, exploring themes of: appropriate questioning; eliminativism; language-games and grammar; and productivity, explanation, and a science of learning. We conclude by discussing ways consistent with Wittgenstein’s thought to go on in thinking about education. (shrink)

Unemployment is a major concern of societies and people around the world. In addressing this phenomenon, the literature has suggested a change in unemployed people’s perceptions of this transition period. In this paper, we apply a differential approach to explore the concept of unemployment normalization, an individual emotional regulation process. The results show how the global socioeconomic context and some individual and psychological variables influence the normalization of unemployment. Thus, the age of the person but also work involvement, coping strategies, (...) locus of control, and level of self-esteem have indirect differential effects, mediated by unemployment normalization dimensions, on unemployed people’s perceived health. Only neuroticism has a direct link to subjective well-being. These results offer a new understanding of the perception of unemployment and are also discussed in the area of career and vocational counseling. (shrink)

The failed criterion of logical truth proposed by Carnap in the Logical Syntax of Language was based on the determinateness of all logical and mathematical statements. It is related to a conception which is independent of the specifics of the system of the Syntax, hints of which occur elsewhere in Carnap’s writings, and those of others. What is essential is the idea that the logical terms are invariant under reinterpretation of the empirical terms, and are therefore semantically determinate. A certain (...) objection to Carnap’s version of the invariance conception has been repeated several times in the literature. It is based on Gödel incompleteness, which is puzzling, since Carnap’s Syntax is otherwise quite careful to take account of Gödel. We show here that, in fact, the objection is invalid and is based on a confusion about determinacy. Sorting this out is worthwhile not only for the purpose of better understanding Carnap’s thinking in the Syntax, though. The invariance conception is also related to recent work in the philosophy of logic regarding “logicality”—the characterization of logical concepts—following a proposal of Tarski’s. It is even connected to some very recent developments in the foundations of mathematics. (shrink)

The fundamental duality theories relating algebra and geometry that were discovered in the mid-twentieth century can also be applied to logic via its algebraization under categorical logic. They thereby result in known and new completeness theorems. This idea can be taken even further via what is sometimes called “categorification” to establish a new connection between logic and geometry, a glimpse of which can also be had in topos theory.

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) so as to shed new light on the relevant strengths and limits of higher-order logic. (shrink)

Contributions from an international team of experts revisit and update the concept of linguistic ecology in order to critically examine current theoretical approaches to language contact. Language is understood as a part of complex socio-historical-cultural systems, and interaction between the different dimensions and levels of these systems is considered to be essential for specific language forms. This book presents a uniform, abstract model of linguistic ecology based on, among other things, two concepts of Edmund Husserl's philosophy. It considers the individual (...) speaker in the specific communication situation to be the essential heuristic basis of linguistic analysis. The chapters present and employ a new, transparent and accessible contact linguistic vocabulary to aid reader comprehension, and explore a wide range of language contact situations in Europe, Africa, the Middle East, Latin America, Asia and the Pacific. This book will be fascinating reading for students and researchers across contact linguistics and cultural studies. (shrink)

In 2016, a Cincinnati Zoo worker shot and killed a Western lowland gorilla to protect a three-year-old boy who had fallen into the animal’s enclosure. This incident involves a variant of the classical trolley problem, one in which the death of a human being on the main track might be avoided by selecting an alternate track containing a member of an endangered species. This problem raises two important questions for environmental ethics. First, what, if anything, imbues a human child with (...) greater value than a member of a critically endangered species? Second, is it ethical for zoos to house species such as gorillas? With regard to the first question, at a minimum, it is not obvious whether a human child or a gorilla has the greater value. With regard to the second question, an appeal can be made to Japanese environmental philosophy, particularly the ethical paradigm of kyōsei and the aesthetics of Yuriko Saitō. Members of endangered species have intrinsic value, which entails human obligations to protect the species as a whole and minimize harm to its specimens. (shrink)

THE RANDLORDS by Geoffrey Wheatcroft. New York: Atheneum, 1986. 314 pp., $17.95. CAPITALISM AND APARTHEID: SOUTH AFRICA, 1910?1984 by Merle Lipton. Totowa, N.J.: Rowman & Allanheld, 1985. 400 pp., $19.95. THE ECONOMICS OF THE COLOUR BAR by W. H. Hutt. London: Institute of Economic Affairs, 1964.

We are currently developing a short, online ethics course that attempts to teach students why well-intentioned people act unethically and what students can do to decrease the likelihood that they will find themselves in the middle of an ethical crisis in the future. Most of the well-known case studies in professional ethics textbooks concern ethical failures that do not involve difficult ethical choices. When our students read these case studies, it is not difficult for them to determine what went wrong (...) or what the players in the cases ought to have done. And students don’t need to read Kant or Mill to make these determinations. Importantly, the agents who acted wrongly usually seem to be ethical people. Rarely do cases involve moral monsters. In a similar vein, most cases of research misconduct are cut-and-dry, and most of our students who plagiarize did not enroll in the course intending to plagiarize. Rather, something went wrong. Professional ethics classes neglect to address an important practical question: Why do well-intentioned people find themselves in the middle of ethical crises and how can our students decrease the likelihood that they will find themselves in an ethical crisis in the future? We are developing a course that answers this question. Our course is built around the findings of John Dorris and other situationists. After teaching our students about situationism, we teach them a two-level theory of cognition that may help them avoid acting unethically. (shrink)

This article assesses online newspapers in Europe from a media evolutionary perspective, ten years after the introduction of the World Wide Web. Comparing print and online front pages of 51 newspapers in 14 countries in 2003, we argue that online newspapers complement print newspapers in modest ways. Online, publishers put more emphasis on service information, offer additional news items, that nonetheless report on similar topics in similar ways, and add personal interactivity, content selectivity and real-time news to the print news (...) offering. One subset of online newspapers charges for services, and offers more content and personal interactivity. Another, partly overlapping subset offers more original news; in a short and anonymous format. Overall, however, online newspapers in Europe make up a heterogeneous group, suggesting that online newspapers still have to find their definite form and role in the European news market. (shrink)

This paper provides a reading of Nishitani’s philosophy of culture. It argues that the advent of nihilism is the logical conclusion of what will be called the “fracturing of culture” in which philosophy and religion lose their creative force to revitalize a cultural tradition as the sense of being-in-time that forms the historical life of a historical world. Section two sets out the paradoxical nature of Nishitani’s philosophy of culture as both a transcendental and existential project. Section three draws attention (...) to the fact that a concept of culture always belongs to a concrete culture as part of its own understanding of itself. Section four interprets the advent of nihilism in terms of a crisis of culture that ensues from the loss of the existential understanding of history that grounds a cultural world, that forms the standpoint of a historical event of worlding, the historical life that we are. Section five examines Nishitani’s project in terms of the reappropriation of a lost tradition through an existential receptive reinterpretation of Buddhism. Section six argues that for Nishitani the advent of nihilism comes about as a result of the negation of myth by science and thus the overcoming of nihilism comes about through the recollection of myth. Section seven determines the nature of myth for Nishitani through a comparison with Cassirer’s and Heidegger’s understanding of myth. Three elements of myth are focused on: 1) myth as an existential confrontation with being-in-the-world; 2) this existential being-in-the-world forms an existential being-in-time; and 3) myth is the position of the imagination, a thinking by means of form-images [keizō 形像] (Bild, image). Section seven considers the difference between Cassirer’s and Nishitani’s respective accounts of myth. Section nine examines the nature and function of philosophy, science, and religion in terms of their relation to myth and in terms of how they understand interdependent origination. Section, ten ends the paper by considering what is called the “fracturing” of culture and the advent of nihilism. (shrink)

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using (...) regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally Cartesian closed categories. We also show how to interpret first-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specifically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic first-order logic, in the sense that a formula from the fragment is derivable in intuitionistic first-order logic if, and only if, its interpretation in dependent type theory is inhabited. As a consequence, a modified double-negation translation into type theory (without bracket types) is complete, in the same sense, for all of classical first-order logic. (shrink)

Although there is much feminist work that has examined the intersection of gender and neoliberalism, critical work on men and masculinities remains underdeveloped in this area. This article suggests that complexity theory is a crucial resource for a critical analysis of the ways in which masculinities contribute to the ongoing maintenance of neoliberal socio-economic systems. Critical work on neoliberalism and capitalist economics has recently been drawn to complex systems theory, as evidenced by the work of scholars such as Sylvia Walby, (...) William Connolly and Brian Massumi. Their work produces important insights into neoliberalism, but does not develop a sustained reflection on the place of men and masculinities in this domain. In order to develop a critical account of the relation of masculinity to complexity, the article draws on the work of Judith Butler and Bonnie Mann. It suggests that Butler’s theorising on precariousness contains important resources for understanding how hegemonic masculinities are positioned in relation to the complexity of neoliberal systems, as illustrated in Mann’s concept of ‘sovereign masculinity’. Finally, drawing on two different examples of the enactment of masculinities in neoliberal contexts, the article argues that hegemonic forms of masculinity can be understood as technologies for the amelioration of the complexities and insecurities generated by neoliberal markets. (shrink)

Masturbation is a neglected topic in debates around biopower and biopolitics. This article takes Michel Foucault’s recasting of the idea of a regulatory, population-level form of biopower in terms of ‘mechanisms of security’ as its starting point for an investigation into the ways in which bodies enter into and are reshaped by biopolitical discourses on masturbation. While the notion of security faded from view in favour of Foucault’s better known focus on governmentality, this article argues that there is value in (...) recovering the concept of security in the context of a genealogy of modern bodies. Specifically, it explores the possibility that a biopolitical perspective on security operates not only above, but also below the disciplining of individual masturbating bodies. The article proceeds, initially, via an examination of contemporary studies of masturbation, arguing that they largely neglect the material dynamism of bodies. The main focus, however, is on rereading some of the key works in the historical anti-masturbation literature from a complexity perspective. It is shown that these texts engage with a ‘population’ of vital forces and affects that must be regulated if life is to remain secure, and which circulate below the level of individual bodies in relation to a complex milieu. Finally, the article claims that men’s bodies appear as crucial sites of biopolitics, and that normative forms of masculinity can be regarded as interventions into embodiment that are designed to nullify or regulate complexity. (shrink)

The issue of representing access control requirements continues to demand significant attention. The focus of researchers has traditionally been on developing particular access control models and policy specification languages for particular applications. However, this approach has resulted in an unnecessary surfeit of models and languages. In contrast, we describe a general access control model and a logic-based specification language from which both existing and novel access control models may be derived as particular cases and from which several approaches can be (...) developed for domain-specific applications. We will argue that our general framework has a number of specific attractions and an implication of our work is to encourage a methodological shift from a study of the particulars of access control to its generalities. (shrink)

This interactive teaching workshop explored what it means to “teach from the heart.” It adopted the format of the wisdom circle to ask participants to share peak teaching experiences so that they could reflect on what their stories reveal about their inner selves as teachers. The hope was that, by learning how to speak with their “authentic” voices, participants could gain the insight and courage needed to better connect with their students as co-learners.

Many political philosophers today think of justice as fundamentally about fairness, while those who defend capital punishment typically hold that justice is fundamentally about desert. In this paper I show that justice as fairness calls for capital punishment because the continued existence of murderers increases unfairness between themselves and their victims, increasing the harm to murdered persons. Rescuing murdered persons from increasing harm is prima facie morally required, and so capital punishment is a prima facie duty of society and sentencing (...) judges.1. (shrink)

One of the great virtues of LEGO is that it has the potential to make any one of us a Master Builder. In The LEGO Movie, Wyldstyle and Batman present a case study in the value of precision in language. If your basic two‐by‐four brick is the "uncarved block", LEGO makes "carved" ones too: cockpits, irregular minifig heads, all those cool bits. In the case of the LEGO brick, the less it's like a toy, the better we can play with (...) it. According to the Daoists, government is no different: the less it does, the better it works. One of the most telling differences between ancient Greek philosophy and ancient Chinese philosophy lies in their attitudes toward imprecision. LEGO Master Builders understand this. This is the Daoist model of government: commit yourself to the goal, be willing to be flexible, and voila, you've freed yourself of the tyranny of the instruction booklet. (shrink)

Abstract:Watsuji Tetsurō's model of human existence (ningen sonzai) and his ethical principle of selfless solicitude (kokorozukai) imply not only a broadly permissive position on reproductive rights but a clearer vision of pregnancy and the fetus, and also a deeper moral critique of the anti-abortion movement.