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

A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems, as well as numerous examples and exercises.

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)

A Brief History of Analytic Philosophy: From Russell to Rawls presents a comprehensive overview of the historical development of all major aspects of analytic philosophy, the dominant Anglo-American philosophical tradition in the twentieth century. Features coverage of all the major subject areas and figures in analytic philosophy - including Wittgenstein, Bertrand Russell, G.E. Moore, Gottlob Frege, Carnap, Quine, Davidson, Kripke, Putnam, and many others Contains explanatory background material to help make clear technical philosophical concepts Includes listings of suggested further readings (...) Written in a clear, direct style that presupposes little previous knowledge of philosophy. (shrink)

Although The Structure of Scientific Revolutions is one of the most influential books of this century, its author, Thomas Kuhn, is notorious for disavowing most of the consequences wrought by his text. Insofar as these consequences have appeared "radical" or "antipositivist," this article argues that they are very misleading, and that Kuhn's complaints are therefore well placed. Indeed, Kuhn unwittingly succeeded where Daniel Bell's The End of Ideology tried and failed, namely, to alleviate the anxieties of alienated academics and defensive (...) policy-makers by teaching them that they could all profit from solving their own paradigmatic puzzles. The influence of tructure is traced from the philosophy of science into the social sciences and science policy. Special attention is paid to the import of the General Education in Science curriculum at Harvard, in which Kuhn taught for most of the period prior to writing tructure. Harvard President James Conant had designed this curriculum in order to keep "pure science" in the good favor of the American public, in whose eyes it suffered after the use of the atomic bomb. While Conant was keen to stress the distinctiveness of science from other social practices, Kuhn's model seemed to provide a blueprint for reconstituting any practice as a science. This enabled potentially antiscientific academics to become scientists themselves, thereby neutralizing any radical challenges to the ends of scientific inquiry. The article concludes by reconstructing some of the inchoate possibilities for radical critique that Kuhn's success preempted, and by making some suggestions for how they may be recovered in the present academic environment. (shrink)

This paper is the second 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)

An exploration of the production, transmission, and mutation of affective tonality—when sound helps produce a bad vibe.

The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Per Martin-Löf into homotopy theory and higher-dimensional category theory.

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)

In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure. Specifically, basic categories of classes and categories of classes are axiomatized and shown to provide models of the aforementioned set theories. Finally, models of these theories are constructed in the category of (...) ideals. (shrink)

We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full (...) axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo—Fraenkel set theory (IZF). (shrink)

This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting (...) framework is intended as a setting for the logical and categorical study of relative computability. (shrink)

This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in [2] by introducing a (...) new and perhaps more natural notion of ideal, and in the class theory of part three. (shrink)

The development and publication of Diagnostic and Statistical Manual of Mental Disorders, fifth edition produced a peak in mainstream media interest in psychiatry, and a large and generally critical set of scientific commentaries. The coverage has focused mainly on the expansion of some categories, and loosening of some criteria, which together may lead to more people receiving diagnoses, and accompanying accusations of the medicalisation of normal living. Instructions given to members of DSM-5 work groups appear to have encouraged this.1 This (...) has not been the only source of disquiet. The influence of the pharmaceutical industry on the members of the appointees to DSM diagnostic work groups, which constructed the new rubrics, has caused concern,2 as remuneration and insurance coverage in the USA depends partly on DSM diagnosis, and the pharmaceutical industry has been accused of encouraging the development of new diagnoses as a way to increase profitability.3 Societal impact has been an explicit part of the development of DSM-5, as outlined in the article in this issue.4 Reasons for decisions by DSM-5 workgroups have included enabling sufferers to receive treatment which would not be available without a DSM diagnosis , or …. (shrink)

We develop a vision of consumer responsibility in purchasing decisions in light of the Coalition of Immokalee Workers’ boycotts. These boycotts succeeded in convincing large fast food companies and national supermarket chains to pay tomato growers a penny more per pound, to improve working conditions and wages for pickers. The C.I.W. efforts to generate consumer support eschewed claims associated with rule-based obligations in favor of appeals more typically associated with virtue and caring ethics. The strategies encouraged consumers to understand the (...) plight of tomato pickers and to extend concern in an effort to improve the world. These strategies are associated more with encouragement to contribute to the social good rather than claims that in refusing to help, consumers would fail to fulfill an obligation. Insights from virtue ethics and caring ethics are offered as a model for a broader account of consumer social responsibility. (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)

ABSTRACT The influence of Cassirer's work on Bakhtin's writings from the 1930s has been studied in some detail but scholars have not examined Bakhtin's early work, Toward a Philosophy of the Act (K filosofii postupka), in connection with Cassirer's philosophy. The article first reveals how attuned Bakhtin was with the intellectual Zeitgeist not only of his own times, but also that of the 20th century. The uncanny intellectual harmony between the ideas of Bakhtin and Cassirer can be seen at the (...) very beginning of Bakhtin's career. The two thinkers are united in their reception, transformation, and attempt to reconcile two antithetical philosophical positions dominant at the beginning of the 20th century: the transcendental philosophy of Kant, Neo-Kantianism, and the Lebensphilosophie of Simmel, Bergson, and Heidegger. Bakhtin and Cassirer were alarmed by the cultural pessimism and potential nihilism inherent in the position of Lebensphilosophie. Next, the author shows ways in which Bakhtin's and Cassirer's ideas resonate with those of the later 20th century Jacques Derrida. (shrink)

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)

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)

Throughout the text, the reader is reminded of the contribution of the Christian faith to matters of development and ethics.

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)

Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the (...) literature for details. Each of the three steps considered below is based on more recent logical research than the preceding one. The first step from sets to types is essentially the familiar idea of set theoretic semantics for a syntactic system, i.e. giving a model; we take a brief glance at this step from the current point of view, mainly just to fix ideas and notation. The second step from types to categories is known to categorical logicians as the construction of a “syntactic category”; we give some specifics for the benefit of the reader who is not familiar with it. The third step from categories to sets is based on quite recent work, but captures in a precise way an intuition from the early days of foundational studies. With these pieces in place, we can then draw some conclusions regarding the differences between the three schemes, and their relative merits. In particular, it is possible to state more precisely why the methods of category theory are more appropriate to philosophical structuralism. (shrink)

Negative attack ads used in political compaigns have been attacked as being unethical because they contribute to voter cynicism and apathy and as being manipulative. From a rhetorical perspective these advertisements serve a positive societal function by creating alternative rhetorical visions that can contribute to the marketplace of ideas.

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular (constant) quantity, and the properties of such quantities are then akin to, and derived from, the..

The [lambda]-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of [lambda]-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is [lambda]-definable. These results subsume earlier ones using cartesian closed categories, as well (...) as those employing so-called Henkin and Kripke [lambda]-models. (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.

The notion of a continuously variable quantity can be regarded as a generalization of that of a particular quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the field of real numbers in many ways, but instead form a ring. Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets . In this (...) expository paper, such applications are explained to the non-specialist. Some recent results are mentioned, including a new completeness theorem for higher-order logic. (shrink)

ulations and show that they are definable by a trivial kind of coequation— namely, over one "color". We end with an example of a covariety which is not closed under bisimulations.