Category mistakes are sentences such as 'Green ideas sleep furiously' or 'Saturday is in bed'. They strike us as highly infelicitous but it is hard to explain precisely why this is so. Ofra Magidor explores four approaches to category mistakes in philosophy of language and linguistics, and develops and defends an original, presuppositional account.

Ci sono molti tipi di cataloghi d’arte. Alcuni presentano collezioni museali o raccolte private, altri le opere d’arte espo- ste in mostre temporanee, altri l’opera (completa o parziale) di qualche artista. Alcuni cataloghi listano soltanto le opere d’arte, altri mostrano alcuni aspetti dei contesti espositivi in cui le opere sono presentate al pubblico – installazioni permanenti di musei, o allestimenti temporanei in gallerie d’arte, per esempio. Tutti i cataloghi d’arte sono strumenti per comunicare al pubblico informazioni: i cataloghi possono essere (...) costituiti da semplici liste di autori e titoli di opere, materiali utilizzati per comporle e anni in cui sono state create, ma anche da volumi con saggi che contestualizzano le opere catalogate, o da volumi che, tramite gli accostamenti visivi di alcune opere alle altre e una grafica studiata, ne fanno risaltare certi aspetti, presentandoci le opere sotto particolari punti di vista. Di solito, i cataloghi d’arte presentano immagini delle opere di cui trattano. Generalmente, i contenuti di un catalogo d’arte sono descritti grazie all’uso di categorie che individuano, per esempio, i media che accomunano certe opere (dipinti, sculture, fotografie, installazioni, e così via), i periodi storici in cui le opere sono state prodotte, e gli stili artistici che le caratterizzano (classico, ellenistico, barocco, rococò, e così via). Categorie artistiche come quelle di frequente riportate nei cataloghi sono oggetto di un saggio del filosofo Kendall Walton, Categories of Art [1970], che avanza una spiegazione acuta ed efficace del ruolo che le nostre conoscenze rivestono nell’apprezzamento delle opere d’arte. In questo capitolo ricostruirò alcuni argomenti di Walton. Il mio obiettivo è mostrare come una migliore comprensione di che cosa siano le categorie artistiche e di come riconosciamo che un’opera appartiene a una certa categoria artistica possa aiutarci a gettare luce sul potenziale comunicativo dei cataloghi d’arte. (shrink)

Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.

A descoberta da “intuição categorial” atesta que há um apreender simples não só do que se dá na percepção sensorial, mas também há, junto com essa, um apreender simples do “categorial”. Com outras palavras, a descoberta da “intuição categorial” atesta que, na percepção cotidiana, ou seja, na percepção concreta da coisa, em toda experiência, se dá também uma percepção do que na filosofia se costumou chamar de “categoria”. Isso significa, ainda, que há atos em que consistências ideais se mostram a (...) si mesmas, sem que sejam criações destes atos, funções de pensamento, produtos do sujeito. Entretanto, é preciso explicitar melhor o que é apreendido nessa intuição e como algo deste modo de ser é intuído. O que se pretende com esta exposição é realizar tal explicitação no horizonte de pensamento de Heidegger, em que fenomenologia se mostra como ontologia. Isso implica discutir também importantes temas, como “intencionalidade”, “intuição e expressão”, “verdade e ser”. (shrink)

Category mistakes are sentences such as ‘Colourless green ideas sleep furiously’ or ‘The theory of relativity is eating breakfast’. Such sentences are highly anomalous, and this has led a large number of linguists and philosophers to conclude that they are meaningless (call this ‘the meaninglessness view’). In this paper I argue that the meaninglessness view is incorrect and category mistakes are meaningful. I provide four arguments against the meaninglessness view: in Sect. 2, an argument concerning compositionality with respect to category (...) mistakes; in Sect. 3 an argument concerning synonymy facts of category mistakes; in Sect. 4 concerning embeddings of category mistakes in propositional attitude ascriptions; and in Sect. 5 concerning the uses of category mistakes in metaphors. Having presented these arguments, in Sect. 6 I briefly discuss some of the positive motivations for accepting the meaninglessness view and argue that they are unconvincing. I conclude that the meaninglessness view ought to be rejected. (shrink)

Foundational ontologies, central constructs in ontological investigations and engineering alike, are based on ontological categories. Firstly proposed by Aristotle as the very ur- elements from which the whole of reality can be derived, they are not easy to identify, let alone partition and/or hierarchize; in particular, the question of their number poses serious challenges. The late medieval philosopher Dietrich of Freiberg wrote around 1286 a tutorial that can help us today with this exceedingly difficult task. In this paper, I (...) discuss ontological categories and their importance for foundational ontologies from both the contemporary perspective and the original Aristotelian viewpoint, I provide the translation from the Latin into English of Dietrich's De origine II with an introductory elaboration, and I extract a foundational ontology–that is in fact a single-category one–from this text rooted in Dietrich's specification of types of subjecthood and his conception of intentionality as causal operation. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an (...) adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

The notion of 'natural kinds' has been central to contemporary discussions of metaphysics and philosophy of science. Although explicitly articulated by nineteenth-century philosophers like Mill, Whewell and Venn, it has a much older history dating back to Plato and Aristotle. In recent years, essentialism has been the dominant account of natural kinds among philosophers, but the essentialist view has encountered resistance, especially among naturalist metaphysicians and philosophers of science. Informed by detailed examination of classification in the natural and social sciences, (...) this book argues against essentialism and for a naturalist account of natural kinds. By looking at case studies drawn from diverse scientific disciplines, from fluid mechanics to virology and polymer science to psychiatry, the author argues that natural kinds are nodes in causal networks. On the basis of this account, he maintains that there can be natural kinds in the social sciences as well as the natural sciences. (shrink)

This is the first volume on category theory for a broad philosophical readership. It is designed to show the interest and significance of category theory for a range of philosophical interests: mathematics, proof theory, computation, cognition, scientific modelling, physics, ontology, the structure of the world.

How do we know what kinds of things constitute knowledge or justified belief? Naturalized epistemology is committed to denying a priori insight into the kinds of kinds that are and are not knowledge or justification makers. By contrast, it is argued here that knowledge of these matters is a priori knowledge of a special kind. Such knowledge may be called “categorial.” The dialectical give and take between categorial and naturalized epistemology is pursued, before endorsing an argument that breaks the standoff (...) in favor of categorial epistemology. In particular, an argument is given for a certain kind of mathematical skepticism that is entirely a priori. The skeptical argument turns on categorial claims about actuality. Responses are considered before defending a method of categorial dumbfounding in certain circumstances. This yields a positive argument for the categoriality of fundamental epistemic principles. The categorial rationalist conclusions are embraced and some consequences noted. (shrink)

This paper establishes a connection between structure sensitive categorial inference and classical modal logic. The embedding theorems for non-associative Lambek Calculus and the whole class of its weak Sahlqvist extensions demonstrate that various resource sensitive regimes can be modelled within the framework of unimodal temporal logic. On the semantic side, this requires decomposition of the ternary accessibility relation to provide its correlation with standard binary Kripke frames and models.

Radical Ontic Structural Realism (ROSR) claims that structure exists independently of objects that may instantiate it. Critics of ROSR contend that this claim is conceptually incoherent, insofar as, (i) it entails there can be relations without relata, and (ii) there is a conceptual dependence between relations and relata. In this essay I suggest that (ii) is motivated by a set-theoretic formulation of structure, and that adopting a category-theoretic formulation may provide ROSR with more support. In particular, I consider how a (...) category-theoretic formulation of structure can be developed that denies (ii), and can be made to do work in the context of formulating theories in physics. Keywords: structural realism, category theory, general relativity.. (shrink)

In this paper we adopt a category-theoretic viewpoint in order to analyze the semantics of complementarity for quantum systems. Based on the existence of a pair of adjoint functors between the topos of presheaves of the Boolean kind of structure and the category of the quantum kind of structure, we establish a twofold complementarity scheme which constitutes an instance of the concept of adjunction. It is further argued that the established scheme is inextricably connected with a realistic philosophical attitude, although (...) substantially different from the classical one. (shrink)

The concept of an ontological category is central to metaphysics. Metaphysicians argue about which category of existence an object should be assigned to, whether one category can be reduced to another one, or whether there might be different equally adequate systems of categorization. Answers to these questions presuppose a clear understanding of what precisely an ontological category is, and Jan Westerhoff now provides the first in-depth analysis. After examining a variety of attempted definitions, he proceeds to argue for a new (...) understanding of ontological categories, according to which they are systematizations of our knowledge of the world rather than essential characteristics of the world itself. Metaphysicians will find his work highly stimulating. (shrink)

Religious-secular distinctions have been crucial to the way in which modern governments have rationalised their governance and marked out their sovereignty – as crucial as the territorial boundaries that they have drawn around nations. The authors of this volume provide a multi-dimensional picture of how the category of religion has served the ends of modern government. They draw on perspectives from history, anthropology, moral philosophy, theology and religious studies, as well as empirical analysis of India, Japan, Mexico, the United States, (...) Israel-Palestine, France and the United Kingdom. (shrink)

La notion de «catégorie» traverse de manière centrale l�histoire de la philosophie depuis sa première formulation par Aristote. Elle tient une place importante dans la logique développée par Husserl. Cet article a pour visée de montrer comment, au-delà d�une simple exposition d�ontologie formelle, les catégories relèvent de la constitution transcendantale. Comme tout objet, réel ou idéel, les catégories sont corrélées à un ensemble structuré d�actes intentionnels qui peut rendre compte de leur émergence et de leur thématisation explicite. Cela permettra de (...) montrer que la problématique de la constitution chez Husserl, que l�on situe à partir du tournant transcendantal de 1907, est déjà bien amorcée à l�époque des Recherches logiques. (shrink)

Category mistakes are sentences such as ”The number two is blue’ or ”Green ideas sleep furiously’. Such sentences are highly infelicitous and thus a prominent view claims that they are meaningless. Category mistakes are also highly prevalent in figurative language. That is to say, it is very common for sentences which are used figuratively to be such that, if taken literally, they would constitute category mistakes. In this paper I argue that the view that category mistakes are meaningless is inconsistent (...) with many central and otherwise plausible theories of figurative language. Thus if the meaninglessness view is correct, the theories in question must each be rejected, and conversely, if any of the theories in question is correct, the meaninglessness view must be wrong. The debates concerning the semantics of figurative language and concerning the semantic status of category mistakes are closely connected. (shrink)

The author of Categories We Live By replies to critics Linda Martín Alcoff, Judith Butler, and Abraham Sesshu Roth.

"What Categories Are About," a paper for a Philosophy Department Colloquium, Michigan State University, September 1998.

This paper challenges the traditional metaphysical assumption that properties can be located in spacetime and examines the ontological implications of categorizing spacetime itself. By introducing a two-category ontology, we argue that attempts to locate properties in spacetime represent a category mistake, as this notion fails to address the fundamental categorization of spacetime. We propose that the notion of instantiation should be conceptually separated from spatiotemporal location, thereby allowing for properties that are Platonic yet immanent. Furthermore, we reevaluate the debate between (...) Platonism and Aristotelianism, advocating that Platonic properties can possess a form of immanence previously thought to be exclusive to Aristotelian frameworks. This reformation not only underscores the need for categorizing spacetime but also reshapes our understanding of the instantiation and immanence of properties within ontological dependence frameworks. (shrink)

Category-based induction is an inferential mechanism that uses knowledge of conceptual relations in order to estimate how likely is for a property to be projected from one category to another. During the last decades, psychologists have identified several features of this mechanism, and they have proposed different formal models of it. In this article; we propose a new mathematical model for category-based induction based on distances on conceptual spaces. We show how this model can predict most of the properties of (...) this kind of reasoning while providing a solid theoretical foundation for it. We also show that it subsumes some of the previous models proposed in the literature and that it generates new predictions. (shrink)

This paper offers a close reading of Aristotle’s account of categories in Topics I.9. It consists of four sections. The first argues that the categories introduced in Topics I.9 are different from those introduced in Categories 4. In particular, the first category of Topics I.9, the category of essence , is different from the firstcategory of Categories 4, the category of substance .The second section contains the main proposal of this paper: I shall argue that the (...) category of essence is characterized by Aristotle as including all terms which are the subject of some essential predication, i.e., all terms which possess an essence. In Topics I.9, Aristotle accepts essential predications such as ‘whiteness is colour’ as well as ‘man is animal’. Thus, the category of essence includes non-substance terms such as ‘whiteness’ as well as substance terms such as ‘man’. The third section proposes a solution to a problem concerning the status of differentiae in the Topics. The problem ist hat, according to the Topics, differentiae are predicated in the essence but do not signify essence . The solution will be that differentiae function as the predicate of essential predications but do not belong to the category of essence.The fourth section investigates the nine non-essence categories of Topics I.9 . I shall argue that these include paronymic non-substance terms such as ‘white’ and ‘coloured’, whereas the corresponding nominal forms such as ‘whiteness’ and ‘colour’ belong to the category of essence. (shrink)

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried (...) out relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo. (shrink)

There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of objects, or formally by the (...) existence of identity morphisms. We show that it is perfectly possible to get rid of the identity of objects by definition, but the formal identity of objects remains as an essential element of the theory. This can be achieved by defining category exclusively in terms of morphisms and identity morphisms and, analogously, by defining category theory entirely in terms of functors and identity functors. With objects and categories eliminated, we focus on the “philosophy of arrows” and the roles various identities play in it. This perspective elucidates a contrast between “set ontology” and “categorical ontology”. (shrink)

Since it is only five years since the publication of our paper, "Geographical categories: An ontological investigation" (Smith and Mark 2001), it seems somewhat strange to be making retrospective comments on the piece. Nevertheless, the field is moving quickly, and much has happened since the article appeared. A large number of papers have already cited the work, which suggests that there is a seam here that people find worthy of being mined. In this short essay, we first review the (...) paper and attempt to assess its significance from the perspective of our current work. We then put the paper in the context of our individual and joint works, which led up to it, and summarize our research trajectories since the paper appeared, pointing out what some of this reveals about spatial ontology in general. We conclude with some remarks on the future of ontological research in geographical information science. (shrink)

Analytical categories of scientific cultures have typically been used both exclusively and universally. For instance, when styles of scientific research are employed in attempts to understand and narrate science, styles alone are usually employed. This article is a thought experiment in interweaving categories. What would happen if rather than employ a single category, we instead investigated several categories simultaneously? What would we learn about the practices and theories, the agents and materials, and the political-technological impact of science (...) if we analyzed and applied styles, paradigms, and models simultaneously? I address these questions in general and for a specific case study: a brief history of systematics. (shrink)

1 Recognition Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Classical Categorial Grammar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Classical (...) Categorial Grammar. Examples . . . . . . . . . . . . . . . . . . . . . 5 4 Logic Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5 Lambek calculus. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Lambek calculus. Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7 Lambek calculus. Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 8 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 9 Residuated and Galois Connected Functions . . . . . . . . . . . . . . . . . . 14 10 Interpretation of the Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11 Nonveridical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 12 Dutch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 13 Classification of NPIs in Dutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 14 Antilicensing Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.. (shrink)

IN A PRESIDENTIAL ADDRESS I believe the speaker is allowed more latitude than in a more ordinary speech. There is more freedom to explore and perhaps even preach. So I am going to do a bit of both. My chapter and verse, some of you will be surprised to know, is a passage from the preface to Foucault’s The Order of Things, in which he argues that categories are a matter of invention. This text has had enormous impact on (...) the issue I wish to address today, and in many ways has helped to define it and to establish as definitive, in the minds of many of our contemporaries, the view that categories are invented. (shrink)

As a metaphysical theory, radical ontic structural realism is characterised mainly in terms of the ontological primacy it places on relations and structures, as opposed to the individual relata and objects that inhabit these relations/structures. The most popular criticism of ROSR is that its central thesis is incoherent. Bain attempts to address this criticism by arguing that the mathematical language of category theory allows for a coherent articulation of ROSR’s key thesis. Subsequently, Wüthrich and Lam and Lal and Teh have (...) criticised Bain’s arguments and claimed that category theory fares no better than set theory in coherently articulating the main ideas of ROSR. In this paper, we defend Bain’s main arguments against these critiques, and attempt to elaborate on the sense in which category theory can be seen as providing a coherent articulation of ROSR. We also consider the relationship between ROSR and Categorical Quantum Mechanics. (shrink)

Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other (...) branches of contemporary mathematics. If such a specification suffices, then a category-theoretical approach will be highly appropriate. But if sets have a richer `nature' than is preserved under isomorphism, then such an approach will be inadequate. (shrink)

Analytic Philosophy, EarlyView. The method of paraphrasing away apparent ontological commitments is a familiar tool for trimming one's ontology. Even so, I argue that aiming to avoid commitment via paraphrase is unjustified. The reason is the standard motivations for paraphrase rest on implicit yet faulty principles regarding ontological categories and categorization- or so I argue. These results also provide indirect support for a permissivist approach to ontology.

The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology (...) in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science. (shrink)

The present investigation reformulates a few Brentanian ideas concerning what is mental. In particular, an attempt to define the categorial structure implicit in the notion of consciousness and in that of inner perception, keeping in mind their connections with external perception and with unconscious, is outlined. Within the mental field is observed a formal violation of some elementary rules of ontology and mereology, and such violation can be interpreted in terms of an infinite multiplicity of the mental field itself.

This paper is an essay in what Austin (_Proc Aristotel Soc_ 57: 1–30, 1956–1957) called "linguistic phenomenology". Its focus is on showing how the grammatical features of ordinary causal verbs, as revealed in the kinds of linguistic constructions they can figure in, can shed light on the nature of the processes that these verbs are used to describe. Specifically, drawing on the comprehensive classification of English verbs founds in Levin (_English verb classes and alternations: a preliminary investigation_, University of Chicago (...) Press, Chicago, 1993), I divide the forms of productive causal processes into five classes, corresponding to Aristotle's ontological categories: there are processes that cause change in location, in state, and in quantity; and that lead to the creation and destruction of substances. These broader categories are then subdivided into other ones, corresponding to different Levin classes, according to further differences in the causal processes they involve. I conclude by discussing the relevance of this argument to research in metaphysics and experimental philosophy. (shrink)

Linnebo and Pettigrew present some objections to category theory as an autonomous foundation. They do a commendable job making clear several distinct senses of ‘autonomous’ as it occurs in the phrase ‘autonomous foundation’. Unfortunately, their paper seems to treat the ‘categorist’ perspective rather unfairly. Several infelicities of this sort were addressed by McLarty. In this note I address yet another apparent infelicity.

The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the Curry-Howard derivational semantics, and structure-preservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural language (...) form and meaning. (shrink)

El objetivo de este trabajo es dar una mirada a la manera en que Roig opera en los ámbitos categoriales. Nuestro propósito es observar cómo el autor critica, confecciona, articula, utiliza, crea y entrelaza categorías, es decir, hemos querido hacer un alto en los procedimientos estructurales de su filosofar, en sus operaciones. Sostenemos como hipótesis que el rearme categorial al que Roig apela en múltiples pasajes de su obra implica, como operación de base, un cambio en el reconocimiento del lugar (...) de enunciación, y que éste se juega en parte importante en el empadronamiento de las categorías. Distinguimos al menos cinco operaciones categoriales en su trabajo reflexivo, las cuales pueden rastrearse a lo largo de toda su obra: (1) rescate, (2) crítica (3) reemplazo, (4) reelaboración e (5) invención. The aim of this work is to give a glance to the way in which Roig operates at categorical levels. Our intention is to observe how the author criticizes, makes, articulates, uses, creates and interlaces categories, that is to say, we wanted to take a break in the structural procedures of his philosophising, in his processes. We support as hypothesis that the categorical rearmament to that Roig appeals in multiple passages of his work involves, as a base operation, a change in the recognition of the place of enunciation, and that it plays an important part in the registration of the categories. We distinguish at least five categorical operations in his reflective work, which can be traced back throughout all his work: (1) rescue, (2) critique, (3) replacement, (4) re-elaboration and (5) invention. (shrink)

An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences. Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. Information is inherently dynamic; the same (...) ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines. Written in an engaging and straightforward style, and assuming little background in mathematics, the book is rigorous but accessible to non-mathematicians. Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs—categories in disguise. After explaining the “big three” concepts of category theory—categories, functors, and natural transformations—the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions. Category Theory for the Sciences is intended to create a bridge between the vast array of mathematical concepts used by mathematicians and the models and frameworks of such scientific disciplines as computation, neuroscience, and physics. (shrink)

Social categories play a central role in inquiry. Some authors have argued that social categories can only play this role because they have a particular metaphysical status, such as a connection to natural kinds or to comparatively joint-carving properties. This reflects the broadly realist idea that categories that play important roles in inquiry do so for metaphysical reasons. In this paper I argue that such metaphysical views of social categories cannot accommodate ‘empty’ social categories, cases (...) in which social categories that cannot have the metaphysical features attributed to them by such accounts still play a central role in inquiry. I defend an alternative approach: context-dependent naturalness, an analogue of metaphysical naturalness that concerns context-dependent, rather than metaphysical, structure. (shrink)

Categories play a major role in contemporary metaphysics. They have not only been invoked in a number of philosophical theories but are themselves objects of epistemological and metaphysical scrutiny. In this article, we will discuss the following questions: How do we know when something belongs to a certain category? Is there a fundamental category of the world? Can we give a satisfactory account of the number of categories and the completeness of systems of categories? Are categories (...) the genuine subjects of ontology? (shrink)

One well known problem regarding quantifiers, in particular the 1storder quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial (...) languages generated by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functorfunction defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)