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)

This volume is about ontological categories. The categories of an ontology are designed to classify all existents. They are crucial and characterize an ontology.

The project of Categories We Live By is to offer a metaphysics of social categories. The strategy is to give a theory of social properties of individuals. The main components of the theory are a conferralist framework for properties; an account of social meaning; and an account of social construction; accompanying is also an account of social identity. This theory can be applies to offer concrete conferralist proposals of categories such as sex, gender, race, disability, religion, and LGBTQ categories. This (...) précis describes the main components (conferralist framework, social meaning, social construction, social identity) briefly, but leaves discussions of applications for another time. (shrink)

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

For more than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, category theory, linguistics, theoretical physics, logic and proof theory. This Festschrift was put together on the occasion of his 90th birthday. The papers in it give a good picture of the multiple research areas where the impact of Jim Lambek's work can be felt. The volume includes contributions by prominent researchers and by their students, showing how Jim Lambek's ideas keep inspiring upcoming (...) generations of scholars. (shrink)

Western philosophy has been dominated by the concept or the idea—the belief that there is one sovereign notion or singular principle that can make reality explicable and bring all that exists under its sway. In modern politics, this role is played by ideology. Left, right, or center, political schools of thought share a metaphysics of simplification. We internalize a dominant, largely unnoticeable framework, oblivious to complex, plural, and occasionally conflicting or mutually contradictory explanations for what is the case. In this (...) groundbreaking work, Michael Marder proposes a new methodology for political science and philosophy, one which he terms “categorial thinking.” In contrast to the concept, no category alone can exhaust the meaning of anything: categories are so many folds, complications, respectful of multiplicity. Ranging from classical Aristotelian and Kantian philosophies to phenomenology and contemporary politics, Marder's book offers readers a theoretical toolbox for the interpretation of political phenomena, processes, institutions, and ideas. His categorial apparatus encompasses political temporality and spatiality; the revolutionary and conservative modalities of political actuality, possibility, and necessity; quantitative and qualitative approaches to the study of political reality; the meaning of political relations; and various senses of political being. Under this lens, the political appears not as a singular concept but as a family of categories, allowing room for new, plural, and often antagonistic ideas about the state, the people, sovereignty, and power. (shrink)

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.

In this essay, I challenge Charles Mills’s use of the category of moral personhood for advancing a robust anti-racist political critique in nonideal circumstances. I argue that the idea of the moral equality of persons is necessary but insufficient for reparative justice. I enrich the normative basis of political critique to include: (1) a clarification of what the public recognition of moral personhood can legitimately entail as a requirement of justice enforceable by the state, especially with respect to economic reforms (...) that advance equal opportunity and (2) a conception of non-alienated labor that assails identity-based occupational segregation in the labor market. These additional components do not exhaust the plausible bases for political critique, but enrich it in a way that the idea of moral equality alone cannot. (shrink)

The proposed book presents an overview of select theories in the classical Vaiśeṣika system of Indian philosophy, such as the concept of categories, creation and existence, atomic theory, consciousness and cognition. It also expounds in detail the concept of dharma, the idea of the highest good and expert testimony as a valid means of knowing in Vaiśeṣika thought. Some of the major themes discussed are the religious inclination of Vaiśeṣika thought towards Pasupata Saivism, the affiliation of the Vaiśeṣika System to (...) the basic foundations of Indian philosophical thought, namely Veda and Yoga, and their insights into science, hermeneutics and metaphysics. In addition, this book includes recent Sanskrit commentaries on key Vaiśeṣika texts and provides a glimpse of Vaiśeṣika studies across the world. Overall, this book enunciates the Vaiśeṣika view from original sources and is an important work for Vaiśeṣika studies in current times for serious students as well as researchers.. (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)

We are women, we are men. We are refugees, single mothers, people with disabilities, and queers. We belong to social categories and they frame our actions, self-understanding, and opportunities. But what are social categories? How are they created and sustained? How does one come to belong to them? Ásta approaches these questions through analytic feminist metaphysics. Her theory of social categories centers on an answer to the question: what is it for a feature of an individual to be socially meaningful? (...) In a careful, probing investigation, she reveals how social categories are created and sustained and demonstrates their tendency to oppress through examples from current events. To this end, she offers an account of just what social construction is and how it works in a range of examples that problematize the categories of sex, gender, and race in particular. The main idea is that social categories are conferred upon people. Ásta introduces a 'conferralist' framework in order to articulate a theory of social meaning, social construction, and most importantly, of the construction of sex, gender, race, disability, and other social categories. (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)

The social categories to which we belong—Latino, disabled, American, woman— causally influence our lives in deep and unavoidable ways. One might be pulled over by police because one is Latino, or one might receive a COVID vaccine sooner because one is American. Membership in these social categories most often falls outside of our control. This paper argues that membership in social categories constitutes a restriction on human agency, creating a situation of non-ideal agency for many human individuals. -/- However, there (...) are ways to resist the causal influence of social categories, and certain socially marginalized groups can be understood as attempting to do just this. I discuss two instances of social category resistance: gender pronouns and the rights of trans individuals. I suggest that the intentional declaration of gender pronouns (“she/her” or “they/them”) can be understood as an attempt to resist the causal powers of nonconsensual social categorization. Similarly, one among many reasons to support the rights of trans individuals is that their self-declarations of gender identity can be viewed as a reclamation of agency in the face of causal constraints imposed by socially defined and imposed gender categories. This lesson can be generalized to people belonging to a broad range of marginalized groups. (shrink)

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

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.

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)

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)

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)

Classifying is a fundamental operation in the acquisition of knowledge. Taxonomic theory can help students of cognition, evolutionary psychology, ethology, anatomy, and sociobiology to avoid serious mistakes, both practical and theoretical. More positively, it helps in generating hypotheses useful to a wide range of disciplines. Composite wholes, such as species and societies, are “individuals” in the logical sense, and should not be treated as if they were classes. A group of analogous features is a natural kind, but a group of (...) homologous features is not. Imposing hypotheses justified only on the basis of nominalist, realist, phenomenalist, or conceptualist metaphysics upon the neurophysiology of organisms or upon the causes of behavior exemplifies the “psychologist's fallacy” of William James. Levels should be distinguished from their members and from classes of levels, and the ontological status of entities ranked at levels should be made clear. It is important not to confuse such categories as substance and process with one another. Several genetical terms, such as “gene” and chromosome,” are even more equivocal than has been realized. Discussions about units of selection, behavior, and thinking suffer from the ambiguity of a “unit of” an entity. An important source of misunderstanding about natural selection is the habit of treating it as an “agent”: in an important sense, natural selection does not “act” at all. (shrink)

『논리연구』에서 후설이 탐구하는 범주적 직관은 단순히 감성적인 방식으로 이루어지지 않는 직관이다. 그리고 거기에서 등장하는 범주적 재현(Repräsentation) 이론은 범주적 직관이 재현이라는 이론이다. 『논리연구』에서 ‘재현’과 ‘파악(Auffassung)’은 외연이 같은 말이므로, 범주적 재현 이론은 범주적 직관이 파악이라는 이론이 된다. 그런데 20여년 후????논리연구????제2판 제2분책 서문에서 후설은 범주적 재현 이론을 더 이상 인정하지 않는다고 말한다. 하지만 후설이 실제로 이 이론을 포기하는가 하는 점은 논란의 여지가 있다. 나는 후설이 이 이론을 포기하지 않거나, 적어도 포기해서는 안 된다고 주장한다. 범주적 재현 이론은 『경험과 판단』 같은 후기의 저작에서도 채택되는 것처럼 (...) 보이며, 무엇보다도 범주적 직관을 이해하기 위하여 필수적이기 때문이다. 이 논문에서 나는 범주적 직관과 범주적 재현 이론을 자세하게 설명한 후, 범주적 재현 이론이 포기되거나 수정될 필요가 있다고 주장하는 투겐트하트(E. Tugendhat)와 로마(D. Lohmar)의 견해를 각각 비판한다. 이와 더불어 명확해지는 것은, 범주적 재현 이론을 통해 이해된 범주적 직관 개념이 선술어적 경험으로부터 술어적 인식의 발생을 규명하는 데 결정적인 역할을 한다는 점이다. (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)

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)

A key figure in the history of Aristotelianism, Porphyry (AD 232/3 - c. 305) was born in Tyre and was a student of Longinus' in Athens and of Plotinus' in Rome. In his commentary on the Categories, Porphyry provided an authoritative interpretation of a notoriously controversial work. Commentators on Aristotle had disagreed fundamentally over whether the Categories was a work of logic, concerning simple terms or the simple concepts they represent, or a metaphysical work addressing the classification by genera of (...) simple entities or concepts. Approaching the Categories as the first of Aristotle's works in logic, Porphyry adopted an earlier Peripatetic view of the Categories as being concerned principally with terms or linguistic items, which he calls "predicates." He successfully defended the Categories against Plotinus, arguing that while it seemed to attack orthodox Platonism by denying the separation of forms and the ontological priority of the universal, it was in fact entirely compatible with Platonism. Because of Porphyry's intervention, the Categories came to serve as a basic textbook of logic for subsequent Neoplatonist teaching. His commentary influenced the Arabic tradition, and Boethius drew heavily on it as well. The full text of Porphyry's commentary was lost, but the extant version is available here in Steven K. Strange's new translation. Ancient philosophers, historians of philosophy, classicists, and medievalists will welcome its appearance in English. (shrink)

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)

Now available in paperback, this acclaimed book introduces categories and elementary toposes in a manner requiring little mathematical background. It defines the key concepts and gives complete elementary proofs of theorems, including the fundamental theorem of toposes and the sheafification theorem. It ends with topos theoretic descriptions of sets, of basic differential geometry, and of recursive analysis.

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)

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.

This volume completes, starting from chapter 6, the commentary by the young Philoponus on Aristotle's Categories, chapters 1-5 of which were previously published in this series. This ancient commentary was the first work in the Aristotelian syllabus after a general introduction to Aristotle by the same author. It is influenced by an extant short anonymous record of his teacher Ammonius' lecturees on the same work, but Philoponus' commentary is two and a half times as long as that anonymous record, and (...) includes special contributions of Philoponus' own, for example in philology, Christian theology and in disagreements with Aristotle. This English translation of Philoponus' work is the latest volume in the Ancient Commentators on Aristotle series and makes this philosophical work accessible to a modern readership. The translation is accompanied by an introduction, comprehensive commentary notes, bibliography, glossary of translated terms and a subject index. (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)