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)

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)

In this chapter I analyze the concept of self-consciousness in order to uncover its conceptual structure. The conclusions of this analysis describe some of the necessary categories of conscious experience. The concept of the self, the concept of consciousness, the concept of objectivity, the temporal distinctions between past, present, and future, and finally the idea of natural regularities, are found to be necessary categories for conscious experience, and hence describe the fundamental cognitive structure of self-conscious beings.

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)

This study of the interpretations of Aristotle's "Categories" in the thirteenth century provides an introduction to some main themes of medieval philosophical ...

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)

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)

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.

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)

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)

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)

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 general topic of this book is the theory of categories, its sources, meaning and development. The inquiry can be seen to proceed on two levels. On one, the history of the theory is traced from its alleged genesis in Aristotle, through its main subsequent stages of Kant and Hegel, up to a kind of consummation in two of its prominent twentieth century adherents, Alfred North White head and Nicolai Hartmann. Special attention has been paid to that aspect of the (...) Hegelian conception of the categorial analysis from which the principle of coherence emerged. On the second, deeper level, however, everything starts with Whitehead's metaphysical system, the central part of which con sists of a fascinating, though highly intricate, web of categorial notions and propositions. The historical perspective becomes a means for untangling that web. I am indebted to a number of people for advice, comment and criticism of various parts of this book. My greatest thanks go to my teachers and colleagues Nathan Rotenstreich, Nathan Spiegel, Yaakov Fleischman, as well as to the late Shmuel Hugo Bergman and Pepita Haezrachi. of this book was published in 1967 by An earlier, Hebrew version the Bialik Institute of Jerusalem. I am grateful to Mr Yehoshua Perel, Mr Arnold Schwartz and to my wife Varda for their cooperation in rendering the extensively revised text of the book into readable English. I also owe great appreciation to Miss Liat Dawe for an accurate and painstaking word-processing of the text. (shrink)

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)

A study of Jim Jones and the Peoples Temple. It features a prologue that considers the meaning of the tragedy for a post-Waco, post-9/11 world.

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)

This article briefly reviews categorization models in both cognitive psychology and cognitive phonology in order to set the background for a psycholinguistically plausible account of the classification of the allophones involved in category overlaps (i.e., the overlapping areas between phoneme categories) and in the so-called positions of neutralization. In addition, the traditional proposals of both Bloomfieldian phonemics (i.e., phonetic similarity) and the Prague School (i.e., archiphonemes) are discussed and an alternative proposal is offered. The latter claims that phonological theory should (...) be informed by psycholinguistic evidence obtained by experimental methods. This line of research will progressively make cognitive phonology a functional phonological theory consistent with the way speakers classify the sounds of their language and will contribute to theoretical progress in phonological theory. (shrink)

The categories of acting and undergoing are not really examined in the Aristotelian treatise. This article aims at showing how the anonymous author of the Book of six principles analyses them in trying to fill this void. By doing so, the article underlines how this analysis philosophically relates to some technical problems discussed in the neo-platonician exegetic tradition of Aristotle’s Categories. It makes reference to some thir-teenth- and fourteenth- century commentaries on the Book of six principles both to present how (...) this text was interpretated and to give a glance at how the conceptions of these two categories came to evolve. (shrink)

Our knowledge of natural categories includes beliefs not only about what is true of them but also about what would be true if the categories had properties other than (or in addition to) their actual ones. Evidence about these beliefs comes from three lines of research: experiments on category-based induction, on hypothetical transformations of category members, and on definitions of kind terms. The 1st part of this article examines results and theories arising from each of these research streams. The 2nd (...) part considers possible unified theories for this domain, including theories based on ideals and norms. It also contrasts 2 broad frameworks for modal category information: one focusing on beliefs about intrinsic or essential properties, the other focusing on interacting causal relations. (shrink)

The concept of an ontological category is central to metaphysics. Metaphysicians argue about which category 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, an issue which is rarely addressed; Jan Westerhoff presents the first in-depth analysis both of the use made of ontological categories in the metaphysical literature, (...) and of various attempts at defining them. He also develops a new theory of ontological categories which implies that there will be no unique system, and that the ontological category an object belongs to is not an essential property of that object. Systems of ontological categories are structures imposed on the world, rather than reflections of a deep metaphysical reality already present. All metaphysicians should find Westerhoff's book highly stimulating. (shrink)

Kant famously claims that the table of the categories of freedom does not require explanation. Kant interpreters have been baffled by this claim, and the disagreement among the increasing number of studies in more recent years suggests that the table is not as straightforward as Kant took it to be. In this article I want to show that a coherent interpretation of the table depends essentially on a clarification of what have been taken to be three fundamental ambiguities in Kants (...) text is, I argue, rooted in a hybrid conception of practical rationality assumed by his interpreters. I believe the task of disambiguating the table in all three cases can be completed. But it will require spelling out Kant’s moral cognitivism in such a way that he emerges as holding what I will call a unitary account of practical rationality. (shrink)

Kendall Walton’s “Categories of Art” seeks to situate aesthetic properties contextually. As such, certain knowledge is required to fully appreciate the aesthetic properties of a work, and without that knowledge the ‘correct’ or ‘true’ aesthetic properties of a work cannot be appreciated. The aim of this paper is to show that the way Walton conceives of his categories and art categorization is difficult to square with certain kinds of aesthetic experience—kinds of experience that seems to defy this claim of category-dependence (...) for aesthetic properties. The argument will be advanced for category-free aesthetic experience by considering Barry C. Smith’s account of wine-tasting and his description of his wine drinking ‘epiphany’. (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)

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. A part of the so-called Gelfand–Naimark–Segal (GNS) construction is identified as an isomorphism of categories, relating states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories.

Inductive reasoning is fundamental to human cognition, yet it remains unclear how we develop this ability and what might influence our inductive choices. We created novel categories in which crucial factors such as domain and category structure were manipulated orthogonally. We trained 403 4–9-year-old children to categorise well-matched natural kind and artefact stimuli with either featural or relational category structure, followed by induction tasks. This wide age range allowed for the first full exploration of the developmental trajectory of inductive reasoning (...) in both domains. We found a gradual transition from perceptual to categorical induction with age. This pattern was stable across domains, but interestingly, children showed a category bias one year later for relational categories. We hypothesise that the ability to use category information in inductive reasoning develops gradually, but is delayed when children need to process and apply more complex category structures. (shrink)

Many theories of kind representation suggest that people posit internal, essence-like factors that underlie kind membership and explain properties of category members. Across three studies (N = 281), we document the characteristics of an alternative form of construal according to which the properties of social kinds are seen as products of structural factors: stable, external constraints that obtain due to the kind’s social position. Internalist and structural construals are similar in that both support formal explanations (i.e., “category member has property (...) P due to category membership C”), generic claims (“Cs have P”), and the generalization of category properties to individual category members when kind membership and social position are confounded. Our findings thus challenge these phenomena as signatures of internalist thinking. However, once category membership and structural position are unconfounded, different patterns of generalization emerge across internalist and structural construals, as do different judgments concerning category definitions and the dispensability of properties for category membership. We discuss the broader implications of these findings for accounts of formal explanation, generic language, and kind representation. (shrink)

In Choosing Well, Andreou proposes that categorial appraisals play a central role in determining the rationality of an agent with “disordered” preferences. In this paper, I examine whether this notion can play this role in the context of cyclical preferences generated by the pattern exemplified in Quinn’s “rational self-torturer” choice situation.

Mathematical categories provide an abstract and general framework for logic and mathematics. As such, they could be used by philosophers in all the basic fields of the discipline: semantics, epistemology and ontology. In this paper, we present the basic definitions and notions and suggest some of the ways categories are starting to infiltrate formal philosophy.

Philosophy for children is an educational movement that includes diverse goals that are not always clearly articulated by theorists and practitioners. In order to navigate the multitude of aims found in the philosophy for children literature I propose distinguishing between the following categories of goals: aims of education; educational goals of philosophy for children ; goals of a community of philosophical inquiry ; goals of the facilitator; and goals of the children. The definitions of these various types are given along (...) with some examples of how the distinctions between them can be of aid in understanding the approaches, criticisms, and practice of philosophy for children. (shrink)

This paper addresses the role of categories and dimensions in the classification of psychopathology. While psychopathology does not sort itself out neatly into natural categories, we do find rough, symptom-based groupings that, through refinement, become diagnostic categories. Given that these categories suffer from comorbidity, uncertain boundaries, and excessive “unspecified disorder” diagnoses, there has been a move toward refining the diagnoses with dimensional measures. The paper traces efforts both to improve the diagnostic categories with validators that allow at least partial validity (...) and to introduce dimensional measures into the diagnostic manual. Drawing from the philosophical pragmatism of Charles Sanders Peirce, William James, and John Dewey, which emphasizes the practical, effect-sensitive consequences of a theory along with an emphasis on empirical evidence and the progressive, probabilistic character of knowledge, the paper argues that these efforts must be guided both by scientific validity and clinical utility. (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.

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.

Peirce’s method of categorial development reveals the experimental nature of phenomenology, of metaphysics, and of the relation between their respective claims. The phenomenological categories of Firstness, Secondness, and Thirdness come to light as an interrelated set of meanings, abductively generated as a tool for focusing on the richness of experience in order to elicit its illusive, “intangible” but pervasive textures, “traits” “tones or tints”. The move from experiential claims to metaphysical claims is an imaginative extension via analogy. In the development (...) of his metaphysical categories, there is an exaggeration of the experimental method by which we have meaningful everyday experience. There is an exaggeration of the metaphorical, imaginative, creative features of the meanings which arise out of past experience though abductive fixations of experience, and which legislate for the analysis of future experience. Further, there is an exaggerated attentiveness to what appears in experience, to its pervasive features or textures, an attentiveness which both founds the categories and serves to verify their adequacy. The claims of his experimental phenomenology, and hence the claims of the metaphysics which it grounds, are fallibilistic and open to alternative categorial possibilities. (shrink)

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and response time (...) are expected near the boundaries between categories. In contrast to previous work that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers. (shrink)

Facets and general categories used in bibliographic classification have been based on a disciplinary organization of knowledge. However, facets and categories of phenomena independent from disciplines can be identified similarly. Phenomena can be classified according to a series of integrative levels (layers), which in turn can be grouped into the major strata of form, matter, life, mind, society and culture, agreeing with Nicolai Hartmann’s ontology. Unlike a layer, a stratum is not constituted of elements of the lower ones; rather, it (...) represents the formal pattern of the lower ones, like the horse hoof represents the shape of the steppe. Bibliographic categories can now be seen in the light of level theory: some categories are truly general, while others only appear at a given level, being the realization of a general category in the specific context of the level: these are the facets of that level. In the notation of the Integrative Level Classification project, categories and facets are represented by digits, and displayed in a Web interface with the help of colours. (shrink)

Seminal work by Knobe, Prasada, and Newman (2013) distinguished a set of concepts, which they named “dual‐character concepts.” Unlike traditional concepts, they require two distinct criteria for determining category membership. For example, the prototypical dual‐character concept “artist” has both a concrete dimension of artistic skills, and an abstract dimension of aesthetic sensibility and values. Therefore, someone can be a good artist on the concrete dimension but not truly an artist on the abstract dimension. Does this analysis capture people's understanding of (...) cornerstone social categories, such as gender, around which society and everyday life have traditionally been organized? Gender, too, may be conceived as having not only a concrete dimension but also a distinct dimension of abstract norms and values. As with dual‐character concepts, violations of abstract norms and values may result in someone being judged as not truly a man/woman. Here, we provide the first empirical assessment of applying the dual‐character framework to people's conception of gender. We found that, on some measures that primarily relied on metalinguistic cues, gender concepts did indeed resemble dual‐character concepts. However, on other measures that depicted transgressions of traditional gender norms, neither “man” nor “woman” appeared dual‐character‐like, in that participants did not disqualify people from being truly a man or truly a woman. In a series of follow‐up studies, we examined whether moral norms have come to replace gender role norms for the abstract dimension. Implications for the evolution of concepts and categories are explored. (shrink)

The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show (...) is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning. (shrink)

We highlight connections between accessible categories and abstract elementary classes , and provide a dictionary for translating properties and results between the two contexts. We also illustrate a few applications of purely category-theoretic methods to the study of AECs, with model-theoretically novel results. In particular, the category-theoretic approach yields two surprising consequences: a structure theorem for categorical AECs, and a partial stability spectrum for weakly tame AECs.

This article argues for two related theses. First, it defends a general thesis: any kind of necessity, including metaphysical necessity, can only be known a priori. Second, however, it also argues that the sort of a priori involved in modal metaphysical knowledge is not related to imagination or any sort of so-called epistemic possibility. Imagination is neither a proof of possibility nor a limit to necessity. Rather, modal metaphysical knowledge is built on intuition of philosophical categories and the structures they (...) form. (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.