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.

This paper explores varieties of scientific structuralism. Central to our investigation is the notion of `shared structure'. We begin with a description of mathematical structuralism and use this to point out analogies and disanalogies with scientific structuralism. Our particular focus is the semantic structuralist's attempt to use the notion of shared structure to account for the theory-world connection, this use being crucially important to both the contemporary structural empiricist and realist. We show why minimal scientific structuralism is, at the very (...) least, a powerful methodological standpoint. Our investigation also makes explicit what more must be added to this minimal structuralist position in order to address the theory-world connection, namely, an account of representation. (shrink)

Recent semantic approaches to scientific structuralism, aiming to make precise the concept of shared structure between models, formally frame a model as a type of set-structure. This framework is then used to provide a semantic account of (a) the structure of a scientific theory, (b) the applicability of a mathematical theory to a physical theory, and (c) the structural realist’s appeal to the structural continuity between successive physical theories. In this paper, I challenge the idea that, to be so used, (...) the concept of a model and so the concept of shared structure between models must be formally framed within a single unified framework, set-theoretic or other. I first investigate the Bourbaki-inspired assumption that structures are types of set-structured systems and next consider the extent to which this problematic assumption underpins both Suppes’ and recent semantic views of the structure of a scientific theory. I then use this investigation to show that, when it comes to using the concept of shared structure, there is no need to agree with French that “without a formal framework for explicating this concept of ‘structure-similarity’ it remains vague, just as Giere’s concept of similarity between models does ...” (French, 2000, Synthese, 125, pp. 103–120, p. 114). Neither concept is vague; either can be made precise by appealing to the concept of a morphism, but it is the context (and not any set-theoretic type) that determines the appropriate kind of morphism. I make use of French’s (1999, From physics to philosophy (pp. 187–207). Cambridge: Cambridge University Press) own example from the development of quantum theory to show that, for both Weyl and Wigner’s programmes, it was the context of considering the ‘relevant symmetries’ that determined that the appropriate kind of morphism was the one that preserved the shared Lie-group structure of both the theoretical and phenomenological models. (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)

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", (...) or turning meta-mathematical analyses of logical concepts into "philosophical" ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down. (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)

In this paper I argue that category theory ought to be seen as providing the language for mathematical discourse. Against foundational approaches, I argue that there is no need to reduce either the content or structure of mathematical concepts and theories to the constituents of either the universe of sets or the category of categories. I assign category theory the role of organizing what we say about the content and structure of both mathematical concepts and theories. Insofar, then, as the (...) structuralist sees mathematics as talking about structures and their morphology, I contend that category theory furnishes a framework for mathematical structuralism. (shrink)

Structural realism has rapidly gained in popularity in recent years, but it has splintered into many distinct denominations, often underpinned by diverse motivations. There is, no monolithic position known as ‘structural realism,’ but there is a general convergence on the idea that a central role is to be played by relational aspects over object-based aspects of ontology. What becomes of causality in a world without fundamental objects? In this book, the foremost authorities on structural realism attempt to answer this and (...) related questions: ‘what is structure?’ and ‘what is an object?’ Also featured are the most recent advances in structural realism, including the intersection of mathematical structuralism and structural realism, and the latest treatments of laws and modality in the context of structural realism. The book will be of interest to philosophers of science, philosophers of physics, metaphysicians, and those interested in foundational aspects of science. (shrink)

Feferman argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro, we can be structuralists all the way down ; (...) we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up. (shrink)

The focus of this paper is the recent revival of interest in structuralist approaches to science and, in particular, the structural realist position in philosophy of science . The challenge facing scientific structuralists is three-fold: i) to characterize scientific theories in ‘structural’ terms, and to use this characterization ii) to establish a theory-world connection (including an explanation of applicability) and iii) to address the relationship of ‘structural continuity’ between predecessor and successor theories. Our aim is to appeal to the notion (...) of shared structure between models to reconsider all of these challenges, and, in so doing, to classify the varieties of scientific structuralism and to offer a ‘minimal’ construal that is best viewed from a methodological stance. (shrink)

I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of (...) the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones. (shrink)

Critical notice of C. Pincock's Mathematics and Scientific Representation (2012).

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, (...) or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be algebraic structuralists all the way down. (shrink)

I argue that if we distinguish between ontological realism and semantic realism, then we no longer have to choose between platonism and formalism. If we take category theory as the language of mathematics, then a linguistic analysis of the content and structure of what we say in and about mathematical theories allows us to justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study. Insofar as this analysis relies on a distinction between ontological and semantic realism, (...) it relies also on an implicit distinction between mathematics as a descriptive science and mathematics as a descriptive discourse. It is this latter distinction which gives rise to the tension between the mathematician qua philosopher. In conclusion, I argue that the tensions between formalism and platonism, indeed between mathematician and philosopher, arise because of an assumption that there is an analogy between mathematical talk and talk in the physical sciences. (shrink)

This book goes far beyond its title. Landry indeed surveys current definitions of “mathematical platonism” to show nothing like them applies to Socrates in Plat.

LandryElaine, * ed. Categories for the Working Philosopher. Oxford University Press, 2017. ISBN 978-0-19-874899-1 ; 978-0-19-106582-8. Pp. xiv + 471.

Part philosophical meditation, part cultural critique, The Body in Pain is a profoundly original study that has already stirred excitement in a wide range of intellectual circles. The book is an analysis of physical suffering and its relation to the numerous vacabularies and cultural forces--literary, political, philosophical, medical, religious--that confront it. Elaine Scarry bases her study on a wide range of sources: literature and art, medical case histories, documents on torture compiled by Amnesty International, legal transcripts of personal injury (...) trials, and military and strategic writings by such figures as Clausewitz, Churchill, Liddell Hart, and Kissinger, She weaves these into her discussion with an eloquence, humanity, and insight that recall the writings of Hannah Arendt and Jean-Paul Sartre. Scarry begins with the fact of pain's inexpressibility. Not only is physical pain enormously difficult to describe in words--confronted with it, Virginia Woolf once noted, "language runs dry"--it also actively destroys language, reducing sufferers in the most extreme instances to an inatriculate state of cries and moans. Scarry analyzes the political ramifications of deliberately inflicted pain, specifically in the cases of torture and warfare, and shows how to be fictive. From these actions of "unmaking" Scarry turns finally to the actions of "making"--the examples of artistic and cultural creation that work against pain and the debased uses that are made of it. Challenging and inventive, The Body in Pain is landmark work that promises to spark widespread debate. About the Author: Elaine Scarry is Associate Professor of English at the University of Pennsylvania. (shrink)

Part philosophical meditation, part cultural critique, The Body in Pain is a profoundly original study that has already stirred excitement in a wide range of intellectual circles. The book is an analysis of physical suffering and its relation to the numerous vocabularies and cultural forces--literary, political, philosophical, medical, religious--that confront it.Elaine Scarry bases her study on a wide range of sources: literature and art, medical case histories, documents on torture compiled by Amnesty International, legal transcripts of personal injury trials, (...) and military and strategic writings by such figures as Clausewitz, Churchill, Liddell Hart, and Kissinger, She weaves these into her discussion with an eloquence, humanity, and insight that recall the writings of Hannah Arendt and Jean-Paul Sartre.Scarry begins with the fact of pain's inexpressibility. Not only is physical pain enormously difficult to describe in words--confronted with it, Virginia Woolf once noted, "language runs dry"--it also actively destroys language, reducing sufferers in the most extreme instances to an inarticulate state of cries and moans. Scarry analyzes the political ramifications of deliberately inflicted pain, specifically in the cases of torture and warfare, and shows how to be fictive. From these actions of "unmaking" Scarry turns finally to the actions of "making"--the examples of artistic and cultural creation that work against pain and the debased uses that are made of it. Challenging and inventive, The Body in Pain is landmark work that promises to spark widespread debate. (shrink)

Introduction: Losing our heads -- Kristeva and Benjamin: melancholy and the allegorical imagination -- Kenotic art: negativity, iconoclasm, inscription -- To be and remain foreign: tarrying with l'inquietante etrangete alongside Arendt -- And Kafka -- Sublimating maman: experience, time, and the re-erotization of existence in -- Kristeva's reading of Marcel Proust -- The "Orestes Complex": thinking hatred, forgiveness, Greek tragedy, and the -- Cinema of the "thought specular" with Hegel, Freud, and Klein -- Conclusion: forging a/head.

Assesses Irish republicanism's strategic process of moderation, from violence to peace and power.

"--J.M.Coetzee "Here is a writer almost magically summoning up the world through words and ideas, in a new way, and so guiding the reader, lovingly, to receive the treasures and accept the pleasures of this book as naturally as breathing.

Perceptions of Medieval Manuscripts takes as its starting point an understanding that a medieval book is a whole object at every point of its long history. As such, medieval books can be studied most profitably in a holistic manner as objects-in-the-world. This means readers might profitably account for all aspects of the manuscript in their observations, from the main texts that dominate the codex to the marginal notes, glosses, names, and interventions made through time. This holistic approach allows us to (...) tell the story of the book's life from the moment of its production to its use, collection, breaking-up, and digitization--all aspects of what can be termed 'dynamic architextuality'. The ten chapters include detailed readings of texts that explain the processes of manuscript manufacture and writing, taking in invisible components of the book that show the joy and delight clearly felt by producers and consumers. Chapters investigate the filling of manuscripts' blank spaces, presenting some texts never examined before, and assessing how books were conceived and understood to function. Manuscripts' heft and solidness can be seen, too, in the depictions of miniature books in medieval illustrations. Early manuscripts thus become archives and witnesses to individual and collective memories, best read as 'relics of existence', as Maurice Merleau-Ponty describes things. As such, it is urgent that practices fragmenting the manuscript through book-breaking or digital display are understood in the context of the book's wholeness. Readers of this study will find chapters on multiple aspects of medieval bookness in the distant past, the present, and in the assurance of the future continuity of this most fascinating of cultural artefacts. (shrink)

Le langage, comme moyen d’expression de la croyance religieuse, entretient un malentendu : il apparaît comme un discours sur des vérités d’une supériorité telle que le croyant peut s’autoriser à faire des jugements d’ordre éthique ou esthétique envers – ou à l’encontre – d’autres personnes. Pourtant, loin d’être un discours logique sur des faits de connaissance, le discours sur la croyance religieuse est un jeu de langage qui manifeste une adhésion à une forme de vie.

In the present research, we validated a new scale developed from self-determination theory to assess the functional meaning of cash rewards offered in the workplace. According to SDT, rewards can take on different meanings based on the way they are perceived by individuals. In a series of three studies in different socioeconomic contexts, we replicated the two-factorial structure of the scale measuring respectively workplace cash rewards’ informative and controlling meanings. In Study 1, we validated the English version of the scale (...) by exploring and then confirming its two-factor structure with two English-speaking employee samples. We further replicated its two-factor structure in a French-speaking employee sample of employees in Study 2 and in a Greek-speaking employee sample in Study 3, allowing us to validate its French and Greek version. Results from our three studies show how distinct meanings attributed to cash rewards, i.e., informative or controlling, relate differently to autonomous and controlled forms of motivation based on SDT. These findings suggest that workplace cash rewards differently influence employees’ motivation depending on whether they are perceived as informative or controlling, thus providing empirical evidence for the theoretical and practical implications of SDT’s concept of functional meaning of cash rewards. Our research contributes to the assessment and understanding of employees’ experience of workplace cash rewards and provides empirical evidence that the concept of the functional meaning of cash rewards is a distinct concept from other money-related concepts such as subjective pay satisfaction, performance-contingent rewards, and financially contingent self-worth. (shrink)

This chapter contains sections titled: Introduction The Dissertation Proposal Shift to the Critique of Teleology Kant's Organicism and Critique of Teleological Judgment Goethe's Aesthetic Philosophy of Nature Multiple Purposivenesses Individuation Rationality and Purposiveness The Legacy of the Dissertation Project in Nietzsche's Later Work.