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.

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)

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)

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)

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)

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)

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 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)

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

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)

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)

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.

Recenzja książki: Elaine Landry, _Categories for the Working Philosopher_, Oxford University Press, Oxford, 2017, ss. xiv+471.

It may be argued that the language of mathematics is about the category\nof sets, although the definite article requires some justification.\nAs possible worlds of mathematics we may admit all models of type\ntheory, by which we mean all local toposes. For an intuitionist,\nthere is a distinguished local topos, namely the so-called free topos,\nwhich may be constructed as the Tarski–Lindenbaum category of intuitionistic\ntype theory. However, for a classical mathematician, to pick a distinguished\nmodel may be as difficult as to define the notion of (...) truth in classical\ntype theory, which Tarski has shown to be impossible. Author Keywords:\nAuthor Keywords: Type theory; Categorical logic; Category of sets;\nTopos theory *1 I wish to thank Elaine Landry for organizing a symposium\nat Brock University in 1996, at which an earlier version of this\npaper was presented, and Phil Scott for helpful discussions. Support\nfrom the Social Sciences and Humanities Research Council of Canada\nis herewith acknowledged. (shrink)

Recent works about ecumenical systems, where connectives from classical and intuitionistic logics can co-exist in peace, warmed the discussion on proof systems for combining logics. This discussion has been extended to alethic modalities using Simpson’s meta-logical characterization: necessity is independent of the viewer, while possibility can be either intuitionistic or classical. In this work, we propose a pure, label free calculus for ecumenical modalities, nEK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {nEK}$$\end{document}, where exactly one logical operator figures (...) in introduction rules and every basic object of the calculus can be read as a formula in the language of the ecumenical modal logic EK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EK}$$\end{document}. We prove that nEK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {nEK}$$\end{document} is sound and complete w.r.t. the ecumenical birelational semantics and discuss fragments and extensions. (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)

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)

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)

Most scholars have focused on The Second Sex and Simone de Beauvoir’s fiction, concentrating on gender issues but ignoring her broader emancipatory vision. Though Beauvoir’s political thinking is not as closely studied as her feminist works, it underpinned her activism and helped her navigate the dilemmas raised by revolutionary thought in the postwar period. In Emancipatory Thinking Elaine Stavro brings together Beauvoir’s philosophy and her political interventions to produce complex ideas on emancipation. Drawing from a range of work, including (...) novels, essays, autobiographical writings, and philosophic texts, Stavro explains that for Beauvoir freedom is a movement that requires both personal and collective transformation. Freedom is not guaranteed by world historical systems, material structures, wilful action, or discursive practices, but requires engaged subjects who are able to take creative risks as well as synchronize with existing forces to work towards collective change. Beauvoir, Stavro asserts, resisted the trend of anti-humanism that has dominated French thinking since the 1960s and also managed to avoid the pitfalls of voluntarism and individualism. In fact, Stavro argues, Beauvoir appreciated the impact of material, socio-economic, institutional forces, without forgoing the capacity to initiate. Applying Beauvoir’s existential insights and understanding of embodied and situated subjectivity to recent debates within gender, literary, sociological, cultural, and political studies, Emancipatory Thinking provides a lens to explore the current political and theoretical landscape. (shrink)

Just Business provides the first comprehensive, reasoned framework for resolving questions of business ethics and corporate governance. Innovative, accessible, and global in scope, its powerful Ethical Decision Model can be used to manage the ethical problems of business as they arise in all their complexity and variety. Just Business combines business realism with philosophical rigor, and demonstrates that it is not necessary to emasculate or to adulterate business for business to be ethical. The book benefits from Elaine Sternberg's extensive (...) experience as an academic philosopher, an international investment banker, and head of successful businesses. She is now Principal of a London-headquartered consultancy firm, and Research Fellow in Philosophy at the University of Leeds. (shrink)

Elaine Kuzmeskus retraces historical sâeance sessions and retells firsthand accounts of spirits reaching from beyond the veil so that we, the living, can better understand what happens in these encounters and why.

I examine the relation between logic and nature in terms of ‘reflection’, the word that Hegel uses at the end of theEncyclopaedia Logicto describe the self-sundering or externalization of the idea into nature. Although nominally the term ‘reflection’ seems to denote a uniquely mental process and is often used so by Hegel in his early critique ofReflexionsphilosophie, in his later writings it also has an irreducibly ontological significance. Hegel describes logic's opening-out to nature as a movement of ‘reflection’ [Widerschein] and (...) he follows Kant in describing the shift from the finite to the infinite in the relation between nature and thought as one of reflective judgement. Although Hegel generally considers reflection to be uniquely concerned with finite cognition and the constitution of finite things, I argue that in his embrace of reflective judgement he sees a key role for reflection in the relation of logic, nature and spirit. (shrink)

A EVOLUÇÃO DO CONCEITO DE FAMÍLIA NO ÂMBITO DO ORDENAMENTO JURÍDICO BRASILEIRO.

Much has been said about intuitionistic and classical logical systems since Gentzen’s seminal work. Recently, Prawitz and others have been discussing how to put together Gentzen’s systems for classical and intuitionistic logic in a single unified system. We call Prawitz’ proposal the Ecumenical System, following the terminology introduced by Pereira and Rodriguez. In this work we present an Ecumenical sequent calculus, as opposed to the original natural deduction version, and state some proof theoretical properties of the system. We reason that (...) sequent calculi are more amenable to extensive investigation using the tools of proof theory, such as cut-elimination and rule invertibility, hence allowing a full analysis of the notion of Ecumenical entailment. We then present some extensions of the Ecumenical sequent system and show that interesting systems arise when restricting such calculi to specific fragments. This approach of a unified system enabling both classical and intuitionistic features sheds some light not only on the logics themselves, but also on their semantical interpretations as well as on the proof theoretical properties that can arise from combining logical systems. (shrink)

Retail banking is facing many challenges, not least the loss of its customers’ trust and loyalty. The economic crisis is forcing banks to examine their relationships with stakeholders and to offer greater reassurance that their brand promises will be delivered. More than ever, banks need to stand for something positive and valued by stakeholders. One way to achieve this is through paying more attention to brand values. Our article explores how values are adopted by employees within a bank. When employees (...) ‘live’ their brand’s values, their behaviour during customer interactions reflects this, encouraging the strengthening of customer relationships. Specifically, we test the relationship between leadership style, employee commitment, and the adoption of values. Data was collected from a survey of 438 branch employees in a leading Irish retail bank. The study found that a structured and directive leadership style was effective at encouraging the adoption of the bank’s values. Moreover, when employees are committed to the organisation, this has a significant impact on their adoption of values. Thus, this study supports the literature which suggests that leadership and commitment are prerequisites for values adoption. (shrink)