El contexto de Pandemia inesperado y letal por Presencia de COVID-19 ocurrido en el presente milenio, pone en evidencia que la humanidad no estaba preparada para tamaño acontecimiento. El presente artículo evidencia que las ciudades como sistemas de hábitat humano, son precarios a nivel de organización, ordenamiento urbano, confort y seguridad; acompañado por entidades públicas y privadas ineficientes en la gestión urbana. En este contexto, el artículo expone el caso de la ciudad metropolitana de Huancayo, cuyas condiciones de ordenamiento urbano (...) y resiliencia a la situación de emergencia vivida fue precaria y altamente ineficiente para responder a la emergencia. (shrink)

The project of this Précis de philosophie de la logique et des mathématiques (vol. 1 under the direction of F. Poggiolesi and P. Wagner, vol. 2 under the direction of A. Arana and M. Panza) aims to offer a rich, systematic and clear introduction to the main contemporary debates in the philosophy of mathematics and logic. The two volumes bring together the contributions of thirty researchers (twelve for the philosophy of logic and eighteen for the philosophy of mathematics), specialists (...) in the history or philosophy of logic or mathematics. Each volume consists of ten chapters; each chapter is about forty pages, is independent from the others and deals with a particular philosophical question about logic or mathematics. The objective is to offer to the French-speaking reader a reference book. On many of the issues addressed, this Précis offers the first clear and thorough synthesis that is available in French. -/- This book is intended for third year / master's students, but also for teachers of philosophy in secondary or higher education, for logicians and mathematicians willing to take a philosophical look at the fundamental concepts of their discipline, and for any reader interested in the basic issues discussed in the philosophy of logic and mathematics. Each chapter is self-contained, although some basic training in logic (or in mathematics, as the case may be) is required. The book should be looked at as a comprehensive framework for contemporary discussions in the philosophy of logic and mathematics, while also giving some guidance on the disciplinary content required by these discussions. -/- The first three chapters of the present volume (on the philosophy of mathematics) are devoted to the history of the philosophy of mathematics: from antiquity to the modern era, and from this period to the foundational crisis of the nineteenth century, to the twentieth century. Next, four chapters deal with the crucial questions for the philosophy of mathematics of the twentieth century: the opposition and/or comparison of set theory and category theory as foundational frameworks for mathematics; mathematical constructivism; the analysis of calculability; and Benaceraff's problem. The two following chapters are focused on the philosophy of mathematical practice, by treating the notion of ideals of proof, in particular explanation and purity, and the notion of informal proof and the use of visual artifacts in mathematical argumentation. Finally the last chapter deals with the applicability of mathematics, including the role of probability. -/- In Chapter 1, Sébastien Maronne and David Rabouin present the history of the philosophy of mathematics from antiquity to the modern era. The goal is to show that the philosophy of mathematics has a history as ancient as mathematics and philosophy itself, a history largely continuous with the latter, often by staying out of sync with respect to the former. In Chapter 2, Sébastien Gandon discusses the question of the foundations of mathematics from Kant to the end of the nineteenth century. By tying this question to Kant, he is able to show that the discussion of the foundations of mathematics that occupied a large part of these two centuries has much older roots. Accent is placed on the differences between the framework proposed by Kant for making sense of classical mathematics and the use made by Frege and then Russell of logical tools for giving a different reading of arithmetic and real analysis. The link between between these different conceptions and the evolution of mathematics itself is also underlined. -/- Chapter 3, written by Hourya Benis-Sinaceur and Mirna Džamonja, deals with the evolution of mathematics in the twentieth century, notably the structuralist turn of Dedekind, E. Noether, and Bourbaki, and certain more recent developments. They devote themselves, among other things, to explaining how these developments can be understood as responses to and extensions of questions posed during the foundational debate of the immediately preceding era. -/- In Chapter 4, Jean-Pierre Marquis and Jean-Jacques Szczeciniarz discuss the different foundational options coming from set theory and category theory. They discuss, among other things, Lawvere's program to replace ZF with an axiomatization of the category of sets, the latter judged better adapted to the needs of contemporary mathematics. Different reactions, as much philosophical as technical, raised by this program are also taken into account. -/- Constructive mathematics are the object of Chapter 5, in which Gerhard Heinzmann and Mark van Atten examine in a critical and comparative manner a variety of constructivisms, including intuitionism, constructive type theory, predicativism, and finitism. -/- In Chapter 6, Guido Gherardi and Maël Pegny present computability theory, stressing its philosophical consequences. They discuss the basic concepts of this theory, for the computability of both the natural numbers and the real numbers. They then tackle the theory of computational complexity. -/- In Chapter 7, Andrea Sereni and Fabrice Pataut present Benacerraf's problem, opposing every possible account of mathematical knowledge to the availability of a theory of truth founded on an abstract ontology of mathematical objects. They give an overview of the original problem and its reformulation by Hartry Field, and reconstruct the debate that this problem has raised, by meeting at once the major opposing positions today of the analytic affiliation of the philosophy of mathematics, as much those on the platonist side as on that of nominalism. -/- In Chapter 8, Valeria Giardino and Yacin Hamami treat certain crucial aspects of the philosophy of mathematical practice. They deal in particular with the notion of an informal proof and on the role of artifacts in mathematical reasoning. They consider how such proofs, making use of these means, can contribute to the understanding of theorems and mathematical theories, thanks also to the role of visualisation and the aid of computer tools. -/- Why do mathematics often give several proofs of the same theorem? This is the question that Andrew Arana raises in Chapter 9, introducing the notion of an epistemic ideal and discuss two such ideals, the explanatoriness and purity of proof. -/- Chapter 10 is devoted to the applicability of mathematics in the study of empirical phenomena. After summarizing the history of this question, going back to Plato and Aristotle and passing by Mill and Kant, Daniele Molinini and Marco Panza reconstruct the contemporary debate, in relation, among other things, to questions raised in the philosophy of science. -/- Two appendices complete the volume. In the first Frédéric Patras treats the French tradition in philosophy of mathematics of the 20th century. In the second, Maria Carla Galavotti discusses the notion of probability and its use in science. (shrink)

Providing a novel multi-disciplinary theorization of memory politics, this insightful Handbook brings varied literatures into a focused dialogue on the ways in which the past is remembered and how these influence transnational, interstate, and global politics in the present. With case studies from Africa, East and Southeast Asia, Europe, South America, and the United States, the Handbook focuses on the political features of historical memory in international relations. Chapters examine key concepts of memory politics, including accountability, commemoration and memorialization, the (...) Europeanization of memory, and the politics of trauma and victimhood, as well as analyzing different sites of memory, from the human body and memorial sites to media, film, and television. It also answers essential questions such as who and what determines the relevance of the past in the present; how does memory become a political question; and what are the political effects and ethical implications of its mobilization? Exploring the links between the politics of memory, international ethics, law, and diplomacy, this stimulating Handbook will be essential reading for students and scholars of politics and international relations, cultural studies, history, and transitional justice. Its discussion of notable agents and practices of memory politics will also be beneficial for practitioners working in human rights, politics, and public policy. (shrink)

Sound and listening are intrinsically linked to how we experience and engage with places and communities. This guide invites landscape architects and urban designers to become soundscape architects and offers practical advice on sound and listening applicable to each stage of a design project: from reading the environment to intervening on it. This book foregrounds listening as an affective mediator between subjects and multispecies environments, and a vehicle to think and conceptualise environmental design beyond prevailing visual and human-centred modes. The (...) guide expands landscape architects' and urban designers' tools and skills to assess existing soundscapes, predict how those soundscapes will be altered through their designs, consider sound as a creative and active part of the design process, and envisage how users might perceive and be affected by those soundscapes as they evolve in time. The volume sits in the interface of research and practice, and incorporates theoretical, methodological and creative contributions from acoustic ecology, ecoacoustics, bioacoustics and sound art. Each of the design stages is illustrated through project examples that demonstrate the many advantages of incorporating attentive listening and sound into Landscape Architecture and Urban Design Practice. The book shows how incorporating listening and sounding as part of the design process promotes slow and subtle ways of practice, adds social and ecological value through the reduction of noise pollution and by monitoring the health of habitats, and enables the design of soundscapes that complement the character and design intent of a scheme and elicit joy and wonder. The book will be of interest to landscape architects, together with other design professionals such as urban designers, architects, geographers and engineers, who play a primary role in the composition of the soundscape. (shrink)

During the last few months, several pioneer genome‐wide transcriptomic, proteomic and metabolomic studies have revolutionised the understanding of bacterial biological processes, leading to a picture that resembles eukaryotic complexity. Technological advances such as next‐generation high‐throughput sequencing and high‐density oligonucleotide microarrays have allowed the determination, in several bacteria, of the entire boundaries of all expressed transcripts. Consequently, novel RNA‐mediated regulatory mechanisms have been discovered including multifunctional RNAs. Moreover, resolution of bacterial proteome organisation (interactome) and global protein localisation (localizome) have unveiled an (...) unanticipated complexity that highlights the significance of protein multifunctionality and localisation in the cell. Also, analysis of a complete bacterial metabolic network has again revealed a high fraction of multifunctional enzymes and an unexpectedly high level of metabolic responses and adaptation. Altogether, these novel approaches have permitted the deciphering of the entire physiological landscape of one of the smallest bacteria, Mycoplasma pneumoniae. Here, we summarise and discuss recent findings aimed at defining the blueprint of any prokaryote. (shrink)

In this Chapter, Maria Kronfeldner discusses whether psychological essentialism is a necessary part of dehumanization. This involves different elements of essentialism, and a narrow and a broad way of conceptualizing psychological essentialism, the first akin to natural kind thinking, the second based on entitativity. She first presents authors that have connected essentialism with dehumanization. She then introduces the error theory of psychological essentialism regarding the category of the human, and distinguishes different elements of psychological essentialism. On that basis, Kronfeldner (...) connects historical, socio-psychological, and philosophical insights in order to show that although essentialism can act as a catalyst for dehumanization, it is not necessary for it. Examples relate to dehumanization in the context of colonialism and evolutionary thinking, to the history of dehumanizing women from Aristotle to 19th-century craniology, and to contemporary self-dehumanization and ‘lesser mind’ attribution. (shrink)

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

Maria Kronfeldner’s Preface and Introduction to the Routledge Handbook of Dehumanization maps the landscape of dehumanization studies. She starts with a brief portrayal of the history of the field. The systematically minded sections that follow guide the reader through the resulting rugged landscape represented in the Handbook’s contributions. Different realizations, levels, forms, and ontological contrasts of dehumanization are distinguished, followed by remarks on the variety of targets of dehumanization. A discussion on valence and emotional aspects is added. Causes, functions, (...) and consequences of dehumanization, and the prospects for reducing or undoing it, are introduced. The systematic overview closes with a discussion of some important theoretical complexities that arise in studying dehumanization. After these systematic sections, the scholarly work on dehumanization gets situated in the broader intellectual landscape of debates about the ‘human’ in the humanities and social sciences. The Introduction ends with some notes on scope, limitations, and intended readership of the Handbook. (shrink)

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case on which (...) to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

In his 1978 paper “Mathematical Explanation”, Mark Steiner attempts to modernize the Aristotelian idea that to explain a mathematical statement is to deduce it from the essence of entities figuring in the statement, by replacing talk of essences with talk of “characterizing properties”. The language Steiner uses is reminiscent of language used for proofs deemed “pure”, such as Selberg and Erdős’ elementary proofs of the prime number theorem avoiding the complex analysis of earlier proofs. Hilbert characterized pure proofs as those (...) that use only “means that are suggested by the content of the theorem”, a characterization we have elsewhere called “topical purity”. In this paper we will examine the connection between Steiner’s account of mathematical explanation and topical purity. Are Steiner-explanatory proofs necessarily topically pure? Are topically pure proofs necessarily Steiner-explanatory? Answers to these questions will shed light on the general question of the relation between purity and explanatory power. (shrink)

Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...) discuss its epistemological significance. (shrink)

Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them.

Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.

"Dr. Montessori was par excellence the great interpreter of the child; and though she herself has passed on from the scene of her labours her work will still go on."-- Westminster Cathedral Chronicle One of the landmark books in the history of education--and one of the least expensive editions now available--this volume describes a new system for educating youngsters. Based on a radical concept of liberty for the pupil and highly formal training of separate sensory, motor, and mental capacities, the (...) system enabled youngsters to master reading, writing, and arithmetic rapidly and substantially. Included are discussions of scientific pedagogy; discipline; the importance of proper diet, gymnastics, and manual labor; and many other topics. Unabridged reprint of the classic 1912 edition. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

Testimonio que recoge algunas experiencias y reflexiones de la artista visual costarricense María Rapela en Alemania y de cómo se involucró en la gestión cultural, coordinando el Fieber Festival, una plataforma colaborativa y auto-gestionada de mujeres artistas migrantes iberoamericanas en Berlín entre el 2011 y el 2017.

Food systems transformations require coherent policies and improved understandings of the drivers and institutional dynamics that shape (un)sustainable food systems outcomes. In this paper, we introduce the Chilean National Organic Agriculture Law as a case of a policy process seeking to institutionalize a recognized pathway towards more sustainable food systems. Drawing from institutional theory we make visible multiple, and at times competing, logics (i.e., values, assumptions and practices) of different actors implicated in organic agriculture in Chile. More specifically, our findings (...) identify five main institutional transformative logics underpinning the interests and actions of organic actors. However, we find that the Law was not motivated by these logics and did not advance them. Rather, the Law was designed to support a market niche targeted to elite consumers and to reinforce agricultural exports. As a result, the Law constrains rather than enables the practice of organic agriculture and access to organic food by consumers, especially at the domestic level. We note that attention to institutional logics in the analysis of food systems, and specifically food system transformation, is relevant to more comprehensive assessments of the transformational potential of food systems policies. We conclude that there is a need to further consider and make visible the way in which different drivers (i.e., laws) are constituted through and by diverse, and often competing, institutional logics. (shrink)

In this paper we consider three potential simplifications to a result of Solovay’s concerning the Turing degrees of nonstandard models of arbitrary completions of first-order Peano Arithmetic (PA). Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤T X of an “appropriate” Scott set and there is a family of functions (tn)n∈ω, ∆0 n(X) uniformly in n, such that lim tn(s) s→∞.

Silence is reserved for what cannot be verbally expressed. The well-known Wittgensteinian quote summarizes an established understanding of the relationship between language and silence: because language is not enough to account for reality and thinking, it must be transcended by other means of expression, like music or silence. But what if the opposite is the case and silence is not the extension but the precondition of language, the ultimate source of meaning? This paper explores how this is the phenomenological and (...) Derridean philosophical framework of Coetzee’s novels, a literary universe created from the will to signify and build singular meanings. (shrink)

A striking feature of atrocities, as seen in genocides, civil wars or violence against certain racial and ethnic groups, is the attempt to dehumanize – to deny and strip human beings of their humanity. Yet the very nature of dehumanization remains relatively poorly understood. The Routledge Handbook of Dehumanization is the first comprehensive and multidisciplinary reference source on the subject and an outstanding survey of the key concepts, issues and debates within dehumanization studies. Organized into four parts, the Handbook covers (...) the following key topics: (I) The history of dehumanization from Greek Antiquity to the Twentieth century, contextualizing the oscillating boundaries, dimensions, and hierarchies of humanity in the history of the ‘West’. (II) How dehumanization is contemporarily studied with respect to special contexts: as part of social psychology, as part of legal studies or literary studies, and how it connects to the idea of human rights, disability and eugenics, the question of animals, and the issue of moral standing. (III) How to tackle its complex facets, with respect to the perpetrator’s and the target’s perspective, metadehumanization and selfdehumanization, rehumanization, social death, status and interdependence, as well as the fear we show towards robots that become too human for us. (IV) Conceptual and epistemological questions on how to distinguish different forms of dehumanization and neighboring phenomena, on why dehumanization appears so paradoxical, and on its connection to hatred, essentialism, and perception. -- Essential reading for students and researchers in philosophy, history, psychology, and anthropology this Handbook will also be of interest to those in related disciplines such as politics, international relations, criminology, legal studies, literary studies, gender studies, disability studies, or race and ethnic studies, as well as readers from social work, political activism, and public policy --- Table of Contents: Introduction. 1. Mapping dehumanization studies. – Maria Kronfeldner. Part I. Oscillating boundaries, dimensions, and hierarchies of humanity in historical contexts. 2. Dehumanization Before the Columbian exchange – Siep Stuurman. 3. 'Humanity' and its Limits in Early Modern European Thought – László Kontler. 4. Enlightenment Humanization and Dehumanization and the Orang-utan – Silvia Sebastiani. 5. Dehumanizing the Exotic in Living Human Exhibitions – Guido Abbattista. 6. Dehumanizing Strategies in Nazi Ideology and their Anthropological Context – Johannes Steizinger. 7. Theorizing the Inhumanity of Human Nature, 1955-1985 – Erika Lorraine Milam. Part II: Further special contexts of dehumanization. 8. The Social Psychology of Dehumanization – Nick Haslam. 9. Dehumanization and the Loss of Moral Standing – Edouard Machery. 10. Dehumanization and the Question of Animals – Alice Crary. 11. Dehumanization, Disability, and Eugenics – Robert A. Wilson. 12. Dehumanization and Human Rights – Marie-Luisa Frick. 13. Dehumanization by Law – Luigi Corrias. 14. Dehumanisation in Literature and the Figure of the Perpetrator – Andrea Timár. Part III: The complex facets of dehumanization. 15. Dehumanization and Social Death as Fundamentals of Racism – Wulf D. Hund. 16. How Status and Interdependence Explain Different Forms of Dehumanization – Susan T. Fiske. 17. Exploring Metadehumanization and Self-dehumanization from a Target Perspective – Stéphanie Demoulin, Pierre Maurage, Florence Stinglhamber. 18. The Dehumanization and Rehumanization of Refugees – Victoria M. Esses, Stelian Medianu, Alina Sutter. 19. Motivational and Cognitive Underpinnings of Fear of Social Robots that become ‘Too Human for Us’ – Maria Paola Paladino, Jeroen Vaes, Jolanda Jetten. Part IV: Conceptual and epistemological questions regarding dehumanization. 20. Objectification, Inferiorization and Projection in Phenomenological Research on Dehumanization – Sara Heinämaa and James Jardine. 21. Why Dehumanization is Distinct from Objectification – Mari Mikkola. 22. On Hatred and Dehumanization – Thomas Brudholm and Johannes Lang. 23. Dehumanization, the Problem of Humanity, and the Problem of Monstrosity – David Livingstone Smith. 24. Psychological Essentialism and Dehumanization – Maria Kronfeldner. 25. Could Dehumanization Be Perceptual? – Somogy Varga. (shrink)

El artículo expone la evolución de las expectativas acerca del futuro y destino final del hombre tanto a nivel individual como colectivo. Hasta el siglo XVIII predominaban las preocupaciones referidas al destino personal de cada hombre y el significado de la muerte. Tras las revoluciones que acabaron con el antiguo régimen, empezó a dominar el interés por el futuro de la humanidad en su conjunto. Desde mediados del siglo XX hay una preocupación creciente ante el peligro de que la especie (...) humana desaparezca y sea suplantada por otro tipo de entidades, tal vez desarrolladas por el propio hombre. El transhumanismo se ha presentado como una respuesta, basada en la ciencia, a las ansias de supervivencia y mejora del hombre. No obstante, sus propuestas adolecen de varias debilidades, algunas de ellas posiblemente irreparables. (shrink)

Conceptions of evil have changed dramatically over time, and though humans continue to commit acts of cruelty against one another, today we possess a clearer, more moral way of analyzing them. In _Narrating Evil_, María Pía Lara explores what has changed in our understanding of evil, why the transformation matters, and how we can learn from this specific historical development. Drawing on Immanuel Kant's and Hannah Arendt's ideas about reflective judgment, Lara argues that narrative plays a key role in helping (...) societies acknowledge their pasts. Particular stories haunt our consciousness and lead to a kind of examination and dialogue that shape notions of morality. A powerful description of a crime can act as a filter, helping us to draw conclusions about what constitutes a moral wrong, and public debates over these narratives allow us to construct a more accurate picture of historical truth, leading to a better understanding of why such actions are possible. In building her argument, Lara considers Greek tragedies, Shakespeare's depictions of evil, Joseph Conrad's literary metaphors, and movies that portray human cruelty. Turning to such philosophers and writers as Jürgen Habermas, Walter Benjamin, Primo Levi, Giorgio Agamben, and Ariel Dorfman, Lara defines a reflexive relationship between an event, the narrative of the event, and the public reception of the narrative, and she proves that the stories of perpetrators and sufferers are always intertwined. The process of disclosure, debate, and the public fashioning of collective judgment are vital methods through which we make sense not only of new forms of cruelty but of past crimes as well. _Narrating Evil_ describes the steps of this process and why they are a crucial part of our attempt to build a different, more just world. (shrink)

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.1 (2003) 131-132 [Access article in PDF] Christia Mercer. Leibniz's Metaphysics: Its Origins and Development. New York: Cambridge University Press, 2001. Pp. xiii + 528. Cloth, $80.00. Christia Mercer's massive study is aimed at unearthing the hidden roots of Leibniz's metaphysics by placing the German philosopher back in the intellectual context within which his thought first took shape. In so doing she stresses (...) the fundamental importance of Leibniz's early years for the development of his philosophy. In particular she convincingly argues for the key role played by Leibniz's early exposure to the thought of eclectic thinkers (including his teachers) who were trying to build a new philosophical synthesis centered around the key tenets of Christian theology and combining elements of Aristotelianism, Platonism, and (in at least some cases) the new mechanical philosophy of nature.One of Mercer's central theses is that the methodological and metaphysical commitments that Leibniz developed during his youth formed the bedrock of his mature philosophy (23). More precisely, her bold claim is that Leibniz's core metaphysics (including the complete concept account of substance) was constructed in 1668-71, re-examined and slightly developed in 1672-79, and then used for the rest of his long philosophical career (18, 470). The key inspirational forces driving the development of Leibniz's philosophy, in her view, had even earlier origins. In the mid-1660s Leibniz did not yet have a fully formulated metaphysics; but he already possessed very definite philosophical objectives which were to guide the development of his metaphysics and form the unstated assumptions underlying his mature thought (40). Mercer identifies these unstated assumptions as a "Metaphysics of Method," a "Metaphysics of Substance," and a "Metaphysics of Divinity," all of which she finds fully developed in his early thought (468-9).According to Leibniz's "Metaphysics of Method" (examined in part I), there exists a truth beneath the prominent philosophical schools which can and ought to be discovered. This conciliatory methodology was Leibniz's answer to the political, religious, and philosophical chaos around him. He was deeply convinced that his brilliantly original eclectic metaphysics would provide the foundation of philosophical, theological, and even political peace. Leibniz's "Metaphysics of Substance," Mercer argues in part II, melded an Aristotelian approach to substance with a mechanical physics and originated at the very outset of his intellectual carrier in an attempt to solve specific theological problems (e.g., those of the incarnation and resurrection). Platonism in turn offered the material for Leibniz's "Metaphysics of Divinity" (explored in part III), that is, for his conception of the relationship between God and creatures. The fourth and final part of Mercer's book takes her analysis of Leibniz's writings up to 1679 and describes the laying of the foundations of his mature metaphysics. She concludes not only that Leibniz proposed the doctrine of pre-established harmony prior to his departure for Paris in March 1672, but also, more generally, that the other tenets of Leibniz's mature thought, including his theory of truth, grew naturally out of his early metaphysics (see, in particular, 2, 15-8, 47, 52, 169-70, 250-2, 300-1, 472).Throughout the book Mercer describes her interpretation as "startling," "unfamiliar," "dramatic," and "surprising," and fears that "readers may balk" at her conclusions since "so much of Leibniz's thought has escaped us for so long" (see, for instance, 1, 9, 471). Some readers may feel, on the contrary, that she has underestimated the extent to which several of her central theses—such as the importance of the earliest stages of Leibniz's philosophical development in general and of eclecticism, theological problems, and Platonism, as well as Aristotelianism within that development in particular—are already reflected in the [End Page 131] best recent traditions of international Leibniz scholarship. But her book undoubtedly breaks new ground and remains an important contribution to an exciting field which cannot and will not be ignored. While international Leibniz scholarship may not absorb some of this book's more... (shrink)

This paper aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way as the (...) set of normative discourses regulating mathematical practices and the ways in which mathematicians promote those practices. To clarify our proposal, we present two case studies from 17th century France. First, we exemplify a case of mathematical hygiene proper: Descartes' algebraic geometry and Newton's subsequent criticism of it, a case of (im)purity in mathematics. Then, we compare Descartes' and Newton's mathematical hygiene with verbal hygiene from this period, as exemplified by the work of the grammarian Claude Favre de Vaugelas (Ayres-Bennett 1987). We argue that these early modern normative discourses on mathematics and language respectively can be seen as emanating from a common socio-political program: the development of a new bourgeois intellectual class. We conclude that the study of mathematical hygiene has the potential to yield new understandings of the social aspects of mathematical practice, and that similarities between mathematical and verbal hygiene at certain time periods, such as 17th century France, open up a new area of inquiry at the borders of linguistics and the philosophy of mathematics. (shrink)

A paper about cross-cultural and cross-racial loving that emphasizes the need to understand and affirm the plurality in and among women as central to feminist ontology and epistemology. Love is seen not as fusion and erasure of difference but as incompatible with them. Love reveals plurality. Unity–not to be confused with solidarity–is understood as conceptually tied to domination.

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

An account of mathematical understanding should account for the differences between theorems whose proofs are “easy” to discover, and those whose proofs are difficult to discover. Though Hilbert seems to have created proof theory with the idea that it would address this kind of “discovermental complexity”, much more attention has been paid to the lengths of proofs, a measure of the difficulty of _verifying_ of a _given_ formal object that it is a proof of a given formula in a given (...) formal system. In this paper we will shift attention back to discovermental complexity, by addressing a “topological” measure of proof complexity recently highlighted by Alessandra Carbone ( 2009 ). Though we will contend that Carbone’s measure fails as a measure of discovermental complexity, it forefronts numerous important formal and epistemological issues that we will discuss, including the structure of proofs and the question of whether impure proofs are systematically simpler than pure proofs. (shrink)

A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.