We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

One of the most influential scientific treatises in Cauchy's era was J.-L. Lagrange's Mécanique Analytique, the second edition of which came out in 1811, when Cauchy was barely out of his teens. Lagrange opens his treatise with an unequivocal endorsement of infinitesimals. Referring to the system of infinitesimal calculus, Lagrange writes:Lorsqu'on a bien conçu l'esprit de ce système, et qu'on s'est convaincu de l'exactitude de ses résultats par la méthode géométrique des premières et dernières raisons, ou par la méthode analytique (...) des fonctions dérivées, on peut employer les infiniment petits comme un instrument sûr et commode pour abréger et simplifier les demonstrations.Lagrange's renewed enthusiasm for .. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.

The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementarity rather than a rivalry of the approaches.

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...) interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (shrink)

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...) Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

Die Rassentheorie, die Geschichtsphilosophie, die Ästhetik und die Naturteleologie haben eine Gemeinsamkeit: In all diesen Themengebieten entwickelte Immanuel Kant ein Denken der Zweckmässigkeit. Die Fokussierung auf diesen Strang macht eine Verbindung sichtbar, die von seinen frühen Schriften zu den unterschiedlichen »Rassen« der Menschen hin zur Kritik der Urteilskraft und damit zu seiner Selbstreflexion über die kritische Philosophie reicht. Karin Hostettler arbeitet das mit diesem Denken verbundene Othering und die damit einhergehende Selbstaffirmation heraus und zeigt so die Selbstverortung der kritischen (...) Philosophie in einer kolonialen Episteme auf. (shrink)

In this tribute to Steven T. Katz on the occasion of his seventieth birthday, Michael Zank and Ingrid Anderson present sixteen original essays written by senior and junior scholars in comparative religion, philosophy of religion, modern Judaism, and theology after the Holocaust, fields of inquiry where Steven Katz made major contributions over the course of his distinguished scholarly career. The authors of this volume, specialists in Jewish history, especially the modern experience, and Jewish thought from the Bible to (...) Buber, offer theoretical and practical observations on the value of the particular. Contributions range from Tim Knepper's reevaluation of the ineffability discourse to the particulars of the Settlement Cookbook, examined by Nora Rubel as an American classic. (shrink)

The Pale God examines the relationship between secularism and religious tradition. It begins with a description of the secular options as expressed by Israeli intellectuals, and describes how these options have led to a dead end. A new option must be sought, and one of the key sources for this option is the works of Spinoza. The author explains that unlike Nietzsche, who discussed "the death of God," Spinoza tried to undermine the authority of religious virtuosos and establish the image (...) of a rational "Pale God." Such changes could channel religious tradition to the basic principles of secular political rule. The author demonstrates that the secular option is inherent in Israeli society, fits the type of secularism that Zionism instilled in the Jewish people, and complements the traditional trends deeply rooted in that society. (shrink)

Bringing Human Rights Education to US Classrooms presents ten research-based human rights projects powerfully implemented in a range of U.S. classrooms, from elementary school through community college and university. In these classrooms, the students--primarily young people of color who have experienced or witnessed human rights abuses such as discrimination and poverty--are exposed for the first time to thinking about their own lives and the world through an empowering human rights lens. Unique in integrating theory and classroom practice, and in addressing (...) human rights issues with special relevance for communities of color in the US, Katz and Spero provide indispensable guidance for those studying and teaching human rights. (shrink)

This book traces and explores the evolution of taste from a design perspective: what it is, how it works and what it does. Karin Tehve examines taste primarily through its recursive relationship to media. This ongoing process changes the relationship between designers and the public, and our understanding of the relationship of individuals to their social contexts. Through an analysis of taste, design is understood to be an active constituent of social life, not as autonomous from it. This book (...) reclaims a term long dismissed from interior design and unveils taste's role as a powerful social and political agent within systems of aesthetics, affecting both its producers and consumers. Each chapter discusses a taste concept or definition, analyzes its reciprocal relationship with media, and explores its implications for interior design. Illustrated with 70 images, taste's relationship to media is viewed through a variety of different lenses, including books, photography, magazines, internet, social media and algorithms. Written for students and scholars of interior design and related design fields, this book will be a helpful resource for all those interested in the question of taste, and is an invitation to produce and consume all media critically. (shrink)

This book offers an interdisciplinary approach to religion and psychology, including a compelling denouement that reveals new narratives about longstanding rumours in the early history of the psychoanalytic movement. This volume demonstrates that the first generation of psychoanalysts succeeded in writing themselves into the history of religious thought and sacralizing the origins of psychoanalysis.

Yeshayahu Leibowitz (1903–1994) was an Israeli philosopher and scientist. For decades, his thinking and persona were the embodiment of a Judaism in Israel. Getting to know him is getting to know a great Israeli thinker and also invites a window into the life of Israel and its problems.

This chapter contains sections titled: I The Nature of Philosophy of Education and Conceptual Analysis II Peters' Conception of Education and some of its Normative Implications III Peters' Indeterminacy of Educational Aims and the Need for Sustained Dialogue References.

Neo-Scholastic metaphysics plays an outstanding role in 17th century philosophy. The beginnings of the modern scientific view of the world originate from precisely the same time. With reference to the anatomist Francis Glisson (ca. 1597-1677), this study investigates the significance of school metaphysics for the inception of a new form of natural philosophy. It shows that Glisson developed a retrospective theory of substance from his identity as a scientist. In so doing, it casts a surprising light on the formative intellectual (...) conditions governing the emergence of the sciences in the modern age. (shrink)

How does science create knowledge? Epistemic cultures, shaped by affinity, necessity, and historical coincidence, determine how we know what we know. In this book, Karin Knorr Cetina compares two of the most important and intriguing epistemic cultures of our day, those in high energy physics and molecular biology. The first ethnographic study to systematically compare two different scientific laboratory cultures, this book sharpens our focus on epistemic cultures as the basis of the knowledge society.

In Conversation with Karen Barad: Doings of Agential Realism is an accessible introduction to Karen Barad's agential realist philosophy. The authors take on a unique approach to involve the readers in in/formal conversations between Karen, postgraduates, and researchers at a research event held in 2017 at Cape Town, South Africa.

Cultural accounts of scientific ideas and practices have increasingly come to be welcomed as a corrective to previous—and still widely held—theories of scientific knowledge and practices as universal. The editors caution, however, against the temptation to overgeneralize the work of culture, and to lapse into a kind of essentialism that flattens the range and variety of scientific work. The book refers to this tendency as culturalism. The contributors to the volume model a new path where historicized and cultural accounts of (...) scientific practice retain their specificity and complexity without falling into the traps of culturalism. They examine, among other issues, the potential of using notions of culture to study behavior in financial markets; the ideology, organization, and practice of earthquake monitoring and prediction during China's Cultural Revolution; the history of quadratic equations in China; and how studying the "glass ceiling" and employment discrimination became accepted in the social sciences. Demonstrating the need to understand the work of culture as a fluid and dynamic process that directly both shapes and is shaped by scientific practice, Cultures without Culturalism makes an important intervention in science studies. Contributors. Bruno Belhoste, Karine Chemla, Caroline Ehrhardt, Fa-ti Fan,Kenji Ito, Evelyn Fox Keller, Guillaume Lachenal, Donald MacKenzie, Mary S. Morgan, Nancy J. Nersessian, David Rabouin, Hans-Jörg Rheinberger, Claude Rosental, Koen Vermeir. (shrink)

Dieses Buch untersucht die Art und Weise, wie die Musikschriftsteller Robert Schumann, E. T. A. Hoffmann und Ludwig Tieck versucht haben, den musikalischen Eindruck von Werken wie Beethovens Neunter Symphonie oder Mozarts Don Giovanni zu beschreiben. Der Autor Moritz Katz zeigt auf, wie diese Beschreibungen notwendigerweise unvollständig sind und wie die Sprache der Musik sich der Beschreibung entzieht. Dieses Buch wird Musikliebhaber und Studenten der Musik- und Literaturgeschichte ansprechen. This work has been selected by scholars as being culturally important, (...) and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how individuals have valued generality and how they have worked with specific types of "general" entities, (...) procedures, and arguments. (shrink)

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, (...) contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)

Gottfried Wilhelm Leibniz leitete die Bibliothek der Hannoveraner Welfen vierzig Jahre lang bis zu seinem Tod 1716.

Attempts to penetrate behind Plotinus' statements to the problem that faced his philosophy to show its function in bridging the inconsistencies that arise when existential considerations are not clearly distinguished.

Humboldts texttheoretischer Ansatz, dessen Einfluß auf die moderne (Text-)Linguistik größer ist als bisher wahrgenommen, wird im ersten Teil dieses Bandes erstmals zusammenhängend nachgewiesen.

This new translation of The Upanishads is at once delightfully simple and rigorously learned, providing today's readers with an accurate, accessible rendering of the core work of ancient Indian philosophy. The Upanishads are often considered the most important literature from ancient India. Yet many academic translators fail to capture the work's philosophical and spiritual subtlety, while others convey its poetry at the cost of literal meaning. This new translation by Vernon Katz and Thomas Egenes fills the need for an (...) Upanishads that is clear, simple, and insightful - yet remains faithful to the original Sanskrit. As Western Sanskrit scholars who have spent their lives immersed in meditative practice, Katz and Egenes offer a unique perspective in penetrating the depths of Eastern wisdom and expressing these insights in modern yet poetic language. Their historical introduction is suited to newcomers and experienced readers alike, providing the perfect entry to this unparalleled work. (shrink)

"How Jewish is modern Jewish philosophy? The question at first appears nonsensical, until we consider that the chief issues with which Jewish philosophers have engaged, from the Enlightenment through to the late 20th century, are the standard preoccupations of general philosophical inquiry. Questions about God, reality, language, and knowledge have been as much concern to Jewish thinkers as they have been to others. In this textbook, which surveys the most prominent thinkers of the last three centuries, Claire Katz situates (...) modern Jewish philosophy in the wider cultural and intellectual context of its day, indicating how broader currents of British, French and German thought influenced its practitioners. But she also addresses the unique ways in which being Jewish coloured their output, suggesting that a keen sense of particularity enabled the Jewish philosophers to help define the whole modern era."--Page 4 of cover. (shrink)

Once shunned, interdisciplinarity has now become fashionable-but badly in need of unpacking. The transfer of ideas from one field to another requires full understanding of the ways in which they are used and understood, where they come from and where they are going. This book calls attention to certain linguistic tools that served scholars in the past and that are still relevant today-if properly employed. It offers a roadmap that inter-relates problems within and between the sciences-human and natural. Throughout, the (...) book interpolates excerpts from scholarly texts that light the road. Spelling out how research may go beyond the mere assembly of facts, it invites the reader to join a sequence of steps that highlights the processes which problem solving entails. Ruth Katz (Ph.D. Columbia University) is one of the founding members of the Department of Musicology at the Hebrew University of Jerusalem. Upon receiving the prestigious Israel Prize in 2012, her work was cited for shedding light on such things as the interdisciplinary processes implicated in "the origin of opera, the history of Western notation, the relationship of stylistic change to the history of ideas, music's place in philosophy, and the role of musical aesthetics in cognition." She founded the Excellence Program at the University and co-founded the Laboratory for the study of music. Her most recent book, hailed as "magisterial," is entitled A Language of Its Own: Sense and Meaning in the Making of Western Art Music (University of Chicago Press, 2009). It is an analysis of the decisions that underlay the crafting and understanding of stylistic change. (shrink)