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)

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.

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)

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)

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)

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)

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.

Researchers from diverse disciplines, including organismal and cellular physiology, sports science, human nutrition, evolution and ecology, have sought to understand the causes and consequences of the surprising variation in metabolic rate found among and within individual animals of the same species. Research in this area has been hampered by differences in approach, terminology and methodology, and the context in which measurements are made. Recent advances provide important opportunities to identify and address the key questions in the field. By bringing together (...) researchers from different areas of biology and biomedicine, we describe and evaluate these developments and the insights they could yield, highlighting the need for more standardisation across disciplines. We conclude with a list of important questions that can now be addressed by developing a common conceptual and methodological toolkit for studies on metabolic variation in animals. (shrink)

In early 2011, news emerged that United States authorities had begun to apply injections of pentobarbital, a substance provided by Danish pharmaceutical company Lundbeck, when executing capital punishments. Lundbeck reported to be appalled by such unintended usage of pentobarbital, which is licensed for treatment of refractory forms of epilepsy and for usage as an anaesthetic.The human rights NGOs Reprieve and Amnesty International urged Lundbeck to ensure that pentobarbital was not made available to U.S. authorities for use in capital punishments. Lundbeck (...) argued that complete halting of provision of pentobarbital would be detrimental to patients suffering from conditions of severe epilepsy for whom the substance was developed and intended for treatment for life-threatening epileptic seizures. Lundbeck’s Danish headquarters also argued that controlling the provision of pentobarbital was out of Danish Lundbeck’s hands because the substance is produced at Lundbeck’s plant in the United States. Eventually, after talks with Amnesty International and Reprieve, Lundbeck in July 2011 announced that it would seek to put a stop to using pentobarbital in capital punishment through the introduction of a “drop-ship” program. (shrink)

The article discusses the dynamics of currencies, of media of exchange in a perspective of longue durée. It explores the concept of ›toxic media‹ and of an ›ecology of practices‹ by tracing discussions on cacao as money and as consumable since the 15th century. Furthermore, historically specifi c versions of general purpose money, such as the Spanish Silver-Peso and the U.S. dollar are compared with to self-restricting currencies: currencies that are more tightly knit into the »web of life« ( Jason (...) Moore), into local economies and ecologies and into cultural systems. The article determines toxic effects of generalized media of exchange in the Capitalocene and discusses the interplay of situated money-regimes with generalized, globalized media of exchange. (shrink)

During the first years of organized agricultural research in Mexico in the 1940s, two agencies ran separate programs for corn improvement. The Rockefeller Foundation's Office of Special Studies and the Mexican government's Office of Experiment Stations carried out research on corn with distinct aims and methods. That they differed strongly is well established in the literature. Many authors have discussed a Rockefeller Foundation program that reportedly emphasized hybrid corn, a technical choice that embodied a preference for assisting wealthy farmers who (...) could afford hybrid corn and the necessary agricultural inputs. Conversely, these authors attribute to the Mexican program a focus on open-pollinated corn, which presumably manifested its concern for Mexico's small farmers who saved seed. This article argues that the reverse was true. The Rockefeller Foundation program initially was committed to a type of improved open-pollinated corn while the Mexican program proceeded with a strict, U.S.-style hybrid program. I try to illuminate each group's priority on yield, as well as additional significant, and complicating, breeding priorities that precluded any collaboration between the two. (shrink)

Relational, selfless, caring, polite, nice, and kind are not how we imagine a woman giving birth in U.S. culture. Rather, we picture her as screaming, yelling, self-centered, and demanding drugs or occasionally as numbed and passive from pain-killing medication. Using in-depth interviews with women about their labor and childbirth, the author presents data to suggest that white, middle-class, heterosexual women often worry about being nice, polite, kind, and selfless in their interactions during labor and childbirth. This finding is important not (...) only because it contradicts the dominant cultural image of the birthing woman but because it reveals that an internalized sense of gender plays a role in disciplining women and their bodies during childbirth. The feminist sociological literatures on birth are concerned with how women and their bodies are controlled, yet they have overlooked this other dimension of control that is not institutional but is a product of how gender is internalized. (shrink)

The U.S. National Institute of Allergies and Infectious Diseases and the National Institute of Mental Health have a new research priority: inclusion of terminally ill persons living with HIV in HIV cure-related research. For example, the Last Gift is a clinical research study at the University of California San Diego for PLWHIV who have a terminal illness, with a prognosis of less than 6 months. As end-of-life HIV cure research is relatively new, the scientific community has a timely opportunity to (...) examine the related ethical challenges. Following an extensive review of the EOL and HIV cure research ethics literature, combined with deliberation from various stakeholders and our experience with the Last Gift study to date, we outline considerations to ensure that such research with terminally ill PLWHIV remains ethical, focusing on five topics: 1) protecting autonomy through informed consent, 2) avoiding exploitation and fostering altruism, 3) maintaining a favorable benefits/risks balance, 4) safeguarding against vulnerability through patient-participant centeredness, and 5) ensuring the acceptance of next-of-kin/loved ones and community stakeholders. EOL HIV cure-related research can be performed ethically and effectively by anticipating key issues that may arise. While not unique to the fields of EOL or HIV cure-related research, the considerations highlighted can help us support a new research approach. We must honor the lives of PLWHIV whose involvement in research can provide the knowledge needed to achieve the dream of making HIV infection curable. (shrink)

The under-representation of women on boards is a heavily discussed topic—not only in Germany. Based on critical mass theory and with the help of a hand-collected panel dataset of 151 listed German firms for the years 2000–2005, we explore whether the link between gender diversity and firm performance follows a U-shape. Controlling for reversed causality, we find evidence for gender diversity to at first negatively affect firm performance and—only after a “critical mass” of about 30 % women has been reached—to (...) be associated with higher firm performance than completely male boards. Given our sample firms, the critical mass of 30 % women translates into an absolute number of about three women on the board and hence supports recent studies on a corresponding “magic number” of women in the boardroom. (shrink)

"Der vorliegende Band hebt sich aus der wachsenden Zahl der Publikationen zur Spatantike und namentlich zur spatantiken Philosophie schon durch die Originalitat des behandelten Themas hervor, das eine Forschungsluecke schlieat. Die Beitrage von Wissenschaftlern verschiedener europaischer Nationen, die als Spezialisten fuer die Spatantike gelten konnen, bieten sowohl einzeln als auch in der Zusammenstellung einen echten Forschungsfortschritt." Plekos "a a valuable contribution. The volume also shows, as the product of predominantly young scholars, that the future of scholarship in the area of (...) ancient and Patristic philosophy is in good hands." Vigiliae Chritianae Inhalt: S. Follinger: Philosophische Theorien in Arnobius' apologetischer Argumentation J. Althoff: Zur Epikurrezeption bei Laktanz C. Riedweg: Philosophische Argumentationsstrukturen in Julians Contra Galilaeos D. J. O'Meara: Epicurus Neoplatonicus R. Thiel: Stoische Ethik im neuplatonischen System (Simplikios' Kommentar zu Epiktets Enchiridion) M. Erler: Hellenistische Philosophie im Platonismus der Spatantike M. Bettetini: Ai limiti della materia, tra neoplatonismo e cristianesimo K. Schlapbach: Ciceronisches und Neuplatonisches in den Proomien von Augustin, Contra Academicos 1 und 2 S. Harwardt: Die Gluecksfrage der Stoa in Augustins De beata vita C. Horn: Augustinus ueber Tugend, Moralitat und das hochste Gut T. Fuhrer: Zum wahrnehmungstheoretischen Hintergrund von Augustins Glaubensbegriff C. Oser-Grote: Virtus Romana und Virtus Christiana - ein romischer Wertbegriff bei Prudentius J. Opsomer / C. Steel: Proclus' Doctrine on the Origin of Evil K. Pollmann: Fiktionalitat in der Philosophie des Hellenismus und in der Spatantike U. Eigler: Strukturen und Voraussetzungen zum Erhalt von philosophischem Wissen in der Spatantike Stellenregister. (shrink)

Moraal was sinds mensenheugenis het exclusieve domein van filosofie en theologie, totdat bioloog Charles Darwin in The Descent of Man (1871) als eerste opperde: ‘Elk dier dat met uitgesproken sociale instincten is toegerust (…) zou onvermijdelijk een moraal of geweten verwerven zodra zijn intellectuele vermogens even goed of bijna even goed ontwikkeld zouden zijn als die van de mens.’ Pas een eeuw later kwam het onderzoek naar moraal goed op gang in empirische wetenschapsdisciplines als biologie, ethologie, culturele antropologie, psychologie en (...) neurowetenschappen. De getemde mens laat zien hoe ver de fascinerende inzichten die dat onderzoek al heeft opgeleverd, zijn doorgedrongen in Vlaanderen en Nederland. Meer dan 140 bekende en onbekende personen met uiteenlopende culturele en levensbeschouwelijke achtergronden laten in dit boek hun gedachten gaan over de vraag waar onze moraal vandaan komt. Primatoloog Frans de Waal wijdt aan deze centrale vraag een uitvoerig essay, terwijl schooldirecteur en expert radicalisering Karin Heremans een stem laat horen uit de multiculturele onderwijspraktijk. Filosoof Guido Vanheeswijck en moraalwetenschapper Dirk Verhofstadt geven niet alleen zelf antwoord op de vraag waar onze moraal vandaan komt, maar lazen alle bijdragen en schreven naar aanleiding daarvan een slotessay. (shrink)

This book collects essays from the 2006 and 2007 International Philosophy Colloquia Evian, centred around a central problem in the philosophy of mind: the relationship between the human faculty of sensory experience and the faculty of conceptual reflection, that is self-consciousness. Containing articles by philosophers of eight nationalities, in three languages (English, French, German), and of "analytical" as well as "continental" provenance, it beautifully represents the spirit of the colloquia. Authors include Joshua Andresen (AU Beirut), Valérie Aucouturier (Kent U / (...) U Paris I), Karin de Boer (KU Leuven), Santiago Echeverri (U Genève), Roberto Farneti (LU Bolzano), Tim Henning (JLU Giessen), Felix Koch (Columbia U), Christophe Laudou (Madrid), David Lauer (FU Berlin), Jason Leddington (Bucknell U), Nicolas Monseu (UC Louvain), Soraya Nour (HU Berlin), Hans Bernhard Schmid (U Wien), Henning Tegtmeyer (U Leipzig). (shrink)

Quine's metalogical 'quasiquotation' is formally added to classical first-Order logic; the resulting system lq is stronger and more natural than all former systems of quotational logic. Lq contains object-Variables ranging over the universe u and expression-Variables ranging over the set e of all expressions of lq; e is a subset of u. Object-Quantifiers are referential, Expression-Quantifiers are substitutional; only the latter ones bind into quasiquotations. Lq contains its own syntactic metatheory and arithmetics. Natural proofs of godel's and tarski's theorems are (...) given for lq. (shrink)