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.

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

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)

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)

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)

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)

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)

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)

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.

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.

In this eye-opening look at the doctor-patient decision-making process, physician and law professor Jay Katz examines the time-honored belief in the virtue of silent care and patient compliance. Historically, the doctor-patient relationship has been based on a one-way trust -- despite recent judicial attempts to give patients a greater voice through the doctrine of informed consent. Katz criticizes doctors for encouraging patients to relinquish their autonomy, and demonstrates the detrimental effect their silence has on good patient care. Seeing (...) a growing need in this age of medical science and sophisticated technology for more honest and complete communication between physician and patients, he advocates a new, informed dialogue that respects the rights and needs of both sides. In a new foreword to this edition of The Silent World of Doctor and Patient , Alexander Morgan Capron outlines the changes in medical ethics practice that have occurred since the book was first published in 1984, paying particular attention to the hotly debated issues of physician-assisted suicide and informed consent in managed care. (shrink)

This article analyzes the question of how the size of bribes should impact criminal sanctions. In contrast to the commonly held view that punishment should increase with the size of the bribe, we argue to the contrary: that the punishment of the bribee should decrease with the size of the bribe. Our conclusion is based both on a philosophical argument and an economic argument. We argue that all else being equal, as an agent’s reservation price for selling public interests decreases, (...) the culpability of the agent willing to receive a bribe increases. In addition, from an economic perspective, the expected social harm of an official acting with a low reservation price for bribes is much greater than one acting with a high reservation price: both the susceptibility of being bribed as well as the potential for social harm is much greater when the reservation price is low. (shrink)

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 radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how to prove (...) the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof. (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 feminist media studies, the growing body of research on media production has indicated that journalism remains divided along gender lines. The purpose of this study is to address the lack of relevant multi-method research on gender inequality in journalism. To assess the structural position of women in the journalistic workforce, the authors conducted a large-scale survey of journalists in Belgium. The survey results were explored in more depth by conducting qualitative interviews with 19 female journalists. The analysis confirms the (...) existence of all traditional barriers that women journalists experience. The added value of this study is that it enumerates several additional difficulties, and offers insight into the strategies that the respondents use to deal with gender-related career obstacles. These strategies were associated with the degree of flexibility in the newsroom for journalists with children, the choice to work part-time or freelance, and the use of new communication technologies. (shrink)

The ArgumentTo highlight speculative trends specific to the mathematical tradition that developed in China, the paper analyzes an excerpt of a third-century commentary on a mathematical classic, which arguably contains a proof. The paper shows that the following three tasks cannot be dissociated one from the other: (1) to discuss how the ancient text should be read; (2) to describe the practice of mathematical proof to which this text bears witness; (3) to bring to light connections between philosophy and mathematics (...) that it demonstrates were established in China. To this end the paper defines its use of the word “proof” and outlines a program for an international history of mathematical proof. It describes the sense in which the text conveys a proof and shows how it simultaneously fulfills algorithmic ends, bringing to light a formal pattern that appears to be fundamental both for mathematics and for other domains of reality. The interest in transformations that mathematical writings demonstrate in China at that time seems to have been influenced by philosophical developments based onThe Book of Changes (Yi-jing), which the excerpt quotes. This quotation within a mathematical context makes it possible to suggest an interpretation for a rather difficult philosophical statement. (shrink)

L'oeuvre de Pic de la Mirandole offre un terrain d'expansion très large à la notion de métamorphose en même temps qu'elle l'investit au profit de la notion de dignité humaine cristallisée autour d'une liberté créatrice qui est tout d'abord libération de la pensée.

Isaac Newton, in popular imagination the Ur-scientist, was an outstanding humanist scholar. His researches on, among others, ancient philosophy, are thorough and appear to be connected to and fit within his larger philosophical and theological agenda. It is therefore relevant to take a closer look at Newton’s intellectual choices, at how and why precisely he would occupy himself with specific text-sources, and how this interest fits into the larger picture of his scientific and intellectual endeavours. In what follows, we shall (...) follow Newton into his study and look over his shoulder while reading compendia and original source-texts in his personal library at Cambridge, meticulously investigating and comparing fragments and commentaries, and carefully keeping track in private notes of how they support his own developing ideas. Indeed, Newton was convinced that precursors to his own insights and discoveries were present already in Antiquity, even before the Greeks, in ancient Egypt, and he puts a lot of time and effort into making the point, especially, and not incidentially, in the period between the first and the second edition of the Principia. A clear understanding of his reading of the classic sources therefore matters to our understanding of its content and gestation process. In what follows we will confine ourselves to the classical legacy, and investigate Newton’s intellectual intercourse with it. (shrink)

In this chapter, Dahan Katz defends relational conceptions of retribution and desert. She clarifies the ways in which such relational conceptions avoid major worries associated with retributive theory, while addressing further worries that arise distinctively with respect to such an approach. In doing so, Dahan Katz provides further defense of the response-retributive theory of punishment that she has proposed elsewhere, while defending a wider set of views within the retributive tradition.

This paper offers a response retributive theory of punishment, taking the role of the punisher as well as the relations between the parties to punishment to be central to retributive justification. It proposes that punishment is justified in terms of the ethics of appropriate response, and more precisely, in terms of the duty agents have to dissociate from the devaluation inherent in the culpable wrongdoing of others. The paper demonstrates that on such account, while the harm and suffering involved in (...) punishment are rightly imposed, they are not good, grounding a further duty to mitigate our retributive practices. The result is a moderate vision of retributive justification, which has the resources to justify, not only criminal punishment, but punishment as imposed across a wide variety of relations: private, institutional and political. (shrink)

We develop an evidence-based theoretical account of how widely shared cultural beliefs about gender, race, and class intersect in interpersonal and other social relational contexts in the United States to create characteristic cultural “binds” and freedoms for actors in those contexts. We treat gender, race, and class as systems of inequality that are culturally constructed as distinct but implicitly overlap through their defining beliefs, which reflect the perspectives of dominant groups in society. We cite evidence for the contextually contingent interactional (...) “binds” and freedoms this creates for people such as Asian men, Black women, and poor whites who are not prototypical of images embedded in cultural gender, race, and class beliefs. All forms of unprototypicality create “binds,” but freedoms result from being unprototypical of disadvantaging rather than advantaging statuses. (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)

In this paper, I address the theory of excuse, or more precisely, exculpatory excuse, and the question of what it is that justifies the category of excuse. I address different potential grounds for the law of excuse, which are often run together in ways that confound rather than clarify, focusing on the role of blamelessness and unfairness of expectations in the theory of excuse.

It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical (...) philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the co-incidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato’s Theory of Forms and Aristotle’s metaphysics and logic. Plato’s and Aristotle’s systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a ‘principle of contradiction’ is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing “space” as the seat of stability, and “time” as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion ‘separable system’, related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of ‘non-locality’ as related to it, indicating the need to re-think the notions system, entity, as well as the implications of the operation ‘measurement’, which is seen here as an application of classical logic (including its ontological consequences) on the material world. (shrink)

The social licence to operate concept is little developed in the academic literature so far. Deployment of the term was made by the United National Guiding Principles on Business and Human Rights and the UN ‘Protect, Respect and Remedy’ Framework, which apply SLO as an argument for responsible business conduct, connecting to social expectations and bridging to public regulation. This UN guidance has had a significant bearing on how public regulators seek to influence business conduct beyond Human Rights to broader (...) Corporate Social Responsibility concerns. Drawing on examples of such public regulatory governance, this article explores and explains developments towards a juridification of CSR entailing efforts by public regulators to reach beyond jurisdictional and territorial limitations of conventional public law to address adverse effects of transnational economic activity. Through analysis of an expansion of law into the normative framing of what constitutes responsible business conduct, we demonstrate a process of juridification entailing a legal framing of social expectations of companies, a proliferation of law into the field of business ethics, and an increased regulation by law of social actors or processes. (shrink)

Human science researchers tend to be targeted for critique on the grounds that their approach is too individualistic to take due cognisance of societal and political influences. What is accordingly advocated is that the phenomenological and so-called romantic theories should be abandoned in favour of analytic or continental theories that have as their main focus the system, the group, the society, and the various influences of the social world on the existential reality of the individual.Without trying to invalidate these social (...) science strategies, this paper attempts to show that it is not necessary to surrender phenomenology in order to understand not only the individual, but also the social world in which individuals live. It is argued that the desired goal of a less individualistic human science’s theoretical basis can still be founded in phenomenology, in that Merleau-Ponty’s philosophy, which has its origin in Husserlian phenomenology, provides us with an adequate ontology for understanding human existence more comprehensively. Merleau-Ponty’s ontological philosophy elucidates the in-between world, that structure of existence where the individual cannot be separated from her/his world context. In his exploration of the reversibility of existence, Merleau-Ponty demonstrates that there is no ontological gulf between the individual and the social world. Instead, the world is ‘in’ the individual as much as the individual is ‘in’ the world. With this phenomenological epistemology, it is argued, it is possible to generate research that is capable “of more than a frozen existence”, as Merleau-Ponty puts it. (shrink)

This book provides an exciting and diverse philosophical exploration of the role of practice and practices in human activity. It contains original essays and critiques of this philosophical and sociological attempt to move beyond current problematic ways of thinking in the humanities and social sciences. It will be useful across many disciplines, including philosophy, sociology, science, cultural theory, history and anthropology.

"Thinking about thinking", the topic of this section of our volume, leads us into an Escherian web of self-referential interconnections — which nevertheless make sense, at least if we do not shy away at once in the face of paradox, for fear of inconsistency. How do our critical faculties enter the picture? Adding a few conceptual dimensions to the merely linearly causal one may help us to shed light on the issue, and may help us also to understand some of (...) the basic strategies "Western" thought has developed throughout its historical pathway in its conscious dealings with the 'real'. Of course, in a few pages it is impossible to do justice to this complicated subject. But I'll try to give the reader an as clear as possible description of the, in my view, crucial issues, pointing out the ways they resonate into the many fields of interest represented in the intended readership, and leave it to the reader to dig deeper through the given references. (shrink)

Investigates the intellectual affinities of Adorno and Nietzsche, culminating in a discussion of their readings of Wagner, who serves as a medium and supplement for their critiques of modern culture.

Classical conceptual distinctions in philosophy of education assume an individualistic subjectivity and hide the learning that can take place in the space between child and adult. Grounded in two examples from experience I develop the argument that adults often put metaphorical sticks in their ears in their educational encounters with children. Hearers’ prejudices cause them to miss out on knowledge offered by the child, but not heard by the adult. This has to do with how adults view education, knowledge, as (...) much as child, and is even more extreme when child is also black. The idea is what Miranda Fricker calls ‘epistemic injustice’ which occurs when someone is wronged specifically in their capacity as a knower. Although her work concerns gender and race, I extrapolate her radical ideas to child. Awareness of the epistemic injustice that is done to children and my proposal for increased epistemic modesty and epistemic equality could help transform pedagogical spaces to include child subjects as educators. A way forward is suggested that involves ‘cracking’ the concept of child and a different non-individualised conception of education. (shrink)

Tracing efforts to control unwanted sound--the noise of industry, city traffic, gramophones and radios, and aircraft--from the late nineteenth to the late twentieth century.