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)

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 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.

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)

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)

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)

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.

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 _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, (...) that numbers are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

In _Realistic Rationalism_, Jerrold J. Katz develops a new philosophical position integrating realism and rationalism. Realism here means that the objects of study in mathematics and other formal sciences are abstract; rationalism means that our knowledge of them is not empirical. Katz uses this position to meet the principal challenges to realism. In exposing the flaws in criticisms of the antirealists, he shows that realists can explain knowledge of abstract objects without supposing we have causal contact with them, (...) that numbers are determinate objects, and that the standard counterexamples to the abstract/concrete distinction have no force. Generalizing the account of knowledge used to meet the challenges to realism, he develops a rationalist and non-naturalist account of philosophical knowledge and argues that it is preferable to contemporary naturalist and empiricist accounts. The book illuminates a wide range of philosophical issues, including the nature of necessity, the distinction between the formal and natural sciences, empiricist holism, the structure of ontology, and philosophical skepticism. Philosophers will use this fresh treatment of realism and rationalism as a starting point for new directions in their own research. (shrink)

In light of the sharp linguistic turn philosophy has taken in this century, this collection provides a much-needed and long-overdue reference for philosophical discussion. The first collection of its kind, it explores questions of the nature and existence of linguistic objects--including sentences and meanings--and considers the concept of truth in linguistics. The status of linguistics and the nature of language now take a central place in discussions of the nature of philosophy; the essays in this volume both inform these discussions (...) and lay the groundwork for further examination. (shrink)

BackgroundOne of the next frontiers in HIV research is focused on finding a cure. A new priority includes people with HIV (PWH) with non-AIDS terminal illnesses who are willing to donate their bodies at the end-of-life (EOL) to advance the search towards an HIV cure. We endeavored to understand perceptions of this research and to identify ethical and practical considerations relevant to implementing it.MethodsWe conducted 20 in-depth interviews and 3 virtual focus groups among four types of key stakeholders in the (...) United States (PWH, biomedical HIV cure researchers, HIV clinicians, and bioethicists) to obtain triangulated viewpoints because little was known about the ethics of this topic. Each group was queried as to ethical considerations, safeguards, and protections for conducting HIV cure-related research at the EOL to ensure this research remains acceptable.ResultsAll four key stakeholder groups generally supported HIV cure-related research conducted at the EOL because of the history of altruism within the PWH community and the potential for substantial scientific knowledge to be gained. Our informants expressed that: (1) Strong stakeholder and community involvement are integral to the ethical and effective implementation, as well as the social acceptability of this research; (2) PWH approaching the EOL should not inherently be considered a vulnerable class and their autonomy must be respected when choosing to participate in HIV cure-related research at the EOL; (3) Greater diversity among study participants, as well as multi-disciplinary research teams, is necessitated by HIV cure-related research at the EOL; (4) The sensitive nature of this research warrants robust oversight to ensure a favorable risk/benefit balance and to minimize the possibility of therapeutic misconception or undue influence; and (5) Research protocols should remain flexible to accommodate participants’ comfort and needs at the EOL.ConclusionBecause of the ethical issues presented by HIV cure-related research at the EOL, robust ethical safeguards are of utmost importance. The proposed ethical and practical considerations presented herein is a first step in determining the best way to maximize this research’s impact and social value. More much inquiry will need to be directed towards understanding context-specific and cultural considerations for implementing EOL HIV cure research in diverse settings. (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)

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)

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)

One theme of this special issue is an incitement to reconsider the relationship between pragmatism and critical realism. While their advocates sometimes come into conflict, there are also clearly b...

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.

The Posthuman Child combats institutionalised ageist practices in primary, early childhood and teacher education. Grounded in a critical posthumanist perspective on the purpose of education, it provides a genealogy of psychology, sociology and philosophy of childhood in which dominant figurations of child and childhood are exposed as positioning child as epistemically and ontologically inferior. Entangled throughout this book are practical and theorised examples of philosophical work with student teachers, teachers, other practitioners and children (aged 3-11) from South Africa and Britain. (...) These engage arguments about how children are routinely marginalised, discriminated against and denied, especially when the child is also female, black, lives in poverty and whose home language is not English. The book makes a distinctive contribution to the decolonisation of childhood discourses. Underpinned by good quality picturebooks and other striking images, the book's radical proposal for transformation is to reconfigure the child as rich, resourceful and resilient through relationships with (non) human others, and explores the implications for literary and literacy education, teacher education, curriculum construction, implementation and assessment. It is essential reading for all who research, work and live with children. (shrink)

How do scientists generate knowledge in groups, and how have they done so in the past? How do epistemically motivated social interactions influence or even drive this process? These questions speak to core interests of both history and philosophy of science. Idealised models and formal arguments have been suggested to illuminate the social epistemology of science, but their conclusions are not directly applicable to scientific practice. This paper uses one of these models as a lens and historiographical tool in the (...) examination of actual scientific collectives. It centres on the analysis of two episodes from the history of photosynthesis research of the late nineteenth- to mid-twentieth centuries, which display a wide and coordinated intellectual diversity similar to Kitcher’s “division of cognitive labour”. The concept, I argue, captures important aspects of the photosynthesis research communities, but the underlying process unfolded in ways that differ from the model’s assumption in interesting ways. The paper unravels how the self-organised interplay of cooperation and competition, and the dynamics of individual and collective goals within scientific communities were influential factors in the generation of knowledge. From there, some thoughts are developed on how historical and philosophical approaches in the analysis of science can productively interact. (shrink)

Philosophy with children (P4C) 1 presents significant positive challenges for educators. Its 'community of enquiry' pedagogy assumes not only an epistemological shift in the role of the educator, but also a different ontology of 'child' and balance of power between educator and learner. After a brief historical sketch and an outline of the diversity among P4C practitioners, epistemological uncertainty in teaching P4C is crystallised in a succinct overview of theoretical and practical tensions that are a direct result of the implementation (...) of P4C in mainstream education. These recurring pedagogical tensions in my practice as P4C teacher, teacher educator and mentor of teacher educators cause disequilibrium that opens up rich opportunities for philosophy of education in supporting novice P4Cers. Disequilibrium is a positive force that opens up a space in which educators need to reflect upon their values, their beliefs about learning and teaching, and ultimately encourages educators to rethink their own role. Plato's metaphor of the stingray highlights the role of the P4C teacher educator as model of the P4C teacher in any setting: 'to numb and to be numbed'. The P4C community and its institutions need to address the questions arising from these pedagogical tensions; and this needs to be done with integrity, that is, in communities of enquiry that include children. If not, in the long term, a more instrumental version of P4C may prevail. (shrink)

Dementia is highly prevalent and incurable. The participation of dementia patients in clinical research is indispensable if we want to find an effective treatment for dementia. However, one of the primary challenges in dementia research is the patients’ gradual loss of the capacity to consent. Patients with dementia are characterized by the fact that, at an earlier stage of their life, they were able to give their consent to participation in research. Therefore, the phase when patients are still competent to (...) decide offers a valuable opportunity to authorize research, by using an advance research directive. Yet, the use of ARDs as an authorization for research participation remains controversial. In this paper we discuss the role of autonomous decision-making and the protection of incompetent research subjects. We will show why ARDs are a morally defensible basis for the inclusion of this population in biomedical research and that the use of ARDs is compatible with the protection of incompetent research subjects. (shrink)

The use of black box algorithms in medicine has raised scholarly concerns due to their opaqueness and lack of trustworthiness. Concerns about potential bias, accountability and responsibility, patient autonomy and compromised trust transpire with black box algorithms. These worries connect epistemic concerns with normative issues. In this paper, we outline that black box algorithms are less problematic for epistemic reasons than many scholars seem to believe. By outlining that more transparency in algorithms is not always necessary, and by explaining that (...) computational processes are indeed methodologically opaque to humans, we argue that the reliability of algorithms provides reasons for trusting the outcomes of medical artificial intelligence. To this end, we explain how computational reliabilism, which does not require transparency and supports the reliability of algorithms, justifies the belief that results of medical AI are to be trusted. We also argue that several ethical concerns remain with black box algorithms, even when the results are trustworthy. Having justified knowledge from reliable indicators is, therefore, necessary but not sufficient for normatively justifying physicians to act. This means that deliberation about the results of reliable algorithms is required to find out what is a desirable action. Thus understood, we argue that such challenges should not dismiss the use of black box algorithms altogether but should inform the way in which these algorithms are designed and implemented. When physicians are trained to acquire the necessary skills and expertise, and collaborate with medical informatics and data scientists, black box algorithms can contribute to improving medical care. (shrink)

This book argues that the development of German philosophy from Kant, through post-Kantian German Idealism, to the thought of Franz Rosenzweig, was largely driven by the perceived promise of Kant's philosophy for solving the conflict of reason, but also by its perceived shortcomings in solving this conflict.

Our understanding of the nature and processing of figurative language is central to several important issues in cognitive science, including the relationship of language and thought, how we process language, and how we comprehend abstract meaning. Over the past fifteen years, traditional approaches to these issues have been challenged by experimental psychologists, linguists, and other cognitive scientists interested in the structures of the mind and the processes that operate on them. In Figurative Language and Thought, internationally recognized experts in the (...) field of figurative language, Albert Katz, Mark Turner, Raymond W. Gibbs Jr., and Cristina Cacciari, provide a coherent and focused debate on the subject. The book's authors discuss a variety of fundamental questions, including: What can figures of speech tell us about the structure of the conceptual system? If and how should we distinguish the literal from the nonliteral in our theories of language and thought? Are we primarily figurative thinkers and consequently figurative language users or the other way around? Why do we prefer to speak metaphorically in everyday conversation, when literal options may be available for use? Is metaphor the only vehicle through which we can understand abstract concepts? What role do cultural and social factors play in our comprehension of figurative language? These and related questions are raised and argued in an integrative look at the role of nonliteral language in cognition. This volume, a part of Counterpoints series, will be thought-provoking reading for a wide range of cognitive psychologists, linguists, and philosophers. (shrink)

Healthcare collectives, such as patient organizations, advocacy groups, disability organizations, professional associations, industry advocates, social movements, and health consumer organizations have been increasingly involved in healthcare policymaking. Such collectives are based on the idea that individual interests can be aggregated into collective interests by participation, deliberation, and representation. The topic of collectivity in healthcare, more specifically collective representation, has only rarely been addressed in bioethics. This symposium, entitled: “Collective Representation in Healthcare Policy” of the Journal of Bioethical Inquiry draws attention (...) to this understudied topic from a variety of disciplines, within a variety of socio-cultural contexts. We draw attention to important ethical, cultural, and social questions, and into the practices, justifications for, and implications of collective representation of patients in healthcare policy. (shrink)

The authors of the paper ‘Advance euthanasia directives: a controversial case and its ethical implications’ articulate concerns and reasons with regard to the conduct of euthanasia in persons with dementia based on advance directives. While we agree on the conclusion that there needs to be more attention for such directives in the preparation phase, we disagree with the reasons provided by the authors to support their conclusions. We will outline two concerns with their reasoning by drawing on empirical research and (...) by providing reasons that contradict their assumptions about competence of people with dementia and the importance of happiness in reasoning about advance directives of people with dementia. We will draw attention to the important normative questions that have been overstepped in their paper, and we will outline why further research is required. (shrink)

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)

A collection of articles, some of which appeared previously. Partial contents:.

Some philosophers claim that young children cannot do philosophy. This paper examines some of those claims, and puts forward arguments against them. Our beliefs that children cannot do philosophy are based on philosophical assumptions about children, their thinking and about philosophy. Many of those assumptions remain unquestioned by critics of Philosophy with Children. My conclusion is that the idea that very young children can do philosophy has not only significant consequences for how we should educate young children, but also for (...) how adults should do philosophy; and that further research is urgently needed. (shrink)

The amount of realist positions put forward by philosophers of religion and theologians is impressive. one can certainly doubt whether there is a need for yet another alternative. However, most realist positions employed in studies on religion fall prey to Hilary Putnam’s criticism against metaphysical realism. This gives rise to a dilemma that I aim at solving by introducing yet another realist position, namely non-metaphysical realism.

Volume 19, Issue 4, August 2020, Page 309-313.

Recension av boken _Gravstenar berättar. Judiskt liv i Stockholm 1775-1875 _.

In this essay I propose a new measure of social welfare. It captures the intuitive idea that quantity, quality, and equality of individual welfare all matter for social welfare. More precisely, it satisfies six conditions: Equivalence, Dominance, Quality, Strict Monotonicity, Equality and Asymmetry. These state that i) populations equivalent in individual welfare are equal in social welfare; ii) a population that dominates another in individual welfare is better; iii) a population that has a higher average welfare than another population is (...) better, other things being equal; iv) the addition of a well-faring individual makes a population better, whereas the addition of an ill-faring individual makes a population worse; v) a population that has a higher degree of equality than another population is better, other things being equal; and vi) individual illfare matters more for social welfare than individual welfare. By satisfying the six conditions, the measure improves on previously proposed measures, such as the utilitarian Total and Average measures, as well as different kinds of Prioritarian measures. (shrink)