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.

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)

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)

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.

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)

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.

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)

Because we tend to study fight books in isolation, we often forget how difficult it is to understand the precise place they occupy in the sociocultural and historical fabric of their time, and spill the many clues they inevitably contain on their owner, their local society, their precise purpose. In order to unlock that information, we need to study them in their broader sociocultural and historical context. This requires a background and research skills that are not always easily accessible to (...) everyone. To illustrate the point, in this paper I show in some detail what is required to make sense of the claim that Aristotelian philosophy and science influenced the medieval fight books in relevant ways, and that understanding this influence helps us to better understand the fight books per se. I give an outline of the general historical framework, and apply it to a testcase: Talhoffer’s Thott 290 2° Ms., with some interesting results. My hope is that this framework may be of some use to other researchers in HEMA Studies who want to dig deeper in sources of interest to them. (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.

THE immense popularity which Bergson's philosophy enjoys is sometimes cast up against him, by those who do not agree with him, as a reproach. It has been suggested that Berg-son's writings are welcomed simply because they offer a theoretical justification for a tendency which is natural in all of us but against which philosophy has always fought, the tendency to throw reason overboard and just let ourselves go. Bergson is regarded by rationalists almost as a traitor to philosophy, or as (...) a Bolshevik inciting the public to overthrow what it has taken years of painful effort to build up. (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...

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)

Chomsky's criticism of Bloomfieldian structuralism's conception of linguistic reality applies equally to his own conception of linguistic reality. There are too many sentences in a natural language for them to have either concrete acoustic reality or concrete psychological or neural reality. Sentences have to be types, which, by Peirce's generally accepted definition, means that they are abstract objects. Given that sentences are abstract objects, Chomsky's generativism as well as his psychologism have to be given up. Langendoen and Postal's argument in (...) The Vastness of Natural Languages to show that there are more than denumerably many sentences is flawed. But, with the view that sentences are abstract objects, the flaws can be corrected. Once psychologism and generativism are abandoned, the revolution against Bloomfieldian structuralism can be brought to completion and linguistics can be put on a sound philosophical basis. (shrink)

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)

Understanding between care providers and patients with stroke and aphasia: a phenomenological hermeneutic inquiry The present study illuminates the understanding in communication between formal care providers and patients with stroke and aphasia. Five care providers and three such patients participated in the study. Video recordings were made during conversations about pictures (n = 15), and the care providers were also interviewed (n = 15) after the video‐recorded conversations. A phenomenological hermeneutic method of interpretation of the interview text was used. The (...) findings showed that a range of conditions for ‘understanding and being understood’ in the communication on the part of the care providers exists. These different conditions are: lacking both knowledge and understanding; having knowledge but not necessarily accompanied by understanding; and being in understanding. Within the condition ‘being in understanding’, the care providers create a feeling of at‐homeness in a relaxed atmosphere and thus have the opportunity to be in ‘understanding and being understood’ together with the patient. The condition ‘being in understanding’ appears in connection with the care providers’ creating of a ‘calm liturgy of caring’ by mediating humility and calm vitality affects to the patients, and further, when needed, being present on the level of mystery, i.e. caring communion. (shrink)

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

Recently, Weisbecker and Goswami presented the first comprehensive comparative analysis of brain size, metabolic rate, and development periods in marsupial mammals. In this paper, a strictly energetic perspective is applied to identify general mammalian correlates of brain size evolution. In both marsupials and placentals, the duration or intensity of maternal investment is a key correlate of relative brain size, but here I show that allomaternal energy subsidies may also play a role. In marsupials, an energetic constraint on brain size in (...) adults is only revealed if we consider both metabolic and reproductive rates simultaneously, because a strong trade‐off between encephalization and offspring production masks the positive correlation between basal metabolic rate and brain size in a bivariate comparison. In conclusion, starting from an energetic perspective is warranted to elucidate relations between ecology, social systems, life history, and brain size in all mammals. (shrink)

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)

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)

While literature is often used as a source of examples and illustrations across disciplines, literary studies tends to be underrepresented in interdisciplinary exchanges. Perhaps the reason lies in a lack of understanding what actually is the expertise of literary studies and how this can be useful in interdisciplinary settings. In this article, I propose to outline the expertise of literary scholars through concepts of 4E cognition and to devise a proposal for how such expertise could successfully shape the epistemic common (...) ground of social cognition of experts in interdisciplinary dialogue. Literature involves metacognition centrally through its language style, the design of the narrative and its links to other texts, and literary scholars have the expertise in formulating exactly how this works – in a non-mimetic way – through the analysis and interpretation of literary texts. This very particular expertise and practice of literary scholars enables literary texts to be proposed as a boundary object in interdisciplinary dialogues through a shared epistemic common ground. For this argument, I build on earlier theoretical work in 4E cognition and predictive processing and my experience running interdisciplinary workshops on that model. (shrink)

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)