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 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)
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 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.
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.
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)
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 philosophy for children curriculum was specially written by Matthew Lipman and colleagues for the teaching of philosophy by non-philosophically educated teachers from foundation phase to further education colleges. In this article I argue that such a curriculum is neither a necessary, not a sufficient condition for the teaching of philosophical thinking. The philosophical knowledge and pedagogical tact of the teacher remains salient, in that the open-ended and unpredictable nature of philosophical enquiry demands of teachers to think in the moment (...) and draw on their own knowledge and experience of academic philosophy. Providing specialist training or induction in the P4C curriculum cannot and should not replace undergraduate and postgraduate degrees in academic philosophy at universities. However, although for academic philosophers the use of the P4C curriculum could be beneficial, I will argue that its use poses the risk of wanting to form children into the ideal ‘abnormal’ child, the thinking child—the adult philosopher’s child positioned as such by the Lipman novels. The notion of narrativity is central in my argument. With the help of two picturebooks—The Three Pigs by David Weisner and Voices in the Park by Anthony Browne—I illustrate my claim that philosophy as ‘side-shadowing’ or meta-thinking can only be generated in the space ‘in between’ text, child and educator, thereby foregrounding a ‘pedagogy of exposure’ rather than ‘teacher proof’ texts. (shrink)
Katz denies that organisms created in a lab as part of a de-extinction attempt will be authentic members of the extinct species, on the basis that they will lack the original species’ defining biological and evolutionary history. Against Katz, I note that an evolutionary lineage is conferred on an organism through its inheriting genes from forebears already possessed of such a lineage, and that de-extinction amounts to a delayed, human-assisted reproductive process, in which genes are inherited from forebears (...) long dead. My conclusion is that de-extinct organisms can perfectly well have an ancient evolutionary history, contrary to what Katz claims. (shrink)
Written by one of the instrumental figures in environmental ethics, Nature as Subject traces the development of an ethical policy that is centered not on human beings, but on itself. Katz applies this idea to contemporary environmental problems, introducing themes of justice, domination, imperialism, and the Holocaust. This volume will stand as a foundational work for environmental scholars, government and industry policy makers, activists, and students in advanced philosophy and environmental studies courses.
Quine's holistic empiricist account of scientific inquiry can be characterized by three constitutive principles: *noncontradiction*, *universal revisability* and *pragmatic ordering*. We show that these constitutive principles cannot be regarded as statements within a holistic empiricist's scientific theory of the world. This claim is a corollary of our refutation of Katz's [1998, 2002] argument that holistic empiricism suffers from what he calls the Revisability Paradox. According to Katz, Quine's empiricism is incoherent because its constitutive principles cannot themselves be rationally (...) revised. Using Gärdenfors and Makinson's logic of belief revision based on epistemic entrenchment, we argue that Katz wrongly assumes that the constitutive principles are *statements* within a holistic empiricist's theory of the world. Instead, we show that constitutive principles are best seen as *properties* of a holistic empiricist's theory of scientific inquiry and we submit that, without Katz's mistaken assumption, the paradox cannot be formulated. We argue that our perspective on the status of constitutive principles is perfectly in line with Quinean orthodoxy. In conclusion, we compare our findings with van Fraassen's  argument that we should think of empiricism as a stance, rather than as a doctrine. (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)
In Probability Designs, Karin Kukkonen presents the predictive processing model of cognition as a means of exploring narrative structure and reader experience. Utilizing the literary canon of various cultures, Kukkonen combines theory and cognitive science to analyze how reader expectation and prediction shape literature, and how literature accomplishes cognitive feats that determine the human capacity for free, exploratory thought.
Classical conceptual distinctions in philosophy of education assume an individualistic subjectivity and hide the learning that can take place in the space between child (as educator) and adult (as learner). 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 (black) 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 nonindividualised conception of education. (shrink)
The anthropological approach is the central focus of this study. Laboratories are looked upon with the innocent eye of the traveller in exotic lands, and the societies found in these places are observed with the objective yet compassionate eye of the visitor from a quite other cultural milieu. There are many surprises that await us if we enter a laboratory in this frame of mind... This study is a realistic enterprise, an attempt to truly represent the social order of life (...) in laboratories and institutes of research, just as they are. By bringing the philosophical issues to the surface as matters not of prejudgement but as matters of concern, Karin Knorr-Cetina has developed the first really positive challenge to the philosophy of science since the days of paradigms and internal definitions of meanings. (shrink)
BackgroundOur human societies and certainly also medicine are more and more permeated with technology. There seems to be an increasing awareness among bioethicists that an effective and comprehensive approach to ethically guide these emerging biomedical innovations into society is needed. Such an approach has not been spelled out yet for bioethics, while there are frequent calls for ethical guidance of biomedical innovation, also by biomedical researchers themselves. New and emerging biotechnologies require anticipation of possible effects and implications, meaning the scope (...) is not evaluative after a technology has been fully developed or about hypothetical technologies, but real-time for a real biotechnology.Main textIn this paper we aim to substantiate and discuss six ingredients that we increasingly see adopted by ethicists and that together constitute “ethics parallel research”. This approach allows to fulfil two aims: guiding the development process of technologies in biomedicine and providing input for the normative evaluation of such technologies. The six ingredients of ethics parallel research are: disentangling wicked problems, upstream or midstream ethical analysis, ethics from within, inclusion of empirical research, public participation and mapping societal impacts, including hard and soft impacts. We will draw on gene editing, organoid technology and artificial intelligence as examples to illustrate these six ingredients.ConclusionEthics parallel research brings together these ingredients to ethically analyse and proactively or parallel guide technological development. It widens the roles and judgements from the ethicist to a more anticipatory and constructively guiding role. Ethics parallel research is characterised by a constructive, rather than a purely critical perspective, it focusses on developing best-practices rather than outlining worst practice, and draws on insights from social sciences and philosophy of technology. (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)
In her book Epistemic Injustice: Power and the Ethics of Knowing , Miranda Fricker introduces the helpful notion of “identity prejudice” as “a label for prejudices against people qua social type” . She focuses on race, class and gender, and Michael Hand in his article What Do Kids Know? A response to Karin Murris is indeed correct when he states that I have applied her arguments to age as a category of epistemic exclusion.I argue that among the usual contenders (...) of epistemic prejudices, we also need to be cognisant of adults’ implicit and explicit assumptions and prejudices about child and childhood. However, Hand incorrectly uses Fricker’s ideas to reject my proposal to include child as being on the receiving end of epistemic injustice. A close reading of some passages about stereotyping will show why this is the case and why his own claim that “children typically are immature, ill-informed and endearing” will turn out to .. (shrink)
In this paper we offer an appreciation and critique of patient-led care as expressed in current policy and practice. We argue that current patient-led approaches hinder a focus on a deeper understanding of what patient-led care could be. Our critique focuses on how the consumerist/citizenship emphasis in current patient-led care obscures attention from a more fundamental challenge to conceptualise an alternative philosophically informed framework from where care can be led. We thus present an alternative interpretation of patient-led care that we (...) call ‘lifeworld-led care’, and argue that such lifeworld-led care is more than the general understanding of patient-led care. Although the philosophical roots of our alternative conceptualisation are not new, we believe that it is timely to re-consider some of the implications of these perspectives within current discourses of patient-centred policies and practice. The conceptualisation of lifeworld-led care that we develop includes an articulation of three dimensions: a philosophy of the person, a view of well-being and not just illness, and a philosophy of care that is consistent with this. We conclude that the existential view of well-being that we offer is pivotal to lifeworld-led care in that it provides a direction for care and practice that is intrinsically and positively health focused in its broadest and most substantial sense. (shrink)
This study explores the discourse of social entrepreneurs and their audiences in pitch situations. Adopting a practice perspective on social entrepreneurship, we videotaped 49 pitches by social entrepreneurs at five different events in two incubators in Germany and Switzerland. Our analysis of the start-ups’ pitches and the audience’s questions and comments as well as of interview data elucidates the nuances of social and business discourse that social entrepreneurs and their audiences draw upon. Our analysis shows how many social entrepreneurs mobilize (...) a discursive repertoire that is familiar to their business-oriented audience while others predominantly draw on a social discourse. We identify separating, mixing, and combining as key strategies that allow social entrepreneurs to dance between the two. We discuss how the intertextual reproduction of concepts, objects, and subject positions contains both enabling and constraining elements, which results in an ethical dilemma for social entrepreneurs: Should they re-package their social impact story in a business discourse to connect with their audience? (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)
This article analyses the relationship between the concept of single aspect similarity and proposed measures of similarity. More precisely, it compares eleven measures of similarity in terms of how well they satisfy a list of desiderata, chosen to capture common intuitions concerning the properties of similarity and the relations between similarity and dissimilarity. Three types of measures are discussed: similarity as commonality, similarity as a function of dissimilarity, and similarity as a joint function of commonality and difference. Relative to the (...) desiderata, it is found that a measure of the second type fares the best. However, rather than recommend this measure alone as a measure of similarity, it is suggested that there are at least three separate concepts of single aspect similarity, corresponding to the three types of measures. In light of this proposal, three of the eleven measures (and variants of these) are deemed acceptable. (shrink)
Libertarians believe certain things about rights and responsibilities, about when one person is to be held responsible for invading the rights of another. Libertarians also believe certain things about consent, about when someone should be held to a contract he has entered into. What they don't realize is that the first set of beliefs doesn't mix well with the second set of beliefs—that their intuitions about rights and responsibilities quite simply don't square with their intuitions about consent. Or so I (...) shall be trying to show in this essay. (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)
Corporate sustainability is an important strategy and value orientation for marketing, but scarce research addresses the organizational drivers and barriers to including it in companies’ marketing strategies and processes. The purpose of this study is to determine levels of commitment to corporate sustainability in marketing, processes associated with sustainability marketing commitment, drivers of sustainability marketing at the functional level of marketing, and its organizational context. Using survey data from 269 managers in marketing, covering a broad range of industries in Sweden (...) and Denmark, we took a structural modelling approach to examine construct relationships, mediation, and moderation effects. Overall, the findings show that marketing capabilities associated with the innovation of new products, services, and business models constitute a strong driver to leverage sustainability marketing commitment. In conjunction with insights into processes related to the enactment of sustainability marketing, this result indicates that companies’ marketing departments have a propensity to drive corporate sustainability. The study provides substance to the idea of aligning substantive marketing capabilities closer to dynamic capabilities. Accordingly, the study reveals that reliance on market orientation alone does not lead to greater sustainability commitment. (shrink)
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)