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.
First century Chinese, fifth century Indian, and Arabic documents from the 9th century onwards, contain similar tabular procedures to extract square and cube roots on place-value numeration systems. Moreover, an 11th century Chinese astronomer, Jia Xian, as well as al-Samaw'al, a 12th century Arab mathematician, extracted roots of higher order with the so-called Ruffini-Horner procedure. This article attempts to define a textual method to organize this corpus, by distinguishing relevant criteria for identifying similarities and differences from a historical as well (...) as conceptual point of view. The first part analyses three different states of the descriptions of algorithms in China between the 1st and the 11th centuries, all of which exhibit a definite historical stability. The rewriting which allows one to proceed progressively from one state to the next shows a uniformity in the components of the algorithm, which culminates in procedures of the type Ruffini-Horner. Textual criteria demonstrate a greater affinity of certain algorithms, such as those described by Kūshyār ibn Labbān with Chinese rather than with Indian texts, which are in turn closer to algorithms described by al-Khwārizmī. Criteria of the same kind link the algorithms of Jia Xian and al-Samaw'al on the one hand, and those of Kūshyār and al-Samaw'al on the other. Les documents chinois, depuis le I er siècle, indiens, depuis le V e siècle, et arabes, depuis le IX e siècle, contiennent des procédures tabulaires similaires pour l'extraction de racines carrées et cubiques avec des systèmes de numération positionnels. Par ailleurs tant Jia Xian, astronome chinois du XI e siècle, qu'al-Samaw'al, mathématicien arabe du XII e siècle, ont extrait des racines de degré plus élevé par la procédure dite de Ruffini-Horner. L'article tente de définir une méthode textuelle pour organiser ce corpus, en y distinguant des axes pertinents qui permettent de dégager similarités et différences, d'un point de vue tant historique que conceptuel. Une première partie analyse trois états différents des descriptions d'algorithmes entre le I er siècle et le XI e siècle en Chine, qui présentent chacun une stabilité historique certaine. La réécriture qui fait passer d'un état au suivant laisse émerger progressivement une uniformité dans les composantes de l'algorithme, laquelle culmine avec des procédures du type Ruffini-Horner. Des critères textuels font apparaître une affinité plus grande de certains algorithmes, tels ceux décrits par Kūshyār ibn Labbān, avec les textes chinois qu'avec les textes indiens, plus proches, eux, des algorithmes décrits par al-Khwārizmī. Des critères de même nature lient, d'une part, les algorithmes de Jia Xian et d'al-Samaw'al, d'autre part les algorithmes de Kūshyār et d'al-Samaw'al. (shrink)
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)
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)
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)
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)
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 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)
The paper develops a conception of marital love as a complex recognitive relation, which I articulate by juxtaposing it against other recognitive relations that figure in Hegel's theory of modern civil society (i.e., respect and esteem). Drawing on Hegel's early writings, I argue that, if love is to provide its unique sort of recognition, it must obtain between “living beings who are equal in power”—a peculiar form of equality that I name (drawing on Stanley Cavell's work) “dynamic equality.” I conclude (...) that it is by Hegel's own lights that we should reject his notorious conception of the sexual difference. However, I also offer reasons why, from Hegel's early 19th century perspective, he could consider the following two conditions as compatible: (1) equality within marriage and (2) sexual hierarchy outside marriage, namely, in civil society. (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)
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)
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)
Fodor and katz criticize cavell's position on the relation between ordinary language philosophy and empirical investigations of ordinary language, In "must we mean what we say?," _inquiry, Volume 1, Pages 172-212, And "the availability of wittgenstein's later philosophy," "philosophical review", Volume 71, Pages 67-93. Cavell holds that disagreements between ordinary language philosophers over grammar and semantics are in no sense empirical. Fodor and katz show that ordinary language philosophers are engaged in empirical investigation. (staff).
_ Source: _Volume 62, Issue 1, pp 26 - 68 Ontological separation plays a key role in Aristotle’s metaphysical project: substances alone are ontologically χωριστόν. The standard view identifies Aristotelian ontological separation with ontological independence, so that ontological separation is a non-symmetric relation. I argue that there is strong textual evidence that Aristotle employs an asymmetric notion of separation in the _Metaphysics_—one that involves the dependence of other entities on the independent entity. I argue that this notion allows Aristotle to (...) prevent the proliferation of substance-kinds and thus to secure the unity of his metaphysical system. (shrink)
Some have recently argued that the current generation dominates future generations by causing long-term climate change. They relate these claims to Philip Pettit and Frank Lovett's neorepublican theory of domination. In this paper, I examine their claims and ask whether the neorepublican conception of domination remains theoretically coherent when the relation is between current agents and nonoverlapping future subjects. I differentiate between an ‘outcome’ and a ‘relational’ conception of domination. I show how both are theoretically coherent when extended to posterity (...) but only if we make different definitional and normative choices than those made by Pettit and Lovett. (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)
In the first decades of the twentieth century, the process of photosynthesis was still a mystery: Plant scientists were able to measure what entered and left a plant, but little was known about the intermediate biochemical and biophysical processes that took place. This state of affairs started to change between the two world wars, when a number of young scientists in Europe and the United States, all of whom identified with the methods and goals of physicochemical biology, selected photosynthesis as (...) a topic of research. The protagonists had much in common: They had studied physics and chemistry to a high level; they used physicochemical methods to study the basic processes of life; they believed these processes were the same, or very similar, in all life forms; and they were affiliated with institutions that fostered this kind of study. This set of cognitive, methodological, and material resources enabled these protagonists to transfer their knowledge of the concepts and techniques from microbiology and human biochemistry, for example, to the study of plant metabolism. These transfers of knowledge had a great influence on the way in which the biochemistry and biophysics of photosynthesis would be studied over the following decades. Through the use of four historical cases, this paper analyzes these knowledge transfers, as well as the investigative pathways that made them possible. (shrink)
This essay is an attempt to analyze, classify and illustrate different scholarly approaches to the Sanskrit philosophical commentaries as reflected in some influential and especially thoughtful studies of Indian philosophy; at the same time it highlights some specific features involving commentary and annotation in general, drawing from results of studies on commentaries conducted in other disciplines and fields, such as Classical and Medieval Studies, Theology, and Early English Literature. In the field of South Asian Studies, philosophical commentaries may be assessed (...) from various overlapping and not always exclusive points of view, such as preservation of otherwise lost historical information, historical authenticity and reliability, interpretational innovation, spiritual or experiential insight, philosophical creativity, intellectual liveliness, doxographic intent, degree of incidentality, expository breadth and explanatory depth. The essay provides numerous examples taken from classical to early modern philosophical literature, especially of the Brahminical and Buddhist traditions, and also discusses their diverging perception by modern scholars and interpretators. (shrink)
In Buddhism, Meditation and Free Will: A Theory of Mental Freedom , Rick Repetti explains how the dynamics of Buddhist meditation can result in a kind of metacognition and metavolitional control that exceeds what is required for free will and defeats the most powerful forms of free will skepticism. This article argues that although the Buddhist path requires and enhances the kind of mental and volitional control Repetti describes, the central dynamic of the path and meditation is better understood as (...) a process of habituation. This not only involves the dis‐identification from mental and emotional content that Repetti discusses—and is commonly emphasized in modern presentations of mindfulness or insight (vipassanā ) meditation—but also a transformation of the heart that is effected through the complementary psychological and somatic qualities associated with calm abiding (samatha ) and concentration (samādhi ) and emphasized in the Pali Nikāyas and commentaries. (shrink)