We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

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

One of the most influential scientific treatises in Cauchy's era was J.-L. Lagrange's Mécanique Analytique, the second edition of which came out in 1811, when Cauchy was barely out of his teens. Lagrange opens his treatise with an unequivocal endorsement of infinitesimals. Referring to the system of infinitesimal calculus, Lagrange writes:Lorsqu'on a bien conçu l'esprit de ce système, et qu'on s'est convaincu de l'exactitude de ses résultats par la méthode géométrique des premières et dernières raisons, ou par la méthode analytique (...) des fonctions dérivées, on peut employer les infiniment petits comme un instrument sûr et commode pour abréger et simplifier les demonstrations.Lagrange's renewed enthusiasm for .. (shrink)

The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (shrink)

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.

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)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

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.

Research indicates the essentiality of dignity as a vital component for quality of life, reconfirming the emphasis on dignity preservation in the international code of nursing ethics. Applying Noblit and Hare’s meta-ethnography, the aim of the study was to develop a theory model by synthesizing 10 qualitative articles from various cultural contexts, exploring nurse and allied healthcare professional perception/practice concerning dignity-preserving dementia care. “Advocating the person’s autonomy and integrity,” which involves “having compassion for the person,” “confirming the person’s worthiness and (...) sense of self,” and “creating a humane and purposeful environment,” was identified as a primary foundation for dignity-preserving dementia care. “Balancing individual choices among persons no longer able to make sound decisions, against the duty of making choices on behalf of the person,” which involves “persuasion” and/or “mild restraint,” was considered a crucial aspect in certain situations. “Sheltering human worth—remembering those who forget” was identified as a comprehensive motive and core value within dignity-preserving dementia care. (shrink)

In recent years, the EU legislation on genetically modified crops has come under severe criticism. Among the arguments are that the present legislation is inconsistent, disproportionate, obsolete from a scientific point of view, and vague in terms of its scope. In this paper, the EU GM legislation is analysed based on five proposed criteria: legal certainty, non-discrimination, proportionality, scientific adaptability, and inclusion of non-safety considerations. It is argued that the European regulatory framework does not at present satisfy the criteria of (...) legal certainty, non-discrimination, and scientific adaptability. Two ways of reforming the present legislation toward greater accommodation of the values expressed through the proposed criteria are briefly introduced and discussed. (shrink)

In _The Play within the Play: The Enacted Dimension of Psychoanalytic Process_ Gil Katz presents and illustrates the "enacted dimension of psychoanalytic process." He clarifies that enactment is not simply an overt event but an unconscious, continuously evolving, dynamically meaningful process. Using clinical examples, including several extended case reports, Gil Katz demonstrates how in all treatments, a new version of the patient’s early conflicts, traumas, and formative object relationships is inevitably created, without awareness or intent, in the here-and-now (...) of the analytic dyad. Within the enacted dimension, repressed or dissociated aspects of the patient’s past are not just remembered, they are re-lived. Katz shows how, when the enacted dimension becomes conscious, it forms the basis for genuine and transforming experiential insight. (shrink)

The Search for the Legacy of the USPHS Syphilis Study at Tuskegee is a collection of essays from experts in a variety of fields seeking to redefine the legacy of the infamous Tuskegee Syphilis Study. The essayists place the legacy of the study within the evolution of racial and ethnic relations in the United States. Contributors include two leading historians on the study, two former United States Surgeons General, and other prominent scholars from a wide range of fields.

The present paper aims to trace back Kant’s account of the schematism of the pure understanding in the Critique of Pure Reason to the Dissertation. I do so by discussing Kant’s understanding of sensible cognition in view of his assessment of metaphysics. I argue, first, that Kant in both texts aims to defend metaphysics against skeptical attacks by discarding those of its elements he considers unwarranted and, second, that this undertaking hinges on his account of concepts that function as the (...) sensible condition of cognition. Yet whereas Kant argues in 1770 that metaphysics must be purely intellectual, he in 1781 draws on his earlier account of sensible concepts to argue, against the Wolffians, that determining intelligible objects by purely intellectual means does not amount to cognition proper. (shrink)

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

The author argues non-human, sentient animals have aggregation-trumping rights by explaining why and how they should be included in the scope of Kantian contractualism. He explains that the beings to whom we owe duties—who can be wronged by our treatment—are all those with the capacity for first-person, subjective experience; i.e., all sentient beings. To determine what duties we owe to such beings, we should reflect on the principles for the general regulation of behavior that could be hypothetically justified to their (...) imaginary perfectly reasonable counterparts; i.e., even though animals actually cannot understand or reflect on the reasons we have for treating them in a particular way, burdening them unjustifiably is wrong to them. The author argues this inclusive contractualist theory can explain all the distinctive moral phenomena that T. M. Scanlon’s approach does and so is a more attractive contractualist moral theory. (shrink)

Although Levinas talks about ethics as a response to the other, most scholars assume that this "response" is not something tangible—it is not an actual giving of food or providing of shelter and clothing. But there is evidence in Levinas's own writings that indicate he does intend for a positive response to the Other. In any event, while he acknowledges that the other is the sole person I wish to kill, killing the other, within an ethical framework would be a (...) violation of that response. The failure to respond to the other ethically requires us to ask if Levinas's project needs an educational philosophy or a model of moral cultivation to supplement it. This essay explores this question by putting into conversation Levinas's ethical project and his interest in Jewish education with John Dewey's philosophy of education and its relationship to the political community. This exploration will help us see what this field of research might offer in promoting the cultivation of ethical response as Levinas envisions it and what its limits are. (shrink)

In several places, Levinas identifies the problem that concerns him as a “ crisis of humanism.” This problem finds its seeds in modernity but comes to fruition in the inhumanities of the 20 th century. Like his philosophical predecessors, Levinas offers an educational model as a solution to a problem he has identified. But this model--Jewish education—is uniquely different from those offered by those who came before him. This essay examines Levinas‘s interest in Jewish education as a solution to this (...) crisis in humanism and considers what the implications of this solution are for his project as a whole. (shrink)

In the Critique of Pure Reason and elsewhere, Kant uses the term ‘object’ in various ways and often without clearly signaling its different meanings. As a result, it is hard to gauge the extent to which Kant’s account of the object of cognition breaks new ground. In this article, I take the Critique to establish what is required to generate an object of cognition per se soleley by examining the various ways in which the human mind can objectify the content (...) of its representations. To clarify this endeavor, I distinguish four different ways in which the term ‘object’ is used in the Critique, namely, to refer to (1) material things, (2) the content of any type of thought, (3) the mind-immanent correlate of a cognition in the broad sense of the term, and (4) the mind-immanent correlate of a cognition in the strict sense of the term. On my reading, the fourth meaning of the term ‘object’ is key to the Transcendental Deduction and completes Kant’s unprecedented conception of the cognitive acts by means of which the human mind produces objects of cognition out of a manifold of representations. (shrink)

The study of philosophy has no leading edge. Scholars may fruitfully explore past eras and superceded theories, revise their views of historic figures, modify inadequate theories, defend successful yet overlooked ideas, salvage the wheat from the chaff. A novel defense of previously discredited arguments could lead to new insights, and this is so even if that defense proved ultimately unsuccessful. But, I believe, one can profitably defend some perhaps too hastily condemned view only if one meanwhile keeps a watchful eye (...) on contemporary problems and theorizing. Fred Wilson's monograph, Explanation, Causation and Deduction, attempts just such a defense, while steadfastly ignoring much of contemporary theory. The lack of a sympathetic ear to recent critics of empiricism is the largest single failing of this attempt, and it is this failing which has prevented Prof. Wilson from seeing the poverty of his position. (shrink)