Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure

Stevin numbers and reality

We explore the potential of Simon Stevin's numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with arXiv:1104.0375, arXiv:1108.2885, arXiv:1108.420

HIV infection and severe malnutrition: a clinical and epidemiological study in Burkina Faso.

OBJECTIVE: To define a clinical profile indicative of HIV infection in a population of severely malnourished children in Burkina Faso. A total of 433 children (average age, 19 months) were recruited at the Sanou Souro National Hospital, Bobo Dioulasso, Burkina Faso. RESULTS: Sixty-three per cent presented with marasmus, 13% with kwashiorkor and 24% with both forms of malnutrition. The prevalence of HIV infection in children aged over 12 months was 13.8%, with a marked predominance of HIV-1 (95.8%). Mother-to-child transmission was proven in 77% of the cases; in 10% of the observed paediatric AIDS cases, transmission may have occurred through multi-injections with contaminated equipment. Marasmus was the form of malnutrition most frequently associated with HIV (P < 0.001); its severity was exacerbated by HIV infection. Adenopathy (P < 0.0001), oral candidiasis (P < 0.0006), skin disorders (P < 0.01) and hepatomegaly (P = 0.01) appeared to be significantly related to HIV infection. Discriminant analysis revealed that the presence of adenopathies was the strongest indicator symptom of HIV infection. Multivariate analysis revealed that a clinical profile of marasmus, adenopathies and oral candidiasis (specificity, 82%) was indicative of HIV infection in this population. The short-term clinical prognosis was poor and usually led to the death of the child when seropositive (P < 0.001). CONCLUSIONS: Among children exhibiting severe malnutrition, HIV-positive children are distinguished by a high horizontal transmission rate, a high specific clinical profile and a very poor prognosis

Genome sequence of an Enterobacter helveticus strain, 1159/04 (= LMG 23733), isolated from fruit powder

We report the draft genome sequence of Enterobacter helveticus strain LMG 23733, isolated from fruit powder. The draft genome assembly for E. helveticus strain LMG 23733 has a size of 4,635,476 bp and a G+C content of 55.9%

Estudio sobre las praxeologías que se proponen estudiaren un curso universitario de cálculo

En este trabajo se analizan las organizaciones que se proponen estudiar en un curso de cálculo universitario relativas a las nociones de límite y continuidad funcional. Desde la Teoría Antropológica de lo Didáctico se analizó el material editado por lo profesores destinado a estudiantes universitarios. Los principales resultados indican que se propone el estudio de tareas aisladas, que no conducen a la elaboración y validación de elementos tecnológicos. De esta manera, se evidencia la organización de los saberes en dos niveles: uno teórico y otro práctico, donde este último no tiene incidencia para la conformación del primero. Esto genera una inadecuada interpretación del conocimiento científico, reduciendo su estudio a organizaciones matemáticas desarticuladas

Teaching and Learning of Calculus

This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process. As a complement to the main text, an extended bibliography with some of the most important references on this topic is included. Since the diversity of the research in the field makes it difficult to produce an exhaustive state-of-the-art summary, the authors discuss recent developments that go beyond this survey and put forward new research questions

Ten Misconceptions from the History of Analysis and Their Debunking

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.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545