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

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

Primordialists and Constructionists: a typology of theories of religion

This article adopts categories from nationalism theory to classify theories of religion. Primordialist explanations are grounded in evolutionary psychology and emphasize the innate human demand for religion. Primordialists predict that religion does not decline in the modern era but will endure in perpetuity. Constructionist theories argue that religious demand is a human construct. Modernity initially energizes religion, but subsequently undermines it. Unpacking these ideal types is necessary in order to describe actual theorists of religion. Three distinctions within primordialism and constructionism are relevant. Namely those distinguishing: a) materialist from symbolist forms of constructionism; b) theories of origins from those pertaining to the reproduction of religion; and c) within reproduction, between theories of religious persistence and secularization. This typology helps to make sense of theories of religion by classifying them on the basis of their causal mechanisms, chronology and effects. In so doing, it opens up new sightlines for theory and research

Healing in the Sámi North

There is a special emphasis today on integrating traditional healing within health services. However, most areas in which there is a system of traditional healing have undergone colonization and a number of pressures suppressing tradition for hundreds of years. The question arises as to how one can understand today’s tradition in light of earlier traditions. This article is based on material collected in Sámi areas of Finnmark and Nord-Troms Norway; it compares local healing traditions with what is known of earlier shamanic traditions in the area. The study is based on 27 interviews among healers and their patients. The findings suggest that although local healing traditions among the Sámi in northern Norway have undergone major transformations during the last several hundred years, they may be considered an extension of a long-standing tradition with deep roots in the region. Of special interest are also the new forms tradition may take in today’s changing global society

Hyperfocusing in Schizophrenia: Evidence From Interactions Between Working Memory and Eye Movements

Recent research suggests that processing resources are focused more narrowly but more intensely in people with schizophrenia (PSZ) than in healthy control subjects (HCS), possibly reflecting local cortical circuit abnormalities. This hyperfocusing hypothesis leads to the counterintuitive prediction that, although PSZ cannot store as much information in working memory as HCS, the working memory representations that are present in PSZ may be more intense than those in HCS. To test this hypothesis, we used a task in which participants make a saccadic eye movement to a peripheral target and avoid a parafoveal nontarget while they are holding a color in working memory. Previous research with this task has shown that the parafoveal nontarget is more distracting when it matches the color being held in working memory. This effect should be enhanced in PSZ if their working memory representations are more intense. Consistent with this prediction, we found that the effect of a match between the distractor color and the memory color was larger in PSZ than in HCS. We also observed evidence that PSZ hyperfocused spatially on the region surrounding the fixation point. These results provide further evidence that some aspects of cognitive dysfunction in schizophrenia may be a result of a narrower and more intense focusing of processing resources. Keywords: schizophrenia, working memory, hyperfocusing, saccadic eye movements, distraction This study tests a recent hypothesis about the nature of impaired attention and working memory (WM) in schizophrenia Although the idea that PSZ have impaired attention is common (see review b