393 research outputs found

    Ten Misconceptions from the History of Analysis and Their Debunking

    Full text link
    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

    Full text link
    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

    Cauchy's infinitesimals, his sum theorem, and foundational paradigms

    Full text link
    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. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc

    Volcanic Gases:Silent Killers

    Get PDF
    This is the accepted manuscript. The final version is available at http://link.springer.com/chapter/10.1007%2F11157_2015_14.Volcanic gases are insidious and often overlooked hazards. The effects of volcanic gases on life may be direct, such as asphyxiation, respiratory diseases and skin burns; or indirect, e.g. regional famine caused by the cooling that results from the presence of sulfate aerosols injected into the stratosphere during explosive eruptions. Although accounting for fewer fatalities overall than some other forms of volcanic hazards, history has shown that volcanic gases are implicated frequently in small-scale fatal events in diverse volcanic and geothermal regions. In order to mitigate risks due to volcanic gases, we must identify the challenges. The first relates to the difficulty of monitoring and hazard communication: gas concentrations may be elevated over large areas and may change rapidly with time. Developing alert and early warning systems that will be communicated in a timely fashion to the population is logistically difficult. The second challenge focuses on education and understanding risk. An effective response to warnings requires an educated population and a balanced weighing of conflicting cultural beliefs or economic interests with risk. In the case of gas hazards, this may also mean having the correct personal protection equipment, knowing where to go in case of evacuation and being aware of increased risk under certain sets of meteorological conditions. In this chapter we review several classes of gas hazard, the risks associated with them, potential risk mitigation strategies and ways of communicating risk. We discuss carbon dioxide flows and accumulations, including lake overturn events which have accounted for the greatest number of direct fatalities, the hazards arising from the injection of sulfate aerosol into the troposphere and into the stratosphere. A significant hazard facing the UK and northern Europe is a “Laki”-style eruption in Iceland, which will be associated with increased risk of respiratory illness and mortality due to poor air quality when gases and aerosols are dispersed over Europe. We discuss strategies for preparing for a future Laki style event and implications for society
    corecore