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Herkansing voor infinitesimalen?

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Abstract

A New Chance for Infinitesimals?

This article discusses the connection between the Zenonian paradox of magnitude and probability on infinite sample spaces. Two important premises in the Zenonian argument are: the Archimedean axiom, which excludes infinitesimal magnitudes, and perfect additivity. Standard probability theory uses real numbers that satisfy the Archimedean axiom, but it rejects perfect additivity. The additivity requirement for real-valued probabilities is limited to countably infinite collections of mutually incompatible events. A consequence of this is that there exists no standard probability function that describes a fair lottery on the natural numbers. If we reject the Archimedean axiom, allowing infinitesimal probability values, we can retain perfect additivity and describe a fair, countable infinite lottery. The article gives a historical overview to understand how the first option has become the current standard, whereas the latter remains ‘non-standard’.
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Keywords: history and philosophy of mathematics; infinitesimals; infinity; probability theory

Document Type: Research Article

Publication date: December 1, 2018

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  • Het Algemeen Nederlands Tijdschrift voor Wijsbegeerte (ANTW), opgericht in 1908, is een filosofisch wetenschappelijk tijdschrift dat oorspronkelijke artikelen publiceert van auteurs uit het Nederlands-Vlaamse taalgebied. Het ANTW richt zich op een breed scala van deelgebieden, stromingen, discussies en perioden van de filosofie. Het ANTW brengt vier nummers per jaar uit: twee focusnummers waarin steeds één omvangrijk artikel centraal staat van een prominente Nederlandse of Vlaamse filosoof, gevolgd door commentaren en een afsluitende repliek van de focusauteur; een regulier nummer met onderzoeksartikelen, overzichtsartikelen en congresverslagen; en een themanummer met artikelen rond een tevoren vastgesteld thema.

    The Dutch language journal Algemeen Nederlands Tijdschrift voor Wijsbegeerte (ANTW), founded in 1908, is a philosophical, academic journal that publishes original articles from authors in the Netherlands and Belgium. ANTW focuses on a broad variety of subjects, trends, discussions and periods in philosophy. ANTW publishes four issues each year: Two Focus Issues that contain one large article from a prominent Dutch or Flemish philosopher (Focus Article), followed by six shorter comments and a closing reply by the focus Author. One regular issue covering research articles, overview articles and congress reports. And one themed issue in which circa six articles focus on certain subjects relevant in philosophy.
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