![]() ![]() However, such claims hinge upon a conflation of Leibnizian notions of bounded infinity and unbounded infinity, a distinction emphasized by early Knobloch. Rabouin and Arthur claim that Leibniz considers infinities as contradictory, and that Leibniz' definition of incomparables should be understood as nominal rather than as semantic. ![]() We analyze Arthur's comparison and find it rife with equivocations and misunderstandings on issues including the non-punctiform nature of the continuum, infinite-sided polygons, and the fictionality of infinitesimals. While many scholars (e.g., Ishiguro, Levey) opt for a default Weierstrassian framework, Arthur compares LC to a non-Archimedean framework SIA (Smooth Infinitesimal Analysis) of Lawvere-Kock-Bell. Recent Leibniz scholarship has sought to gauge which foundational framework provides the most successful account of the procedures of the Leibnizian calculus (LC). We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations of real-valued measures. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. The most relevant scenario involves algebraic structures that extend the field of rational numbers hence, it is possible to associate to the filter integral an upper and lower standard part, which can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. ![]() This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. We introduce a notion of integration defined from filters over families of finite sets. ![]()
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