![]() ![]() ![]() We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory's teacher degli Angeli, and Gregory's books at Venice, in the late 1660s. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? as well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an "unrigorous" practice, in fact finds close procedural proxies in modern infinitesimal theories. Here Gregory referred to the last or ultimate terms of a series. As a case study, we analyze the historians' approach to interpreting James Gregory's expression ultimate terms in his paper attempting to prove the irrationality of pi. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Keywords: historiography infinitesimal Latin model butterfly model Cauchy's procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson's framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson's framework but scholars working in a Weierstrassian framework bend over backwards either to claim that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Such procedures have immediate hyperfinite analogues in Robinson's framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler's own presentation. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz's infinitesimals. Thus, Leibniz's distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson's framework, while Leibniz's law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. The latter provides closer proxies for the procedures of the classical masters. Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. Of relating inassignable to assignable quantities by means of his The conception of modern infinitesimals as a development of Leibniz's strategy We show, moreover, that Leibniz's system forĭifferential calculus was free of logical fallacies. WeĪrgue that Leibniz's defense of infinitesimals is more firmly grounded thanīerkeley's criticism thereof. Imaginaries, which are not eliminable by some syncategorematic paraphrase. Logical fictions, as Ishiguro proposed, but rather pure fictions, like Leibniz's infinitesimals are fictions, not We argue that Robinson, among others, overestimates the force ofīerkeley's criticisms, by underestimating the mathematical and philosophical Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptlyĭemonstrating the inconsistency of reasoning with historical infinitesimal Tradition inspired Lakatos, Laugwitz, and others to consider the history of the A notableĮxception is Robinson himself, whose identification with the Leibnizian Theory of infinitesimals, require the resources of modern logic thus manyĬommentators are comfortable denying a historical continuity. Robinson's hyperreals, while providing a consistent Infinitesimal calculus of the 17th century and 20th century developments suchĪs Robinson's theory. Many historians of the calculus deny significant continuity between
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