标题： 莱布尼茨的无穷小量：它们的虚构性，它们的现代实现，以及从贝克莱到罗素及其更远的反对者 显示英文标题

标题： Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

摘要： 许多微积分的历史学家否认17世纪的无穷小演算与20世纪的发展（如罗宾逊的理论）之间有重要的连续性。 尽管罗宾逊的超实数提供了一致的无穷小理论，但它们需要现代逻辑资源的支持；因此，许多评论员倾向于否认历史上的连续性。 一个值得注意的例外是罗宾逊本人，他对莱布尼茨传统的认同启发了拉卡托斯、劳格维茨等人以更积极的态度看待无穷小的历史。 尽管他同情莱布尼茨的观点， 罗宾逊认为贝克莱对无穷小演算的批评恰当地展示了用历史上无穷小量推理时的一致性问题。 我们认为罗宾逊等人高估了贝克莱批评的影响，低估了莱布尼茨可用的数学和哲学资源。 莱布尼茨的无穷小量是虚构，但不是伊希古罗提出的逻辑虚构，而是纯粹的虚构，如同虚数一样，不能通过某种量化短语来消除。 我们论证莱布尼茨对无穷小量的辩护比贝克莱的批评更为坚实。 此外，我们还表明莱布尼茨的微分演算系统没有逻辑谬误。 我们的论证加强了这样一种观点：现代无穷小量是莱布尼茨策略的一种发展，即通过他的齐次性超越定律将不可赋值量与可赋值量联系起来。 显示英文摘要

摘要： 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.