莱布尼茨的无穷小：它们的虚构性，它们的现代实现，以及它们的敌人
Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
摘要：
许多微积分历史学家否认17世纪和20世纪微积分发展之间的重要连续性，如罗宾逊理论。罗宾逊的超现实，虽然提供了一个一致的无穷小理论，需要现代逻辑的资源;因此，许多评论家舒适否认历史的连续性。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激发了拉卡托斯、劳格维茨和其他人以更有利的眼光看待无穷小的历史。尽管他的莱布尼兹同情，罗宾逊认为伯克利的批评的微积分恰当地表明不一致的推理与历史的无穷小的大小。我们认为，罗宾逊，除其他外，高估了力量伯克利的批评，低估了数学和哲学资源提供给莱布尼茨。莱布尼茨的无穷小是虚构的，不是石黑浩所说的逻辑虚构，而是纯粹的虚构，就像是不可消除的抽象，不能被某些连系语法的释义所消除。我们认为，莱布尼茨的辩护无穷小更坚定地比伯克利的批评。此外，我们表明，莱布尼茨的系统微分学是免费的逻辑谬误。我们的论点加强了现代无穷小的概念，作为莱布尼茨的战略的发展，通过他的先验齐性定律，将不可分配的数量与可分配的数量联系起来。
英文摘要：
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.
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