无穷小、想象、理想和虚构
Infinitesimals, Imaginaries, Ideals, and Fictions
摘要：
莱布尼茨受理各种概念的无穷小，考虑他们有时作为理想的东西和其他时间作为小说。但在这两种情况下，他都把无穷小比作虚根。我们同意大多数评论家的观点，莱布尼茨的无穷小是虚构的，而不是理想的东西。然而，我们质疑他们的意见，莱布尼茨的无穷小是最好的理解为逻辑小说，可消除的释义。这种所谓的关于无穷小的共形论概念出现在莱布尼茨的文献中，但那里也有另一种关于无穷小的形式主义的解释。我们认为，形式主义的帐户更好地理解与虚根的类比，更符合莱布尼茨最深刻的哲学信念。形式主义的概念支持罗宾逊和其他人的主张，即非标准分析的哲学基础和莱布尼茨的微积分是从同一块布上剪下来的。
英文摘要：
Leibniz entertained various conceptions of infinitesimals, considering them sometimes as ideal things and other times as fictions. But in both cases, he compares infinitesimals favorably to imaginary roots. We agree with the majority of commentators that Leibniz's infinitesimals are fictions rather than ideal things. However, we dispute their opinion that Leibniz's infinitesimals are best understood as logical fictions, eliminable by paraphrase. This so-called syncategorematic conception of infinitesimals is present in Leibniz's texts, but there is an alternative, formalist account of infinitesimals there too. We argue that the formalist account makes better sense of the analogy with imaginary roots and fits better with Leibniz's deepest philosophical convictions. The formalist conception supports the claim of Robinson and others that the philosophical foundations of nonstandard analysis and Leibniz's calculus are cut from the same cloth.
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