Leibniz’s syncategorematic infinitesimals II: their existence, their use and their role in the justification of the differential calculus

IF 0.7 2区 哲学 Q2 HISTORY & PHILOSOPHY OF SCIENCE
David Rabouin, Richard T. W. Arthur
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引用次数: 11

Abstract

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom.

Abstract Image

莱布尼茨的合范畴无穷小Ⅱ:它们的存在、使用及其在微分学论证中的作用
在这篇论文中,我们试图给出一个历史上准确的表述,说明莱布尼茨是如何理解他的无穷小的,以及他是如何证明这些无穷小的使用是合理的。一些作者声称,当莱布尼茨在世纪之交回应罗尔和其他人对微积分的批评时,称它们为“小说”,他脑海中的“小说”的含义与他早期的作品不同,涉及到对它们作为连续体的非阿基米德元素存在的承诺。与此相反,我们表明,到1676年,莱布尼茨已经形成了一种他从未动摇过的解释,根据这种解释,无穷小和无限整体一样,不能被视为存在的,因为它们的概念包含矛盾,即使它们可以被视为在特定条件下存在——他后来将这一概念描述为“合范畴的”。因此,我们不能从无穷小的成功使用中推断出它们的存在。通过对莱布尼茨在1675–1676年的De quadratura中的论点的详细分析,我们表明莱布尼茨已经提出了两种提出无穷小方法的策略,一种是使用有限量,可以使其尽可能小,以使误差小于可分配的误差,从而为零;以及另一种“直接”方法,其中无穷大和无穷小由类似于代数中虚根的虚构引入,并引入到投影几何中的无穷大点。然后,我们展示了在他成熟的论文中,后一种策略(现在被阐述为基于连续性定律)是如何被呈现给微积分的批评者的,因为它在代数和几何的基础上同样是本构的,而且根据公认的标准,它是可证明的严格的,符合阿基米德公理。
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来源期刊
Archive for History of Exact Sciences
Archive for History of Exact Sciences 管理科学-科学史与科学哲学
CiteScore
1.30
自引率
20.00%
发文量
16
审稿时长
>12 weeks
期刊介绍: The Archive for History of Exact Sciences casts light upon the conceptual groundwork of the sciences by analyzing the historical course of rigorous quantitative thought and the precise theory of nature in the fields of mathematics, physics, technical chemistry, computer science, astronomy, and the biological sciences, embracing as well their connections to experiment. This journal nourishes historical research meeting the standards of the mathematical sciences. Its aim is to give rapid and full publication to writings of exceptional depth, scope, and permanence.
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