LEIBNIZ’S WELL-FOUNDED FICTIONS AND THEIR INTERPRETATIONS

JACQUES BAIR, PIOTR B LASZCZYK, ROBERT ELY, PETER HEINIG,

AND MIKHAIL G. KATZ

【摘要】莱布尼茨将“fiction”一词与无穷小联系起来。它们究竟是什么样的“ficitons”一直是学术界争论的话题。Bos和Mancosu的观点与Ishiguro和Arthur形成了鲜明的对比。莱布尼茨在他发表的文章和信件中表达了自己的观点，这使得Bos区分了莱布尼茨工作中的两种方法:(A)一种是利用经典的“穷竭”论点，(B)一种是利用不可分割的无穷小和连续性法则。

特别令人感兴趣的证据来自莱布尼茨的作品《人类理智新论》，以及他与Arnauld, Bignon, Dagincourt, Des Bosses,and Varignon的通信。对证据的仔细研究使我们得出与亚瑟相反的结论。

罗尔提出了反对意见，认为没有理由将几何和分析中的公理和运算从普通域扩展到无穷小微积分，我们分析了这一迄今未被注意的意见，以及索林和莱布尼茨对此的态度。

最新发布的莱布尼茨1705年手稿目前正在识别，阐明了莱布尼茨的不可分割的无穷小的本质。

在1695年的两篇文章中，莱布尼兹明确指出，他不可比的量违反了欧几里德的定义V.4，也就是阿基米德性质，证实了莱布尼兹微积分的非阿基米德解释。

关键词：阿基米德性质；可分配量与不可分配量；欧几里得定义V.4；无穷小；连续性定律；同质性定律；逻辑虚构；纯虚构；同步范畴；转移原理；阿尔诺；比格农；德斯波士；罗尔；索林；瓦里尼翁

Abstract. Leibniz used the term fiction in conjunction with infinitesimals. What kind of fictions they were exactly is a subject of scholarly dispute. The position of Bos and Mancosu contrasts with that of Ishiguro and Arthur. Leibniz’s own views, expressed in his published articles and correspondence, led Bos to distinguish between two methods in Leibniz’s work: (A) one exploiting classical ‘exhaustion’ arguments, and (B) one exploiting inassignable infinitesimals together with a law of continuity. Of particular interest is evidence stemming from Leibniz’s work Nouveaux Essais sur l’Entendement Humain as well as from his correspondence with Arnauld, Bignon, Dagincourt, Des Bosses, and Varignon. A careful examination of the evidence leads us to the opposite conclusion from Arthur’s. We analyze a hitherto unnoticed objection of Rolle’s concerning the lack of justification for extending axioms and operations in geometry and analysis from the ordinary domain to that of infinitesimal calculus, and reactions to it by Saurin and Leibniz. A newly released 1705 manuscript by Leibniz (Puisque des personnes...) currently in the process of digitalisation, sheds light on the nature of Leibnizian inassignable infinitesimals. In a pair of 1695 texts Leibniz made it clear that his incomparable magnitudes violate Euclid’s Definition V.4, a.k.a. the Archimedean property, corroborating the non-Archimedean construal of the Leibnizian calculus.

Keywords: Archimedean property; assignable vs inassignable quantity; Euclid’s Definition V.4; infinitesimal; law of continuity; law of homogeneity; logical fiction; Nouveaux Essais; pure fiction; quantifier-assisted paraphrase; syncategorematic; transfer principle; Arnauld; Bignon; Des Bosses; Rolle; Saurin; Varignon

http://dx.doi.org/10.15330/ms.49.2.186-224​dx.doi.orghttps://arxiv.org/abs/1812.00226​arxiv.orgMatches for: MR=3882551​mathscinet.ams.org