Interpreting the Infinitesimal Mathematics of Leibniz and Euler

【摘要】我们运用Benacerraf对数学本体论和数学实践(或者说数学家在实践中使用的结构)的区分，来考察在Bos、Ferraro、Laugwitz等人的著作中关于17世纪和18世纪无穷小数学的对比解释。历史学公认欧拉的无穷小数学，我们在史学背后探查维尔斯特拉斯的灵魂。Ferraro提出用魏尔斯特拉斯的极限概念来理解欧拉，而Fraser宣称，古典分析是“理解18世纪理论的主要参考点”。与此同时，像Bos和Laugwitz这样的学者试图以一种更忠实于欧拉本人的方式来探索欧拉的方法论、实践和程序。他们在对正弦函数进行无穷乘积分解的原文基础上，分析了欧拉对无穷整数及其相关的无穷乘积的使用。将欧拉的消去律与莱布尼兹的超越的齐次律进行了比较。莱布尼兹的连续性定律在欧拉中也有类似的反映。

我们认为，Ferraro关于欧拉在经典的数量概念下工作的假设，具有这样的特征，即在分析的发展过程中循着阿基米德的传统把欧拉放置在后维尔斯特拉斯的位置上，以及对博斯注意到的双重发展轨道的区别有所模糊。在阿基米德概念框架中解释欧拉，模糊了欧拉工作的重要方面。这样的一个框架被一个语法上更加通用的现代无穷小框架所取代，这个框架为他的推理动作提供了更好的阐述。

关键词：阿基米德公理，无穷乘积，无穷小，连续性定律，齐次性定律，消去律，过程，标准部分原理，本体论，数学实践，欧拉，莱布尼茨

【Abstract】 We apply Benacerraf's distinction between mathematical ontology and mathematical practice(or the structures mathematicians use in practice) to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass's ghost behind some of the received historiography on Euler's infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler's own. Euler's use of infinite integers and the associated infinite products are analyzed in the

context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro's assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler's work. Such a framework is profitably replaced by a syntactically more versatile

modern infinitesimal framework that provides better proxies for his inferential moves.

Keywords： Archimedean axiom Infinite product Infinitesimal Law of continuity Law of homogeneity Principle of cancellation Procedure Standard part principle Ontology Mathematical practice Euler Leibniz

http://dx.doi.org/10.1007/s10838-016-9334-z​dx.doi.org

https://arxiv.org/abs/1605.00455​arxiv.org

Matches for: MR=3663035​www.ams.org