CAUCHY, INFINITESIMALS AND GHOSTS OF DEPARTED QUANTIFIERS

摘要：在莱布尼茨（Leibniz）、欧拉（Euler）和柯西（Cauchy）的工作中基于无穷小的过程已经在魏尔斯特拉斯（Weierstrass）和罗宾逊（Robinson）的框架中得到了解释。后者为传统大师的工作提供了更接近本意的表述。因此，莱布尼茨对可赋值数和不可赋值数的区分，在罗宾森的框架的描述中可以对应标准数和非标准数之间的区分，而莱布尼茨的齐次性定律，及其所包含的可忽略项的等价概念，对应于标准部分的数学形式。魏尔斯特拉斯框架很难提供相应的形式化描述，但自从Ishiguro以来的学者开始致力于寻找消失量的幽灵，以便为莱布尼茨的无穷小提供魏尔斯特拉斯式的解释。欧拉同样也有可忽略项的等价概念，他将其分为两类:几何和算术。欧拉通常使用乘积分解成特定的无穷多个因子，并使用具有无穷指数的二项式公式。这样的过程在Robinson的框架中有直接的超有限类比，而在Weierstrass的框架中，它的重新解释与Euler自己的表述明显不同。柯西以无穷小的形式给出连续性的清晰定义，在鲁滨逊的框架中可以找到无穷小的形式化描述，但在维尔斯特拉斯的框架下工作的学者们在拼命声称柯西是模糊的，或者说他的工作是在追寻消失量的幽灵。柯西在1853年的和定理(连续函数的级数)中的工作，更容易从罗宾逊框架的观点理解，在这个框架中，我们可以利用诸如一致收敛概念的点态定义等工具。作为案例研究，我们分析了Craig Fraser和Jesper Lutzen研究柯西对无穷小分析有何贡献的方法，以及Fraser对莱布尼茨处理无穷小的理论策略的方法。哲学家Ian Hacking等人对情境性和偶然性的重要作用的洞见，可能会破坏Fraser的解释框架。

关键词：史学；无穷小；拉丁模型；蝴蝶模型；连续性定律；本体；实践；柯西；莱布尼茨。

Abstract. Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson’s frameworks. The latter provides closer proxies for the procedures of the classical masters. Thus, Leibniz’s distinction between assignable and inassignable numbers finds a proxy in the distinction between standard and nonstandard numbers in Robinson’s framework, while Leibniz’s law of homogeneity with the implied notion of equality up to negligible terms finds a mathematical formalisation in terms of standard part. It is hard to provide parallel formalisations in a Weierstrassian framework but scholars since Ishiguro have engaged in a quest for ghosts of departed quantifiers to provide a Weierstrassian account for Leibniz’s infinitesimals. Euler similarly had notions of equality up to negligible terms, of which he distinguished two types: geometric and arithmetic. Euler routinely used product decompositions into a specific infinite number of factors, and used the binomial formula with an infinite exponent. Such procedures have immediate hyperfinite analogues in Robinson’s framework, while in a Weierstrassian framework they can only be reinterpreted by means of paraphrases departing significantly from Euler’s own presentation. Cauchy gives lucid definitions of continuity in terms of infinitesimals that find ready formalisations in Robinson’s framework but scholars working in a Weierstrassian framework bend over backwards either to claim

that Cauchy was vague or to engage in a quest for ghosts of departed quantifiers in his work. Cauchy’s procedures in the context of his 1853 sum theorem (for series of continuous functions) are more readily understood from the viewpoint of Robinson’s framework, where one can exploit tools such as the pointwise definition of the concept of uniform convergence. As case studies, we analyze the approaches of Craig Fraser and Jesper L¨utzen to Cauchy’s contributions to infinitesimal analysis, as well as Fraser’s approach toward Leibniz’s theoretical strategy in dealing with infinitesimals. The insights by philosophers Ian Hacking and others into the important roles of contextuality and contingency tend to undermine Fraser’s interpretive framework.

Keywords: historiography; infinitesimal; Latin model; butterfly model; law of continuity; ontology; practice; Cauchy; Leibniz.

https://arxiv.org/abs/1712.00226​arxiv.org

http://matstud.org.ua/texts/2017/47_2/115-144.pdf​matstud.org.ua

http://dx.doi.org/10.15330/ms.47.2.115-144​dx.doi.org

Matches for: MR=3733080​mathscinet.ams.org