We can distinguish between two different ways in which mathematics is applied in science: when mathematics is introduced and developed in the context of a particular scientific application; when mathematics is used in the context of a particular scientific application but it has been developed independently from that application. Nevertheless, there might also exist intermediate cases in which mathematics is developed independently from an application but it is nonetheless introduced in the context of that particular application. In this paper I present a case study, that of the Lagrange multipliers, which concerns such type of intermediate application. I offer a reconstruction of how Lagrange developed the method of multipliers and I argue that the philosophical significance of this case-study analysis is twofold. In the context of the applicability debate, my historically-driven considerations pull towards the reasonable effectiveness of mathematics in science. Secondly, I maintain that the practice of applying the same mathematical result in different scientific settings can be regarded as a form of crosschecking that contributes to the objectivity of a mathematical result.

中文翻译：

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有意和无意的数学：拉格朗日乘数的案例

我们可以区分数学在科学中应用的两种不同方式：在特定科学应用的背景下引入和发展数学；当数学用于特定科学应用的上下文中，但它是独立于该应用开发的。尽管如此，也可能存在一些中间情况，其中数学是独立于应用程序开发的，但它仍然是在该特定应用程序的上下文中引入的。在本文中，我介绍了一个案例研究，即拉格朗日乘子的案例研究，它涉及此类中间应用。我重建了拉格朗日如何开发乘法器的方法，并且我认为这种案例研究分析的哲学意义是双重的。在适用性辩论的背景下，我受历史驱动的考虑将推向数学在科学中的合理有效性。其次，我认为在不同的科学环境中应用相同的数学结果的做法可以被视为一种交叉检查形式，有助于数学结果的客观性。