Mathematics and logic are indispensable in science, yet how they are deployed and why they are so effective, especially in the natural sciences, is poorly understood. In this paper, I focus on the how by analysing Jean Piaget’s application of mathematics to the empirical content of psychological experiment; however, I do not lose sight of the application’s wider implications on the why. In a case study, I set out how Piaget drew on the stock of mathematical structures to model psychological content, namely, the operations of thought involved in reasoning. In particular, I show how operations of thought form structured wholes that initially resisted modelling by either lattices or groups but could be modelled adequately by modifications of these mathematical structures. Piaget coined the term ‘grouping’ for the modified structure, and I conclude that it represents a non-canonical application of mathematics to the empirical content of experimental psychology. I also touch on the role external factors played in Piaget’s development of the grouping. According to the genetic epistemology conceived by Piaget, the origin of intelligence lies in the biological organism and develops in stages over time, and, via the grouping, Piaget established a genetic relationship between two stages of reasoning. I show how this relationship explains why mathematics and logic fit the psychological content of reasoning whilst simultaneously making their successful deployment in the natural sciences more mysterious. Finally, I turn to the explanation Piaget envisaged for the unreasonable effectiveness of mathematics in the natural sciences and consider some consequences for naturalism and Pythagoreanism.

数学和逻辑在科学中是不可或缺的，但它们是如何部署的以及它们为何如此有效，尤其是在自然科学中，却鲜为人知。在本文中，我通过分析让·皮亚杰（Jean Piaget）将数学应用到心理实验的经验内容中，着重探讨了这一点；但是，我并没有忽视应用程序对原因的更广泛影响。在一个案例研究中，我阐述了皮亚杰如何利用现有的数学结构来模拟心理内容，即推理中涉及的思维操作。特别是，我展示了思想的运作如何形成结构化的整体，这些整体最初抵制格子或组的建模，但可以通过修改这些数学结构来充分建模。Piaget 为修改后的结构创造了“分组”一词，我得出结论，它代表了数学对实验心理学经验内容的非规范应用。我还谈到了外部因素在 Piaget 的分组发展中所起的作用。根据皮亚杰构想的遗传认识论，智力起源于生物有机体，随着时间的推移分阶段发展，通过分组，皮亚杰建立了两个推理阶段之间的遗传关系。我展示了这种关系如何解释为什么数学和逻辑适合推理的心理内容，同时使它们在自然科学中的成功部署更加神秘。最后，