It is widely believed that mathematics carries a substantial part of the explanatory burden in science. However, mathematics can also play important heuristic roles of a different kind, being a source of new ideas and approaches, allowing us to build toy models, enhancing expressive power and providing fruitful conceptualizations. In this paper, we focus on the application of dynamical systems theory (DST) within the extended cognition (EC) field of cognitive science, considering this case study to be a good illustration of a general phenomenon. In the paper, we justify both a negative and a positive claim. The negative claim is that dynamical systems theory hardly plays any explanatory role in EC research. We justify our claim by analyzing several accounts of the explanatory role of mathematics and stressing the way mathematical arguments are used in explanations. Our positive claim is that even though, for now, DST has no explanatory power in many of the EC approaches, it still plays an important heuristic role there. In particular, using mathematical notions improves the expressive power of the language and gives a sense of understanding of the phenomena under investigation. The case study of EC allows us to identify and analyze this important role of mathematics, which seems to be neglected in contemporary discussions.

人们普遍认为，数学承担着科学中很大一部分解释负担。然而，数学也可以发挥不同类型的重要启发作用，成为新思想和新方法的来源，使我们能够构建玩具模型，增强表达能力并提供富有成效的概念化。在本文中，我们重点关注动力系统理论（DST）在认知科学的扩展认知（EC）领域中的应用，认为该案例研究很好地说明了一般现象。在本文中，我们对消极和积极的主张都进行了论证。反面的说法是动力系统理论在 EC 研究中几乎不起任何解释作用。我们通过分析数学解释作用的几种说法并强调数学论证在解释中的使用方式来证明我们的主张是正确的。我们积极的主张是，尽管目前 DST 在许多 EC 方法中没有解释力，但它仍然发挥着重要的启发作用。特别是，使用数学概念可以提高语言的表达能力，并给人一种对所研究现象的理解感。 EC 的案例研究使我们能够识别和分析数学的这一重要作用，而这一作用在当代讨论中似乎被忽视了。