The aim of this article is to present elements and discuss the potential of a research program at the intersection between mathematics and heterodox economics, which we call Criticial Mathematical Economics (CME). We propose to focus on the mathematical and model-theoretic foundations of controversies in economic policy, and aim at providing an entrance to the literature as an invitation to mathematicians that are potentially interested in such a project. From our point of view, mathematics has been partly misused in mainstream economics to justify `unregulated markets'. We identify two key parts of CME, which leads to a natural structure of this article: The first part focusses on an analysis and critique of mathematical models used in mainstream economics, like e.g. the Dynamic Stochastic General Equilibrium (DSGE) in Macroeconomics and the so-called ``Sonnenschein-Mantel-Debreu''-Theorems. The aim of the second part is to improve and extend heterodox models using ingredients from modern mathematics and computer science, a method with strong relation to Complexity Economics. We exemplify this idea by describing how methods from Non-Linear Dynamics have been used in Post-Keynesian Macroeconomics', and also discuss (Pseudo-) Goodwin cycles and possible Micro- and Mesofoundations. Finally, we outline in which areas a collaboration between mathematicians and heterodox economists could be most promising, and discuss both existing projects in such a direction as well as areas where new models for policy advice are most needed. In an outlook, we discuss the role of (ecological) data, and the need for what we call Progressive Data Science.

翻译：本文旨在阐述并探讨一个位于数学与非正统经济学交叉领域的研究计划（我们称之为批判性数理经济学，CME）的要素与潜力。我们提议聚焦于经济政策争议背后的数学与模型理论基础，旨在为相关文献提供一个入口，以吸引可能对此类项目感兴趣的数学家。在我们看来，数学在主流经济学中部分被误用于为“无管制市场”提供辩护。我们识别了CME的两个关键部分，这构成了本文的自然结构：第一部分聚焦于对主流经济学所用数学模型（例如宏观经济学中的动态随机一般均衡模型以及所谓的“Sonnenschein-Mantel-Debreu”定理）的分析与批判。第二部分的目标是利用现代数学与计算机科学的要素来改进和扩展非正统模型，这一方法与复杂性经济学有紧密联系。我们通过描述非线性动力学方法如何被用于后凯恩斯主义宏观经济学来例证这一思想，并讨论了（伪）古德温循环以及可能的微观与中观基础。最后，我们概述了数学家与非正统经济学家在哪些领域的合作可能最具前景，并讨论了该方向上的现有项目以及最需要新政策建议模型的领域。在展望中，我们探讨了（生态）数据的作用，以及对我们所称的进步数据科学的需求。