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.

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