When is a mathematical model a good model? Does it have to represent a specific system? Does it have to make predictions? Provide an explanation of the observed behaviors? How can one compare models in mathematics with models in the empirical sciences in terms of their meaning and use? This course will address these questions by offering, on the one hand, an overview of the philosophical and epistemological positions that emerge from contemporary debates on mathematical models and modeling. On the other hand, it will analyze case studies drawn from recent and contemporary practices of mathematical modeling of dynamic and complex systems in different disciplinary fields (physics, biology, social sciences, cognitive sciences, etc.). From the perspective of the recent history of modeling, the course will show how dynamical systems theory itself, as a mathematical theory, has produced criteria intended to make it possible, sometimes a priori, to evaluate the quality of a model (notions of stability, genericity, and structural stability). From a philosophical perspective, it will examine the epistemic status of such notions, their empirical relevance, and their transcendental connotations. Each academic year, these general questions will be explored in relation to a specific topic, which will be presented at the beginning of the course.

This year, we will continue to explore the “how-possibly vs. how-actually” debate in the context of the natural sciences vs. human sciences debate, drawing in part on the works of C.H. Waddington, A. Turing, and R. Thom.