Applying Mathematics to the Natural and Social World

My dissertation explores the roles that mathematics plays in explanations and action-guidance. In particular, I address epistemic and ethical aspects of applying mathematics in empirical and normative inquiry. In Chapter 2, I examine whether mathematical explanations can be analyzed in terms of counterfactual conditionals. I offer three challenges for such analyses and tentatively conclude against countermathematical explanations. In Chapter 3, I extend the epistemic resources of a popular framework for applying mathematics in sciences. In particular, I use two case studies from biology and physics in order to argue that this framework for applying mathematics in the empirical sciences must be complemented by a distinct role, which I name the bridging role. I show how considering the bridging role would capture mathematical practice in the sciences in a more holistic way. In Chapter 4, I investigate and compare the conditions for applying mathematics in the empirical sciences and normative inquiry. Using two case studies from fair algorithmic decisions and the allocation of scarce resources, I identify two typical sets of different conditions for the applicability of mathematics. Finally, in Chapter 5, I explore the ramifications of the arguments from Chapter 4 to a novel field of research known as Explainable Artificial Intelligence.Ph.D