This course has a twofold objective. On the one hand, we will seek to further understand the use of mathematics in science---with cases from both the natural and the social sciences---by focussing on the concept of explanation. Our approach will be based on the now standard (but not entirely uncontroversial) philosophical method of rational reconstruction of the scientific practice in order to isolate the methods of justification that underly (proper) scientific knowledge. This being said, students will be introduced to an area of the philosophical literature on science that refrains from idealizing scientific practice to the extent that philosophy of science becomes a fantastic story about "in principle science''. The problem with such rational reconstructions of the scientific method is that they fail to appreciate the genuinely unavoidable epistemic and computational constraints that undermine a naive approach. Once one recognizes that the devil is in the details indeed, it becomes essential to examine the methods used in actual science to overcome the epistemic and computational limitations in question. This is what will lead us to the study of asymptotic explanation.
On the other hand, students will be expected to improve their working knowledge (that is, their "know-how'' and not merely their "know-that'') of the use of mathematics in science. Thus, students will need to study elements of mathematics in the fields of calculus, linear algebra, statistics, numerical analysis, and perturbation theory. Students will also be asked to learn the programming language Matlab (and maybe Maple), and to typeset their work in LaTeX. Homework assignments with progressively increasing levels of difficulty will be the main pedagogical aid to acquiring these skills.