Applicability, Idealization, and Mathematization

In this thesis I provide a study of the applicability of mathematics. My starting point is the account of applicability offered in Hartry Field’s book Science without numbers and arising from the nominalistic project carried out therein. By examining the limitations and shortcomings of Field’s account, I develop a new one. My account retains the advantages and insights of Field’s and avoids its difficulties, which are essentially due to its being incomplete and too restrictive. Field’s account is incomplete because it does not deal with the nature and use of idealization in science. Field only describes how mathematics is applied to highly idealized physical theories (e.g. ones containing postulates which are untestable or contradicted by experiment) but he does not explain how idealization arises and why idealized theories are relevant to the actual experimental investigation of empirical phenomena. I offer such an explanation for an elementary scientific theory to which the more complex examples discussed by Field can be reduced. Even in presence of an analysis of idealization, Field’s account of applicability remains problematic. The reason is that it characterizes the role of mathematics in applications in a very restrictive way, which neglects some of its most important uses. I show this by looking at several examples of applications. I then employ the resulting analysis of how mathematics enters them to give a characterization of applicability which does not suffer of the restrictiveness affecting Field’s. This characterization encompasses Field’s but also extends to applications he cannot adequately describe. I thus complete and extend Field’s account of applicability, reaching a more comprehensive and realistic alternative.