Arguments for the effectiveness, and even the indispensability, of mathematics in scientific explanation rely on the claim that mathematics is an effective or even a necessary component in successful scientific predictions and explanations. Well-known accounts of successful mathematical explanation in physical science appeals to scientists’ ability to solve equations directly in key domains. But there are spectacular physical theories, including general relativity and fluid dynamics, in which the equations of the theory cannot be solved directly in target domains, and yet (...) scientists and mathematicians do effective work there (McLarty 2023, Elder 2023). Building on extant accounts of structural scientific explanation (Bokulich 2011, Leng 2021), I argue that philosophical accounts of the role of equations in scientific explanation need not rely on scientists’ ability to solve equations independently of their understanding of the empirical or experimental context. For instance, the process of formulating solutions to equations can involve significant appeal to information about experimental contexts (Curiel 2010) or of physically similar systems (Sterrett 2023). Working from a close analysis of work in fluid mechanics by Martin Bazant and Keith Moffatt (2005), I propose an account of heuristic structural explanation in mathematics (Einstein 1921, Pincock 2021), which explains how physical explanations can be constructed even in domains where basic equations cannot be solved directly. (shrink)

Philosophers of science have long been concerned with the question of what a given scientific theory tells us about the contents of the world, but relatively little attention has been paid to how we set out to build theories and to the relevance of pre-theoretical methodology on a theory’s interpretation. In the traditional view, the form and content of a mature theory can be separated from any tentative ontological assumptions that went into its development. For this reason, the target of (...) interpretation is taken to be the mature theory and nothing more. On this view, positions on ontology emerge only once a theory is to hand, not as part of the process of theory building. Serious attention to theory creation suggests this is too simple. In particular, data collection and experimentation are influenced both by theory and by assumptions about the entities thought to be the target of study. Initial reasoning about possible ontologies has an influence on the choice of theoretical variables as well as on the judgments of the methodology appropriate to investigate them. (shrink)