Philosophers of science have been writing about scientific modelling for many decades. The traditional focus on the relationship that models bear to abstract theories, and more recent debates about scientific representation and the ontological status of models, have arguably distracted attention away from an important aspect of modelling practice in mathematized sciences which we focus on here: the role played by approximation methods. In order to help rectify this neglect, and lay some groundwork for an account of the way that approximation methods are implicated in mathematical modelling, we advance two descriptive theses in this paper. Firstly, paralleling claims about models made by advocates of the “models as mediators” slogan (Morgan and Morrison, 1999), we argue that approximations need to be recognised as a distinct species of scientific output, whose construction does not flow automatically from the kinematic and dynamical structures posited by a theory or model, a claim we call the Distinctness thesis. Secondly, we claim that, in at least some cases in modern physics, approximations play an integral role in assigning empirical and physical content to a model, in a sense which is difficult for some extant philosophical approaches to scientific modelling to account for, a claim we call the Content Determination thesis. We support these claims with an analysis of two prominent historical episodes in post-war high energy physics: the development of the Chew and Nambu-Jona-Lasinio models.
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