 We thus see that the problem of the explanatory role of mathematics in science is intimately related to problems of modeling and idealization in science (see Morrison 2015). In turn, understanding how modeling and idealization work is an integral part of addressing the question of how mathematics hooks on to reality, i.e. an account of the applicability of mathematics to reality (see Shapiro 2000, 35 and 217). It would of course be impossible here to describe the major accounts of mathematical applications on offer. However, the discussion on mathematical explanation in the empirical sciences has fruitfully interacted with some of the proposals on offer. One such proposal is the mapping account of mathematical applications offered by Pincock (2007, 2011c, 2012). The mapping account emphasizes the centrality of the existence of a mapping between the mathematical structures and the empirical reality that is the target of the scientific modeling. Critics have claimed that while there are aspects of the applications of mathematics that are correctly captured by the mapping account, there are also aspects related to mathematical idealization and explanation that it does not manage to capture. This led to the inferential conception of applications of mathematics defended by Bueno and Colyvan 2011 (which preserves some of the mapping account defended by Pincock) and to refinements of Batterman’s own position (see, for instance, Batterman 2010a). Pincock 2011a and 2012 replies to some of the criticisms and argues that his account can be extended to handle mathematical idealizations and explanations. More criticisms are raised in Rizza 2013 who argues that examples from social choice theory cannot be handled by mapping accounts such as those defended by Pincock and Bueno and Colyvan.