Structural realism, mathematics, and ontology

Over the last two decades structural realism has been given progressively more elaborated formulations. Steven French has been at the forefront of the development of the most conceptually sophisticated and historically sensitive version of the view. In his book, The Structure of the World (French (2014)), French shows how structural realism, the view according to which structure is all there is (ontic structural realism), is able to illuminate central issues in the philosophy of science: underdetermination, scientific representation, dispositions, natural modality, and laws of nature. The discussion consistently sheds novel light on the problems under consideration while developing insightful and provocative views. In this paper, I focus on the status of mathematics within French's ontic structural realism, and I raise some concerns about its proper understanding vis-d-vis the realist components of the view.