I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of (...) the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones. (shrink)

Category theory has become central to certain aspects of theoretical physics. Bain has recently argued that this has significance for ontic structural realism. We argue against this claim. In so doing, we uncover two pervasive forms of category-theoretic generalization. We call these ‘generalization by duality’ and ‘generalization by categorifying physical processes’. We describe in detail how these arise, and explain their significance using detailed examples. We show that their significance is two-fold: the articulation of high-level physical concepts, and the generation (...) of new models. 1 Introduction2 Categories and Structuralism 2.1 Categories: abstract and concrete 2.2 Structuralism: simple and ontic3 Bain’s Two Strategies 3.1 A first strategy for defending Objectless 3.2 A second strategy for defending Objectless4 Two Forms of Categorical Generalization 4.1 Generalization by duality 4.2 Generalization by categorification 4.3 The role of category theory in physics5 Conclusion. (shrink)