Alfredo Watkins

The aim of this dissertation is to propose and defend a position in the philosophy of mathematics called “immanent structuralism.” This can be contrasted with the standard Platonist view in the philosophy of mathematics, which holds that mathematics studies a unique category of non-physical, abstract entities. Platonism immediately leads to the epistemological problem of how we can know about these entities if they are not part of the physical world. By contrast, immanent structuralism holds that the things mathematics studies are structures or structural patterns. These structures or patterns are like other physical universals in that they can be instantiated by physical systems. Therefore, some of them can be known through ordinary perception and, as I argue, the rest can be built out of these ones. The first half of the dissertation lays out the core of the theory: I discuss what these structural patterns are and how they can constitute the subject-matter of mathematics. I also give an essence-based account of mathematical truth which refers only to these properties. I then argue that this view avoids the epistemological problems with Platonism, since it allows some basic mathematical properties to be literally instantiated in the physical world, making them graspable by perception, while others can be constructed out of these. I claim that this view is better suited to account for the ordinary knowledge of mathematics had by most people. The second half of the dissertation applies this theory to several special topics in the philosophy of mathematics, including mathematical reduction, mathematical treating-as, mathematics and modality, and mathematical explanation. I also discuss why immanent structuralism presents a unique challenge to indispensability arguments in mathematics and to certain parity arguments in other fields of philosophy. The ultimate hope of the dissertation is to show that a better path forward for realists in the philosophy of mathematics is to move away from object-based accounts like Platonism, and instead move toward more Aristotelian, property-based accounts, like the theory presented here.