The present work is largely concerned with a limited number of themes in the philosophy of mathematics. The first is the notion of object as it is deployed in mathematics. I begin in Chapter 1 with a general discussion of the notion of object, not on the whole focused on mathematics. One of the motives of this discussion is to defuse too-high expectations of what the existence of objects of some mathematical type such as numbers would entail. We proceed to discuss issues surrounding the structuralist view of mathematical objects, which has had a lot of currency in the last forty years or so but has much earlier roots. The general idea of this view is that mathematical objects do not have a richer “nature” than is given by the basic relations of some structure in which they reside. The problem of giving a viable formulation developing this idea is not trivial and raises a lot of issues. That is the concern of Chapters 2 and 3. Chapter 2 is mainly devoted to pursuing a program that uses the structuralist idea to eliminate explicit reference to mathematical objects. Along the way, I discuss some questions about nominalism, about second-order logic, and about how structuralism understands the application of mathematics. Some difficulties of the eliminative program call for using modal notions, and their use in mathematics is a subject of Chapter 3.