Eliminative and non-eliminative forms of mathematical structuralism contain insights that every decent philosophical account of mathematical structures has to accommodate. So the challenge is to incorporate the insights in the nature of structure that have been acquired by eliminative and non-eliminative mathematical structuralists and to do better where existing accounts have fallen short. It is in this spirit that I try to connect my account of mathematical structure with elements of leading versions of mathematical structuralism.

For as long as there have been theories of arbitrary objects, many of the paradigmatic examples of arbitrary objects have been drawn from number theory (arbitrary natural numbers, for instance) and geometry (arbitrary triangles, for instance). In this chapter, I take a closer look at some examples of arbitrary objects that are related to number theory. In particular, I investigate the properties of arbitrary natural numbers and the epistemological importance of arbitrary finite strings of strokes, which can play the role of natural numbers, as Hilbert taught us more than a century ago.

In this chapter I introduce the notion of arbitrary object and give an initial discussion of it. I take puzzles surrounding the notion of arbitrary object as key questions that should motivate and inspire the construction of a metaphysical theory of arbitrary objects. The theory that I propose bears resemblance to Kit Fine's theory of arbitrary objects. Moreover, key elements of the theory of arbitrary objects can already be found in the theory of variables in Russell's \emph{Principles of Mathematics}.

This chapter is logical in character. The focus is on the logical properties of one particular generic structure: the generic omega-sequence. I take the perspective that is internal to arithmetic, from which arithmetic investigates \emph{one} structure.

In this final and brief chapter I reflect on problems in arbitrary object theory that remain unsolved and, in my opinion, merit philosophers' and logicians' attention.

In this chapter I develop a conception of some particular mathematical structures (such as the natural number structure and graph theoretic structures) that is based on the notions of arbitrary and generic systems.

In this chapter I discuss how a conception of random variables can be developed within the framework of arbitrary object theory. As before, the discussion is example oriented. Moreover, the natural numbers again play a special role: particular attention is given to the way in which the generic $\omega$-sequence can be seen as a collection of random variables.

This chapter deals with methodological questions. The methodological approach that is advocated in this chapter leaves little room for realism debates in philosophy. In particular, from the next chapter onwards, questions of realism in metaphysics and in philosophy of mathematics are left mostly aside. Instead, I will be doing what Kit Fine calls \emph{naive metaphysics}.

This chapter is of a transitional nature. I turn to the connection between arbitrary object theory and the notion of mathematical structuralism. Arbitrary object theory is put aside for a while, and I concentrate on forms of eliminative and non-eliminative structuralism in the philosophy of mathematics.