The Platonist answer to the question, "What is mathematical language about?", is that it is about abstract individuals (such as zero, the null set, omega, etc.) and abstract relations (successor, membership, group addition, etc.). One way to make this answer precise is to provide a formal, background theory of abstract individuals and abstract relations. I review one such formal theory and explain the special way in which the language and theorems of arbitrary mathematical theories can be interpreted in this formalism. (A full analysis is developed in my paper "Neologicism? An Ontological Reduction of Mathematics to Metaphysics", Erkenntnis, vol. 53, nos. 1-2 (2000), 219-265.)

However, it turns out that the background formalism for abstracta itself is subject to interpretation. The Platonistic interpretation is just one of (at least) four ways of interpreting the theory. I'll explain how one can develop fictionalist, structuralist, and inferentialist interpretations of the formalism. Since each interpretation offers us a clear, but different, answer to our initial question, the resulting analysis not only offers a way to make these philosophies of mathematics more precise, but also unifies them in a new and unsuspected way. (It also has the consequence that no matter how the mathematicians decide to extend mathematics with new axioms or mathematical foundations, the philosopher will have something to say about the mathematical language used in the extension.)