The abstract sciences are those that cannot, and have no need to, negotiate the empirical check. This anyhow is a widely received view of the matter. An abiding question for such theories is this: What sorts of check can they negotiate, and does doing so preserve intuitive presumptions of objectivity and realism? There is a particularly vivid context for posing this question and reflecting on how it might be answered. The context is that of conflict resolution strategies for rival theories.

In a broadly accepted use, with which I concur, objectivity attaches to things when they exist apart from and antecedently to anyone's thought of them; and objectivity attaches to statements or beliefs when they are true, or false, apart from and antecedently to anyone's conceiving of them as so. Realism in turn is always realism about something – about abstract objects, about universals, about material things, and so on. The realisms that absorb us in this book are those that attribute this twofold objectivity to what I am calling abstract theories when they meet certain properly understood conditions of adequacy.