 As we turn to an overview of the current debate, it is therefore important to remember that the use of the terms platonist (for those who affirm the existence of abstract objects) and nominalist (for those who deny existence) is somewhat lamentable, since these words have established senses in the history of philosophy. These terms stood for positions that have little to do with the modern notion of an abstract object. Modern platonists (with a small ‘p’) need not accept any of the distinctive metaphysical and epistemological doctrines of Plato, just as modern nominalists need not accept the distinctive doctrines of the medieval nominalists. Moreover, the literature also contains mention of anti-platonists, many of whom see themselves as fictionalists about abstracta, though this doesn’t help if it turns out that the best analysis of fictions is to regard them as abstract objects. So the reader should therefore be aware that terminology is not always well-chosen and that the terms so used sometimes stand for doctrines that are more restricted than the traditional doctrines that go by the same name. Henceforth, we simply use platonism for the thesis that there exists at least one abstract object, and nominalism for the thesis that the number of abstract objects is exactly zero (Field 1980).