 If the fictionalist thesis is correct, then one demand that must be imposed on mathematical theories is surely consistency. Yet Field adds to this a second requirement: mathematics must be conservative over natural science. This means, roughly, that whenever a statement of an empirical theory can be derived using mathematics, it can in principle also be derived without using any mathematical theories. If this were not the case, then an indispensability argument could be played out against fictionalism. Whether mathematics is in fact conservative over physics, for instance, is currently a matter of controversy. Shapiro has formulated an incompleteness argument that intends to refute Field’s claim (Shapiro 1983).