Abstrakt
Ante rem structuralists endorse descriptive metaphysics. Accordingly, Shapiro takes what mathematicians say seriously (faithfulness constraint) and at the same time does not wish to revise the results they reach (minimalism constraint).
When arithmetic is viewed from this perspective, one reaches the conclusion that a number is a relational object and therefore structuralism is justified. However, difficulties arise once the limits of common arithmetic are crossed.
Then it turns out that numbers are also operated in ways that contradict structuralism. That can lead one either to doubt structuralism as a whole, or to reject descriptivism.
We prefer the latter alternative, whereby we reject descriptivism only partially. It remains valid in the context of well-established mathematical practice, let us say arithmetic; it poses difficulties in spheres that as yet lack clear contours, for instance when mathematicians say 2real=2nat.
Although they understand statements of such type, we think that the way they express them is misleading and often confused. And we think that in such circumstances philosophers have the right to take part in creating more exact means of expression.
Obviously, this proposal weakens Shapiro's minimalism constraint.