This bothers me a lot, probably because I just fail to understand some basic mathematical principle that textbooks think too obvious to mention.

Go to any textbook about thermodynamic potentials and you'll read something like this:

Book:
>Want to evaluate the spontaneity of a system at constant temperature and pressure? You need the Gibbs Energy.
>dG = -SdT + VdP

Me:
>Wait, it can't be constant T and P because both terms would always be zero.

The same argument can be made for other potentials which the textbook says hold a pair of natural variables constant.

This seems like an obvious contradiction to me, but the textbooks never mention it.