Thermodynamic stable equilibrium conditions If I understand correctly (which I clearly don't) the condition of stable equilibrium of a closed system at constant $(P,T)$ is that $G$ is minimum. 
That implies that $dG=0$ and $d^2G≥0$. Since $dG=VdP-SdT$, if the second differential is positive then $\left(\frac{\partial V}{\partial P}\right)_T≥0$, which is false. What am I doing wrong?
EDIT: I would like the answer to be as mathematically rigorous as possible, and with no hand-waving about differentials. I would also appreciate if you could tell me a textbook/article where stability is discussed with some level of rigour.
 A: 
... the condition of stable equilibrium of a closed system at constant $(P,T)$ is that $G$ is minimum. 

This is true, but it is not supposed to mean that equilibrium happens only for special values of $T,P$ for which some $G(T,P)$ has local minimum. Temperature and pressure are understood as given external conditions, the system is not supposed to change their values. Stable equilibrium ordinarily can exist for almost any values of these parameters.
What the statement means is that when we can express Gibbs energy of a thermodynamic system as a function of $T,P$ and some independent variable $x$ describing state of the system, equilibrium state has to have $x$ with value that minimizes $G(T,P,x)$.
For example, thermodynamic state of a puddle made of mixture of liquid water and kitchen salt can be described, in addition to $T,P$, by the amount of salt in dissolved state $n_d$ (assuming salt conservation). The principle of stable equilibrium then implies that in equilibrium, $n_d$ has value for which
$$
\frac{\partial G}{\partial n_d}(T,P,n_d) = 0,
$$
$$
\frac{\partial^2 G}{\partial n_d^2}(T,P,n_d) > 0.
$$
