A number of workers have proposed the fluorite structure for a hypothetical postrutile phase of silica. However, this phase has never been observed, nor has the existence of any other postrutile phase been established unambiguously. Estimates of the density of this phase, based on empirical systematics of analogue compounds and theoretical arguments and calculations, differ widely. We have therefore investigated the equation of state and dynamic and elastic stability of SiO2 (fluor) using a first principles band structure calculation and pair potentials derived from the modified electron gas model. Our analysis indicates that it is unlikely that the density of the fluorite phase exceeds that of stishovite anywhere in the mantle. The theoretical phonon spectrum exhibits dynamic instability at pressures below 170 GPa, suggesting that even if the fluorite phase were produced at high-pressure, it would be unquenchable and hence unobservable in unloaded experimental samples. However, it is also possible that the dynamic instability makes the fluorite phase a stable superionic conductor at lower mantle conditions. The relatively low density, and low-pressure dynamic instability of the theoretical fluorite-structured silica, is most likely due to strong oxygen-oxygen repulsive interactions, coupled with an inherent instability of the simple cubic oxygen sublattice.