In the chemistry setting, a doped semiconductor is a crystal lattice with holes/extra electrons in it. 

In the band/quantum mechanical picture, these holes/extra electrons can be seen as the amount of missing/excess charges from a full band of energy states. 

Therefore, in a pure silicon crystal, with full covalent bonds and thus no holes or extra electrons, the valence band is completely full, and the conducting band is completely empty. 

Which means that Silicon is an insulator, since the Fermi level lies above the valence and below the conducting band. What are the holes in my reasoning?