If a silicon cyrstal is connected to a battery does current flow through the wire?
It seems that there is no current flowing through the crystal because the flow of the holes cancels out the flow of the electrons


This premise isn't valid. 

If you apply an electric field pointing to the left, then holes will drift to the left (producing a conventional current to the left), and electrons will drift to the right (producing a conventional current to the left). 

The hole current and electron current won't cancel out, they'll both be in the same direction.