I have a correction. I went out and bought a tube of Colgate ("Triple Acción! Extra Blancura!") and it does indeed contain silica as an abrasive. (And Colgate is hardly some weird off-brand toothpaste.). The difference is that it’s hydrated silica, which is amorphous rather than crystalline, and is somewhat softer than normal silica. (And, as an extra special bonus, not carcinogenic like normal silica!) This is the same stuff known as "silica gel" that comes in little pouches with your electronics labeled "DESICCANT — DO NOT EAT".

Amorphous hydrated silica is still hard enough to polish glass and abrade your tooth enamel. Chalk is not, although I'm not sure it's hard enough to abrade away your tartar deposits.