In the cutting process, the hardie is the fixed tool that you place the material on and then strike with a hammer.  The term "hardie" comes from the blacksmithing world.  Traditional anvils have a square hole on one end for inserting various tools, including chisels.  This hole is called a "hardy hole".  The mosaicist's hardie looks just like a blacksmith's chisel.  In Italian, this tool is called a "tagliolo", which actually means 'chisel".