Why don't solids fuse upon contact?

In baby physics/chemistry, we imagine that solids are composed of atoms which bind together because of favourable combinations of valence electrons. Later, we blame solidity upon the requirement that no two particles occupy the same energy level. But both of these explanations seem to portray solid matter as universally mutually attractive, which is obviously not observed. Clearly some sort of edge effect is at play here, but what is it? In particular, what happens at the edges when a solid is broken in two?