Does a mother bird know if her egg is alive or dead?

A mother dove has decided to build a nest on a ladder in front of our window. She & ,what we think may be the father bird, have gone back and forth taking turns warming the egg during the entire day. Last night we checked on the bird & she was gone. There were feathers everywhere, sign that a cat must have gotten to her. We saw two small eggs on the ground near the ladder one broken & one in tact. My mom then placed the unbroken egg in the nest with hopes of the mother or father returning. This morning I saw the mother bird sitting in her nest. Was putting the egg back in her nest a good idea? Do birds naturally know the conditions of their eggs?