Homology (mathematics)
The original motivation for defining homology groups is the observation that shapes are distinguished by their holes. Since a "hole" is defined by something that is "not there", this poses the problem of how to define what a hole is, and how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for defining and categorizing holes in a shape. As it turns out, there are some kinds of holes that homology cannot "see" — in which case homotopy groups may be what is needed.