An important thing to remember about abstract art is that abstraction is something that happens in degrees and/or stages. There are also many different types of abstraction: formal, ideological, data abstraction, ... In fact, there are degrees of abstraction for anything we call a concept. To equate Abstract Art with nonobjective art is a strangely narrow view for a painter.

eponymous - the qualities that you enjoy in non-objective paintings are the results of creative approaches which seek to eliminate abstraction entirely. Even the most photorealistic painting is an abstraction of the real, and further degrees of abstraction follow from that initial step. Nonobjective artists employ archetypal shapes (if only the shape of the canvas itself) in an effort to make the works devoid of any real world analogue, and may regard representational works as mere shadows on the wall of Plato's cave. Of course, they never really escape Jungian associations, or concepts such as "psychology of color".

To really appreciate Abstract Art (or any art for that matter), it's important understand the techniques of abstraction, and the degree to which they're employed. Once you start to deconstruct a painting in this manner, the question that stares you in the face is...
"What's this painting about?"

P.S. I get my jollies from nonrepresentational pornography.