Recognizability of the object has little to do with abstract photography. The fundamental distinction for me is whether the photographer is presenting the object or another graphical structure to me. Look at the work of Aaron Siskind where you can identify the peeling paint or a stack of rocks. As that recognition quickly fades, you are left with the surface of the photograph and its contents to enjoy, separated from the object photographed. Or look at the photograph of the side of a building by André Kertész where you clearly recognize that it is a building but you have no reason to focus on the building because the frame is rich with geometric shapes creating a strong photographic structure for you to enjoy. It is quite possible for a photograph to have abstract qualities as well as being read in different ways.