Until mid-19th century, an unfinished artwork was unacceptable for both aesthetic and philosophical reasons. A result of this tendency was that collectors of ancient art (mainly Greek and Roman) would dare restore sculptures by adding missing body parts or objects according to historically accurate or even imagined reconstructions of the objects at hand.
The end of this obsession for complete and perfect sculptures came to an end with the coming of the Parthenon marbles in London. The fragmented marbles had suffered from the passage of time, wars, as well as a big adventure that saw them removed from their original place and transported to a foreign land. The marbles managed to overwhelm the intellectuals of the time who realised that these imperfect, fragmented ruins from the Parthenon were carrying something magic that could make them more appealing than the most beautiful sculptures that could be found in collections of European monarchs. That something magic was history, and more specifically the history of the Athenian city state.
Additionally the development of Classical Archaeology as an independent science led to a demand for minimum interventions on ancient material culture which was proclaimed to be perfect under a Romantic Idealism with its roots in the philosophy of Johann Joachim Winckelmann (1717-1768). As a result, by the 20th century classical sculptures were appreciated for their incompleteness, which would be unimaginable two centuries ago.
The coming of photography played a huge part in changing the attitude towards the complete in art. Photography had managed to finally complete the task of competing with nature better than any artistic medium had done before. Art could no longer remain the same. It had to choose between competing with photography in an impossible race towards realism or look for alternative options. Thus, the cults of the fragmented, the symbolic, the abstract, the esoteric and the expressionist were born.
So what is the mysteriously appealing quality of an incomplete artwork? For many people the answer could be that the incomplete allows us to imagine how things should or could be, letting us access alternative realities and hidden potentialities?
For others there is nothing magic in a fragmented piece of art. Maybe the cult of the incomplete is nothing more than exactly that…a cult. Maybe we feel that we must appreciate such art because we have been taught that we need to appreciate it. This seems to be true to a certain degree with Classical art as the history of its reception indicates.
Of course there is no way of knowing who is right, as the answer is highly subjective. But we should definitely take into account that these possibilities do not necessarily contradict each other. It could be that there is something special about incompleteness and that at the same time it is an acquired taste.