The many definitions approach may be the most interesting one, because
it is operational rather than abstract.  Looking at an analogous situation
in physics, how would you define energy? You can talk about it as the
ability to do work, or as a quantity which is always conserved, but the
first quickly becomes circular and the second makes sense only if you start
enumerating the forms energy can take.  Richard Feynman makes the point
(see volume 1, The Feynman Lectures): "there is a certain quantity, which
we will call energy, that does not change in the manifold changes which
nature undergoes." And later he adds: "It is important to realize that
in physics today we have no knowledge of what energy is." Yet this point
of view does not keep us from identifying the form it takes in various
physical situations, and any perceived vagueness about ultimate definitions
in no way detracts from the central role that energy plays in all of the
sciences.  The analogy to music is relatively unforced.  Bernard Chasan

Professor Bernard Chasan
Physics Department, Boston University