I got into an argument with my friend, which cast confusion on my understanding of density and its relationship to volume. I'm hoping to get some clarity. The argument involved describing density in terms of volume. Let's say you define a sphere in empty space. You choose a point, apply the formula for a sphere, and now you have a sphere. Not a sphere OF anything other than space, just a spacial, theoretical sphere. No particles, massless or otherwise (this is a thought experiment). What is the density of that sphere? Is it zero, or is it undefined? I know density can be defined as p = m/v. But in a theoretical sphere, which HAS volume, should we call mass zero, because there is none? Or is it undefined because a theoretical sphere really isn't related to mass at all? If it IS undefined, does that mean it makes no sense to relate density to volume because density is only a property of mass?
The answer, to me, seems to be that in fact it makes no sense to talk about the density of a massless object. Sorry if I answered my own question, but I would still like clarity. If someone could help guide me through the assumptions I'm making about reality and math and how they relate, I'd really appreciate it.