Scientific realism about mass

While I’ve grown up as a scientific realist, and been trained as one as a philosophy graduate student, and I suppose I still identify as one, I’ve been finding it more difficult to say what scientific realism claims.

For instance, what does it mean to be a realist about mass in a Newtonian context? A naive thought is that for each physical object, there is a positive real number, the mass of the object, which mathematically enters into the laws of nature such as F = ma and F = Gm₁m₂/r². But that seems to commit one to there being some odd objective facts, such as to which objects have the property that the square of their masses is less than their mass—a property that barely seems to make any sense, since normally in physics, we don’t compare masses with squares of masses, as they are measured in different units.

A more sophisticated thought is that there is a determinable mass, and a family of determinates, with various mathematical relations between them, with the family isomorphic with the positive real numbers with respect to the relations, but without necessarily a single isomorphism being privileged. But this more sophisticated thought is much more philosophy than physics: physicists hypothesize entities like forces and particles and the like, but not such entities like determinables and determinates. Indeed, this approach commits one to the denial of nominalism, and surely realism about mass in a Newtonian context shouldn’t commit one to such a controversial metaphysical thesis.

Is there some alternative? Maybe, but I don’t know.