Structural Magnitudes?
The fundamental properties provide a minimal basis for all intrinsic qualities of things. That is, whenever two things are not perfect qualitative duplicates, they differ in the distribution of fundamental properties over their parts; whenever two things do not differ by that distribution, they are perfect qualitative duplicates. It follows that all fundamental properties are intrinsic. But not all intrinsic properties are fundamental: the fundamental properties provide a minimal basis for all qualitaties. Hence there is no fundamental property of having a mass of either 1g or 2g, because instantiation of that property is already determined by the distribution of mass 1g and mass 2g. For the same reason, there is no fundamental property of being the fusion of a round thing and a distinct rectangular thing. By and large, fundamental properties are never logically complex (like A or B) and never structural (determined by the distribution of properties over the parts of their instances).
Consider the mass and shape of this table here. They seem to be structural: the table's mass and shape are entirely determined by the mass and spatiotemporal arrangement of its parts.
This suggests a simple treatment of fundamental quantities, which I'll call "Armstrong's theory". On Armstrong's theory, the only fundamental quantities are basic unit quantities. There is, for instance, only one fundamental mass, instantiated by certain fundamental particles; all other mass properties are structural, determined by the distribution of the unit mass property over an object's parts. Likewise, there is only one fundamental length, or distance; whenever something is 2n long, that's because it has two distinct adjacent parts with length n.
(Merging Armstrong's theory with Lewis's theory of laws, one could arguably translate laws involving quantities into laws only mentioning the fundamental unit quantities, like: "whenever something x is composed of n distinct parts with unit mass, and something y is composed of m distinct parts with unit mass, and x and y are d unit lengths apart...".)
Unfortunately, Armstrong's theory seems to entail that particle physics in its current state is an impossible story about the fundamental nature of a world. If electrons have a mass of 0.0005 and certain quarks one of 0.003, at least those quarks must be composed of further particles according to Armstrong's theory. But isn't it at least conceivable that they are not so composed?
I don't believe this refutes the Armstrong theory, but it does make it look somewhat unattractive. Though it depends on the number of distinct magnitudes physics might assign to fundamental particles. If there were, say, 3 fundamental unit masses, Armstrong's theory could probably deal with that by treating them as 3 distinct fundamental properties related to each other by certain laws. (That each of the three masses is fundamental would mean that something could have two of them at the same time, as fundamental properties are freely recombinable. I think that's acceptable. The relevant possibility here is metaphysical, not epistemic. I gues it's analytic that nothing has two different masses at the same time; but this doesn't entail that it's necessary that nothing has two different masses at the same time.) Real trouble would arise only if there were infinitely many fundamental unit properties.
The only alternative I can see to Armstrong's theory is to accept as primitive that some properties, like the masses, come in ordered families. Brian nicely shows how this could work, by treating magnitudes as relations to numbers and taking those relations as fundamental. (It would be nicer, I think, if we could somehow subsume mass under a general theory of determinables that also works for properties like colours which are certainly not fundamental relations to numbers. But that's only an aesthetic objection.)
Anyway, here's a puzzle: Suppose there is a very heavy kind of fundamental particle with exactly the same mass as this table here. Isn't the mass of this table still completely determined by the distribution of properties over its parts? I would like to say that the table's mass is not fundamental, whereas the mass of the heavy particle is. But how can I say that if it's the very same mass?
Or consider the shape of this letter: i. I'd like to say that that's a structural property: the symbol has it in virtue of having a small round part and a thin rectangular part, suitably arranged. But presumably there are worlds with i-shaped mereological atoms (as there are temporally i-shaped atoms at worlds where endurantism is true). It seems that these atoms have exactly the same shape as the letter above. But then that shape property cannot be structural.
I'm inclined to say that in all such cases, the relevant properties (the table mass, the i shape) are disjunctions of a structural property and a non-structural property. Only the non-structural disjuncts are candidates for fundamental properties. This means that masses are not, in general, fundamental properties at all; only the masses of fundamental particles are.