Interesting stuff. One thing it has me thinking is this: in the OP, I had to try and condense Rosen's presentation by focusing heavily on the idea of a common measure as a way to pedagogically make clear what I/Rosen meant by commensurability. However, one thing Rosen emphasizes - and that I did not mention for economy's sake - is that the idea of commensuribility requires no reference to a common measure at all. That is, insofar as two values can be defined by reference to a common third value, its easy enough to actually get rid of the common value and define the two values in reference to nothing but each other. If A and B are our length, C is our common measure, and n and m are integers:
(1) A = mC, and B = nC, then we can cancel out C such that:
(2) A/B = m/n
(3) B = A(m/n)
So that now, length B is measured in terms of (rational) units of A. This is where the assumption of commensuribility becomes properly insidious because now there is no 'external referent' which acts as a 'real world' mediator between values. Once you define a length in terms of units of another length, what you end up with is a 'closed system' which becomes impossible to get out of (Rosen: "Once inside such a universe, however, we cannot get out again, because all the original external referents have presumably been pulled inside with us. The thesis [of commesurability] in effect assures us that we never
need to get outside again, that all referents have indeed been internalized in a purely syntactic form").
And in fact, this is the real upshot of commensurability: not simply the idea that there is a common measure between things, but that everything inside a formal system can be measured
in terms of other things within that system without remainder. And
this is what the 'disaster' is: the expulsion, if you will, of any external referent which would 'ground' the seeming self-sufficiency of such a self-enclosed system. On the flip side then, what
incommensurability is, ultimately, is the non-identity (or the non-coincidence) of what can be
measured with what can be
constructed (the irrationals again are exemplary: the discovery of the irrational forced us to expand our universe of number so as to make measurement and construction commensurate again, which, as I've tried to point out, simply caused problems further down the line).
So this is all not strictly about measurement
per se, and I have no beef whatsoever with the awesome innovations of measurement theory. Instead, it's about the relation between measurement and the things so measured, and an attempt to delineate the proper bounds of that relation; a spin, if you will, on the idea of a Kantian-inspired 'Critique of Pure Math', in which if measurement is left to it's own devices to see the world only in it's image, you end up with all sorts of transcendental illusions like Zeno's paradox and so on.