Incommensurability is the thesis that world-views which scientific practice has posed throughout the ages are fundamentally different, or not comparable. An example often used a comparison between Einsteinian, Newtonian, and Quantum physics. Newton stated that mass is an entity separate from energy. Einstein's physics posits that mass is a manifestation of energy -- a possible property for energy to take on. Quantum physics, contra both Newtonian and Einsteinian physics, posits (in its first form, at least) that causality is a probabilistic construct, instead of an infinitely deterministic construct.
Usually the incommensurabile thesis is defended by pointing out dramatic changes between scientific systems. Naturally there is wiggle room for what constitutes "dramatic". Generally I take this to mean that the ontological construct of science has changed. So, we have an atomic theory, for instance, and it would not change the ontological structure of chemistry to posit another atom, or another molecule, or even a new way of bonding. There exists atoms and bonds. However, were we to posit that the universe is not composed of atoms, but waves of energy and waves of energy only, and that the atoms we reference are tools in the same sense that a meter is a tool (whereas "length" would be the ontic unit of a meter), then we'd have an instance of incommensurability. " "The universe is atoms and only atoms" and "The universe is energy waves and only energy waves" "can not be true. You have to choose one or the other, or make an adjustment that allows for both.
In playing with the incommensurability thesis, I broke open my Aristotle and wrote the following thought experiment where I interpreted spectroscopic data using Aristotle's theories.
Spectroscopic analysis would be conceived of in an entirely different way within the Aristotelian framework. Nature abhors a void so there aren't any atoms, and... (From Book II of De Anima)
"...to explain what light is.
Now there clearly is something which is transparent, and by 'transparent' I mean what is visible, and yet not visible in itself, but rather owing its visibility to the colour of something else; of this character are air, water, and many solid bodies. Neither air nor water is transparent because it is air or water; they are transparent because each of them has contained in it a certain substance which is the same in both and is also found in the eternal body which constitutes the uppermost shell of the physical Cosmos. Of this substance light is the activity-the activity of what is transparent so far forth as it has in it the determinate power of becoming transparent; where this power is present, there is also the potentiality of the contrary, viz. darkness. Light is as it were the proper colour of what is transparent, and exists whenever the potentially transparent is excited to actuality by the influence of fire or something resembling 'the uppermost body'; for fire too contains something which is one and the same with the substance in question. "
So, the differences one can obtain from a spectroscope could be explained by this transparent substance in activity with different proportionate mixtures of the elements, which is something Aristotle references often in explaining why different things are what they are (I'm just taking a guess here. But I don't think it's fair to infer, using Aristotle's work in a scientific manner, that reference to quantum energy states modeled by operator algebra explains lines on a given spectroscopic measurement). But, even more importantly, this would be the mere material cause, reflecting a samples potentiality. The actuality could only be garnered from what that material would be used for. Suppose it is a medicine. The ratio of elements would be the potential within the substance, and the shape of the sample at the time of the spectroscope would be during its coming-to-be. For the end of medicine is getting well, and when it is used would be its actuality. (I'm pulling from ideas in Aristotle's Physics and Metaphysics as well, here)
In using Aristotle, while I can find a common referent, and I even think that modern theories are better with respect to truth-value, one can come to understand the incommensurability thesis best, I think. This is because science works by inferring to the best explanation within a certain explanatory framework, and inferences, contra arguments, are actions. It's in the use of science that one understands incommensurability best, and not the "logic" of science.