Nothing is more fundamental to physics than measurement. The result of any measurement is a number, a number that is usually referred to some system of units.
The Lumonator, following M J Duff and Jorge Majueijo has lately posted on the status of fundamental constants. One of the most important confirmations of Einstein's special relativity theory was the discovery that measurements of the speed of light yielded the same result independent of the movement of the measurer  in contrast to say, measurements of the speed of water or sound waves. The discovery of a universal relation between energy and frequency of light led to another constant of nature, Planck's constant. Some have speculated that these constants might actually be varying with time.
What caught Motl's attention was Duff's statement that:
The possible time variation of dimensionless fundamental constants of nature, such as the finestructure constant , is a legitimate subject of physical enquiry. By contrast, the time variation of dimensional constants, such as ¯h, c, G, e, k. . . , which are merely human constructs whose number and values differ from one choice of units to the next, has no operational meaning.
Duff caught a lot of guff from the Nature reviewers and Paul Davies for this comment, and Lubos jumps to the defense  not that Duff needs much help  see his answers to his critics at the link, following the paper. I myself was more interested in Duff's quote of the following by Dirac:
The fundamental constants of physics, such as c the velocity of light, h the Planck constant, e the charge and me the mass of the electron, and so on, provide for us a set of absolute units for measurement of distance, time, mass, etc. There are, however, more of these constants than are necessary for this purpose, with the result that certain dimensionless numbers can be constructed from them.
The most famous of these is the fine structure constant but there are more. One of the annoyances of the standard model of particle physics is that it seems to require some 26 of them. It's important to notice that theory is an essential ingredient of any interesting dimensionless constant. It's theory that assigns a universal significance to a dimensionless number.
It's worth noting that any measurement is essentially the determination of a dimensionless number. When you take out your ruler to measure the length of a screw, for example, you are actually determining the ratio of the length of that screw to some standard  helped by the fact that your ruler has numerical values of those ratios conveniently printed on the side. Our first standards of time were based on astronomical phenomena, and it's plausible that some of our first astronomical ideas came from noting the dimensionless ratios between the periods of the day, month, and year.
From our present standpoint, those particular dimensionless numbers are accidental  a consequence of our peculiar history in our particular solar system. From a deeper standpoint we may hope that the same thing can eventually be said about all those dimensionless ratios in the standard model  either that, or that their values will be manifestations of some more fundamental dimensionless ratios.
Let's return briefly to Duff's point: suppose we were in communication with some isolated aliens, with whom we could exchange data, but nothing tied to the physical. Could we usefully exchange information with them about how to measure the speed of light? Not really, because our measurements depend either upon artifacts (like the meter and kilogram of some decades back) or upon other things assumed constant, like the electric charge and Planck's constant. Dimensionless numbers like the fine structure constant or the ratio of electron to muon mass are different. If they reported different values, we could be quite sure that their physics was different from ours.
If their physics was really different, of course our dimensionless ratios might not be theirs. We can imagine a universe in which the lepton spectrum is quite different, or where supersymmetry is exact.
But I started by talking about constancy of the speed of light and Planck's constant  what's up with that? Why doesn't Duff's point apply to those measurements? Because those measurements are referred to actual common artifacts in our present world  they are assumed to be dimensionless ratios between the measured phenomena and and the appropriate artifacts (clock, meter stick, the arms of Michelson's interferometer). In a larger sense, it does apply however. In making our measurement we assume that our standards can be moved about our used in different circumstances without "changing." The ultimate justification for that assumption is that it leads to a simple physics.
As Poincare emphasized a hundred years ago, most of the conventional assumptions of physics  the geometry of space, for example  are ultimately conventional choices made to simplify our physics. At the time he was trying to rationalize Euclidean geometry and relativity, but Minkowski and Einstein soon found a better way that produced a deeper simplification.
