The metre is perhaps not the best possible example; it's no longer defined in reference to a physical object, but rather as "the distance travelled by light in absolute vacuum in 1/299,792,458 of a second." [Wikipedia] and is now purely definitional. Indeed, scientists established the new definition precisely to overcome philosophical challenges such as those Law describes.
Assuming the previous Standard Metre as "the length between two marks on a platinum-iridium bar [in Paris]," Law attempts to answer Wittgenstein's and Kripke's questions: Is the metre by definition the length of the Standard Metre? Or is it that the Standard Metre just so happens to be exactly one "true" metre? In philosophical jargon, is the metre an essential, necessary or analytical property of the Standard Metre, or is it an accidental property of the Standard Metre that it represents exactly one "true" metre?
Law gets the answer right, but his heavy use of philosophical jargon and his appeals to possible worlds and high-level intuitions about length undermine the clarity of his presentation, at least to a lay audience. (Law does not deserve censure for this lack of clarity to the lay audience. He's a professional philosopher writing for other professional philosophers; for all I know he's crystal clear to his intended audience.) We lay people can better understand the concept of measurement by appealing only to this actual world and lower-level intuitions, specifically our intuitions about space and, most importantly, our intuitions about consistency and comparability.
Consistent comparability over space and time is the essence of measurement. I want to predict today whether tomorrow the couch at the store will fit through the door at my home: I want to know that the couch is smaller than the door, and will stay smaller at a different time and place.
A concept of measurement and specifically the definition of metre will be good if I can use the concept reliably make such predictions; a concept of measurement will be bad if my predictions often contradict my actual experience. It is, of course, logically possible that no system of measurement could reliably predict my experience, that all systems of measurement are bad. Happily, we can discover that some systems of measurement are indeed good.
We can start with a very simple form of measurement: Take some "rulers" i.e. objects such as wooden or metal sticks, strips of cotton cloth (which we wash in hot water after each use) or rubber bands, hold each up to various "things", i.e. other objects, and remember the result. We move the things around and measure them repeatedly with one ruler, and see whether repeated measurements match our memory. If the measurement of a thing always matches our memory, we put that thing in the "R" pile; if the measurements differ from our memory, we put them in the "E" pile.
Based on this experiment, we make some curious observations. When we use some kinds of material for our ruler, such as wood or metal, we end up with a bunch of things in the "R" pile, and a bunch of things in the "E" pile. Furthermore, some things always end up in the "R" pile, no matter which of these rulers we use. When we use other kinds of material for our ruler, such as rubber bands or strips of cloth, most things end up in the "E" pile, and there's no consistency about what ends up in the "R" pile. Based on these outcomes, we'll now separate the rulers themselves into an "R" pile and an "E" pile. We can observe that when we measure one ruler with another
"R" ruler, "R" rulers tend to be "R" things, and "E" rulers tend to be "E" things. (Additionally, The appearance of "E" and "R" rulers tend to match the appearance of "E" and "R" things respectively: Bonus!)
I've used letters to designate these piles to highlight the arbitrary, a posteriori nature of the classification, but we now have enough information to coin a couple of words: we'll label rulers and things that usually end up in the "R" pile as "rigid", and rulers and things that usually end up in the "E" pile as "elastic".
We can also form a scientific theory: Objects have an abstract property, which we'll arbitrarily call "length". Rigid objects have a consistent length; elastic objects do not have consistent length. The abstract concept of "length" is itself invariant (i.e. universally consistent): It's not "length" that changes, it is the elastic objects themselves that change length.
We can make our measurements more precise, either by using shorter rulers or by using geometric techniques to make consistent subdivisions of a ruler. But as our measurements become more precise, we notice a curious thing: We end up putting more things and rulers in the "elastic" pile. At a certain degree of precision, everything goes in the elastic pile, both rulers and things!
Something has to give: Should it be our invariant abstract notion of length? Fortunately, we have another observable consistency to rescue us: We can count how many shorter rulers (or how many subdivisions of a longer ruler) it takes before we classify an object as "elastic". Objects will consistently require more or fewer shorter rulers to show elasticity. From this consistency, we can save the notion of invariant abstract length, and attribute all changes to changes of the objects themselves.
But we've begged a metaphysical question here: What if length itself is not invariant, but variable, even widely variable? What if a true "metre" (in the abstract) was yesterday about 1/10,000,000 of the distance from the equator to the north pole through Paris, and tomorrow about 10,000,000 times the distance? Or what if objects vary wildly in length, but they all do so "in sync", with our rulers ? What if yesterday the universe was only a "true" metre across, but today is 10^1000 meters across, with everything shrinking or expanding proportionally? (It's easier to imagine variation across time; it's the same, but harder to imagine, to have variation across space.)
What's important, though, is that, so long as the variation remains uniform, so long as all the true metres or all the objects vary proportionally, the consistencies we've observed will still be consistent. Therefore, we can conclude that "length" or "metre" does not actually name a thing—its grammatical use as a noun notwithstanding—but rather an invariant relationship: Things that I measure the same are the same length.
Both Wittgenstein and Kripke (at least as Law presents their ideas) are wrong in a sense. The marks on the platinum-iridium bar in France does not have the property of being one metre apart at all, neither essentially nor accidentally. Rather, the marks bear an observably stable relationship to the objects in the world. So long as the couch always fits through the door—i.e. so long as they maintain their relationship—I don't care if the door and the couch are "really" one metre wide when I move in and one million metres wide when I move out.
Law's objections, seen in the light of an invariant relationship, become unproblematic. What if, as Law suggests, some technologically sophisticated vandal were to undetectably change the Standard Metre by adding or removing material between the marks? We would quickly notice an inconsistency between measurements performed with instruments calibrated against the Standard Metre before and after the vandalism. The ordinary objects in the world would still maintain invariant relationships to everything but our Standard Metre.
This problem involves a common philosophical technique of assuming a truth which is not knowable (I consider this technique a solecism: It assumes we know something that we are, by definition, unable to know). Our actual experience would be that we suddenly noticed that everything in the world had apparently changed length, while being unaware (at least directly) of the reason for the changed length. Suddenly, everything in the world that we used to measure as 1 metre we now measure as 1.1 metres; everything that was 2 metres is 2.2 meters; everything that was 1.23456x10^6 kilometres is now 1.35802x10^6 kilometres. In short, we can observe that everything in the world has retained its relationship to everything else but the Standard Metre.
We're in a bit of epistemic bind: We don't know for sure whether the "true" metre has changed, the Standard Metre has shrunk, or the entire universe has expanded. But we really don't care that much. The simplest explanation is that the actual bar has shrunk, and no longer represents a Standard Metre. But that conclusion is more a matter of convenience than of discernible truth: It's easier to lengthen our bar by 10% than to rewrite all our old measurements. In Law's terminology, we just switch the definition of "metre" from the length of the Standard Metre to the length of the majority (or mean or median) of the things we used to measure as 1 metre according to the Standard Metre definition. It would have the same net effect (although a lot more work) to re-measure everything using the new Standard Metre. At a metaphysical level, we're agnostic and entirely apathetic about what the "right" definition is: We don't know for certain—nor do we care—which definition has "really" varied, we just want to maintain the universal invariance of the relationships.
Law cleverly introduces the concept of "smedlium":
Imagine a world quite similar to our own that contains large quantities of a metal-like material – let's call it smedlium – which gradually and unpredictably alters in size. All smedlium objects expand and contract in unison. At one o'clock on one particular day all the smedlium objects are 5% larger than they were at mid-day; at two o'clock they are all 10% smaller, and so on. Despite this peculiarity, smedlium remains a useful material. In fact, it is the strongest and most durable material available. One of the inhabitants of this world builds machinery made wholly out of smedlium. The machines are used in situations where their size relative to non-smedlium objects doesn't matter. The smedlium engineer constructs and calibrates a measuring rule made out of smedlium to use when manufacturing such machines. She measures dimensions in ‘S’s, one S being measured against the length of her smedlium measure. Of course, so far as manufacturing smedlium machines is concerned, a smedlium measure is far more useful than is a rule made out of some more stable material, for it allows the smedlium engineer to ignore the changes in size of the object upon which she is working. For example, she knows that, say, if the hole for the grunge lever measured 0.5 S in diameter at one o'clock, then a grunge lever which measures 0.5 S in diameter at two o'clock will just fit into that hole, despite the fact that the hole is now noticeably smaller than it was at one o'clock.Law uses this case to good effect in his own argument, but it's not particularly helpful to the construction of measurements as relationships.
We can tell—using the exact same techniques as we used to distinguish between ordinary rigidity and elasticity—that smedlium things are elastic relative to ordinarily rigid (e.g. steel) rulers and rigid relative to smedlium rulers. (Likewise, smedlium rulers are elastic relative to ordinarily rigid things, and rigid relative to smedlium things.) Although we have to make our concepts of rigidity and elasticity a bit more abstract and relative, there's no fundamental change to the definition. Likewise, as Law notes, changing the bar which represents the Standard Smedlium-Metre has exactly the same effects—and demands the same response—as changing the bar that represents the Standard (Ordinary) Metre. We can even preserve the invariance of our concept of length: Length in the abstract remains constant, and it is the smedlium objects (or the ordinary objects; we're agnostic and unconcerned with the difference) whose length changes.
As I mentioned at the beginning of this essay, the definition of the metre is now specified in terms of the speed of light. Light has the counterintuitive (but very helpful) property of moving in a vacuum at the same speed not only always and everywhere, but also in every reference frame, regardless (I'm sure) of its uniform motion and (I'm pretty sure) its accelerated motion. Even better, as Michelson and Morley showed, we can detect this property of the speed of light without reference to any other standard metric. Because of these properties, we can usefully define the speed of light in a pure vacuum to be absolutely rigid, and measure everything else relative to that metric. The speed of light can thus serve as a truly analytic (but not necessary over all possible worlds or a "rigid designator" per Kripke) definition of length.