Is length an intrinsic property or is it something which is only relative to other lengths? Is an inch an inch? Or is it simply a relation between other (length) phenomena?

Interesting questions. As I understand it, special relativity in physics says that having a particular length isn't intrinsic to an object, because observers in various "inertial frames of reference" can measure different values for the length of an object without any of them being mistaken: the length of an object is always relative to an inertial frame, and no inertial frame is objectively more correct than any other.

As for units of length such as an inch, I'm inclined to say that they're always relative to some physical standard, whether the standard is a single physical object such as a platinum bar or, instead, some physical phenomenon like the path traveled by light in a given period of time (with units of time also being physically defined). In a universe containing no physical standard that defines an inch, nothing has any length in inches even if things have lengths in (say) centimeters when a physical standard exists for the centimeter.

I hesitate a bit in holding this position, because it seems to be denied by the eminent philosopher Saul Kripke in his justly famous book Naming and Necessity. On page 55 of that book, Kripke (as I understand him) says that the meter is an abstract object that has (or is) the same length in all possible worlds, even worlds in which nothing physical exists at all, including any physical standards of measure. I confess I find that view very implausible, assuming I understand it.

It is indeed an interesting question, and in fact it's more than one question.

To begin with, my colleague is correct: in special relativity, length is like velocity in classical mechanics: it's a "frame-dependent" quantity. However, the theory of relativity is also a theory of absolutes; between any two points in space-time there is a quantity called the interval, and it is not frame-dependent. To put it in the jargon of relativity, the space-time of special relativity has a metric -- a generalized "distance function" -- and that metric delivers an unequivocal answer to the question of whether the interval between w and x is equal to the interval between y and z.

But now we have a new question: suppose that relativity says that the interval between w and x is, in fact, the same as the interval between y and z. What kind of fact is that? Suppose that the two intervals have no overlap. Doing business as usual, so to speak, we come up with the answer that the intervals are equal, but we could use a different metric function that gave a different answer, and by making adjustments elsewhere, we could make the physics work out. The physics that made the adjustments might be more unwieldy; it might involve some peculiar "universal forces," for example. However, it's not immediately clear that this shows the usual way of doing things to be ontologically privileged.

The debate we're now describing has to do with the "conventionality of the metric," and some heavyweight thinkers, not least the philosopher of science Hans Reichenbach, have argued that the metric is indeed conventional. That is, it depends on choices we make that could, in principle, have been made in a different way.

This issue has a connection to questions about meters and such. We pick out a meter by reference to the standard meter stick (or at least, that's how we used to do it.) But there's some complexity here. Suppose I take my own meter stick, lay it against the standard meter, and find that they match. While they're in contact, there's no doubt that they have the same length. But what about when they're not? Does my meter stick retain its "real" length when I move it around? (Leave issues of relativity aside for the moment. We could say things in a more complicated way that took them into account, but the issue would not really change.) Or does it contract or expand? Or is there really no absolute metaphysical fact of the matter? The conventionalists would pick this last option. When we consider the whole package of our physics and our measuring devices and our assumptions about forces, we may say that (absent unusual circumstances) the measuring rod has the same length when it's "here" as it does when it's "there." But the conventionalist would insist that saying this ultimately rests on certain stipulations or conventions.

The literature on this topic is complex, as you might imagine. Even though he argues for one side rather than the other, I'd suggest that Reichenbach's Philosophy of Space and Time is a good place to start. It's an engaging book that's more accessible than it might appear at first sight.

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