I have wondered, in some environmental or geomorphic courses, whether the drive to quantify wasn't slightly misplaced; whether it would be possible, and more reliable if less precise, to merely rank qualities without trying to rank them on a metric line. In this sense 'metric' doesn't mean 'meters, centimeters...', but any system in which there are distances that you can add up, and divide in half, etc., as one can meters or feet. There are plenty of systems, called generally 'topology', that describe complicated things/worlds that aren't metric; in which you can say that two things are near but not how near they are. (I gave up on the book Small Pieces Loosely Joined after way, way too much falling-about astonishment that the common users' Internet has more of a topology than a metric, without any sign that we were going to be introduced to the idea of a topology -- or graph, any name will do -- and the many things known about them.)
Then I had the mild shadow of a realization that if you have orders, you might have partial orders; for instance, you and your parents and your grandparents are in 'parent-child' orders with each other, but your maternal and paternal grandparents almost certainly can't be so ordered; none of them are parent or child to the other three. If you go back far enough, your tree of descent certainly does cross itself, though; very possibly with 'legs' of different numbers of generations; even this commonplace example doesn't nail down descriptions right away. Also, mm, ecosystems, they have both spatial and developmental nesting, partially ordered sets (posets) might arise very naturally.
So they do, and here are some examples. The book starts with an even lovelier natural source; you can't rank molecules into a single order of development, but there are obvious ways to put them into directed graphs with partial orders. If A is a sub-molecule of B and C, perhaps because B and C have different side-chains, that's the beginning of an order; and this happens rather a lot with organic molecules, since organisms would naturally rather not build everything from scratch. There might be a molecule D that has the side-chains of both B and C; or that might be impossible; so the 'family tree' of molecule A has a range of possible shapes.
There are metrics on some traits of molecules, e.g. their boiling-points, and there are some mappings from posets to metrics. (Much of this book is thinking about toxicology, for which it's handy to be able to guess from molecular data what the organism or system effects might be.)
A later chapter (,,) is describing biodiversity by posets> They start by pointing out that political contention 'need's a single ordering, and the bulk of the chapter is describing habitat diversity in a part of Pennsylvania. Handy, if you're trying to set policy for watershed protection.
I'm dubious that political matters naturally need a single ordering, though. Market contention surely does -- in fact, would like to translate everything into the single metric of price or price-of-tort. This seems to me to be one of the things we need political systems to avoid; we have alternate rankings, of things you cannot alienate, things you can give away but not sell or buy, things you can sell or buy only if some other characteristic is in play. In a partially-ordered mindset, you need not say that one of the characteristics is more protected than another, even if both are protected with regard to a third. It's hard to live up to this in reality, when we have limited resources and generally end up trying to minimize harm (for which we do need to rank the harms).
For that matter, the political process leads us to try and rank goods; the Democratic primary just concluded was rrrrather an example of that; every political platform with realistic goals must be. Somewhere along here we must hit Arrow's Impossibility Theorem; I am being very lazy about this post, because I am mostly trying to get a lingering urge to procrastinate off my desk, but does the AIT arise from a poset-ness in social goods?
Find in a Library: Partial Order in Environmental Sciences and Chemistry
So wrote clew in Science.