Life Lessons from the Mathematical Notion of Ordering

It is a well-known exercise to show that the field of complex numbers cannot be made into an ordered field. It’s a fun little exercise that showcases one of the unique abilities in math of being able to prove that certain things can’t exist. But this exercise doesn’t get at the whole truth; though the complex numbers don’t have a linear (or total) ordering that’s compatible with its algebraic structure, it does have plenty of partial orderings that do. This is not just a curiosity, either. The partial ordering of the complex numbers obtained by saying z \leq w if w - z is real and positive is quite natural and used quite often, even when it’s not outright stated.

In fact, the further I got into my mathematical studies, the more pervasive partial orders were in comparison to linear orders. In my area, C*-algebras have partial orders and partially ordered abelian groups are important to their study with nary a linear order in sight. They are so pervasive in this area that one story goes that during a lecture, when the lecturer used the phrase “partially ordered group”, a well-known mathematician piped in with a “What do you mean ‘partially ordered group’? You mean it’s not ‘linearly ordered’? That’s called an ‘ordered group’.” Quite the contrast from the exercise in the beginning that presumed that an ordered field meant a linear order!

Since at this point, I’m probably losing most of my non-mathematician readers, let’s have a brief introduction. An ordering is the notion of “less than” or “greater than” between objects. You are probably familiar with the order on the real numbers, but that order has a special property: every two numbers are comparable. In other words, for any two numbers x,y, either: x \leq y or y\leq x, and the only time both statements are true is when x and y are the same number. This is what we’ll call a linear order, just as the real numbers fall into a line based on their order. But there are orderings where not every two objects are comparable.

To take an example, among the movies you watched, you have a preference ordering, where you one movie is “greater than” another if you like it more. I dare say that not every two movies are comparable. I know that I both like Star Wars and 2001: A Space Odyssey, but I don’t like one better than the other. But there are comparisons that can be made; Star Wars: A New Hope is better than Star Wars: A Phantom Menace.

Starting from childhood, I had an aversion to the concept of “favorites”. To be sociable, I would say something, but it always felt dishonest. I didn’t understand what it meant to have a “favorite” food, color, movie, TV show, or whatever. It wasn’t until I learned Zorn’s lemma that I could articulate what my problem was with the concept. When learning Zorn’s lemma, two distinct terms are defined for an ordering: a greatest element is an object that is greater than every other object and a maximal element is an object where no other object is greater than it. When the ordering is linear, these notions are the same. But when you allow incomparability, they’re different ideas. Notice that a greatest element is a Highlander notion, there can only be one. The concept is dominating and competitive. On the other hand, multiple maximal elements can coexist with each other. It’s a more peaceful notion. So it became clear at last: my preferences for movies is not linear, and while there are many maximal movies (which I chose among as my “favorite”), there is no greatest movie.

But it doesn’t stop at preferences, and I’m certainly not unique in noticing this distinction. In many math books now, there are chapter flow charts with which chapters you need for a given chapter. The reader can chart a path through a book that doesn’t follow a linear order. To a lesser extent, this is precisely what footnotes are; when the main text of the book is linearly ordered, but every now and then there are partially ordered footnotes that split off from the linear text. While time flows in a linear fashion, ideas often do not. From one idea, many arise. A concept that is illustrated with the hyperlinked web.

Before I decide to chase the consequences of this idea down every rabbit hole, I ought to finish this blog post. To sum up, there’s a life lesson to gained from studying order structures: there are more partial orderings than linear orderings; and maximal elements abound, while greatest elements are scarce.


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