Today’s A to Z term comes from Reynardo, @Reynardo_red on Twitter, and is a challenge. And the other A To Z posts for this year should be at this link.
Some terms are hard to discuss. This is among them. Mathematicians find commutative things early on. Addition of whole numbers. Addition of real numbers. Multiplication of whole numbers. Multiplication of real numbers. Multiplication of complex-valued numbers. It’s easy to think of this commuting as just having liberty to swap the order of things. And it’s easy to think of commuting as “two things you can do in either order”. It inspires physical examples like rotating a dial, clockwise or counterclockwise, however much you like. Or outside the things that seem obviously mathematical. Add milk and then cereal to the bowl, or cereal and then milk. As long as you don’t overfill the bowl, there’s not an important different. Per Wikipedia, if you’re putting one sock on each foot, it doesn’t matter which foot gets a sock first.
When something is this accessible, and this universal, it gets hard to talk about. It threatens to be invisible. It was hard to say much interesting about the still air in a closed room, at least before there was a chemistry that could tell it wasn’t a homogenous invisible something, and before there was a statistical mechanics that it was doing something even when it was doing nothing.
But commutativity is different. It’s easy to think of mathematics that doesn’t commute. Subtraction doesn’t, for all that it’s as familiar as addition. And despite that we try, in high school algebra, to fuse it into addition. Division doesn’t either, for all that we try to think of it as multiplication. Rotating things in three dimensions doesn’t commute. Nor does multiplying quaternions, which are a kind of number still. (I’m double-dipping here. You can use quaternions to represent three-dimensional rotations, and vice-versa. So they aren’t quite different examples, even though you can use quaternions to do things unrelated to rotations.) Clothing is a mass of things that can and can’t be put on first.
We talk about commuting as if it’s something in (or not in) the operations we do. Adding. Rotating. Walking in some direction. But it’s not entirely in that. Consider walking directions. From an intersection in the city, walk north to the first intersection you encounter. And walk east to the first intersection you encounter. Does it matter whether you walk north first and then east, or east first and then north? In some cases, no; famously, in Midtown Manhattan there’s no difference. At least if we pretend Broadway doesn’t exist.
Also if we don’t start from near the edge of the island, or near Central Park. An operation, even something familiar like addition, is a function. Its domain is an ordered pair. Each thing in the pair is from the set of whatever might be added together. (Or multiplied, or whatever the name of the operation is.) The operation commutes if the order of the pair doesn’t matter. It’s easy to find sets and operations that won’t commute. I suppose it’s for the same reason it’s easier to find rectangular rather than square things. We’re so used to working with operations like multiplication that we forget that multiplication needs things to multiply.
Whether a thing commutes turns up often in group theory. This shouldn’t surprise. Group theory studies how arithmetic works. A “group”, which is a set of things with an operation like multiplication on it, might or might not commute. A “ring”, which has a set of things and two operations, has some commutativity built into it. One ring operation is something like addition. That commutes, or else you don’t have a ring. The other operation is something like multiplication. That might or might not commute. It depends what you need for your problem. A ring with commuting multiplication, plus some other stuff, can reach the heights of being a “field”. Fields are neat. They look a lot like the real numbers, but they can be all weird, too.
But even in a group, that doesn’t have to have a commuting multiplication, we can tease out commutativity. There is a thing named the “commutator”, which is this particular way of multiplying elements together. You can use it to split the original group in the way that odds and evens split the whole numbers. That splitting is based on the same multiplication as the original group. But its domain is now classes based on elements of the original group. What’s created, the “commutator subgroup”, is commutative. We can find a thing, based on what we are interested in, which offers commutativity right nearby.
It reaches further. In analysis, it can be useful to think of functions as “mappings”. We describe this as though a function took a domain and transformed it into a range. We can compose these functions together: take the range from one function and use it as the domain for another. Sometimes these chains of functions will commute. We can get from the original set to the final set by several paths. This can produce fascinating and beautiful proofs that look as if you just drew a lattice-work. The MathWorld page on “Commutative Diagram” has some examples of this, and I recommend just looking at the pictures. Appreciate their aesthetic, particularly the ones immediately after the sentence about “Commutative diagrams are usually composed by commutative triangles and commutative squares”.
Whether these mappings commute can have meaning. This takes us, maybe inevitably, to quantum mechanics. Mathematically, this represents systems as either a wave function or a matrix, whichever is more convenient. We can use this to find the distribution of positions or momentums or energies or anything else we would like to know. Distributions are as much as we can hope for from quantum mechanics. We can say what (eg) the position of something is most likely to be but not what it is. That’s all right.
The mathematics of finding these distributions is just applying an operator, taking a mapping, on this wave function or this matrix. Some pairs of these operators commute, like the ones that let us find momentum and find kinetic energy. Some do not, like those to find position and angular momentum.
We can describe how much two operators do or don’t commute. This is through a thing called the “commutator”. Its form looks almost playfully simple. Call the operators ‘f’ and ‘g’. And that by ‘fg’ we mean, “do g, then do f”. (This seems awkward. But if you think of ‘fg’ as ‘f(g(x))’, where ‘x’ is just something in the domain of g, then this seems less awkward.) The commutator of ‘f’ and ‘g’ is then whatever ‘fg – gf’ is. If it’s always zero, then ‘f’ and ‘g’ commute. If it’s ever not zero, then they don’t.
This is easy to understand physically. Imagine starting from a point on the surface of the earth. Travel south one mile and then west one mile. You are at a different spot than you would be, had you instead travelled west one mile and then south one mile. How different? That’s the commutator. It’s obviously zero, for just multiplying some regular old numbers together. It’s sometimes zero, for these paths on the Earth’s surface. It’s never zero, for finding-the-position and finding-the-angular-momentum. The amount by which that’s never zero we can see as the famous Uncertainty Principle, the limits of what kinds of information we can know about the world.
Still, it is a hard subject to describe. Things which commute are so familiar that it takes work to imagine them not commuting. (How could three times four equal anything but four times three?) Things which do not commute either obviously shouldn’t (add hot water to the instant oatmeal, and eat it), or are unfamiliar enough people need to stop and think about them. (Rotating something in one direction and then another, in three dimensions, generally doesn’t commute. But I wouldn’t fault you for testing this out with a couple objects on hand before being sure about it.) But it can be noticed, once you know to explore.