Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act II
4. On Circles and Giving Up
4.5. What Is Meaningful
4.5.1Coordinates Are Invented Only to Be Ignored
In mathematics courses, one often hears the strange term “coordinate system,” as well as related terms like “Cartesian coordinates” and “polar coordinates.” It’s worth pausing to ask ourselves a question: what on earth are coordinates? Coordinates are often used to talk about a two-dimensional plane, three-dimensional space, and so on, but that doesn’t make it clear what they are or why we need them. This morning, for example, I spent a lot of time walking around inside three-dimensional space, and I never saw a single “coordinate.” However, if I had tried to calculate, say, the distance from my apartment to the hospital, or the angle from my desk to that bird outside who keeps making mating calls in response to someone’s car alarm, then I would quickly find that there are two ways to perform these tasks. First, there’s the qualitative way. A qualitative answer to the question “What’s the current distance between Reader and myself?” would be something like “far” or “pretty close.” However, we might want a more precise answer. How do we make answers more precise? Well, one way would be to say
Reader is (very)n far away from my apartment.
where (very)n stands for the word “very” n times in a row, and n is the number of steps it would take to walk from one location to the other. That would be more precise, but it’s also a tremendous pain. Needless to say, no human language works like that.
Another way to describe geometrical things like distances and angles more precisely is to use numbers. We’ve already done this plenty of times. Using coordinates and numbers to describe geometrical things is a process we’re all familiar with, but this process of assigning coordinates can become so second nature that it’s easy to miss an essential feature of it: the coordinates are not the geometry. Space doesn’t know anything about coordinates. Sometimes, though, we would like to have the extra structure provided by numbers — we would like to be able to add lengths, figure out shortcut distances, and so on — so we imagine assigning numbers (coordinates) to each point of space. However, to do this, we have to make arbitrary choices that are not inherent in the geometry. Once we draw two perpendicular directions on a piece of paper (a two-dimensional “coordinate system”), we immediately gain access to all the artillery of numerical calculations, which we can bring to bear in talking about anything in our two-dimensional space. However! In drawing one coordinate system rather than another one at, say, a slightly different angle, we have made an arbitrary choice. We have introduced more structure than we intended to talk about, so we must then immediately undo this extra structure by declaring the only meaningful quantities to be those that would have been the same under a different choice of this geometrically meaningless structure (i.e., under a different choice of coordinate system).
Therefore, it might appear that coordinates are useless. They happen to be extremely useful, but because of the above considerations, coordinates often find themselves in a very strange position: we invent them, use them, and then pretend they didn’t exist. Coordinates do all the work in our calculations, and get none of the credit.
4.5.2Coordinates and Meaning in Mathematics
I just claimed that space doesn’t know anything about the coordinates we use to describe it. While that’s true of the physical world, it’s not entirely true of mathematics. In mathematics, we’re free to say that coordinates have as much meaning as we want them to have. How we make this decision determines which other quantities will end up being “meaningful.”
For example, we can choose to study a two-dimensional universe where no direction or point is any more special than any other. This is what mathematicians sometimes call an “affine plane.” Since no point or direction is special, the meaningful things in this universe cannot depend on particular points or directions. This world has no “x axis,” “y axis,” or “origin,” so if we use these concepts to aid us in calculations, we must make sure our results do not somehow depend on the particular arbitrary coordinate system we chose.
We could then enrich our universe by choosing to single out one special point, the “origin,” but not single out any special direction. In this second universe, the distance of any particular point from the origin is meaningful, whereas it wasn’t meaningful in the first. However, the angle between any particular point and the positive x axis (should we choose to draw one) is still not a meaningful quantity, because in this second universe, we have declared that no particular direction is special. The axes themselves are simply something we grafted onto our second universe in order to help ourselves calculate things.
Finally, we could choose to move to a third universe, in which one point is special, and one direction is special. We can call the direction “up.” This is effectively the space we’re playing with when we “graph” a function in two dimensions. We’re not just studying a structure-free plane or a world with one special point; we’re studying a plane with a meaningful concept of “up” as well as an origin. In this world, we typically use the distance away from the “x axis,” as measured in the “up” direction, to talk about what a given machine spits out when we feed it a particular number. This universe is where angles as measured from our axes become meaningful for the first time.