Coordinate invariance: a manifesto

What is a coordinate system? It is a way of assigning numbers to any point in a given region. The point of a coordinate system is that it allows us to quantitatively describe what is happening to an extended object (at each point of the object). [TODO that second sentence under-sells the value of coordinate systems.]

Traditionally, a coordinate system is conceived of as a map $R\rightarrow \mathbb {R} ^{n}$ from the region $R$, to tuples of $n$ real numbers. However, since we only ever have finite precision, this is not quite the right way of thinking about things. Indeed, consider a real-life coordinate system, such as the latitude and longitude coordinates on the surface of the Earth. When we want to specify a latitude and longitude of some location, we don't give a pair of real numbers, we give a pair of intervals. We might say that this location is $40.000\pm 0.001$ degrees north, and $12.000\pm 0.001$ degrees east, but we would never (nor should ever) say that this location is precisely 40 degrees north and 12 degrees east. So we see that there is a mismatch between what standard mathematics means by "coordinate systems," and the way that things are actually measured in real life.

Shea Levy would argue, basically, that you can have this infinite extra precision and only ever use the finite part. That is largely true, but I regard the infinite precision as unnecessary conceptual clutter.

Standard mathematics does have a concept that I regard as being more similar to the real-life version of coordinate systems, and that is: finite cell complexes which triangulate a space. A finite cell complex is basically just a collection of intervals.

Whatever the laws of physics are, they ought to be independent of any arbitrary choices that we humans make when measuring. In particular, since a coordinate system is just an arbitrary human choice, the laws of physics must be independent, in some appropriate sense, of our choices of coordinate system. It is precisely this consideration that led Einstein to the Theory of General Relativity.

General Relativity is what we get when we search for a coordinate-invariant theory while using the traditional notion of a coordinate system. But what would we get when looking for a coordinate-invariant theory while using the "finite triangulation" notion of a coordinate system? What sorts of equations could be independent of, or change covariantly with, a choice of finite triangulation?