by Rishidev Chaudhuri
Like the rest of us poor mortals, wandering in constant confusion between things and the names for things, bewitched by language and unable to resist it, mathematicians and physicists are constantly struggling with their representations and yet entirely reliant upon them to grasp the world.
Many of the fundamental intuitions that we start to describe the world with are geometric or spatial: this is a point; this is another point; walk in this direction to get from the first point to the second; this is the path a particle takes. If we want to make this precise, to describe and classify and manipulate and compute, we need to be able to make these statements precise. The simple act of drawing a pair of coordinate axes on a flat surface and using pairs of numbers to describe points is extraordinarily powerful, yoking algebra and symbolic manipulation to geometry and spatial intuition, and it unlocks for us a language within which to watch spatial and temporal processes unfold. Similarly, describing points on the surface of the Earth by pairs of numbers (latitude and longitude are the most common) allows us to specify locations relative to other locations, to calculate distances and trajectories and to describe and communicate quantities that vary across the surface of the Earth, like weather patterns and temperatures.
But in picking a particular representation we've done a certain violence to the geometric structure we started with, by forcing an arbitrary layer of description on top. We might have decided to describe points relative to axes at right angles, like so:
But we could equally well have rotated the axes, or shifted the center, or chosen axes that were at other angles, like so:
Similarly, the standard way to describe points on the surface of the Earth is by their distance from the equator (i.e. latitude) and their distance from a line perpendicular to the equator and passing through Greenwich (longitude), but I could choose to describe places by how far away they are from my house and in which direction relative to some local landmark. And this is how we generally give directions locally.
And so, now that we've introduced a way of describing space, we have to be careful that we don't get led astray by our representations, and that we keep separate the convenient descriptors that we use and the spatial and physical quantities that we're trying to describe. Depending on our system of representation, the particular coordinates attached to London and New York might vary dramatically. But our calculation of the physical distance between them shouldn't depend on how we've chosen to represent them.
Physicists and mathematicians have developed a lot of theory to derive and explain which quantities are physically meaningful (e.g. the distance between London and New York) and which quantities are simply consequences of the particular representation that we have chosen (e.g. the longitude of New York). This is often not trivial. For example, as Einstein famously found, the distance in space between events will be calculated differently by observers moving at different velocities (a form of coordinate dependence), but there is a quantity called the interval that combines the distance and time between events that all observers can agree on.
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