I've been working to understand the geometrical objects and intuition associated with the algebra which is used in the abstract theory which I've studied so far (topology, calculus, linear algebra) and I'm stumbling on the elementary distinction between points and vectors.
As far as I'm, points are just general elements of a set (maybe with some minimal structure such as a topology) whereas vectors can be added and scaled. So far so good. But then I don't see how we can talk about coordinates of points of a space. Take $\mathbb{R}^n$ for example. Seen as a general space (say a manifold), how does it make sense to represent the space in the standard way using coordinate axes etc.? Surely to write ${\bf{p}} = (2,3,5)$ means that I can get to ${\bf{p}}$ by starting from the origin and taking the prescribed step in the 3 directions, right?
If you reject this position vector interpretation of points of $\mathbb{R}^n$, as seems to be the norm, then shouldn't you be required to provide a definition of point that doesn't use coordinates? For instance, in the book A Visual Introduction to Differential Forms and Calculus on Manifolds, the author says:
'even though you are used to thinking of (3, 5) as a point, a point is actually more abstract thanthat. A point exists independent of its coordinates.'
All right, then to what point do the coordinates (2,3,4) correspond? How can you answer this question without reference to coordinates?
Best Answer
It sounds like the author is trying to drive at the difference between vector spaces and affine spaces. The idea here is to imagine the $3$D world we live in. Where is the origin? Obviously, there isn't one. So what does it mean to say that a point lives at the coordinate $(2,3,4)$? It's clear that there is still a notion of "point" in our space, even though there isn't a notion of coordinates!
However, for solving concrete problems, we first choose where we want the origin to be (oftentimes we can make a choice that renders our problem particularly simple) and then we have access to all of our vector space machinery. In fact, it can be really useful to work without coordinates for as long as possible. You only get one shot at choosing where the origin goes, so if you can work without making that choice, it might be a good idea to defer until you have a good idea where the best choice might be for your particular computation.
More generally, we can define manifolds without coordinates. Think of a circle. If you're only interested in the manifold structure, then a circle with radius $1$ centered at the origin, and a circle with radius $5$ centered at some other point are exactly the same, even though the points have different coordinates1. Why should one parameterization be better than the other? Of course a circle (and the points on it) exist independent of our choice of coordinates. You might argue that, in this simple case, we should always use a unit circle centered at the origin, but for more complicated manifolds there isn't a good notion of "best" embedding. Already for a circle in $\mathbb{R}^3$, which plane should we put the circle in? $xy$? $yz$? These all give different coordinates to points on the circle, but obviously the geometric features of the circle don't care at all which one you choose.
So how, then, do we talk about points without talking about coordinates? Well, every manifold is (in particular) a topological space. So we have a bunch of points, which we can talk about abstractly as $x$, $y$, $z$, etc., but we can't necessarily give any individual point a name. This is analogous to working with a vector space before you choose a basis. The points still exist, but you'll get a different name for your point based on which basis you pick. This is still good enough for a lot of purposes, because we can say "Let $x$ be the point at which $f$ is minimized", etc.
Of course, eventually we have to work with a fixed coordinate system to really do computations. That's a really important part of the subject! What's important to remember is that, abstractly, we don't need this coordinate system for the points to exist. And sometimes, the geometry itself becomes more apparent if we work without a coordinate system for a while.
1: I assume we're talking about smooth manifolds, not riemannian ones. If you want riemannian manifolds, then pretend I said both circles have radius $1$.
I hope this helps ^_^