Comment by somethingsome

A rough idea and simplification:

Suppose you have a bunch of 2D points, without coordinates, they exist because you say so, they can represent anything you want.

But you can't do a lot with those points, you may be interested in knowing their distances. To do that, you create some reference system, i.e. 2 non parallel axis and you set a unit on each, for example one could have one centimeter and the other one meter.

Now by placing the reference system on one particular point for example, you can 'identify' each other point on that scale.

With this correspondance, you can uniquely map each point to a coordinate and each coordinate to a point in space, this allows you to measure distances for example.

Notice that the choosen coordinates didn't really matter nor the direction of the axis. But as the rest of the 2D world can be mapped to them, everything is coherent.

Now if you create a novel axis system with another initial point and both axis with 1cm on each, you can find a transformation that transform your first system into the second, this transformation allows you to transform any other point in the new coordinates system.

So what is a position exactly? I would say it's the identification of some objects by an arbitrarly chosen referential system. What are the numbers? They corresponds to an arbitrary chosen unit of measure.

I hope this will give you more tought matter :)