There's no issue with the notation; I think you've misunderstood the mathematical idea. Consider a more familiar algebraic object, a real number, x. This can model a length, an area, volume, time, time interval, temperature, weight, speed, physical constant, geometric ratio, fractional dimension, etc...

In mathematics, we abstract by forgetting about what the things are, and retain information about how they behave, and about what abstract properties they satisfy. The insight is that 2d locations and 2d displacements have the same abstract properties, which are modeled by a certain algebraic object: 2-vectors.

Thanks for the explanation. Makes sense. Something else I noticed, vector notation does not specify a coordinate system. V = (1, 2) is just an array of two numbers. The cartesian coordinate interpretation is a choice we make. Correct?

Yes, the keyword here is 'basis'. You represent a vector by giving two pieces of data, (1) an ordered list of coordinates, and (2) a basis. The vector is then a linear combination of the basis elements, and the coordinates tell you how to form that linear combination.

For example, let's use the standard Cartesian basis consisting of unit vectors e1, e2, e3 (which point north, east, and up, informally speaking). If our vector v is given by the coordinates (3,4,8) (with respect to the standard basis), then this means that v = 3 * e1 + 4 * e2 + 8 * e3.

If the coordinates were given with respect to a different set of basis vectors, then you would take the linear combination using those vectors instead. Note the similarity of how a basis works to how a base system works representing numbers. Using base 10, the 'coordinates' of the number 348 mean that 348 = 3 * 100 + 4 * 10 + 8 * 1. Using a different base, say base 9, they would instead mean 348 = 3 * 81 + 4 * 9 + 8 * 1.

> You represent a vector by giving two pieces of data, (1) an ordered list of coordinates, and (2) a basis.

Ok, I understand. But as used in computer languages, a vector can be simply 2 numbers. No coordinates or basis are implied. That's what I meant.

Aha, yes. Computer languages borrowed the word 'vector', but they have basically nothing to do with the mathematical structure from linear algebra. It's best to keep them completely separate in your mind.

So in math, when we say "vector" coordinate system is a given, as you explained?

If a coordinates are given, then they will be given with respect to a basis. However, it's entirely possible to do things more abstractly without introducing coordinates and bases to begin with, for example:

https://en.wikipedia.org/wiki/Tensor_(intrinsic_definition)

My advice would be not to get stuck on these “philosophical” questions, if your goal is to actually learn math, and instead just press on and keep learning and solving real problems. Eventually the fog will dissolve by itself, and these kinds of questions will seem to you either naive or devoid any real substance, or just uninteresting compared to everything else that you have learned.

No. This advice does not apply to me. I don't want to learn mathematics. I'm more interested in learning parts of mathematics that interest me at the moment. And I think philosophy comes before mathematics. Or mathematics is the philosophy of quantities. Both philosopnies are based on definitions.