[Tainnor]

> So I had to repair my intuition again: a vector is a local linearization of something that might be macroscopically, and the linearity is what makes it possible to add and scalar-multiply it. And also that there is functionally no difference between composing vectors with addition or multiplication, they're just notations.

Except none of this is true of vectors in general, although it might be true of very specific vector spaces in physics that you may have looked at. Matrices or continuous functions form vector spaces where you can add any vectors, no matter how far apart. Maybe what you're referring to is that differentiability allows us to locally approximate nonlinear problems with linear methods but that doesn't mean that other things aren't globally linear.

I also don't understand what you mean by "no difference between composing vectors with addition or multiplication", there's obviously a difference between adding and multiplying functions, for example (and vector spaces in which you can also multiply are another interesting structure called an algebra).

That's the problem if you just go from intuition to intuition without caring about the formalism. You may end up with the wrong understanding.

Intuition is good when guided by rigour. Terence Tao has written about this: https://terrytao.wordpress.com/career-advice/theres-more-to-...

The vector space axioms in the end are nothing more than saying: here's a set of objects that you can add and scale and here's a set of rules that makes sure these operations behave like they're supposed to.

[ajkjk]

You have unfortunately written off everything I said by assuming the most naive understanding of it.

The general theme is that I am interested in the metaphysical concept of vectors, not the thing that human mathematicians have labeled vectors. The universe doesn't care if you write ax+by or x^a y^b, hence addition vs multiplication is just a choice of coordinate system. And matrices and functions are vector spaces sure, but out in the world, when they show up in modeling things, they are local linearizations of curved things. Every linear algebra is (inevitably) a local point in a nonlinear one, as far as I can tell. Not in a formal sense, but in the sense that when you go out into the world and find them, it turns out to be the case.

The general theme is: I don't want to spend my life mastering the rigor of these simplistic models so that I can do it intuitively (in Tao's sense); I want to use them to learn intuition of the things that they are simplistic models of, and then master that.

[Tainnor]

> The general theme is that I am interested in the metaphysical concept of vectors, not the thing that human mathematicians have labeled vectors.

That's fine but then you shouldn't be surprised that you can't read higher level mathematics textbooks, because those are not about metaphysics.

The rest of what you wrote is... just not true. Matrices are used in plenty of areas where they are not mere approximations (e.g. cryptography), and spaces of functions remain vector spaces even when the functions themselves are not linear (because there's a difference between requiring that f(x+y)=f(x)+f(y) and that (f+g)(x) = f(x)+g(x)).