Why are the basis unit vectors |0⟩ and |1⟩ written as column vectors [1, 0] and [0, 1], respectively?

Question

I am starting to learning quantum computing and am currently reading Quantum Computing: A Gentle Introduction. As I started reading into the first section after the Introduction, it talks about basis vectors |0⟩ = |↑⟩ and |1⟩ = |→⟩ along with this figure:

However, what confuses me is why |0⟩ = |↑⟩ and |1⟩ = |→⟩ are, respectively, represented as \begin{pmatrix}1 \\ 0\end{pmatrix} \begin{pmatrix}0 \\ 1\end{pmatrix}

Shouldn't |↑⟩ be represented as \begin{pmatrix}0 \\ 1\end{pmatrix}

and shouldn't |→⟩ be represented as \begin{pmatrix}1 \\ 0\end{pmatrix}, since those appear to be their heads' coordinates on the circle?

Answer 1

What's in a name? That which we call a rose
By any other name would smell as sweet;

Shakespeare

Which physical state or one on the sphere we call $|0\rangle$ and which we call $|1\rangle$ has no impact whatsoever on any sort of analysis, as the labels 0 and 1 are completely arbitrary.

You can repeat any analysis with 0 and 1 switched and get the same result.