Why do we think of ${\mathbb{R}}^2$ as a plane? It seems like every one assumes there's an obvious answer to this question but I just can't see it. ${\mathbb{R}}^2$ is just defined as the set of all 2-tuples of real numbers i.e.
${\mathbb{R}}^2 := \{(a,b)| a,b \in\mathbb{R} \}$
The $xy$ - plane clearly has a lot more structure in it than just being a set of points. There is a definitive order to the arrangement of points. We can define curves on it (maybe even closed curves with some enclosed area). We can talk about distances between 2 points & so on. So where does this additional structure come from & how does that relate to the set ${\mathbb{R}}^2$ .

There is a lot of additional structure, yes. However, we usually "construct" this structure on top of the "raw plane" $\mathbb R^2$


*

*The notion of distance is a metric

*Closeness is a topology

*Curves / solutions of polynomials are a variety

*...


However, we always start with $\mathbb{R}^2$ and add structure on top of it. The choice of structure depends on what you want to study currently!
