As a low-level theoretical physics student, my hands are always dirty with abstract mathematics, and whenever I get the chance I try to dive deep. Recently I had an Advanced Mathematical Physics course following Nakahara. And while dwelling over the concepts, I find it really pleasing to start from the bare minimum and start building structure. However, sometimes it appears that people along the way throw in concepts that are ment to be understood or met later. E.g., I don't understand how do we speak of charting a circle given by relation $x^2 + y^2 = 1$ while we still know nothing about coordinates? On the same footing, how do you describe $\mathbb{R}^n$ just knowing that it's a set of n-tuples?
EDIT: My questions come from a place of not understanding. So I am asking implicitly for showing me where I'm looking at things wrong. So please tell me how should I look at these abstract creatures
$\mathbf{R}^n$ is characterized or defined up to a natural isomorphism by the universal property of products:
To be precise, $\mathbf{R}^n$ is not merely a set, but a set $\mathbf{R}^n$ together with $n$ functions
$$\pi_1,\ldots,\pi_n\colon\mathbf{R}^n\to\mathbf{R}$$ with the following universal property: If $X$ is a set and $f_i\colon X\to\mathbf{R}$ are some functions, then there exists exactly one function $f\colon X\to \mathbf{R}^n$ such that $f_i=\pi_i\circ f$ for all $i\in\{1,\ldots,n\}$.
For $X\in\mathbf{R}^n$ and $i\in\{1,\ldots,n\}$ we write $X_i:=\pi_i(X)$ and $(X_1,\ldots,X_n):=X$.
Definition of addition and scalar multiplication:
Consider the product $\mathbf{R}^n\times\mathbf{R}^n$ with projections $\Pi_{1,2}\colon\mathbf{R}^n\times\mathbf{R}^n\to\mathbf{R}^n$. By the universal property of $\mathbf{R}^n$ there exists a unique function $A\colon \mathbf{R}^n\times\mathbf{R}^n\to \mathbf{R}^n$ with the property $\pi_i\circ A=(\pi_i\circ\Pi_1)+(\pi_i\circ\Pi_2)$, i.e. $$(X+Y)_i=X_i+Y_i$$ for all $i$.
Consider the product $\mathbf{R}\times\mathbf{R}^n$ with projections $\Pi_1\colon\mathbf{R}\times\mathbf{R}^n\to\mathbf{R}$. By the universal property of $\mathbf{R}^n$ there exists a unique function $M\colon \mathbf{R}\times\mathbf{R}^n\to \mathbf{R}^n$ with the property $\pi_i\circ M=\Pi_1\cdot(\pi_i\circ\Pi_2)$, i.e. $$(\lambda\cdot X)_i=\lambda\cdot X_i$$ for all $i$.
So far, we have defined a vector space structure.
Definition of the inner product:
The standard inner product is defined as follows: \begin{align} \mathbf{R}^n\times\mathbf{R}^n&\to\mathbf{R}\\ (x,y)&\mapsto\sum_{i=1}^n\pi_i(x)\cdot\pi_i(y) \end{align} The inner product defines a norm and the norm defines a metric, which allows to assign a distance to each pair of points.