Chapter 4 of Michael Spivak's Calculus is a swift overview of the classic visual representation of functions with a domain in $\mathbb R$ and a range in $\mathbb R$. In particular, each ordered pair, denoted as $\{ x, f(x) \}$, that is a member of this function is visually depicted through the use of a horizontal axis, which corresponds to the first component of the ordered pair, and a vertical axis, which corresponds to the second component of the ordered pair. Importantly, each axis is a straight line, and the axes intersect at $90^{\circ}$ angles. There is/are certainly more structure/regulations regarding the invocation of a Cartesian Coordinate System (rules of plotting, etc), but you get the point. The summarized theme of this fourth chapter is effectively:
Depicting $\mathbb R \to \mathbb R$ functions in this aforementioned manner will be useful in guiding problem solving for future chapters.
Having completed all of the exercises in the first seven chapters of this book, I certainly agree! Representing $\mathbb R \to \mathbb R$ functions in this manner consistently produces trustworthy intuition on how to navigate my formal arguments. Well...at least "trustworthy" in the sense that I arrive at an answer the solution manual agrees with!
However, the more I think about it, the less I understand why this is the case (i.e. why is it the case that these visual representations...and the rules that govern such Cartesian representations...orient my arguments so well?). Arguably, the manner in which one decides to convert a symbolic/syntactic entity like "$\{ x, f(x) \}$" into a visual object is completely arbitrary.
It seems to me that any of the below visual representations of dual axes, each perhaps with its own unique plotting rule-system, could potentially be valid (where $\color{\orange}{\text{orange}}$ represents the axis of the first component, $\color{\black}{\text{black}}$ represents the axis of the second component, and the $\color{\purple}{\text{purple}}$ tick mark denotes $0$):
Could these alternative axes (with alternative plotting rules) also offer appropriate correspondences to the formal reasoning needed in Real Analysis? Or is there something particularly unique about the Cartesian Coordinate System that makes it the most well-suited at informing logical arguments needed in Real Analysis?
"Could these alternative axes also offer appropriate correspondences ?"
Yes, I see these two cases at least:
1st case If you already have a "ruling" of the 2D space (be it linear or curvilinear) by a double family of curves, any global transform:
$$T:\mathbb{R^2} \to \mathbb{R^2}$$
will generate a "curvilinear ruling" on which you can "track" interesting features just like you can do it with a "linear ruling".
A typical case is when $T$ is issued from a complex transform (think to $\mathbb{R^2} \approx \mathbb{C}$), the classical term being "conformal representation", and I refer you to an answer of mine where the complex derivative is explained in this way.
Remark : the initial "ruling" can a polar ruling by radials $\theta$=const. and circles $|z|=r$, sometimes called "radar-type ruling".
2nd case: I would like also to attract your attention on a special category of diagrams using curvilinear coordinates, called nomograms ; they are usually conceived for 3 connected variables, say $x,y,z$, for example Pression, Volume and Temperature for gases. This kind of tool based on alignments is able
to provide a numerical evaluation of one of the parameters as a fonction of the 2 others in any order $z$ as a function of $x,y$, but as well $x$ as a function of $y,z$, etc.
to convey an understanding of "situations", in the broad sense of this term, with a categorization for example in terms of diagnosis.
A certain number of examples, in the domain of health, can be found in this dowloadable document.
I can find other interesting nomograms in different applied domains if you are interested.