Why is everything geometrical modeled on $\Bbb R$?

Question

The reals naturally arise when discussing limiting processes of rational numbers like trying to compute roots or $\pi$. Similarly I have read that axioms of Euclidean geometry (I mean those not explicitly relying on $\Bbb R$ a priori) naturally lead to the real numbers. So there is no doubt in their motivation or in their usefulness.

Nevertheless I am wondering why they are so prominent in every geometric kind of situation. To be more explicit let me draw the following analogy: In algebra it is common to consider some property of the ring of integers and turn this into an abstract definition (think integral domain, factorial domain, principal ideal domain, euclidean domain etc.). I find it particularly interesting that in these settings properties we know and love in $\Bbb Z$ continue to hold, because the corresponding abstract definition perfectly encapsulates the essence of the properties (eg. prime=irreducible in factorial domains). On the analytic / geometric side however the most abstract things derived from the reals I know of are topological spaces and sigma-algebras. Yet as soon as we try to work with them we immediately come back to the reals by forcing connections to $\Bbb R$ (metric spaces have a distance function evaluating in $\Bbb R$, similarly normed vector spaces have norms evaluating in $\Bbb R$, measures evaluate in $\Bbb R$, path-connectivity and homotopy are based on the unit interval $[0,1]\subseteq \Bbb R$ etc. etc.). Compared to the algebraic side mentioned before, this feels rather unaxiomatic to me. Why don’t we assume that evaluations take place in totally ordered groups or something along these lines? Why do we pick the unit interval in $\Bbb R$ and not some totally ordered lattice? Is it just convenience, are the more general / abstract definitions unnecessary by some sort of Lefschetz-principle or is this purely coincidental?

TLDR Everything geometrical (metric spaces, path-connectedness etc.) is inherently based on $\Bbb R$. Why is that and why don’t we see more axiomatic versions.

As always thank you very much for your time, patience and considerations.

Answer 1

There is an axiomatic characterization of the reals: they are the unique complete, ordered, archimedic field. All of these properties are important for what we classically think of as geometry. The field operations correspond to classical constructions done with compass and straightedge. The ordering allows us to identify which of three points on a line is between the other two. Completeness guarantees that lines are connected. And the Archimedic property guarantees a certain sense of finiteness: if we go along a line in fixed increments, we will eventually pass any given point. So the reals are a very natural object to capture the essence of what a line is, and in geometry, lines are fundamental.

Of course, there are other ways to do geometry. For instance, we can define affine planes over arbitrary fields, even if they aren't ordered. For instance, the affine plane over $\mathbb F_2$ consists of four points, and its lines consist of two points. But it misses a lot of the classical properties of geometry. If we want the most important properties of classical geometry to hold (basically the ones mentioned above), we have to model it on the real line - or some other way which is then equivalent to a model based on the real line.

Answer 2

There is actually a characterization of real numbers based on geometric axioms, totally independent from the algebraic approach with field extensions [Will add a reference, e.g. Prof. Degen, Stuttgart de]

So, to some degree geometry is real.