The familiar number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ all have "natural constructions", which indicate, why they are mathematically interesting.
For example, equipping $\mathbb{N}$ with the usual sucessor function and the constant $0$, it can be described as the initial $(0,1)$-Algebra. Or, if we want it to be an additive monoid, it is the free monoid on some one-point set. Since monoids are something very elementary and can be described purely categorically in many equivalent ways, this gives an idea, why considering the natural numbers might be interesting.
Now the forgetful functor $\mathbf{Groups}\longrightarrow\mathbf{Monoids}$ has a left adjoint sending $\mathbb{N}$ on the additive group of integers $\mathbb{Z}$. Alternatively, $\mathbb{Z}$ is the initial ring. If one questions that considering rings is interesting, we could reply that they are just monoids in the "natural" category of abelian groups with the "natural" tensor product making it a monoidal category.
From $\mathbb{Z}$ to $\mathbb{Q}$ it's not far, since the rationals are the image of $\mathbb{Z}$ under the left adjoint of the embedding $\mathbf{Fields}\longrightarrow\mathbf{Domains}$.
(One could go one step further and move on to the algebraic numbers as the field-theoretic completion of the rationals.)
My question now is: Why do we consider the reals $\mathbb{R}$, from a structural point of view? It is clear to me (or at least I don't feel I have the right to question it) that considering real numbers in physics, finance, etc. is necessary, because it provides good models for our reality.
I rather wonder, whether there are mathematical reasons making the reals interesting. The only description of the real numbers by some universal property that I am aware of is that they are the Cauchy-Completion of the rationals as a metric space – but since the definition of a metric space already depends on some notion of the reals, this is just a cheap trick. (One could of course define a metric space as a set $X$ with a set-function into some archimedian ordered field, satisfying the usual axioms, but this is a little artificial, I think.) Also note that it is almost impossible to do Algebra, Set Theory or Graph Theory without knowing what the natural numbers are, whereas one can prove many results of Algebra and Topology without ever coming across the reals.
I hope you can provide some ideas, showing why the reals numbers considered as a topological space/a field/a group/an ordered set are interesting in conceptual (i.e. category-based) mathematics. Of course, if one questions that the real numbers are interesting, one also has to question complex numbers $\mathbb{C}$ and other concepts based on these notions (like almost all of Analysis, Differential topology, etc.), so I am well aware of the fact, that I should not refuse the real numbers as old-fashioned even in case, I don't receive many answers.
Edit: Of course, my question is implicitly based on my strong belief that mathematical interestingness and categorical interestingness are equivalent concepts. As suggested by some of the comments, I would like to rephrase my question: How can I deduce from the mathematical properties of the real numbers that they are mathematically interesting, and thus that they are the optimal formalization of our intuitions of geometry and infinitesimal operations?
Best Answer
I think you are not asking the question you mean to ask. The question you mean to ask is something like "what kind of universal properties does $\mathbb{R}$ satisfy?" which is very different from "why should mathematicians care about $\mathbb{R}$?" Of course the answer to that question is to model lots of phenomena of obvious mathematical interest, e.g. differential equations and manifolds.
Here is one: $\mathbb{R}$ is the terminal archimedean field. (But unlike the example of $\mathbb{N}$ I don't consider this the last word on why the real numbers are interesting. This doesn't really explain why we use the real numbers to model Euclidean space, for example.)