Category-theoretic description of the real numbers

Solution 1:

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.)

Solution 2:

An interesting alternative approach to defining the reals was discussed on the category theory mailing list many years ago: http://comments.gmane.org/gmane.science.mathematics.categories/1319

Let $S$ be the ring of functions $s: \mathbb{N}\rightarrow \mathbb{N}$ such that the function of two variables $(m,n) \mapsto s(m+n) - s(m) - s(n)$ is bounded. Addition is element-wise and multiplication is functional composition. Let $I$ consist of the bounded sequences. Then $\mathbb{R} = S/I$.

So $S$ includes these things that are almost but not quite homomorphisms. This is a little surprising as defining approximate things (ie. actually approximate by a finite amount rather than infinitesimally) and composing them often leads to things getting badly behaved. So there's something special about $\mathbb{R}$ and its relationship with $\mathbb{N}$.

Solution 3:

There is a characterisation of the reals purely in terms of its order structure which explains its ubiquity as a model in mathematics and physics: a totally ordered, Dedekind complete set which has a countable order dense subset but no largest or smallest element. This is the basis of many results which show that natural orderings which arise there are induced by a real- valued function. This is one of the central problems of the theory of measurement and explains how one can pull the reals out of a hat from a system of axioms which do not contain the concept of number explicitly. Examples are: entropy (from the ordering "adiabatically accessible" on the states of a thermodynamical substance), temperature (hotter than), price in economics from the relationship "worth more than" (under the generic name of "utility function") and of course in the synthetic approach to axiomatic euclidean geometry (using the ordering induced by the "in between" axiom). The important point is that these concepts in the physical world can be experimentally verified in the ordinal sense without recourse to a numerical scale.

Solution 4:

There are various topological characterizations of the topological space $\mathbb{R}$. See MO/76134. For example it is the unique connected, locally connected separable regular space, such that deleting any point gives two components. To some extent this is exactly what we want the continuum to be.

But the question seems to be aimed more at algebraic characterizations. In this case, it would be nice to give a characterization of $\mathbb{R}$ within the category of fields. This fits nicely into my question SE/634010 if the category of fields is rigid (still unsolved), which would give a categorical characterization of any field. At least, we have a characterization of those fields which are elementary equivalent to (i.e. satisfy the same sentences as) $\mathbb{R}$. These are the real closed fields, which have various purely algebraic characterizations. A very short one is that $F$ is real closed if $F^{alg} / F$ is a proper finite extension.

Solution 5:

There is certainly a broad consensus that the real numbers are extremely useful and moreover categorical. However, they are not "the optimal formalization of our intuitions of geometry and infinitesimal operations" as you put it. The hyperreal numbers are a better candidate for this task since they literally contain infinitesimal numbers, and moreover allow functions to be naturally extended.

In more detail, the hyperreals are the only theory within ZFC that give a theory of infinitesimals useful in analysis. Other theories don't even provide natural extensions of functions like the sine to the extended domain. Furthermore, a hyperreal field is definable in ZF. There are two main competitors: the surreals and the Smooth Infinitesimal Analysis. The former cannot even extend the sine function. The latter requires intuitionistic logic and does not work in ZFC.

Keisler has a unique characterisation of a hyperreal line in terms of a natural system of axioms. In a different direction, the group around Di Nasso has also provided such characterisations.

Since the OP mentioned both the real numbers and infinitesimals in his question, it is worth mentioning that Edward Nelson developed a more versatile set-theoretic foundation called internal set theory (which is however a conservative extension of ZFC) where numbers behaving like infinitesimals can be found in the real number field itself. These are still the usual real numbers so in particular they enjoy all the properties of categoricity that we all rave so much about.