Is our interest in $\mathbb{R}$ "historical"?

Question

In the process of a topology course, it occurred to me that a number of concepts are defined with reference to $\mathbb{R}$ and standard subsets thereof. For instance, we consider metrics, which are of course maps into $\mathbb{R}$ (specifically the non-negatives, but still). We devote a chapter of a book to the concept of path-connectedness, i.e. a way of expressing (part of) a space as a continuous image of a closed real interval. We define separation axioms according to the ability to separate particular sets with continuous maps into closed intervals.

A lot of concepts in topology are defined with reference to particular subspaces of $\mathbb{R}$. But move to other fields, say measure theory, and the entire branch is defined by functions into $\mathbb{R}$.

I suppose my question is: To what extent is our interest in objects "founded on" $\mathbb{R}$ simply a matter of our massive familiarity with $\mathbb{R}$, and to what extent do these concepts draw on characteristics really constitutive of $\mathbb{R}$? Put another way, do we tend to consider real-based objects because the reals are "intuitive", or because $\mathbb{R}$ (or its relevant subsets) has particularly special properties that we don't really find in other (truly distinct*) objects?

*By "truly distinct", I mean to preempt answers along the lines of, "Well you can construct this thing which isn't the set $\mathbb{R}$ per se, but is still for all intents and purposes the exact same thing as $\mathbb{R}$."

Answer 1

I think you can definitely try and make a distinction between fields of mathematics that are "founded on $\mathbb{R}$" in a strong sense and those that aren't (but in which our familiarity with $\mathbb{R}$ and its cousins might play and important role in terms of providing examples, intuition, driving questions, etc).

Consider the following fields:

1. Set theory. I would argue that the real numbers play no foundational role in set theory. You can define and investigate the notions of cardinality, partial orders, etc without even bothering to construct the real numbers. Of course one can say that our familiarity and interest in the real numbers is the reason set theory was created in the first place and problems like the continuum hypothesis (which involve the real numbers) served as important foundational problems in the field, but from a mathematical point of view, I would say that the interesting set-theoretic questions that involve the real numbers can be viewed as "applications" of set theory to specific situations and that the core of set theory has nothing to do with the real numbers.
2. Group theory or any abstract algebra field. Again, one can investigate the abstract properties of groups without knowing anything about the real numbers. The familiarity with the real numbers can give us important examples of groups (such as Lie groups) but again, results about such groups can be viewed as applications of group theory to $\mathbb{R}$-founded objects.
3. Smooth manifolds. Here, we are working with objects that are modelled locally on $\mathbb{R}^n$ and we are generalizing our ability to do calculus over the reals to this more abstract setting. This is a field that in my opinion is "strongly founded on $\mathbb{R}$". You can of course try to abstract away the properties that make the theory work (what do you need in order to do calculus, etc) and people have done that resulting in fields that are much less $\mathbb{R}$-founded but investigating smooth manifolds remains very $\mathbb{R}$-founded.

Now, regarding general topology, I would like to argue that while it has many applications to objects that are "founded on $\mathbb{R}$" and fields that are founded on $\mathbb{R}$ in a strong sense (such as manifold theory), it is, not, per se, a field that is founded on $\mathbb{R}$ and the real numbers don't play a particularly important role.

The basic players in topology are defined using an abstract family of axioms (very much like in group theory). By imposing additional restrictions (separation axioms, again, defined in terms of the basic operations), we can single out specific classes of topological spaces. For example, one might define a family of spaces that are regular, Hausdorff and have countably locally finite basis and investigate their properties. It turns out by the Nagata-Smirnov metrization theorem that such spaces are precisely the topological spaces that admit a metric but a priori we can investigate the topological properties of such spaces without introducing a metric at all. Choosing a metric on such a space can be considered as introducing an "axillary" data that helps us (as we are familiar with the real numbers and properties of distance in Euclidean spaces) to analyze the family and describe its topological properties.

Regarding path-connectedness, the amazing answer of Eric to this question shows that the notion of path-connectedness can also be defined without introducing the interval $[0,1]$ and using the interval to define a path can be considered as introducing "axillary" data that helps us to visualize, give intuition and analyze path-connected spaces.

Of course, the history went the other way and we care about paths because we visualize them as generalization of paths in an Euclidean space and we care about seperation axioms because we care about metric spaces and want to understand which parts that hold in metric spaces can be "abstracted away" but once the abstraction has been done, the real numbers stop playing a foundational role.

Answer 2

Euclid's book on Geometry describes lines, planes, and shapes such as circles and triangles in the plane, relation between the lengths of sides. This is 2300+ year-old mathematics. Lines play an important role from then on (even earlier). Numbers were initially thought of as lengths or areas etc. So real line was the fist familiar model well understood. So measure theory trying to generalize length, and topology all have real lines and planes as the basic model.

Still one cannot say our understanding of Euclidean spaces was deep. the three usual conditions for metric: non-negativity, symmetry, and triangle inequality were not strong enough to exclude non-archimedian metric spaces, a highly non-intuitive one (if two open balls intersect then one is contained in another!). But non-archimedean ones are useful is another matter.

Answer 3

The reason that reals are so irreplaceable in mathematics is that, up to isomorphism of all defining structures (addition, multiplication, and linear ordering, archimedean property), there is no other such object.