TITLE: Why does the distance between two members of a metric space map to a real number?
QUESTION [6 upvotes]: Why does the distance between two members of a metric space map to a real number? The set of real numbers is itself (along with the Euclidean distance function) a metric space - in which case, real numbers are used to describe a property of real numbers (themselves).  How is that possible? (How can you use something to describe itself?)﻿

REPLY [7 votes]: Requiring the distance function on the reals to be real-valued is no more circular than requiring that the sum of two reals be real.

That we use the real numbers for distance reflects the typical interests of geometers; prior examples that the notion of metric space was meant to generalize had real distances, and the kinds of spaces many people like to study are suitably measured by real distances. 
There's even a prior reason why the reals make a good target: they are the unique linearly ordered, complete, Archimedean abelian group. (e.g. see Wikipedia)
Specialized applications might prefer a different definition of metric; e.g. see Wikipedia again.