What does it mean for a number to be "larger" than another number? In everyday life we have a clear notion of what it means for something to be larger than something else. Usually we would evaluate something's size based on it's volume. However, is there a formal method for extending this everyday notion of size to numbers and the realm of mathematics in general?
 A: Generally we mean the latter, that $3 > -10$ because $3$ is numerically greater than $-10$. I don't have to get away from the Euclidean metric for this to start to complicate. Consider the points $(2,3)$ and $(3,2)$ in $\mathbb{R}^2$. Which one is larger? Or, should I say they are of the same "size" because they have the same distance from the origin? 
I believe you're confusing the notions of ordering (given two elements, I can tell you which is greater) and size. Certainly in a metric space (Euclidean or not) I will have a concept of distance, but I'm not sure how you're using the word "size" to relate this to your question. 
A: On some sets there exists an ordering $<$ which satisfies
(1) $a \not < a$
(2) $a<b$ and $b<c \implies  a<c$
When we say $a$ is larger $b$ than we typically mean $a>b$. It would be unusual to say larger when dealing with metric spaces, we would probably say further and as in your example say $-10$ is further from $0$ than $3$ if $m(-10,0) >m(3,0)$ where $m$ is a metric.
A: Taking your question to the absolute level of generality, you can define any reordering of the integers that you want and then just use the usual metric on that reordered set of integers, and you will obtain a unique ordering defined by that metric (up to the direction of the ordering).
