Is there even a point in defining the notion of a 'metric' (as opposed to a metric space), etc.?

Question

When we define a new mathematical structure, we generally double up on definitions. We define structures (think: metric spaces, partially ordered sets, etc.) and also the ingredients that they're built up from (think: metrics, partial orders, etc.)

My question: are the ingredients even worth naming?

Consider that

(1) Structure-preserving maps go between structures, not ingredients, and

(2) Neither structures nor ingredients have the edge with respect to the ease with which new definitions can be created, e.g.

• A partial ordering $\leq$ (carrier $X$) is linear iff either $x \leq y$ or $y \leq x$ for all $x,y \in X$.
• A partially ordered set (carrier $X$, ordering $\leq)$ is linear iff either $x \leq y$ or $y \leq x$ for all $x,y \in X$.

(3) Structures are more general than ingredients because they're allowed to include multiple sorts. e.g. A vector space includes two distinct sorts, scalars and vectors.

So to repeat my question, are ingredients even worth naming? Why not just leave it at 'a metric space is an ordered pair $(X,d)$ such that...' without adding 'a metric is a function $d$ such that...'?

Answer 1

The reason it's meaningful to define "metric" is that a space can have more than one metric, so we need a word to call those different functions.

The reason it's meaningful to define "metric space" is to distinguish those spaces that admit a metric from those that don't, without necessarily having to spell out what that metric is.

Answer 2

I don't know about you but I refer to the "metric" by name a lot even when it's the only one. It seems perverse to have something ubiquitous with no name. Imagine someone asks "What's $d$?" - what should you say? People would instantly adopt names for convenience anyway, why not standardize?

More mathematically perhaps, you often have e.g. two metrics on the same set. In your language, I'm only really supposed to talk about whole structures, but this situation clearly separates out the ingredients. Then one might reasonably talk about properties of these two metrics on this one set, not of these two unrelated structures.

Further, one might consider categorizations of possible metrics on a set. Here you are really intuitively dealing with a space of functions from the space to $\mathbb R$, not a space of structures. I would want to talk about "all possible complete metrics on $X$", not "all possible metric spaces of the form $(X,d)$". The former is more economical and efficient and in agreement with what you're doing, at least to my mind.