Most of the remarks in this thread of messages are sharp and deep observations when it comes to the concept of proximity.Distance is a whole different issue and becomes a solid definition concept when ones defines  clearly the objects concerned and the metric used.In the next days allow me to suggest reading an upcoming paper of mine with  a title starting with the "funny" statement "A new proximity definition...". .Funny  in the sense of the timing that  i entered in this thread exactly when  I had taken the affirmative answer of the Jouurnal of Mathematics and Computer Science.

As the old Greek philosophers would  say " ΈΡΩΣΘΕ",

You are right. As far as I know there is no objective (measurable) definition of proximity. So... anything you invent is fine as long as it decreases monotonically with the increase of distance. You might want to start with something as simple as proximity=1/distance.

Dear Fernando,

Thanks for the lead and your input.

Best Regards,

Bernard

This is a great question with many possible answers.

In addition to the incisive remarks by @Fernando Soares Schlindwein and @Shian-Loong Bernard Lew, there is a bit more that can be observed about proximity vs. distance.

Before I venture into the mathematical distinction between proximity and distance, I suggest that we first consider the problem from a philosophical perspective.   What  I point out next is a bit like the situation for those that share our thoughts on RG (often in being philosophically close to each other) and yet there are great geographic distances between us (physically, we are far apart). 

This philosophic perspective on proximity vs. distance can be found in

See Section 5.2.2 (Distance vs. Proximity of Addresses), starting on page 108.   Physically speakers or writers can be physically far apart and yet proximal (near) in their hearts.

S. Alomary, Cognitive uterances: A Qur'anic perspective, Ph.D. thesis, University of Salford, UK, 2011:

http://site.iugaza.edu.ps/salomary/files/2012/04/Shaban-ALOMARY-PhD-Thesis-2011.pdf

From a mathematical perspective, we can define proximity without consider distance and we can define a metric (distance measure) without considering proximity.   For proximity,  the simplest approach is the one introduced by E. Cech during his Byrno seminar on proximity spaces from 1936 to 1939 (later put into book from Cech's seminar notes by Katetov and Frolik.    Let $X$ be a nonempty set and let $A,B,C$ be subsets of $X$.    A  relation $\delta$ is a proximity relation, provided the relation satisfies the following axioms.

1.   $A \delta B$ implies $A$ is not empty and $B$ is not empty.

2.   $A \cap B \neq \emptyset$ implies $A \delta B$.

3.   $A \delta B$  implies $B \delta A$.

4.   $A \delta (B \cup C)$ if and only $A \delta B$ or $B \delta A$.

$A$ and $B$ are proximal (near), provided axioms 1 to 4 are satisfied.    For more about this, see

http://www.encyclopediaofmath.org/index.php/Proximity_space

A nonnegative function d(x,y) describing the distance points x and y in a given set X  is a metric, provided d satisfies the triangle inequality, is symmetric and the d(x,x) = 0.   For more about this, see

http://mathworld.wolfram.com/Metric.html

It is true the notions of proximity and metric and be combined to define a metric proximity.   However, you can see that we can have the one without the other.    The two notions are independent of each other.

It is also good to mention that proximity as a description of nearness between sets/subsets while distance in most cases is between points, although there is a notion of distance between sets and elements can be considered as singletons to fit in to the definition of proximity as well. 

For the sake of historical perspective, aside from Dear James reference, see  http://ads.adsrvmedia.net/imp6996?a=38491251&size=1024x768&ci=1&context=c48581004

@Dejenie A. Lakew: It is also good to mention that proximity as a description of nearness between sets/subsets while distance in most cases is between points, although there is a notion of distance between sets and elements can be considered as singletons to fit in to the definition of proximity as well. 

Excellent observations!    It should also be mentioned that there are forms of distance.   In keeping with your second point, there is the Hausdorf distance

\[

d(x, A) = inf\{ ||x-a||: a \in A\}, \mbox{where} ||x-a|| \mbox{is, e.g., Euclidean distance}.

\]

and \u{C}ech ditance

\[

D(A,B) = inf\{||a-b||: a \in A, b \in B\}.

\]

Dear James,

Thanks for your thoughts and pointers to the philosophical and mathematical flip-side of proximity and distance. Truly distance is not an obstacle when there is proximity of thoughts! I also notice an emergent theme in both the discreteness and the continuity of the two constructs.

Dear Dejenie,

Thanks for the set-point delineation introduced to define distance and proximity. I am currently thinking also along that line from a question related to narratives and linguistics. Link to RG question forum below.

https://www.researchgate.net/post/How_useful_is_Wittgensteins_idea_of_family_resemblance_Familienaehnlichkeit_for_narrative_construction

Most of the remarks in this thread of messages are sharp and deep observations when it comes to the concept of proximity.Distance is a whole different issue and becomes a solid definition concept when ones defines  clearly the objects concerned and the metric used.In the next days allow me to suggest reading an upcoming paper of mine with  a title starting with the "funny" statement "A new proximity definition...". .Funny  in the sense of the timing that  i entered in this thread exactly when  I had taken the affirmative answer of the Jouurnal of Mathematics and Computer Science.

As the old Greek philosophers would  say " ΈΡΩΣΘΕ",

I see the term 'distance' as being closely tied with the idea of a Metric or Pseudometric space.  Proximity is lose.

@Joseph Alexander Brown: Proximity is lose

What does the term "lose" mean?   Perhaps you meant to write something else.

@George Stoica: …Proximity needs not be quantified...

Yes, if we define proximity (nearness) of sets in terms of the nonempty intersection of sets, then distance does not enter into the picture.   In fact, in traditional proximity space theory, the axioms for a proximity relation on a nonempty set X are given without assuming that X is a topological space.

Proximity is more a topological concept, and is related to "neighborhood". If we have space with metrics, proximity can be viewed as a ball of certain radius. But this ball can be deformed without the destruction of the neighborhood. Topological space even does not require metrics.

Distance can be defined between any pair of points (elements) of vector space.

In both cases we need to work in metric space. Metrics can be defined in many ways, both for finite and infinite dimensional spaces. A simple example of metrics in R^n is Euclidean metric. In an infinite dimensional space metrics can also be introduced; see Hilbert and Banach spaces; http://mathworld.wolfram.com/HilbertSpace.html

Dear Yuri,

Thanks for the fresh and alternative perspectives that you have contributed to the question.I especially find the idea of spheres and variations to the idea of " spheres of influence" insightful. Since in our day-to-day experiences this conjures in our minds the notion of neighbourhoods and "gravitational" attractions towards a nucleus. Also grateful for  the distinction between metric spaces and topological spaces in light of the question.

Best Regards