Distance in a metric space, the chessboard

A metric space is a set or collection of elements possessing a function called a metric.

    Please bear with me as I am only an amateur mathematician.

This function, eg.   d(x,y)  , yields a result defined as the distance between the elements x and y.

The distance is 0 ( zero ) if the elements are identical ,  x = y .

Consider the first rank of the board : squares a1 , b1 , c1 ...

    distance function  d( a1, a1 ) = 0    by definition

                                  then   d( a1 , b1 ) = 1

                                             d( a1 , c1 ) = 2

Fallacy of "squares between" equals distance.  Many writers confuse or at least interchange these, even      the great Dvoretsky ( perhaps the problem is the  English translation ).

For example I recently came across the following descriptions of knight-check-shadow :

    " the king restricting the knight at a distance of one square diagonal or two squares on a rank or file"

Actually the diagonal distance = 2 , although there is only one square in between.  

The rectilinear distance = 3 , although two squares "between".

Kings in opposition have a distance of 2 .

In the 8 years I've been reading chess web sites and books I've come across this confusion

countless times.  I'm sure the authors know what they mean, but it can be quite confusing to the

reader, esp. when the writer shifts back & forth from "between" to "distance" in the same

paragraph.  Is it unreasonable to request uniform, consistent terminology ?

Mathematics is older than chess and the 64 square board is older than the game.

This must be an example of what set you off:

http://godofblunder.weebly.com/knight-check-shadow.html

 If this is what you are talking about, I see your point (although it wouldn't bother me to the same extent).

I hope you will post links to any of the  countless examples you come across.

Thanks for your response.

  Your're correct, the source was godofblunder.weebly.com/knight-check-shadow.html

I have not kept a record of all places I've seen this type of inconsistency. Consider Dvoretsky's Endgame Manual , Chapter 1 Pawn Endgames , Section on Opposition : 

    "to get the opposition" means to achieve this standing of the kings one square apart with the opponent to move ...

Here the word apart is suspect. Don't know what was in the original Russian. The first definition of

apart that came up :  'apart'  of two or more people or things separated by a distance ; at a specified

distance from each other ...  ( italics are mine ).

Consider kings-in-opposition on squares e1 & e3 , and assume the unit of measurement on a chessboard is a square.  How far apart are integers 1 and 3 ?

On maps in North America the distance from one city to another is give from center to center ,

not edge to edge.  Consider stacked rooks on d1 & d2.  What is their distance ? Answering 'zero' is

contrary to the fundamental concepts of a metric space.

I'm sure Mark Dvoretsky & other authors know what they're talking about ; they know far more chess

than I ever will.  But I've repeatedly had difficulty following deep annotations because of the fuzzy terms used describing distance.

Fallacy of  "squares between" equals distance. 

Taken from Wikipedia,

In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L_∞ metric^[1] is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension.^[2] It is named after Pafnuty Chebyshev.

It is also known as chessboard distance, since in the game of chess the minimum number of moves needed by a king to go from one square on a chessboard to another equals the Chebyshev distance between the centers of the squares, if the squares have side length one, as represented in 2-D spatial coordinates with axes aligned to the edges of the board.^[3] For example, the Chebyshev distance between f6 and e2 equals 4.

three paths, same amount of moves. in chess the fastest way is not always straight line