Definition of distance?
Firstly let me start by saying that I understand that "distance" is one of those things that is defined differently in different contexts. I know that there is no 'the-definition' of "distance" but it's supposed to be a measurement that measures how far apart two objects are.
However since the definition varies depending on the context, I suppose it's only reasonable that we have some "ground rules" for what "qualifies" as a distance in any given context in general.
For example, one ground rule could be that the distance from one thing to the second thing should be the same as the distance from the second thing to the first. Another ground rule could be that it must always be non-negative.
While looking around I found out that in Mathematics there is a generalized notion of "distance" called Metric. (mathematics)
While I understand the other properties that the metric must satisfy, I do not understand why should a distance satisfy the triangle inequality? In cases, say in the context of Physics, if I define my distance as "the length of a specific path travelled between two points" see here for example then this definition of distance would not satisfy the triangle inequality.
So to summarise my question, is triangle inequality a necessary "ground rule" while defining distance in general?
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$\begingroup$if I define my distance as "the length of a specific path travelled between two points"
That's not a good definition of distance, if you ask me. Or, at best, it is an incomplete definition. The distance between two points should be dependent only on the two points, not on the path taken to get from one to another. So, in your definition, you must also specify how you get the specific path for a given pair of points.
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