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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. https://en.m.wikipedia.org/wiki/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 https://en.m.wikipedia.org/wiki/Distance#Overview_and_definitions 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?

William
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    Yes. Triangle inequality says that if you want to go from $x$ to $y$, going directly from $x$ to $y$ is better than going from $x$ to another point $z$, and then going from $z$ to $y$. Whatever path you allow you, whatever the way you estimate the "length", this seems really natural. – TheSilverDoe Sep 10 '20 at 12:02
  • The triangle inequality is the formulation of the idea that the shortest distance between 2 points is a "straight line". – SeraPhim Sep 10 '20 at 12:02
  • @SeraPhim Describing a metric space with "geometric" properties is dangerous... – TheSilverDoe Sep 10 '20 at 12:03
  • @TheSilverDoe which is why I used quotation marks. – SeraPhim Sep 10 '20 at 12:04
  • @SeraPhim I know :) I just add the precision for other readers. – TheSilverDoe Sep 10 '20 at 12:04
  • @TheSilverDoe that's fair! I should've emphasised that my description is not a precise one – SeraPhim Sep 10 '20 at 12:06
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    "distance" does not need to satisfy the triangle inequality, for example if you say your "distance" is a "Semimetric" instead of a "Metric". (read further down the wikipedia article you linked) – Vepir Sep 10 '20 at 12:14
  • There is an even more general notion generalising the notion of metric space (a space equipped with a distance), i.e., the concept of uniform space – Jean Marie Sep 10 '20 at 12:14
  • I see the question a little bit philosophical: if a definition is useful, then it is worth using. Nothing prevents to using different definitions of distance in different contexts, and I personally don't find useful to try to find general properties of distance, without context. – Vincenzo Tibullo Sep 10 '20 at 12:22
  • @TheSilverDoe It seems natural, sure but only when I define distance as some form of generalized "Euclidean distance" (or straight line distance). I'm hoping for some kind of "umbrella-definition" for "any" kind of distance *in general* if you catch my drift... – William Sep 10 '20 at 12:23
  • @enzotib Hey, I get it but you know how there are instances where we don't define what "something" is but rather we explain what do we want our "something" to do or have (for eg, by listing down it's properties or something). My motivation is the same. – William Sep 10 '20 at 12:34
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    Regarding your first ground rule, see Examples of non symmetric distances and Are there any “spaces” that violate symmetry of metric spaces? Regarding your second ground rule, see On Lattice Valued Metrics and google searches such as "complex valued metric" and "quaternion+valued+metric". – Dave L. Renfro Sep 10 '20 at 13:16
  • No comment on the concept of uniform space I mentionned ? – Jean Marie Sep 11 '20 at 21:52
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    @JeanMarie Oh no, it was quite helpful. Thank you. But I already have my answer now, all of these comments were very insightful and after Dave's comment, I've come to conclusion that it's simply best to leave it at "defined differently in different contexts". – William Sep 17 '20 at 14:19

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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.

5xum
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  • In topology define a "metric" as a function from pairs of points to the non-negative numbers. That is, the distance between point p and point q is the non-negative number f(p, q) where – user247327 Sep 10 '20 at 12:23
  • That was only an example but fair enough, let's say I do define some kind of "path" whose length will be my measure of how far apart $2$ points are, then I don't see why the length of these paths must satisfy triangle inequality? – William Sep 10 '20 at 12:29
  • @William You don't define a path, you have to define a path for each pair of points.. – 5xum Sep 10 '20 at 15:01
  • @5xum You are not answering my actual question but fair enough. Let's just say the space I have has only three points, $ \text{S} = {A, , B, C}$ and I define paths from each point to the other and back as $P(A,B), P(A,C), P(B,C)$ such that the length of these paths are the distances between the given points. Then I don't see why they should satisfy the traingle inequality. – William Sep 17 '20 at 14:13
  • If the $P(A,C) > P(A,B)+P(B,C)$, then how can we call $P(A,C)$ the actual distance from $A$ to $C$? Why can't a particle move from $A$ to $B$, and then to $C$? It will cover a shorter distance from $A$ to $C$ that way, right? – 5xum Sep 17 '20 at 14:25
  • @5xum it will cover a shorter distance, sure! But who ever said "particles can only travel from one point to the other if the path chosen, is the shortest"? Why can't it choose a longer path? How does the "length" of a path decide whether something can pass through it or not? You are making no sense, sir :'( – William Sep 17 '20 at 15:18
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    If I can get from A to B in 10 km, but there is an alternative path from A to B that is 5km long, I don't see the point in saying that A and B are 10km apart... I mean, look, call your "thing" whatever you want, but if it doesn't satisfy triangle inequality, I fail to see any reason to call it a distance. – 5xum Sep 17 '20 at 19:27