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It's been argued here on MSE that adding points doesn't make sense (e.g. Ray Toal's answer here: What is the difference between a point and a vector?). The reasoning was that points can be thought of as locations, and vectors can be thought of as displacements. So, adding e.g. Moscow and Los Angeles shouldn't make sense.

But upon considering the fact that the difference between two points is a vector, I came up with the following counterexample. Let $M=[m_1,m_2]$ be the location of Moscow and let $L=[l_1,l_2]$ be the location of Los Angeles. The difference of the locations is a vector, i.e. $$L-M=[l_1,l_2]-[m_1,m_2]=(l_1-m_1,l_2-m_2)$$ (notice the change of brackets). Now, adding $L$ and $M$: $$L+M=L-(-M)=[l_1,l_2]-(-[m_1,m_2])=[l_1,l_2]-[-m_1,-m_2]=(l_1+m_1,l_2+m_2)$$ which clearly is something. Also, ignore Earth's curvature.

So, why do people say that one can subtract two points but not add two points? Do they forget that addition can be turned into subtraction?

UraUra
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    $-M=0-M$ requires fixing the origin, which is exactly what you want to avoid. If you fix the origin, then all linear combinations of points make sense but if you prefer to rely only on the affine structure, then only linear combinations with zero sum of coefficients are well-defined. – fedja Jan 11 '20 at 21:27
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    Let me add for the record that I appreciate that you placed your examples on Earth rather than on an abstract mathematical plane, because on a sphere the point $-M$ does have a natural meaning: it is the antipode! – Vincent Jan 11 '20 at 21:34
  • So $L+M=L-(-M)$ is false when the origin is not fixed? Also, what is affine structure? I'm not familiar with that concept. – UraUra Jan 11 '20 at 21:35
  • @Vincent Of course, I planned to but then forgot to mention that we ignore Earth's curvature. – UraUra Jan 11 '20 at 21:36

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In an Euclidian metric, the difference between two points is invariant of the location of the origin, while it is not for addition. Thus considering displacement is ok, but adding to get two length depends heavily on the origin which isn't useful.

PackSciences
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I would say that the statement "$L + M = L - (-M)$" is not so much false as it is undefined: when we talk about the plane as a bunch of points then there isn't a natural 'partner point' $-M$ for every point $M$. (Fedja above offers a way out by writing $-M = 0 - M$ but that would make $-M$ a vector rather than a point).

Geometrically speaking there is no 'standard' $-M$ for every $M$. If you want to define it by something like '$-M$ is the point you get by rotating $M$ 180 degrees around the origin' then you run into the trouble the other other two answers are talking about: it singles out one arbitraty point (the origin) in your plane and makes it really important for no particular reason.

Vincent
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  • The origin is already really important, isn't it? The points $L$ and $M$ are defined with respect to the origin. "Go $l_1$ units horizontally from the origin, then $m_1$ units vertically, and that is the desired location." – UraUra Jan 12 '20 at 00:53
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    @UraUra You appear to be thinking of points as coordinate tuples in some distinguished coordinate system. That coordinate system of course has an origin, but the goal here is to have a “coordinate-free” notion of points and vectors. There is no coordinate system in the first place, thus no origin, either. – amd Jan 12 '20 at 02:51