A reply to “A FINAL ARGUMENT AGAINST x’ = x – vt”

In one of his short articles on special relativity, Miles Mathis discusses the equation x’ = x – vt.
x is a spatial coordinate in one coordinate system and x’ is a spatial coordinate in a different coordinate system moving with velocity v with respect to the coordinate system whose coordinate is labeled x. You could imagine two observers, one on a platform and one in a train moving with velocity v relative to the platform. Each observer considers his own position to be the origin of his own coordinate system. Miles writes:

But the origin of x’ is not moving. If the origins of the two coordinate systems were together at t0 , then they are still together, since origins don’t move, by definition. This is just to say that if the train started from the station at t0 , then after time t the train still started from the station, which has not moved. Train stations do not move, just as origins do not move: t0’ and x0’ are still back at the origin, which is still back at the train station.

How origins of two different coordinate systems cannot move with respect to each other “by definition” is unclear to us. Can the two observers described above not set up coordinate system in which they themselves are at the origin? It seems that Miles has misunderstood the galilean principle of relativity.

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