steve waterman wrote:in the Galilean, ( as per their own given ), x is always in Red and x' is always in Blue.
They say that x is the coordinates of some point in the Red coordinate system, and x' is the coordinates of that same point in the Blue coordinate system.
the coordinate values of x' in Blue = the coordinate values of x in Red - d
In the sense that the point at coordinates x' in Blue is the same point as that at coordinates x in Red, sure.
The numerical values of those coordinate will not (necessarily) be the same unless the two coordinate systems are identical, and nobody claims that they will. (Of course they might in some special cases be the same, like how the temperature -40C = -40F.)
They only claim that if you travel some distance (move at some velocity for some time), the same things will end up at different coordinates in your frame of reference than they were at before you moved. This is... so incredibly trivial I can barely believe we're actually arguing about it.
I am running down a street, headed due north, at 4m/s (v = 4m/s N). As the clock strikes noon, there is a reflector dot 20m due north of me (x = 20m N), or more generally, of the origin of the frame of reference centered on where I am at that moment; call the coordinates of that reference frame "red" if you like. I continue running for another 2s (t = 2s), thus traveling a distance of 8m in that time (d = 8m). The reflector dot is at that time 12m due north of me (x' = 12m N), or more generally, of the origin of the frame of reference centered on me at that
moment; call the coordinates of that reference frame "blue" if you like.
To say what the location of the reflector dot in Blue coordinates is relative to its location in Red coordinates, we could say:
x' = x - vt
Plug in the values in our word problem:
12m N = 20m N - 4m/s N * 2s.
And simplify the right to check, starting with reducing it to your "mathematically purer" x' = x - d
12m N = 20m N - 8m N
12m N = (20-8)m N
12m N = 12m N
Hey look, it works out. This is my surprised face.
Nobody is saying that 20m N = 12m N. They're saying that anything that was 20m N of you will be 12m N of you if you move 8m N. If you really think anybody is claiming 20m N = 12m N, well... there's your problem.