steve waterman wrote:equating x with x',

vt = 3

S(2,0) = P

S'(-1,0) = P

therefore,

2 = -1?

NB: I'm going to ignore the error in the number of dimensions in this post, and simply add an extra ",0" to all 2d coordinates in grey.

If x' = x - vt

and vt ≠ 0

then x' ≠ x.

The mistake is in assuming that S(x,y,z) = S'(x',y',z') implies x = x'.

steve waterman wrote:equating point P relative coordinate sets

vt = 3

S(2,0) = P

S'(-1,0) = P

therefore, in the manifold

S(2,0) = S'(-1,0) = P

However, since S(x,y,z) = S'(x',y',z') was given, it is also quite true that S(2,0) = S'(2,0).

No... How? How does S(x,y,z) = S'(x',y',z') mean S(2,0,0) = S'(2,0,0)?

steve waterman wrote:So, S(2,0) = S'(2,0) = S'(-1,0)?

edit - took out the quote, which I had misread. The above is stand-alone.

added -

ucim -

The Galilean has no double prime notation/logic, nor do I.

Hence, why I backed down on any possible M(x,y,z)...no such thing.

The Galilean has no inherent labels. The primes and the double primes and the 'x'es and the 'y's and the 't's and the 'z's have no inherent meaning beyond simply being placeholders.

When we say something like

"Given two systems S(x,y,z) and S'(x',y',z') on a manifold M"

It is the term "S(x,y,z)" that defines x, y, and z, and the term "S'(x',y',z')" that defines x', y', and z'.

It is the statement "Given two systems S(x,y,z) and S'(x',y',z') on a manifold M" as a whole that defines S, S', and M.

The assignment of labels can be done however we wish, although consistency is useful. The labels are not magical.

Statements like "x' = x - v*t" were not discovered written on a cave wall or handed down from above to be decoded. The meaning was discovered first, and then expressed algebraically.

Edit: If two systems are spacially coincident, they are by definition not separated, aren't they?