steve waterman wrote:ucim wrote:What does it mean for a coordinate system to be "coincident"?
Think about the question, and keep in mind Schrollini's first lesson; that is, separating the idea of a manifold from the idea of a coordinate system, and remembering that points live in a manifold, not in a coordinate system,
So, does that make it true? The idea, is that I indeed DISAGREE and I start with a different set of initial premises.
Then you are doing a different kind of mathematics. The stuff in Schrollini's first lesson underlies the abbreviated notation you found and are using for the Galilean. If you reject it, then the notation ceases to actually mean what it is supposed to mean.
steve waterman wrote:given S(x,y,z) and only S(x,y,z)...
Do any points exist "on top of, but not inherent to" ( according to Scrollini ) manifold M?
If we removed manifold M as a thought experiment, what would remain? Would we know exactly where to place S(2,3,4) wrt S(0,0,0)?
I'm not sure what you mean by S(x,y,z), although I know what others often mean by it. But in any case, the answer to your second question ("Do any points..") is no. All the points are in the manifold. The manifold is just a pile of points, with no structure1, no order. It's the only place where the points "are". And if we removed the manifold as a thought experiment, you would have nothing useful... essentially, you'd have the card catalog to a library that does not exist. Your question: Would we know exactly where to place S(2,3,4) wrt S(0,0,0)? is actually backwards, for this reason. S(2,3,4) isn't "placed" anywhere. S(2,3,4) is the card in the card catalog that identifies a specific point in the manifold (book in the library). Without the manifold, there wouldn't be a point for S(2,3,4) to refer to. The same is true of S(0,0,0). No manifold, no point.
Remember also that points are not located "with respect to" other points. They are just piled up in the manifold, jumbled up. A coordinate system gives you the ability to identify them "from outside", so to speak, but if you want to determine distance between points, you need to decide what that means first. That involves a metric. (Key takeaway: a coordinate system, by itself, does not have a metric. You need to explicitly pick one.)
eta: 1not quite true, but close enough for now