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ok, so did the term "space expansion" come from the fact that mathematically describing the motion of galaxies worked better in an expanding coordinate system?
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I think so. In these coordinates, if you calculate the expression for "change in proper distance", you get a very simple and suggestive formula. Like "recession = motion of space + motion through space". This took a life on its own, it seems.
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Is there physical evidence that space is expanding or is it just a result of the math?
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There is ample evidence for redshift being more or less proportional to distance. Which means that the universe may be described well as objects more or less at rest in an expanding coordinate system. I don't know how that pertains to evidence for "expanding space".
Yes.
For example, in special relativity you may have heard that it's tricky to
add velocities that different observervers heve measured. In cosmological coordinates, you add them without correction nonetheless. That's easy and appropriate, but the result definitely has nothing to do with "velocity" in the SR sense. Especially, v<c does not apply. That's just the effect of an unusual coordinate system.
Then, in GR, matter determines geometry. You don't even have that static background on which to base your misunderstandings. Is distance changing, or is the light we took to measure distance delayed by some matter that intervened? There's not way to decide such questions.