I was wondering about symmetries in physics, and have a couple of questions.

The symmetries of space/time translation and rotation are often explained in terms of doing an experiment e.g. 10m away instead of right here and getting the same result, as long as we are in the same physical situation. This has always bugged me a bit, because if these symmetries are exact then in principle, then e.g. everything in the universe would have to be moved 10m (otherwise we could detect a difference in our result), and so who would know we've done anything?

This fits with using geometrical objects to describe things, e.g. tensors in GR. This practice says to me that "we assume these objects exist in their own right, and it doesn't matter what bit of maths we use to describe its position" - it is a statement of belief about the universe, and has little to do with shifting experiments around. Is this a reasonable view?

As I understand it (albeit badly) allowing all coordinate systems to be valid in GR leads to this gauge symmetry, which is something to do with ambiguity in our description of things. Is there some way to fit the gauge symmetry of electrodynamics (and other non spacetime symmetries for that matter) into the rationale above? And is all of this stuff just a strange artefact of using ambiguous maths?

Cheers,

Tom