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but does it fix the value of pi in _another_ euclidean geometry as the same as our pi? I don;t per se see how.
I'm pretty sure it does, because the numeric value logically depends only on the definitions of the abstract entities we are discussing. (I am assuming by "another euclidean" geometry you mean some alternate universe with some flat local region of spacetime).
Also I suppose there is a little semantic confusion here because in a lot of math "pi" is just a constant, i.e it's a value that so happens to correspond to that particular ratio in classic geometry but it's adopted as a constant, and so even in non-euclidean spacetime where the ratio between circumference and diameter is different (depends on the curvature, and on a curved surface pi will vary with diameter I believe relative to the curvature) "pi" the math constant is still the same.