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A mathematician can define as many dimensions as s/he needs for a given problem. That doesn't make the dimensions meaningful to anyone who isn't directly concerned with the math. Some 2d problems can be made much simpler by treating them as 3d problems. That doesn't change the fact that the problem as an abstraction only exists in 2d. Don't know if this is relevant to this discussion-physics is not a strong suit at the moment.
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