What does 'up to' mean in mathematics?

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The term 'up to' in mathematics signifies that multiple solutions exist, but they share a fundamental equivalence. For instance, when stating "the solution is unique up to isomorphisms," it indicates that while several solutions may be present, they are isomorphic to one another. A practical example is the Rubik's Cube, which has 24 distinct configurations, but all can be considered the same when accounting for rotations. Thus, 'up to' emphasizes the equivalence of solutions under certain transformations.

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In the context such as 'up to isomorphism' or 'up to reordering', I've seen it a lot but never understood.

Thanks!
 
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"The solution is unique up to isomorphisms" means that there are several solutions, but they are all isomorphic.

Think of Rubic's cube. There are really 24 solutions (6 different faces can point up, and then you have 4 different face that can point forward). But essentially ther are all the same. You could say "there is one solution to Rubic's cube, up to rotations of the entire cube.".
 

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