What does 'up to' mean in mathematics?

[Firepanda]

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

Thanks!

 

[M Quack]

"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.".