I have begun seeing the phrase "up to" a lot after taking abstract algebra. I usually can figure out what means in context.
For example, $\mathbb Z_4$ equals the set of rotations of a square, up to isomorphism.
I just don't really understand why we use it and what it actually means. The way I interpret it is sort of that, if "isomorphism" were a person, then $\mathbb Z_4$ would equal the rotations of a square, if it were up to them.
Another explanation I've seen on Wikipedia is that $A=B$, up to $x$ means that $A$ would equal $B$ if it weren't for $x$. That just doesn't make sense to me considering the first example I gave.
I hope this isn't too vague of a question, but it seems like the phrase itself is a bit vague. I guess I'm trying to find a casual understanding of how to use the phrase.