Up To

In mathematics, the phrase up to is useful for modeling fundamental concepts within a realm of mathematical inquiry, and can be compared with the phrase "all other things being equal" in other disciplines. It indicates that its grammatical object is some equivalence class, to be regarded as a single entity, or disregarded as a single entity. If this object is a class of transformations (such as "isomorphism" or "permutation"), it implies the equivalence of objects one of which is the image of the other under such a transformation.

If X is some property or process, a translation of "up to X" is "disregarding a possible difference in X". For instance we might follow the statement "an integer can be factored as a product of prime numbers" with "the product is unique up to ordering", meaning the order of the operands is irrelevant, integers and their prime factorization are; or we might say "the solution to an indefinite integral is f(x), up to addition by a constant", meaning that the added constant is not the focus here, the solution f(x) is, and that the addition of a constant is to be regarded as a background, of secondary focus. Further examples concerning up to isomorphism, up to permutations and up to rotations are described below.

In informal contexts, mathematicians often use the word modulo (or simply "mod") for similar purposes, as in "modulo isomorphism".