Up to

In mathematics, the phrase up to 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, the phrase "up to X" means "disregarding a possible difference in X". For instance we might follow the statement "an integer's prime factorization is unique up to ordering", meaning that the prime factorization is unique if we disregard the order of the factors; 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.

Up to eleven

"Up to eleven" or "these go to eleven" is an idiom from popular culture, coined in the movie This Is Spinal Tap, where guitarist Nigel Tufnel proudly demonstrates an amplifier whose volume knob is marked from zero to eleven, instead of the usual zero to ten. The primary implication of the reference is one in which things that are essentially the same are seen as differentiated due to the user's misunderstanding of the underlying operating principles. A secondary reference may be anything being exploited to its utmost limits, or apparently exceeding them. Similarly, the expression "turning it up to eleven" refers to the act of taking something to an extreme. In 2002 the phrase entered the Shorter Oxford English Dictionary with the definition "up to maximum volume".

Original scene from This Is Spinal Tap

The phrase was coined in a scene from the 1984 mockumentary/rockumentary This Is Spinal Tap by the character Nigel Tufnel, played by Christopher Guest. In this scene Nigel gives the rockumentary's director, Marty DiBergi, played by Rob Reiner, a tour of his stage equipment. While Nigel is showing Marty his Marshall guitar amplifiers, he points out one in particular whose control knobs all have the highest setting of eleven, unlike standard amplifiers, whose volume settings are typically numbered from 0 to 10, believing that this numbering actually increases the highest volume of the amp ("It's one louder, isn't it"). When Marty asks why the ten setting is not simply set to be louder, Nigel hesitates before responding blankly again, "These go to eleven."