Talk:Up to

Two meanings:let us discuss
I agree with Taku's changes. Shall we discuss? Oleg Alexandrov 5 July 2005 15:58 (UTC)


 * I agree too. In fact, I'm going to re-apply Taku's changes.  Wikipedia is not a dictionary.  Compare the clumsy cruft that Taku deleted with the excellent Wiktionary entry.  A link to Wiktionary is the right thing here. Jorend 13:09, 10 August 2005 (UTC)


 * I also agree with Taku. --Zero 13:13, 10 August 2005 (UTC)

References needed??
I removed the "no references" tag. It seemed ridiculous to me. I have no idea what kind of reference one should find here. Or would you ask for references in an article that explains what a corollary is? Spaetzle (talk) 08:17, 24 May 2012 (UTC)

Intro needs de-blatherifying
"In mathematics, the phrase up to is useful for modeling fundamental concepts within a realm of mathematical inquiry,"

Er, doesn't every mathematical idea fit this description?

" and can be compared with the phrase "all other things being equal" in other disciplines."

How? I'm pretty sure "up to" means almost the opposite -- "disregarding that the about-to-be-mentioned aspects are different".

" It indicates that its grammatical object is some equivalence class,"

The introduction of a "grammatical object" seems an excruciating contortion.

"to be regarded as a single entity, or disregarded as a single entity. This seems to say nothing useful at all.

I hope someone can shed some light.

Gwideman (talk) 06:05, 14 September 2012 (UTC)


 * I have addressed some of these issues. Feel free to edit the article yourself if you see other potential improvements. 138.16.21.199 (talk) 03:24, 4 December 2012 (UTC)


 * ... and here I am 3.5 years later, attempting an improvement. At least getting rid of the "grammatical object". Gwideman (talk) 17:04, 27 February 2016 (UTC)
 * Some of the examples of the 'meaning' of [equality/equivalence] "up to" suggest that it has a lot of ambiguity and can even mean 'apart from' matters ignored as differences by the relevant 'class' to which an object under discussion is said to relate. So yes, I agree with the poster who said that equal 'up to' can mean almost the opposite. Can anything be done on wikipedia to clarify such obscurantism? Terry0051 (talk) 21:09, 11 June 2022 (UTC)
 * All examples fit into the explanation that is (meanwhile) given in the lead (which may differ considerable from the version in 2012 or 2016). If there are issues with some particular examples, pleas speak up, and I'm willing to try to fix them. - Jochen Burghardt (talk) 13:55, 15 June 2022 (UTC)

Intro figure caption
"20 partitions of a hexagon vertex set with 3 singletons. "

I guess the intent here is partitions with four subsets, three of which are single-element sets, plus one having three elements. It is true that specifying only "3 singletons" nails this down, since the remaining three elements have to be grouped together in a subset because if they weren't then at least one them would form an additional single-element set.

So the "3 singletons" specification is an indirect way of describing the example, made somewhat more obtuse by the coloring which highlights only the one 3-element set explicitly. I have revised the caption to help make all this clear, including the fact that the uncolored single dots indicate single-element subsets.

Note that the figure is still misleading, in that it labels the set of all partitions as "P", whereas customarily "P" is used to represent a single partition, I believe. Gwideman (talk) 16:29, 27 February 2016 (UTC)

Is "up to" ever inclusive, and does it ever imply an ordering of classifications?
The several uses of "up to" that I have seen seem to identify only a single criterion to be ignored in grouping set elements.

However, there are some ways in which the everyday meaning of "up to" trigger some cognitive dissonance, and addressing that dissonance would help understanding.

1. In everyday usage, "up to" is usually inclusive. "You can have up to three cookies", means you can have 0, 1, 2 or 3. It does not mean you can have any number not including 3, or less than 3.

Is it ever used inclusively in a mathematical context?

2. In everyday usage, "up to" implies that there is some ordering. "Mary studied math up to course MAT234", implies that Mary has taken a sequence of courses MAT101, MAT123 and so on, up to (and including) MAT234.

Naively then, the mathematical usage "up to rotation" seems to suggest that there is some progression of distinguishing criteria at work, and those "before" rotation continue to operate, but rotation and those "after" rotation do not. Indeed, where rotation and reflection are involved, a reader might expect that one of those "comes before" the other, so that "up to reflection" might imply that both rotation and reflection are to be ignored.

I'm pretty sure that this would be an incorrect understanding of "up to" in math context, and if so would bear pointing out. Gwideman (talk) 17:27, 27 February 2016 (UTC)

Need a better introduction/overview
The older intro was more clear:


 * 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.


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

maybe just moer cocise