# How did 'up to' evolve to mean 'regardless of', in maths?

Even the OED seems not to have featured it. I couldn't find an explanation on Etymonline.

[Wikipedia:] If X is some property or process, the phrase "up to X" means "disregarding a possible difference in X".

So how did up + to combine to mean thus? This especial mathematical meaning seems to jar with its regular meanings in English (which connote some upper bound).

Footnote: If anyone knows of a better reference than Wikipedia, please advise.

• Mathematical senses are often at odds with common senses. You'll hafta ask a historian of mathematics who's familiar with that particular term; there aren't very many. Commented May 6, 2015 at 18:29
• The moderators recommend identifying the source of links to immune posts against link-rot. Edited your post for this. Commented May 6, 2015 at 18:30
• I'd say it's "equal [for the purpose at hand, with a difference] up to X [which is irrelevant in context]", but I'm no authoritative source. So you're right, it is an upper bound, an upper bound on the difference. Commented May 6, 2015 at 18:39
• You might get more informed answers at History of Science and Mathematics. Commented Jun 5, 2015 at 23:02

To my mind, the "mathematical" meaning of up to quite accords with the "general" meaning, and also with your intuition of an upper bound.

For example, the stipulation that "the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) is only defined up to an additive constant" [Wikipedia], it is natural to perceive the constant differences there can be between the antiderivatives as upper bounds on the difference.

Likewise, the term adequality, used by Fermat, can arguably be read (translated) as equality up to an infinitesimal (i.e., difference by at most an infinitesimal), as hinted here.

Footnote: If anyone knows of a better reference than Wikipedia, please advise.

Math SE does better with their How do I understand the meaning of the phrase “up to~” in mathematics? question (or a possible duplicate). There is a good answer and I will quote the first sentence. For the full experience, I really suggest going over there:

When one says "X is true up to Y", then one means that it is not strictly speaking correct that X is true, and the Y which occurs after the up to clarifies the sense in which X is not quite true.

Taking into account that answer, I find your characterisation "regardless of" not to be so fitting. A simpler example would be to say that and E are the same up to rotation (well maybe not on closer inspection, but they are up to homeomorphism for sure!). In that case, I wouldn't substitute up to for regardless of.

So how did up + to combine to mean thus? This especial mathematical meaning seems to jar with its regular meanings in English (which connote some upper bound).

I'd say that this isn't case. One could see it as follows: up to a certain criterium, two things can be the same, they are the same up to a certain point. For example, suppose we have identical apartments (indistinguishable from the inside) in different locations, then our apartments are the same up to the interior. Should we look further, beyond their upper bound (e.g. the ceiling of our building), then we find they are in fact different.

Indeed, the mathematical definition is broader. We could have the same car up to the colour (which doesn't necessarily have anything to do with topological spaces), meaning our cars have the same specification except for (possibly) their colours. Also, if they are identical with the same colour, they would still be the same up to the colour (at least from a mathematical perspective).