What does "up to" mean?

Question

There are several prior questions of the form "What does 'up to X' mean?" The answers generally focus on "X", which has led some commentators to ask "What does 'up to' mean?" but in answer to that question, key to all the others, I've found no direct answers if not deafening silence.

One way of elaborating, so as to clarify, the question "What does 'up to' mean?" might be:

The phrase "up to" indicates motion along a dimension up to a point and not beyond. So there is more at stake here than talking about the limiting point.

What exactly is this dimension?

Answer 1

"Up to" or "apart from" aren't precisely defined mathematical terms. They're "English padding" to quickly convey what we mean. For example, consider the fundamental theorem of arithmetic:

"Every integer $n\geq 2$ can be written uniquely, up to reordering of the factors, as a product of primes.

or a slightly different phrasing:

"Every integer $n\geq 2$ can be written uniquely, apart from the order of the factors, as a product of primes.

or yet another phrasing:

"Every integer $n\geq 2$ can be written uniquely, modulo the order of factors, as a product of primes.

One typically does not wish to write out the more formal and cumbersome statement which avoids the phrase "up to" or "apart from"

For every integer $n\geq 2$, there exists $k\in\Bbb{N}$, and prime numbers $p_1,\dots, p_k$ such that $n=\prod_{i=1}^kp_i$. Furthermore if $m\in\Bbb{N}$ and $q_1,\dots, q_m$ are primes such that $n=\prod_{j=1}^mq_j$, then $m=k$ and there is a bijection $\sigma:\{1,\dots, k\}\to\{1,\dots, k\}$ such that for every $i\in \{1,\dots, k\}$, we have $p_i=q_{\sigma(i)}$.

In this formulation, we have completely avoided any "informal" language (ok one bit of informality left in the statement is my use of $\dots$, but I think we can side-step this minor detail), and used only "$\forall$ and $\exists$ and $\implies$ and $=$".

So, "up to" in this case is not referring to an "upward motion" along some axis/set or anything like that. So this has little to do with any dort of convergence/limits of any kind.

Answer 2

One extremely vague and informal way to think about it is that equality is the strictest sense in which two objects can be the same. So, we are "going the furthest" in terms of how much we require the two things to be similar, in that they are exactly identical. By saying "up to", we mean that we aren't going that far, and some weaker notion of sameness is enough.

For example, "there is one group of order 3 up to isomorphism" means, okay, there may be many groups of order 3 which aren't equal (e.g. group of rotational symmetries of a triangle, and group of integers mod 3), but they're the same if we "stop at" isomorphism and don't be so strict as to require equality. This is how I always thought of the term "up to", anyway.