What does it mean for something to hold "up to isomorphism"?

Solution 1:

A good example to add. "There is only one group of order 2 up to isomorphism". The group is $G=\{e,a\}$ with $ea=ae=a$ and $a^2=e$. Now you could argue "I have another group, $H=\{e,b\}$ with $eb=be=b$ and $b^2=e$". Your group IS different than mine, but your group is just the renamed version of mine. "Renaming" is isomorphism.

Solution 2:

More generally, "equal up to $\sim$", where $\sim$ is some first-order equivalence relation, means that we have temporarily redefined equality so that $x$ and $y$ are "equal" or "equivalent" if $x\sim y$. A property $P$ "holds up to $\sim$" or "holds modulo $\sim$" if $P$ depends only on the equivalence class.

(By first-order equivalence relation, I mean a formula at the first-order logic level which has the properties of an equivalence relation. If there is a set which contains all of the objects under study, then this technicality may be ignored, and a normal equivalence relation will do.)

Equivalences let us look at sets in a way which hides irrelevant detail. As Timothy Gowers wrote in Mathematics: A Very Short Introduction, "a mathematical object is what it does." In algebra we have isomorphisms, in topology we have homeomorphisms, in differential geometry we have diffeomorphisms, in algebraic topology we have homotopy equivalence, and in category theory we have natural isomorphisms. All of these preserve exactly the structure that can be measured when working with the set by some particular axioms. For instance, two groups are "the same" (equal up to isomorphism) if one cannot tell the difference between them when composing or inverting elements.

One great benefit to using equivalence rather than equality is uniqueness. Otherwise it would be more difficult to express something such as "there is exactly one group with seven elements [up to isomorphism]." Of course there are many groups of seven elements, but all of them are the same as far as being a group is concerned.

A metamathematical aside: when we speak of the basic concept of equality of sets, two sets are considered equal if and only if they have the same elements, and so they are "equal up to extensionality" (though they might "really" be different, but not in any way we care about).

A practical aside: computer programming languages give an interface for determining equivalence between two objects, and there are many ways to approach this. Some options include: do they occupy the same memory location? do they have the same representation? what does their decision procedure for determining equality say? Whatever choice is made determines in which mathematical universe the language computes.