Is it an abuse of language to say "*the* integers," "*the* rational numbers," or "*the* real numbers," etc.?

Solution 1:

How you probably should think about this, most of the time -- and certainly the approach that seems to be most productive for most ordinary mathematics -- is that the integers, real numbers and even complex numbers exist in and of themselves in some "Platonic" sense, independently of what we think of them.

The set-theoretic constructions you find in textbooks construct models of the Platonically existing numbers within a pure set theory. Knowing how to do this -- and in particular knowing that it can be done -- is important and brings with it many technical advantages, but you should not let it trick you into thinking that these set-theoretic models are "what numbers really are". Doing so would make a mockery of the centuries and millennia where mathematicians reasoned about numbers without set theory even having been invented. It would be absurd that Dedekind, Cantor, Zermelo, and other mathematicians working in the 19th and 20th centuries were for the first time discovering what Euclid, Euler, Gauss and so forth had really been talking about.

Thus for ordinary mathematics the most useful approach is to continue thinking that all of the real numbers (and the complex ones too, if you can bring yourself to it) just exist, that you can put them all into a set, and that the rationals and integers are particular subsets of them.

Of course this "naive" view is not sufficient in order to work in axiomatic set theory, logic, or similar foundational areas of mathematics. There you're interested in how well the set-theoretic formalism can capture our naive ideas about integers and real numbers, and perhaps even in to which extent the usual Platonic belief in the existence of numbers can be upheld when one really thinks about it.

It it good to be able to ask such philosophical questions, and to know enough to begin answering them.

All I'm saying here is that it is not productive to let such philosophical uncertainty paralyze you when doing mathematics outside the foundational domain. One of the benefits about knowing about foundational theories is that it allows you to relax and know that whichever philosophical objections one might have to such-and-such argument in everyday mathematics can be handled another day, because we have a good model of the fundamental assumptions that everyday mathematics depends on.

Solution 2:

Is this a matter of learning to no longer think of these sets or structures as unique, but rather unique up to isomorphism? Or unique up to unique isomorphism?

Yes. And depending on what you mean by "the", it's not even an abuse of language!

(this post assumes some familiarity with the ideas of category theory)

The "uniqueness" of the real numbers is a theorem that says if $A$ and $B$ are any two complete, ordered fields, then there is a unique isomorphism $A \to B$.

Now, consider the following two full subcategories of the category of rings:

• the category $\mathcal{A}$ of all complete ordered fields and isomorphisms between them
• the category $\mathcal{B}$ consisting just of the complete ordered field $\mathbb{R}$ and its identity map

The uniqueness theorem above can be rephrased as saying that the categories $\mathcal{A}$ and $\mathcal{B}$ are equivalent.

There is a general philosophy in category theory that equivalence is the "right" notion, rather than equality or isomorphism. If we had a suitable language for doing mathematics, there wouldn't be any distinguishing between $\mathcal{A}$ and $\mathcal{B}$: it would be perfectly correct to speak of the complete ordered field.

From this perspective, that that is not the current state of affairs and instead we have to juggle transporting the mathematics we do along unique isomorphisms is simply an unfortunate accident of history that elevated the notion of identity over that of equivalence and that had an impact on how mathematics was developed.

Solution 3:

It all hangs on the isomorphism theorems at each level. They show that any two sets with the properties of N, Z, Q, R, C are isomorphic. Of course, those are isomorphisms that preserve sums, products, order , completeness, whenever that structures are defined.

Answering your question more directly: Yes, it is an abuse of language, but it is justified because of the isomorphism theorems. Proving those will be an excellent exercise in your understanding of the subject.