"Every thing is what it is, and not another thing"

In general, I'm not keen on reductionism. Bad for the health. That isn't to say we don't know how to reduce $A$s to $B$s (or, reduce theory $T$ in $L$ to theory $T^{\prime}$ in $L^{\prime}$, or, to interpret one structure $\mathcal{A}$ in another $\mathcal{B}$).

One can reduce natural numbers to finite sets (usually, reduce $\mathbb{N}$ to $\omega$.) One can reduce integers to equivalence classes of pairs of natural numbers. For example, identify $\mathbb{Z}$ with the set $\{[(n, m)]: n, m \in \mathbb{N}\}$, where $[(n, m)]$ is the equivalence class of pairs $(k, p)$ such that $n+p = k+m$. The intuitive idea is that the ordered pair $(n,m)$ represents the integer $n-m$. But there are infinitely many such; so we take the equivalence class of all. By a similar method (this time with $\times$ instead of $+$), we get the rationals, $\mathbb{Q}$. One can reduce real numbers to equivalence classes of Cauchy sequences of rationals, or to Dedekind cuts of rationals. One can reduce the ordered pair $(x,y)$ to some set, say $\{\{x\}, \{x, y\}\}$. One can reduce ordered pairs of numbers (and, more generally, finite sequences of numbers) to numbers. See what I did? - I conflated ordered $n$-tuples with finite sequences. No matter: we can reduce either to the other. And relations can be reduced to sets of ordered $n$-tuples and functions to special relations satisfying a uniqueness condition on one argument place. One can reduce ordinals to transitive $\epsilon$-well-founded sets and reduce cardinals to ordinals.

Even so, I don't think numbers are sets; I don't think real numbers are equivalence classes of Cauchy sequences; etc. These reductions are examples of interpretations or encodings - successful ones, but generally non-unique. This is the message of Paul Benacerraf's famous 1965 article, "What Numbers Could Not Be" (Philosophical Review 74). Given the non-uniqueness of these reductions, one can perhaps try and argue that one is special or distinguished, but this is quite implausible (despite the convenience of identifying $\mathbb{N}$ with $\omega$.) Or one can argue that there are no numbers, integers, ordered pairs, etc., and all there are are sets or classes (and so we consider mathematical theories to be convenient definitional extensions of standard set theory, while agreeing that these extensions are not unique). That seems to be Quine's view. Or one can think of these special mathematical entities as sui generis. Or one can think of them structurally (e.g., to be a real number is to be a node in the abstract real number structure). This is Shapiro's and Resnik's view. Or one can be a nominalist (there are no natural numbers, rationals, reals, sets, functions, etc.). This is Field's view.

(Constructivism, as an ontological claim (e.g., Heyting), seems indefensible: if numbers are "mental constructions", then either there are infinitely many mental constructions (one for $0$, one for $1$, and so on) or only finite many numbers. The former contradicts empirical fact and the second contradicts mathematical fact. Perhaps numbers are "possible" mental constructions, in the mind of God? Well, then one might as well just admit that they are not constructions after all. Constructivism, as an epistemological claim, concerning which proofs are legitimate, is not affected by this argument.)

The views are: sets are basic and there's a special reduction of mathematical entities to sets (implausible, I think); there are only sets and classes and all other mathematical entities are introduced by convenient but non-unique definitions (Quine); different kinds of mathematical entities are sui generis; mathematical entities are nodes/places/positions in abstract structures (Shapiro); mathematical entities are mental constructions (Heyting); there aren't any mathematical entities (Goodman, Field).

Of these, I think my preferred view is the sui generis one. Nominalism and constructivism are inconsistent with science. Quine's view is too austere. Structuralism is a bit too weird (the "nodes" in the domain of an abstract structure have an odd status, but maybe structuralism is sui generis-ism in disguise).

So, the natural numbers are just what they are, not another thing, which brings me to Bishop Butler:

If the observation be true, it follows, that self-love and benevolence, virtue and interest, are not to be opposed, but only to be distinguished from each other; in the same way as virtue and any other particular affection, love of arts, suppose, are to be distinguished. Every thing is what it is, and not another thing. (Bishop Joseph Butler 1726, Fifteen Sermons Preached at the Rolls Chapel. Preface.)

Ok, so that's not mathematics, I admit; but, as Tom Lehrer once joked, the idea's the important thing.