Understanding the philosophy behind the axiomatic definition of reals

It seems like a better question to me than some people seem to think. The question of which set we're talking about doesn't matter, but the question of how we know such a structure exists certainly matters. Lemme paraphrase what it seems to me the issue is, in language that people here will understand:

Q: I've read that the reals are a complete ordered field. Ok, one can easily show that any two (Dedekind-)complete ordered fields are isomorphic, so this characterizes the theory of the reals, fine. But how do we know that a complete ordered field exists? After all, if there's no such thing as a complete ordered field then the theory of complete ordered fields seems a little pointless.

A perfectly reasonable mathematical question; no need to snicker about the need for a philosopher. Luckily:

A: There are various methods one can use to show that a complete ordered field exists. One of the best known is via "Dedekind cuts", which you can read about in various places online.

Axiomatic definition like this doesn't tell you “this particular thing is the set of reals”, but rather “we call the reals anything that satisfies the conditions”. It just specifies the interface, not giving any particular implementation.

The point here is that not only some implementation exists, but that it is unique up to isomorphism. This is not part of the definition, it is a theorem that has to be proved. But it justifies the name “the reals” (i.e. using the definite article).

Actually, it doesn't matter which particular set realizes the reals, the structure is important. You can for example look at a simpler case of natural numbers and integers – do you know their actual set theoretic representation? (There are some natural choices, maybe one of them is considered canonical.) Does it really matter? I mean, the structure itself is important, and the fact that it can be realized as a set. But the particular realization is not so important.

Axiomatic definitions are more often used in the situation without uniqueness – vector spaces, metric spaces, topological spaces, Banach spaces – they are axiomatically defined structures/interfaces that abstract over particular instances, so you can formulate some theorems that hold in general.

Your inference "If ... . he would have to turn to the only set that has these properties" is not a mathematical inference and rests on non-mathematical presuppositions. In particular, you simply presuppose that the axiom-system characterizes the real numbers strictly uniquely (up to equality in some universe of sets); this is not the usual point of view nowadays. At most 'uniqueness up to isomorphism of ordered fields' is usually considered, and for that you even forgot to mention a condition: the real numbers are the only complete ordered field, up to isomorphism of ordered fields.

A similar comment is in order in response to your question "Which set is that?" This question presupposes a view which is considered outdated: that any property (like e.g. 'satisfying all the axioms you are currently asking about') would define a set. G. Frege, among others, thought that something similar could be adopted as a principle of thought, but this turned out to be an untenably naive point of view. As user87690 has already hinted at, in usual set-theory there are infinitely-many sets which are isomorphic as fields to the real numbers.

The intuition you evince in " I feel like without a concrete reference of some sort all we're left with is cyclical logic"^(usage) is quite relevant: the Axiom schema of specification arguably arose out of a similar intuition.

Re your "Can anybody help me shed light on the philosophical process of this definition?": the 'philosophical process' is that usually only the isomorphism type is considered essential for mathematical and scientific applications of $\mathbb{R}$, while the set-theoretic realization is considered a, as it were, set-and-forget^(pun) issue of (model-)theoretic interest only. Relevant keywords with which you can learn more are 'axiomatic method' and 'model theory' (where a relevant topic to learn about is categoricity).

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^(pun) Pun intended. The model is usually taken to be a set which for most intents and purposes you then forget.

^(usage) By the way, 'circular logic', or 'circular reasoning' is a much more usual collocation in English.