TITLE: Is most of mathematics independent of set theory?
QUESTION [19 upvotes]: Is most of mathematics independent of set theory?  Reading this quote by Noah Schweber:

most of the time in the mathematical literature, we're not even dealing with sets!

it seems that the answer to my question is "yes". But why? When I read in the mathematical literature, sets appear everywhere – we need them in the definitions of groups, rings, vector spaces, ... However, is there some truth in the quote, and in what sense?

REPLY [11 votes]: Most mathematics can be translated into some suitable set theory such as ZFC. That is certainly true! However, it is totally different from the claim that most mathematics deals with sets. From the point of view within ZFC, of course it is true because in ZFC there are nothing else except sets! But in fact, most ordinary mathematics can be translated into a very very weak system called ACA, which has no internal notion of sets of sets. Furthermore, there are alternative systems such as type theories that some logicians even argue to be more natural than ZFC as a foundation for mathematics. That is naturally a subjective opinion, but the objective fact is that there are indeed different formal systems that can 'do the same thing', so to speak, even though one might 'think' of everything as sets while another 'thinks' that there are urelements.
Ultimately I agree with the main point in Qiaochu Yuan's answer. Namely that mathematics is nearly always based on existing structures that obey some properties. For elementary number theory, we do not care what are the internals of each natural number as long as the collection of natural numbers together with the arithmetic operations on them, as a whole, satisfy the Peano axioms. This abstraction would be a valid reason to say that most of them when we use natural numbers we do not actually deal with sets, since it does not matter even if natural numbers are urelements!
Of course, this mirrors the idea in programming that we write high-level programs and do not deal with CPU instructions in the sense that we do not actually care how our programs are translated to CPU instructions. We only care that our program behaves in a way that is solely determined by the high-level programming language. Take Java for example. On different machines it is necessarily translated into different CPU instructions, but that is done by the Java environment; in our terms the Java language is the abstraction that frees us from 'bare-metal' concerns. In the same way our mathematical axiomatizations (like PA for natural numbers) frees us from foundational concerns.
Finally, there is a huge benefit to working with abstractions than with underlying implementations. In case we ever wish to use a different foundational system for whatever reason, abstraction makes it far easier to identify which parts transfer over without change to the new system. We can in fact precisely reason about such general transfer via interpretability.