I've gone through a bit of an interesting transformation as I've done more and more pure math. When I first began learning about abstract math, especially abstract algebra, I began to think that abstract math is really about structure. When you first see group theory, ring theory, module theory, etc, you start to think that these incredible objects are sort of a "new" type of math, in a sense. These are an abstraction of properties we see in places like arithmetic, symmetries, solutions to polynomial equations, etc, and are attempt to unify all these seemingly disparate ideas.
At a certain point, in my mind, a ring no longer had any connections to numbers, except by the similar rules of formal manipulation. No numbers here, just some arbitrary collection of symbols. Especially when you begin studying things like algebraic geometry and category theory. But now I have begun looking much more seriously into the research that is actually done by mathematicians in areas I am interested in, and it really seems like in a sense, it all comes back to numbers. Algebraic geometry is often studied with arithmetic geometry in mind (I know Langlands program stuff is hot right now), topology seems usually to be thought of in full generality so we can eventually apply it back to C\^n or R\^n, abelian groups are products of Z/nZ, "arbitrary field" in practice really just ends up being C or the algebraic closure of a finite field, etc. In general, it feels like when we study anything algebraic in full generality, the scope of the generality comes down to different types of objects we can construct out of certain sets of numbers.
When people who are not into math say "oh, so you work with numbers?" I have in a sense gone from thinking "well, no, rings and groups for example are much more general" to thinking "well, maybe in a sense, I really am." I'm still an undergraduate and my knowledge is very limited, so I would love to hear perspectives from people with more experience, whether it really does all tie back to numbers in some sense later on, or at least interesting anecdotes about cases where this does or does not happen. Thanks all!
Edit: People seem to be misunderstanding. I am not claiming/asking whether you can prove results without numbers immediately in the picture. I am saying that all this machinery we work with/prove things about, even if not directly related to number theory, seems to have number theory somewhere in the background. Maybe this is just my university, but I have been told by pure algebraic topologists and geometers here who on the surface seem to do work completely unrelated to number theory, things along the line of "the end goal of what we are all studying is Langlands/Riemann hypothesis"