I'm keen for etymologies and hence believe that math means the art of problemsolving or learning successfully. That includes language as a problem domain, and teaching tool for learning, but as modern development would have it, it's about structure and organization, not just in language.
I was just trying to highlight the fact that mathematics as an institution is a process of building up share-able symbolic abstractions; of inventing a "language" one new word at a time.
I'm using "language" here, and above, to refer not to the words used to discuss mathematics (the... "mathematology" of math), but rather to the thing that includes objects like mathematical operators (e.g. "+", "⨯", "∫", "⇒") as its "words." Not the language about mathematics, but rather the language that is mathematics: the ever-growing set of abstract tools with symbolic handles which we've constructed to allow us to manipulate other concepts inside our heads, in rigorous ways where you can trust that if you and another mathematician do the same named mental 'move' to the same source concept, you will both arrive at the same destination concept as a result.
For a cute analogy: you can think of a martial art as a vocabulary of known, precise body movements, that can be taught. You can think of mathematics as a vocabulary of known, precise mental movements, that can be taught. Yes, this makes mathematics an art; but, equivalently, this makes a martial art a language.
To sum up:
• Mathematics is itself a (formal) language. It doesn't really fit in the category of words like "biology"; it fits more in with words like "logic" or "C++".
• To say that someone is "doing" Mathematics just means that they are using that language to achieve a goal; it's about the same as saying that someone is "doing" Python.
• To say someone is a mathematician, is to say that that person works to explore and extend the language of Mathematics, to test its properties and its limits, and to invent new 'words' within it that may then be used by those "doing" Mathematics.
Math is where we invent language to refer to new forms of abstraction themselves: novel possible shapes for our thoughts to take when we think about other things. You can't talk about a new shape by analogy to existing shapes. Nor can you abstract an abstraction in a way that gives you anything more familiar. (Instead, you'll usually get more novel, ontologically-primitive abstractions, like going from monads to arrows, or going from numbers to fields to rings.)
Sometimes disciplines like physics will find a concept that we don't have any mental tools in our toolkit for yet. Then we build some. But we still refer to the process of doing that as doing mathematics—and then we apply that new mathematics back in the problem domain to talk about the new concept.
(For a good example, the formalization of quantum theory in physics, required the creation of infinite dimensional analysis in mathematics. Physicists did most of that work, but the work itself was still mathematics, not physics.)
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Now, other fields do still have a similar problem to mathematics, of historical momentum carrying forward old "things" (again: concepts, relationships, laws) when there are better, clearer "things" that could be used in their place. But when we're not working with pure abstractions—ways of thinking—we can make the effort to compare and contrast old and new "things", and decide that some might be more edifying than others.
It is possible for an especially-gifted physics teacher to write a very accessible physics textbook, because they need only pick all the clearest "things" to demonstrate. That teacher will still be stuck in a given paradigm—a way of thinking, a belief in the worth of some "things" over others, popular in the culture of their discipline at the time and place they worked. But they might be able to (barely) rise above it, if they think hard about the history of their discipline and the paradigms it has gone through in the past, compare-and-contrast those, and synthesize something that isn't quite just the paradigm they're immersed in.
Mathematics is uniquely problematic because it is entirely paradigm. It is a tree of paradigms—each new abstraction only making sense assuming the paradigm it was created in, and then becoming the paradigm for further abstractions still. Every mathematician, all the time, is trying to discover what a particular paradigm—their specialization—can be twisted to accomplish. Not one of us has the brain power to know the total space of things that one of these ways of thinking can be used to express—the problem domains the tool is applicable to—in order to know which tools show more or less promise at being "powerful." We know what we've discovered so far, but we have such an infinitesimal idea of the "space of all possible abstractions" that we could be totally missing some of the best, and using ones that are barely satisfactory.
Point a hypercomputer AI at "solving physics", and it'll spit out a description of the universe that will certainly have more "things" in it than we know about today—but which will still also contain a subset of the "things" we do have. (The most "carving nature at its joints"ing ones, presumably.) Those "things" that get carried over will, of course, be defined much more precisely; their concept-boundaries will be adjusted to include or exclude edge-cases we weren't aware of. But they'll be the same "things."
Point a hypercomputer AI at "solving mathematics"—of giving us the most powerful abstractions possible to solve problems with—and the result might be entirely unlike anything we've invented so far. It might not share any resemblance to current mathematics, beyond a shared fondness for sets. (For a silly hypothetical: it might turn out that there's a better abstraction for thinking about relationships between things than "functions", and that all the rest of mathematics and physics and whatever other problem domains simplify greatly if we think about the relationships in them in terms of that abstraction, instead of trying to specify relationships using the "function" abstraction.)
A way to think about this process: "math" isn't quite a tower of abstract concepts; those only exist in any given mathematician's head, and die with them. Instead, "math" is a name for the novel pieces of language we invent, with the purpose of using them to talk about and (hopefully) share those internal abstract concepts, exposing to the rest of the world concepts that were entirely inaccessible (to everyone but the originator) before that language was introduced. Math is the attempt to communicate never-before-communicated epiphanies; to describe the shapes of never-before-pondered abstractions, with properties only describable using never-before-spoken words.
One thing that this implies, is that the first piece of language that lets us even vaguely point at some particular idea so that we can get to work on analyzing it further, might stick around long after we come up with some clearer or more coherent language, because the former one now has the momentum of historical use behind it.
Math, when done this way, then becomes a precarious pile of "just good enough to survive" conceptualizations, rather than a precise tower of "best-tool-for-the-job" conceptualizations. And that's no good for teaching.