> "math" is a name for the novel pieces of language ...
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.
Sure, literally, yes, the word "mathematics" is not an -ology; it isn't a discipline concerned chiefly with carving up reality and giving words to the results.
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.
Yes, I might have missed that you used "language" as an allegory rather than a metaphor. I find ironic how this reaffirms the stereotype of the divide between language and mathematics, ie. mathematicians being bad at maths and the other way around, when really the combination of both creates a synergistic effect that helps each to surpass the effectiveness of each on its own.
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.