When you do math you invent new language, ie you add notation and describe what they mean. The math is the actual concepts, not the language you create, an other person could have structured his language for this problem very differently yet ultimately describing the same thing.
Edit: To clarify better, the way you solve math is that first you solve the problem, then you figure out how to communicate the solution. This can either use language people created before or you create new language, you can just use English or you can just draw pictures and have no words at all. Either way you are doing maths.
I'd clarify that when people talk about "a different language", they're usually describing the notation used in formal mathematics. But that's a comparatively recent phenomena (most of it from the last century). For most of its history, mathematics was (and still is, most mathematics papers have far fewer uses of the notation than most laypeople would think) communicated with the natural language of its authors and their readers. The problem was that natural language was frequently too imprecise to easily describe things formally (inclusive vs exclusive or is an example), so mathematics writing became stilted and full of jargon and structural formalisms: "consider two values that are natural numbers and neither of which divides the other without remainder and ..." versus "$$ \{x,y \mid x,y \in \mathbb{Z}_+ \ \mathbb{and}\ x \perp y \}$$" (Edit: wait, Hacker News doesn't support MathJax?) Modern mathematics notation is more to simplify clear communication rather than to describe things that can't be described with words. That is, mathematical notation has far more in common with shorthand than it has with Esperanto, and I think that's an important distinction. When I read set-builder notation, what I "hear" in my head is English. (Also, you're begging the question by implying that there is an external concept/thing that the various languages and pictures are referencing)
When people talk about different language, they mean more than just notation. For example, I can write "2 | 4" as "4 = 0 in Z/2Z". There is more than just a notational difference here.
A more non-trivial example: Say you want to say that 2 x 4 = 8. One way is to just write 2 x 4 = 8. Another way is to say that a finite set of cardinality 8 is the product of a finite set of cardinality 2 and cardinality 4 in the category of finite sets (in the sense that 8 satisfies the universal property of 2 x 4).
How is your example different from claiming that the statements "Aaron and Elizabeth" and "My friends whose names begin with vowels" are statements in different languages? I also note that with the exception of the symbols for multiply and equality, your final statement is written entirely in (an admittedly stilted and formal dialect of) English. I'm not saying that Mathematics is anything but beautiful and expressive---but the only way to justify a declaration that it's a distinct language is to do irreparable harm to the meaning of the word "language". And, of course, you make my point in your first sentence. I agree that what we call language is more than just notation. Which is one of several reasons (among them concepts related to things like mutual intelligibility, orthography vs. grammar, native speakers, effortless learning during early childhood, communication as an evolutionarily selectable trait, ...) that I disagree that mathematical statements or reasoning are the same species of thing as British Received English, American Sign Language or even Elvish and Klingon.
Well, if we're allowed to just redefine the terms, you can make and prove whatever claim you wish, but its unsatisfying and ultimately pointless. I'm not sure why you think the above (which, I'm sorry, parses as gibberish for me) is useful given my assertion that mathematics is not the same species of thing as human language, as opposed to whatever the thing you're describing above is.
You don't really "solve" math. You apply math to a concrete model. For example, if you want to build a four-sided pyramid and all you have is a really big straight edge and a really big collapsible compass, you'd probably want to read Euclid first. And if you happen to be doing this 2000 years before Euclid you'd read some of the geometric methods his work was based on. To the extent the surface you're building on is "flat" the axioms and the theorems apply. And so you can square a circle easily using simple tools like cord and stakes. Thus you can build a pyramid with some cordage and a stake.
Doesn't this still depend on what specifically "math" is referring to? Sure there is some underlying concept, but when people use the word math I think they are largely referring to the language that has been/continues to be built up to describe those concepts. Without at least learning the existing language one would have to rebuild a ton of foundations themselves before they could do much of anything. I suppose you could say the same thing about e.g. biology but I don't think it's quite analogous, as terminology is nowhere near as precise and things often don't abstract super nicely.
E.g. the word “carbohydrate”, or it’s equivalent in different languages, describes a concept; but the concept itself exists independently of any language construct that describes it. Similarly, in math, a concept e.g. “prime number” is independent of the many ways and many languages you can use to describe it. It just is.
It's less clear in informal languages that concepts exist apart from the language to describe them. This is one of the great difficulties in translation; different languages conceptualize the world in different (and sometimes incompatible) ways. For example, Latin has a variety of words for relationships of power that don't convey cleanly into English. The Greek word λόγος also doesn't cleanly render in English. Similarly for much of the Tao Te Ching. To what extent our language shapes our conceptualization of the world is currently an open question, but it seems like it's at least some.
These seem like extreme cases that just take more words to translate due to limited overlap of shared experiences. If it never snows in Hawaii, the Hawaiian language may not have a word for it. But snow is a real thing apart from language. I think what is meant by informal language is more the structural rules, which ultimately seem to just be arbitrary patterns of behavior people fell into, instead of sometime deeper.
Those rules too exhibit cross-language conceptual shear. For example, English doesn't really have an equivalent of Latin's subjunctive mood, and our clause boundary system is way weaker. We use punctuation marks in a way that the Romans didn't (they used punctuation words in a much more systematic way than we do). Greek has a separate grammatical number for pairs of things as opposed to our singular and plural. Those rules, in addition to the individual words (perhaps) bespeak a different way of conceptualizing the world.
It actually does snow in Hawaii. People on the big island have skis and snowboards for the few times it snows on top of their mountains. It also snows on Haleakala in Maui which is pretty fun to think about.
The concept of "a number that cannot be written as a product of two other numbers" just is. It might happen in some setting that none exist, but the general concept most certainly just is (and they can be shown to exist among the natural numbers if you insist - but they do not exist in e.g. Z_2 - the binary field).
