This is conflating analytical descriptions of these potential mathematical objects for synthetic ones. Analytical characterizations ("how does this object arise") is what's sought in this kind of research, but a synthetic one ("here's a mini-language that formally describes this mathematical object, and the exactly-stated 'rules of the game' that ensure that you're using it correctly in a logical sense") is all that a high school student needs. High school students don't typically concern themselves with constructing, e.g. the real numbers as formal mathematical objects, but they need to know how to make correct statements in the language of real numbers.

Note that synthetic descriptions are not "informal" in any sense; they're just as watertight logically as analytical ones, they just arrive at the result in a different, and arguably complementary way.