a^2 + b^2 = c^2 is perhaps a poor example, since there are several visual proofs that don’t rely on language. I don’t think this negates your overall point fwiw.

Math is a terrible example. Euclid's "Elements" was a core book in Western math education for millennia, and it uses drawn geometric proofs. The proofs are then described in Greek, but the rigorous definitions are the pictures. A lot of mathematical fields can be reasoned about and explored by imagining shapes and objects. The end results today are written in notation because this is a standard language that helps clarify ideas.

Geometry pictures are absolutely not "rigorous definitions". For 1 thing, pictures of lines aren't finite in extent, and pictures of points are finite in extent.

Some of Euclid's proofs are wrong, because they relied on those non-rigorous picture definitions.

* Book I, Proposition 1 (constructing an equilateral triangle)

* Book I, Proposition 4 (Side-Angle-Side triangle congruence).

* Multiple theorems throughout the Elements rely on the visual "betweenness" of points.

To steelman your point with gowld's point notwithstanding, how do you articulate ideas like infinity without notation? The point is that language is an abstraction for ideas that are difficult to rationalise 'close to the metal'.