I feel that here the authors think about mathematics as just the language and not what it talks about. We know at least since Gödel that mathematics is more tha

I feel that here the authors think about mathematics as just the language and not what it talks about. We know at least since Gödel that mathematics is more than just the formal system. I even remember going to a talk by Gromov and he getting "angry" about a question from the public in the lines of "what if we suppose this extra thing and change this other thing on that theorem", and him saying that it's stupid to think of new theorems for the sake of doing it, that the language of mathematics are like metaphors speaking about something else. In my experience learning math, their claim that "The central hypothesis is that a desirable body of theorems better summarizes the set of all provable statements, for example by having a small description length" is not true. Short != Better, better is what gets me faster to form the correct intuitive idea about the mathematical statement. For example, several times I have experienced the fact of understanding the formal definitions and proofs of a theory, but it's not until I form the correct intuitions (maybe months later), that I truly understand the theory. And it's not until I have the correct intuitions, that I can successfully apply the theory to create meaningful new theorems. Anyways, I understand that one has to start from somewhere and the point of view of the article is more tractable and explicit.