I disagree with what you say about 'formalities' in math, since they're practically the whole subject. It's a question of how they're taught, not whether to teach them. Not teaching the formal part and leaving it all to 'intuition' amounts to letting misconceptions fester with no feasible way to correct them. Even in music, a successful composer or improviser needs to know a lot about viable patterns and rules of thumb, and this knowledge would've been conveyed explicitly as well as by example. (These are not really 'formalities' in a strict sense but they're the closest thing in that domain.)
You might be imagining you think I said about formalities in mathematics, they're more or less essential in making something rigorous and passing it on to others but you'll find that many prolific mathematics start with intuitions, invent their own informal notations, and move toward a formal expression only when some new direction or concept is almost complete.
Maths proceeds faster with those who have been passed good intuitions about lengths, symmetries, and other concepts - these are the fundemental groundings. The formal notations follow to add rigor and to assist in the reasoning and formal proof.
> Even in music, a successful composer or improviser needs to know a lot about viable patterns and rules of thumb, and this knowledge would've been conveyed explicitly as well as by example.
There are entire schools of music that are conveyed entirely by example with no formal music theory whatsoever.
