[greenpeas]

I think you may be confounding different kinds of understanding. There's a big difference between understanding what a statement is saying, versus understanding why a statement must hold true. And understanding why a statement holds doesn't necessarily have to be a formal proof, it just means that you have convinced yourself that some concepts that you are familiar with behave in a way to support the statement. It can actually happen that following a correct formal proof doesn't automatically help you understand why something is true. And vice-versa, it may happen that when you try to formalize your understanding, you discover that you missed some crucial details.

But usually true understanding and formal proofs go hand in hand.

P.S: Though as a disclaimer, I don't know maths beyond the undergraduate level, and I'm sure there must be very complicated concepts that mathematicians don't understand, but can reason formally about, as well as concepts that mathematicians feel that they intuitively understand but can't quite prove.

Additionaly, there's von Neuman saying "Young man, in mathematics you don't understand things. You just get used to them"