The thing is that actually doing mathematics is a process where you take your intuition and find a way to formalize it into definitions and proofs and computati

The thing is that actually doing mathematics is a process where you take your intuition and find a way to formalize it into definitions and proofs and computations. A toddler could understand the difference between a smooth curve and a spiky curve, but to actually get a handle on them computationally, the best tool that we have is Calculus -- and even after decades of people iterating on how to teach it, Calculus is still a mystery to most undergrads who muddle through it. Similarly I could give you some intuition about manifolds by telling you that they're kinda like surfaces that can have higher dimensionality, but until you're able to actually understand and work with the definition of "locally homeomorphic to R^n", you're not going to be able to do anything with them. I think it's bollocks, this idea that mathematicians are obfuscating easy concepts with complex language. The language doesn't exist to get research grants and gatekeep, it's there because you need some succinct way to describe objects and concepts without running through a full definition every time. Obviously you can always improve communication, and from my experience most mathematicians try pretty hard to do that, especially in the classroom, but it's not easy.