I get the feeling that you're stuck in Terry Tao's 'rigourous phase' of mathematical understanding, where everything in the end is a computation and has to be carried out according to a set of rigorous steps and definitions.
I get that, but it does miss a bit the cultural context of how mathematically fluent people use mathematics to communicate with each other. When you're discussing maths with colleagues in front of a blackboard, you're often not really trying to prove anything, but discussing the relationship between mathematical objects. In this context the ambiguity and implication in the notation is almost a requirement, otherwise the communication speed tanks.
Having a mathematical discussion between a group of people all fluent in the context and terminology is a wonderfully fluid thing.
I get that, but it does miss a bit the cultural context of how mathematically fluent people use mathematics to communicate with each other. When you're discussing maths with colleagues in front of a blackboard, you're often not really trying to prove anything, but discussing the relationship between mathematical objects. In this context the ambiguity and implication in the notation is almost a requirement, otherwise the communication speed tanks.
Having a mathematical discussion between a group of people all fluent in the context and terminology is a wonderfully fluid thing.