And somehow a thing that keeps coming into my mind while reading them is the pre-rigorous/​rigorous/​post-rigorous split Terrence Tao talks about, where mathematicians start off just doing calculation and not really understanding proofs, and then they understand proofs through careful diligence, and then they intuitively understand proofs and discard many of the formalisms in their actions and speech.
This is an instance of three levels of mastery.