This is an excerpt from There’s more to mathematics than rigour and proofs of Terrence Tao:
The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years.
The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner. The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. This stage usually occupies the later undergraduate and early graduate years.
The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.
Does this have any stance in the context of cognitive science? A comment in Mathematics Education says that this idea can't be broaden in other fields/jobs in any level of academic, since:
Tao is using the word rigor in a way that's fairly narrow and specific to math. Namely, rigor means proving every statement one makes or uses, in full detail, from some list of axioms. One cannot rigorously work on cars, any more than one can rigorously eat one's dinner. It also does not make sense to talk about "post-rigorous" in the context of solving geometry exercises, because "rigorous" doesn't mean being good at something.
But I still don't understand this. Sure, it does not make sense to say "eat dinner rigorously", but it is possible for a good cook to know the hidden ingredients, the recipe or the taste of it by just looking and smelling the food, and I think this is equivalent to "proving every statement one makes or uses, in full detail, from some list of axioms".
So how is this phenomenon described rigorously in cognitive science?