This is a reduced quote from There’s more to mathematics than rigour and proofs of Terrence Tao (emphasis mine):
- 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.
I ask a person on Reddit, and they say that (emphasis theirs):
- The rigorous stage, which undergrads never get past
- Post-rigorous only happens to the rare person who succeeds in grad school and begins doing actual high-level unguided research.
- There is no post-rigorous stage of grade 12 math, it's simply pre-rigorous end of story. You may very well completely understand it, but that has nothing to do with whether or not you've reached a level of rigor in your thinking.
Why do such stages have specific timestamps but not relative? And is this math-only? If an undergraduate can sketch out a proof of an grade 12 student, why doesn't this mean they are in post-rigorous stage of the grade 12? I think what Tao means simply about the requirements to contribute new knowledge. But if one solves enough problems on Pythagoras triangle and is ready to move to the next topic, can we say that they have reached the post-rigorous stage on that "field"? A vehicle technician may have never been to college, but they can very well know what's wrong with an engine by just passingly hearing its sound.