This is a reduced quote from There’s more to mathematics than rigour and proofs of Terrence Tao (emphasis mine):

  1. 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.
  2. 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.
  3. 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.

  • 5
    $\begingroup$ I would read his essay as a generalization. I don't think he is arguing for crisp differences. Probably just pushing back against those with a fetish for rigor but who have lost the Eulerian ability/interest in discovery and application. $\endgroup$
    – guest
    Apr 16, 2018 at 14:58
  • $\begingroup$ So if one solves enough problems on Pythagoras triangle, then have they reached the post-rigorous stage on that "field" without having a math PhD? $\endgroup$
    – Ooker
    Apr 16, 2018 at 15:13
  • $\begingroup$ Probably not. But I can't articulate why. For what it's worth, Tao's essay is a generalization, not a label of some sort of MRI-to-the-brain detectable phenomenon. $\endgroup$
    – guest
    Apr 16, 2018 at 15:54
  • $\begingroup$ Maybe you will get some better discussion from other people. It is an interesting question. You could also comment on the Tao blog (you linked). May not get a reply, but you can try. $\endgroup$
    – guest
    Apr 16, 2018 at 15:55
  • $\begingroup$ I would think that showing evidence of using the Euclidean concepts in real world problems in surprising ways (not obvious problems) would be what is expected for post-rigor. However, I think the right kind of person (Euler, Feynman, people in general as they have some characteristics of E and F) can do these sort of interesting applications even without the benefit of going through the rigor phase (although I'm not saying it doesn't help). I would also say new math phenomena, but I doubt there are any to be discovered in Euclid's wheelhouse (ground has been worked, hard). $\endgroup$
    – guest
    Apr 16, 2018 at 16:00

1 Answer 1


I don't see any specific timestamps in the piece quoted. There are some vague time frames given, all marked by phrases like "generally" and "usually", which indicates that they're far from universal.

Furthermore, I read those time frames as being intended as clarification - they're there to help readers understand the stages by pointing out when they typically happen. For instance, it helps to understand what Tao means by stage 1 to compare it to our own observations of typical high school and early undergraduate math. But it only helps to compare it to what's typical at those levels - there's no claim that it's a universal description of every mathematical activity by every person at those levels.

  • $\begingroup$ so can the post-rigorous stage happen outside math and lower on grad school? I've added another piece quote to clarify my point $\endgroup$
    – Ooker
    Apr 17, 2018 at 8:30
  • $\begingroup$ @Ooker: I'm not sure what that means ("outside math"? do you mean outside formal school? what does "lower on grad school" mean?) In general, your questions here and your discussion on reddit look like you're asking around whatever your actual question is, and you might get clearer answers if you just asked your question. /u/sleeps_with_crazy is correct that Tao's post-rigorous stage is something more than just having some intuition for problems and being able to sketch proofs correctly without needing to fill in the details. $\endgroup$ Apr 17, 2018 at 16:21
  • $\begingroup$ The discussion there started differently to this question. The question about Tao's article emerged later, and that user stopped answering me so I post it here. My previous comment wants to have an answer about this: "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"?" I want to know if it can be generalized in any field or job and any level $\endgroup$
    – Ooker
    Apr 18, 2018 at 5:14
  • 1
    $\begingroup$ @Ooker 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. $\endgroup$
    – user9833
    Apr 18, 2018 at 14:01
  • 1
    $\begingroup$ @Ooker At most, we could make analogies between math and other fields. For example, in cooking, maybe the analogue of rigor is inventing recipes and understanding what makes a recipe good, rather than simply following recipes. But I hope you can see that this analogy is weak, and we cannot learn anything useful about culinary education by trying to translate Tao's advice on math education via this analogy. $\endgroup$
    – user9833
    Apr 18, 2018 at 16:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.