A familiarity with the pre-rigor/post-rigor ontology might be helpful for reading this post.
University math is often sold to students as imbuing in them the spirit of rigor and respect for iron-clad truth. The value in a real analysis course comes not from the specific results that it teaches — those are largely known to scientifically literate students by the time they take it. Instead, they are asked to relearn all those things from first principles; in so doing, they strip themselves of bad habits they previously learned and are inducted into the skeptical culture of the mathematician. Pedagogical and exam materials usually support this goal, putting emphasis on proof-writing, careful argumentation and attention to detail.
This incentivises the student to cultivate an invaluable attitude of healthy distrust towards their own world-models, which is not as trivial as it sounds. Many of my colleagues dropped out of their degree after learning in their first exams that they were more or less unable to argue why some "obvious" facts are true, or even to articulate what parts of the argument they were missing. A math undergraduate either teaches or selects for people who live in a fruitful, respectful relationship with the tenuousness of their own grasp of reality. Such skills are extremely useful and tend to generalise well to non-math domains.
Unfortunately, this philosophy of math pedagogy is somewhat at odds with goals you might have in educating a cohort of researchers. Your role as an undergraduate is to scrutinise the material you are fed as if it was written by your worst enemy, learning a culture of aggressive dialectical deconstructivism. But it takes two to make a dialectic process. Any idea that is worth deconstructing comes from an inventive mind that advocates for it, even if it is only because shooting down that concept will itself be a generative process.
Undergraduate teaching culture tends to work against this kind of creativity precisely due to its optimisation for instructing the virtue of radical skepticism to the point of pedantry. To support the norm of exactness and respect of minutia, classes frequently rely on concrete, exhaustive reference materials. Some courses even describe the specific content that could be used for an exam, defined up to arbitrarily excruciating levels of detail. Moreover, the learner only rarely comes across exercises that make them engage with unsolved problems or open questions. Successful students are thus able to identify and digest efficiently the knowledge sectioned off as relevant for a course, but they are rarely given affordance to push or even peek past these bounds.
The blogpost by Terrence Tao that I referenced in the introduction refers to this creative, research-taste-shaped stance towards mathematics as "post-rigorous". He highlights how "rigorous" and "post-rigorous" thinking should co-habitate harmoniously inside one's mind. He moreover comments on how rigorous thinking can be mis-used to discard or demean intuitive reasoning, leading to a failure of the dialectic process. Tao diagnoses the same problem as I do but focuses his solutions on what an individual (likely a graduate student) can do to nurture their post-rigorous self. I would instead like to observe that this focus on individual solutions is indicative that math academia has no institutionalised plan for teaching research taste. Whereas radical skepticism is embedded throughout math education in both legible and hidden ways, the canonical way to teach students to develop their creative research abilities seems to involve pairing them with mentors (e.g. PhD supervisors) and hoping that something rubs off.
I am not even close to being the first person to recognise this issue. Imre Lakatos' "Proofs and Refutations" and Donald Knuth's "Surreal numbers" are both attempts to accessibly communicate fundamental insights about post-rigor. Knuth in particular acknowledges in the postscript that his intended purpose in writing the book was to teach some mental motions needed for research mathematics. I'm sure there are many more wonderful published materials that I'm not aware of. However, I don’t see how these insights have meaningfully percolated into the design of institutionalised math education.
A familiarity with the pre-rigor/post-rigor ontology might be helpful for reading this post.
University math is often sold to students as imbuing in them the spirit of rigor and respect for iron-clad truth. The value in a real analysis course comes not from the specific results that it teaches — those are largely known to scientifically literate students by the time they take it. Instead, they are asked to relearn all those things from first principles; in so doing, they strip themselves of bad habits they previously learned and are inducted into the skeptical culture of the mathematician. Pedagogical and exam materials usually support this goal, putting emphasis on proof-writing, careful argumentation and attention to detail.
This incentivises the student to cultivate an invaluable attitude of healthy distrust towards their own world-models, which is not as trivial as it sounds. Many of my colleagues dropped out of their degree after learning in their first exams that they were more or less unable to argue why some "obvious" facts are true, or even to articulate what parts of the argument they were missing. A math undergraduate either teaches or selects for people who live in a fruitful, respectful relationship with the tenuousness of their own grasp of reality. Such skills are extremely useful and tend to generalise well to non-math domains.
Unfortunately, this philosophy of math pedagogy is somewhat at odds with goals you might have in educating a cohort of researchers. Your role as an undergraduate is to scrutinise the material you are fed as if it was written by your worst enemy, learning a culture of aggressive dialectical deconstructivism. But it takes two to make a dialectic process. Any idea that is worth deconstructing comes from an inventive mind that advocates for it, even if it is only because shooting down that concept will itself be a generative process.
Undergraduate teaching culture tends to work against this kind of creativity precisely due to its optimisation for instructing the virtue of radical skepticism to the point of pedantry. To support the norm of exactness and respect of minutia, classes frequently rely on concrete, exhaustive reference materials. Some courses even describe the specific content that could be used for an exam, defined up to arbitrarily excruciating levels of detail. Moreover, the learner only rarely comes across exercises that make them engage with unsolved problems or open questions. Successful students are thus able to identify and digest efficiently the knowledge sectioned off as relevant for a course, but they are rarely given affordance to push or even peek past these bounds.
The blogpost by Terrence Tao that I referenced in the introduction refers to this creative, research-taste-shaped stance towards mathematics as "post-rigorous". He highlights how "rigorous" and "post-rigorous" thinking should co-habitate harmoniously inside one's mind. He moreover comments on how rigorous thinking can be mis-used to discard or demean intuitive reasoning, leading to a failure of the dialectic process. Tao diagnoses the same problem as I do but focuses his solutions on what an individual (likely a graduate student) can do to nurture their post-rigorous self. I would instead like to observe that this focus on individual solutions is indicative that math academia has no institutionalised plan for teaching research taste. Whereas radical skepticism is embedded throughout math education in both legible and hidden ways, the canonical way to teach students to develop their creative research abilities seems to involve pairing them with mentors (e.g. PhD supervisors) and hoping that something rubs off.
I am not even close to being the first person to recognise this issue. Imre Lakatos' "Proofs and Refutations" and Donald Knuth's "Surreal numbers" are both attempts to accessibly communicate fundamental insights about post-rigor. Knuth in particular acknowledges in the postscript that his intended purpose in writing the book was to teach some mental motions needed for research mathematics. I'm sure there are many more wonderful published materials that I'm not aware of. However, I don’t see how these insights have meaningfully percolated into the design of institutionalised math education.