[lisper]

> One can rewrite their books in modern language and notation or guide others to learn it too but I never believed this was the significant part of a mathematician work

There's yer problem right there. Good pedagogy is hard and highly undervalued. IMHO Grant Sanderson (a.k.a. 3blue1brown) is making some of the most significant contributions to math in all of human history by making very complex topics accessible to ordinary mortals. In so doing he addresses one of the most significant problems facing humankind: the growing gap between the technologically savvy and everyone else. That gap is the underlying cause of some very serious problems.

[TrackerFF]

Big fan of him - but I also want to throw out the most obvious name in this space: Sal Khan

Hard to imagine now, but back when he started out, there were really no (to very few!) accessible math tutoring vids on the video platforms. Most of the times you had some universities, like MIT, putting out long-form vids from lectures - but actually having easily digestible 5 min vids like those Khan put out, just wasn't a thing.

[bobajeff]

I like to watch 3blue1brown too but I think it's a bit of an exaggeration to say his topics are accessible to normal folks. From my perspective I think it's more realistic to say he makes videos that shows you the beauty in math without having to understand it really. Which is valuable since most people get turned off on math because tiresome drills and tests hammered into them at school by people with zero interest in it.

[xiaoyu2006]

Quite true. Real math needs practice and calculation to build the intuition and motivation towards the abstraction we want to construct. The video is more of a complementary material to the boring lectures (my prof uses 3b1b videos sometimes).

[susam]

Good pedagogy is a problem even for graduate-level mathematics students and professional mathematicians. The proofs in many graduate-level mathematics textbooks are, in my humble opinion, not really proofs at all. They are closer to high-level outlines of proofs. The authors simply do not show their work. The student then has to put in an extraordinary amount of effort to understand and justify each line. Sometimes a 10-line argument in a textbook might expand into a 10-page proof if the student really wants to convince themselves that the argument works.

I am not a mathematician, but out of personal interest, I have worked with professional mathematicians in the past to help refine notes that explain certain intermediate steps in textbooks (for example, Galois Theory, by Stewart, in a specific case). I was surprised to find that it was not just me who found the intermediate steps of certain proofs obscure. Even professional mathematicians who had studied the subject for much of their lives found them obscure. It took us two days of working together to untangle a complicated argument and present it in a way that satisfied three properties: (1) correctness, (2) completeness, and (3) accessibility to a reasonably motivated student.

And I don't mean that the books merely omit basic results from elementary topics like group theory or field theory, which students typically learn in their undergraduate courses. Even if we take all the elementary results from undergraduate courses for granted, the proofs presented in graduate-level textbooks are often nowhere near a complete explanation of why the arguments work. They are high-level outlines at best. I find this hugely problematic, especially because students often learn a topic under difficult deadlines. If the exposition does not include sufficient detail, some students might never learn exactly why the proof works, because not everyone has the time to work out a 10-page proof for every 10 lines in the book.

Many good universities provide accompanying notes that expand the difficult arguments by giving rigorous proofs and adding commentary to aid intuition. I think that is a great practice. I have studied several graduate-level textbooks in the last few years and while these textbooks are a boon to the world, because textbooks that expose the subject are better than no textbooks at all, I am also disappointed by how inaccessible such material often is. If I had unlimited time, I would write accompaniments to those textbooks that provide a detailed exposition of all the arguments. But of course, I don't have unlimited time. Even so, I am thinking of at least making a start by writing accompaniment notes for some topics whose exposition quality I feel strongly about, such as s-arc transitivity of graphs, field extensions and so on.

[zozbot234]

These days it's easy to just look for the details to any proof on mathlib. Of course a computer checked proof is not always super intuitive for a human, but most of the time it does work quite well.

[LPisGood]

Indeed, pedagogy is important to staving off the end of mathematics.

That sounds dramatic, but it’s really obvious if you think about it. Right now, a person has to study for about 20 years (on average) to make novel contributions in mathematics. They have to learn what’s come before, the techniques, the results, etc. If mathematics continues, eventually it could take 25 years, or 30 years, or even a whole lifetime. At some point, most people will not be able to understand the work that’s been done in any subfield (or the work required to understand a subfield) in a human’s life. I claim this is the logical end of mathematics, at least as a human endeavor.

Now, there will be some results which refine other work and simplify results, but being able to teach a rapidly growing body of literature efficiently will be important to stave off the end of mathematics.

[pertique]

There's a Scott Alexandar story that plays with this exact topic: Ars Longa, Vita Brevis [1]

To your point, I think you're right. I'm not in mathematica, but the value of good pedagogy on shrinking the time it takes to get people to the forefront of any field feels like it's heavily overlooked.

https://slatestarcodex.com/2017/11/09/ars-longa-vita-brevis/

[rustymather]

Seconded. Good pedagogy is like fertilizing the soil- it creates conditions conducive to learning in order to do good research.