What's a mathematician to do? (2010)

To add onto these answers, there were some notable "outsider art" additions to math around this time.

A few months before this post, Futurama contributed a new proof to the mathematical canon (for "The Prisoner of Benda"), resolving the conflict of the episode.

Almost a year after posting this, a 4chan user solved a previously-unsolved superpermutation (combinatorics) problem in a discussion about anime.

I think everyone who has thought about math seriously has felt similarly to the OP. It was impressed upon me early on that there are combinatorically (hah) many combinatorics problems to be solved and that these were just a few.

this guy is resigned to feel worthless compared to other mathematicians (suggesting he become cannon fodder in some type of mathematical sacrifice. i wonder if that analogy even makes sense in the field xD).

but, he desperately wants to become a great mathematician who creates completely original work.

from my experience, people tend to or even want to limit themselves. they think they know the ceiling of their capabilities and it becomes some self fulfilling prophecy.

if you really care about doing something great like this guy does, don't limit yourself. push until you achieve the greatness you want to achieve.

it's like that one saying, aim for the stars and you might land on a cloud. you will be surprised at how capable you actually are

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.

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.

Numberphile doesn't do any education. That's like saying the Discovery Channel is educating a new generation of zoologists.