Who are some good math youtubers out there? Numberphile & 3b1b & Veritasium excluded. Someone I might not run into normally?— Jeremy Kun (@jeremyjkun) August 21, 2021

As opposed to a textbook, real maths is highly non-linear. ​ As we will see throughout the post, personalization (and the engagement inherent in it) is essential to the success of the lecture.] ​ [Before the third, I ask the class whether the first two alone are enough. If I get nods, I draw a random collection of dots and lines, with the lines not at all connected to the dots, and they see we need some statement of incidence.] ​ Since we will always draw constellations as a picture, we can just use the picture as our “function.” ​ Compare this to being given the definitions and propositions in the established mathematical language. To an untrained, uninterested student, this is not only confusing, but boring beyond belief! They don’t have the prerequisite intuition for why the definition is needed, and so they are left mindlessly following along at best, and dozing off at worst.

Unfortunately this sentiment is mirrored among most programmers who claim to be interested in mathematics. Mathematics is fascinating and useful and doing it makes you smarter and better at problem solving. But a lot of programmers think they want to do mathematics, and they either don’t know what “doing mathematics” means, or they don’t really mean they want to do mathematics. ​ Honestly, it sounds ridiculously obvious to say it directly like this, but the fact remains that people feel like they can understand the content of mathematics without being able to write or read proofs. ​ So read on, and welcome to the community. ​ I honestly do believe that the struggle and confusion builds mathematical character, just as the arduous bug-hunt builds programming character. ​ I’m talking, of course, about the four basics: direct implication, proof by contradiction, contrapositive, and induction. These are the loops, if statements, pointers, and structs of rigorous argument, and there is simply no way to understand the mathematics without a native fluency in this language. ​ And so it stands for mathematics: without others doing mathematics with you, its very hard to identify your issues and see how to fix them. ​ And finally, find others who are interested in seriously learning some mathematics, and work on exercises (perhaps a weekly set) with them.

@JadeMasterMath: There are lots of mathematical concepts which don’t have well written resources to learn about them. I think that explaining something in a clear way with a story arc can sometimes be enough. @jeremyjkun: Write about the topics that you learned, where there was a succinct phrase, picture, or idea that suddenly made it clear. Then arrange the whole blog post around getting the reader to that same understanding.

Andrew Wiles, one of the world's most renowned mathematicians, wonderfully describes research like exploring a big mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn't exist without, the many months of stumbling around in the dark that precede them. ​ But more often than not you'll find that by the time you revisit a problem you've literally grown so much (mathematically) that it's trivial.

Slowly, gradually, it dawned on me that what I enjoyed was mathematics. The mathematical aspects of CS were what got me excited and kept me up at night working on projects. ​ The Summer after I graduated, I decided I had too much awesome stuff in my head that nobody wanted to hear me talk about at parties, so I started a blog called Math Intersect Programming ​ If someone offered me this deal to write about math and CS, I would take it in a heartbeat. I would never want to retire.

For the last four years I’ve been working on a book for programmers who want to learn mathematics. It’s finally done, and you can buy it today. The website for the book is pimbook.org, …