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

Fewer than 1 percent of doctorates in math are awarded to African-Americans. Edray Goins, who earned one of them, found the upper reaches of the math world a challenging place.

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.

we are going to see an increasing number of academics thinking not so much about their paper CV, but rather about their [videos].

This situation crops up so often in mathematics that the acronym “TFAE” (for “The Following Are Equivalent”) has become a standard part of a mathematician’s education.

This is why I respect the Functional Programming movement: they get it. Functional Programmers understand that (at a minimum) 50 years of research and refinement is a pretty good thing to stake your data types on. ​ Stick to actual mathematics. You'll have to learn it eventually, you may as well not cloud your own thinking in the process. ​ Mathematics is the simplest and most precise language mankind has ever invented, and you should be able to speak it.

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.

(See PDF in reMarkable for annotations and notes).

But it is quite difficult to find a level of teaching which is comprehensible and at the same time interesting to an entire class with heterogeneous background. The shape of the mathematics education of a typical student is tall and spindly. It reaches a certain height above which its base can support no more growth, and there it halts or fails. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression.

In July 2019, ICMS hosted a workshop on Category Theory.  During the workshop, Eugenia Cheng (School of the Art Institute of Chicago) gave a public lecture entitled Inclusion-Exclusion in mathematics and beyond: who stays in, who falls out, why it happens and what we could do about it. This is a recording of that talkThis talk has captions.  To turn the captions off, press CC on the bottom toolbar.

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.

nLab is to collect the stable bits, organized, hyperlinked, usefully. ​ Such a far-flung structure is highly vulnerable and, once broken, impossible to restore.

When working with a monad, you work in its Kleisli category which is another example of a CCC. The above discussion relating function evaluation to function composition, would then relate Kleisli evaluation (=) to Kleisli composition (=). Woah, is `bind` just monadic function evaluation?

Hi, my name is Eitan. I’m a student of mathematics. Welcome to my personal blog. I intend to post mathematical exposition, but since this is a personal blog I will also post political thought…

You might suppose a great mathematician such as Kolmogorov would be writing about some very complicated piece of mathematics, but his subject was the humble equals sign: what made it a good piece of notation, and what its deficiencies were. ​ even great mathematicians – perhaps, especially, great mathematicians – thought it worth their time to engage in such deepening. ​ I’m still developing the heuristic, and my articulation will therefore be somewhat stumbling. ​ Every piece should become a comfortable part of your mental furniture, ideally something you start to really feel. That means understanding every idea in multiple ways, ​ People inexperienced at mathematics sometimes memorize proofs as linear lists of statements. A more useful way is to think of proofs is as interconnected networks of simple observations. ​ For someone who has done a lot of linear algebra these are very natural observations to make, and questions to ask. But I’m not sure they would be so natural for everyone. The ability to ask the “right” questions – insight-generating questions – is a limiting part of this whole process, and requires some experience. ​ Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will. ​ So, my informal pop-psychology explanation is that when I’m doing mathematics really well, in the deeply internalized state I described earlier, I’m mostly using such higher-level chunks, and that’s why it no longer seems symbolic or verbal or even visual. I’m not entirely conscious of what’s going on – it’s more a sense of just playing around a lot with the various objects, trying things out, trying to find unexpected connections. But, presumably, what’s underlying the process is these chunked patterns.

Learning starts with metacognition, and in order to metacognify (it’s a word) well, your students need you to be transparent in your pedagogical choices. ​ Be honest with yourself about how much time you spend working and how much time you need to work.

Uhlenbeck, who was born in 1942 in Cleveland, was a voracious reader as a child, but she didn’t become deeply interested in mathematics until she enrolled in the freshman honors math course at the University of Michigan. “The structure, elegance and beauty of mathematics struck me immediately, and I lost my heart to it,” ​ Mathematics research had another feature that appealed to her at the time: It is something you can work on in solitude, if you wish.

A thread, on the ever so lovely Bézier curve ✨ pic.twitter.com/2Mo5OjVIJ9— Freya Holmér (@FreyaHolmer) November 17, 2018

“Category Theory is the greatest refactor of mathematics in history.”

You see, it's very different than other branches of math. Rather than being another sibling lined up in the family photograph, it's more like a common gene that unites them in the first place. With this vantage point, it becomes evident that different areas of math share common patterns/trends/structures. This becomes extraordinarily useful when you want to solve a problem in one realm (say, topology) but don't have the right tools at your disposal. By transporting the problem to a different realm (say, algebra), you can see the problem in a different light and perhaps discover new tools, and the solution may become much easier. a "template" for all of mathematics: depending on what you feed into the template, you'll recover one of the mathematical realms. Naturally, then, you're prompted to also ask about relationships between categories. These are called functors. But why stop there? What about the relationships between those relationships? These are called natural transformations. (And yes, you can ask a hierarchy of questions: "What about the relationships between the relationships between the relationships between the...?" This leads to infinity categories. [And a possible brain freeze.]

“Eli’s lectures were always masterpiece of clarity. In one hour, he would set up a theorem, motivate it, explain the strategy, and execute it flawlessly; even after twenty years of teaching my own classes, I have yet to figure out his secret of somehow always being able to arrive at the natural finale of a mathematical presentation at the end of each hour without having to improvise at least a little bit halfway during the lecture. The clear and self-contained nature of his lectures (and his many books) were a large reason why I decided to specialise as a graduate student in harmonic analysis (though I would eventually return to other interests, such as analytic number theory, many years after my graduate studies).”

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, …

“The people you know who seem wicked smart, and who seem to come up with ideas much faster than you possibly could, are often people who have simply thought about the issues for much longer than you have.”