“Who are some good math YouTubers out there? Numberphile and 3B1B and Veritasium excluded. Someone I might not run into normally?”
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
Halfspace #5: The Spiritual Nature of Software Tools
That a good tool is one that hones your form, and into which you can project your uniqueness and soul. Curiously, I can’t think of any “spiritually significant” tools for mathematicians. Unless you count the running joke about Hagoromo chalk, which I can attest is the Rolls Royce of chalk. Perhaps it’s time for someone to build one.
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.
This primer exists for the background necessary to read our post on RSA encryption, but it also serves as a general primer to number theory. Oh, Numbers, Numbers, Numbers We start with some easy de…
Why there is no Hitchhiker’s Guide to Mathematics for Programmers
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.
As a fair warning to the reader, these primers are a bit more terse than what you’d find in your average textbook. I only introduce the bare minimum required to understand the main content po…
“Math Twitter, have any favorite tips for making advanced math accessible to wide audiences?”
@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.
the kinds of emotional intelligence one needs to succeed in a field where you spend almost all of your time understanding nothing. And second, by presenting them with a formal definition, I gave them a common reference point from which they could compare and contrast their own notions. There we had the beginnings of disaster avoidance. As a mathematician, Devlin did nothing unusual. In fact, the most common question a mathematician has when encountering a new topic is, “What exactly do you mean by that word?” The mathematical habit is putting your personal pride or embarrassment aside for the sake of insight.
Mathematicians are chronically lost and confused (and that's how it's supposed to be)
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.
once we see the formal definition below, it will become clear that mathematical (say, first-order logical) statements, together with proofs of implication, form a category. Even though a “proof” isn’t strictly a structure-preserving map, it still fits with the roughly stated axioms above. One can compose proofs by laying the implications out one after another, this composition is trivially associative, and there is an identity proof. Thus, proofs provide a way to “transform” true statements into true statements, preserving the structure of boolean-valued truth. The section on diagram categeories was fantatsic.
Moreover, a universal property jumps right to the heart of why a construction is important. I want to make this point very clear, because most newcomers to category theory are never told this. Category theory exists because it fills a need. Even if that need is a need for better organization and a refocusing of existing definitions. l One hopes, then, that very general theorems proved within category theory can apply to a wide breadth of practical areas. Could it be that there is some (non-categorical) theorem that can’t be proved unless you resort to category-theoretical arguments? In my optimistic mind the answer must certainly be no. Moreover, it appears that most proofs that “rely” on category theory only really do so because they’re so deeply embedded in the abstraction that unraveling them to find non-category-theoretical proofs would be a tiresome and fruitless process.
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, …
