[TeachYourselfCS1] Develop a working understanding of discrete mathematics: - logic - combinatorics and probability - set theory - graph theory (MAKE SURE TO ADD INTO THE MATH PAGE, AS THERE'S NOW A REF ON TS) - a little of the number theory informing cryptography - also, linear algebra can be useful in computer graphics and machine learning [TeachYourselfCS2] Laszlo Lovasz lecture notes Percentages are reversible. Working out 4% of 50 will give you the same result as 50% of 4 NOTE: go through basic mathematics (e.g., addition, multiplication, exponents, decimals) and their core philosophical parsing - don't forget to indicate irrational numbers The Sumerians base-6 system bestowed on us the division of the day into twenty-four hours and of the circle into 360 degrees. More signs were added to the Sumerian system, gradually transforming it into a full script that we today call cuneiform. Other full scripts were developed in China around 1200 BC and in Central America around 1000-500 BC. 0 to 9 are known as Arabic numerals even though they were first invented by the Hindus. But the Arabs get the credit because when they invaded India they encountered the system, understood its usefulness, refined it, and spread it through the Middle East and then to Europe. If doing any mathematical thing at all, find some practical example for which it would be useful. For example: There's a flagpole, and there's a rope that comes down from the top. When you hold the rope straight down, it's a meter longer than the pole, and when you pull the rope out tight, it's 1.75 meters from the base of the pole. How high is the pole? When somebody is explaining something that I'm trying to understand: I keep making up examples. I construct something which fits all the conditions. They would tell me the general problem they were working on, and would begin to write a bunch of equations. "Wait a minute," I would say. "Is there a particular example of this general problem?" "Why yes; of course." "Good. Give me one example." That was for me: I can't understand anything in general unless I'm carrying along in my mind a specific example and watching it go. I have the specific, physical example of what he's trying to analyze, and I know from instinct and experience the properties of the thing. So when the equation says it should behave so-and-so, and I know that's the wrong way around, I jump up and say, "Wait! There's a mistake!" I learned to use one method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me. I used to enjoy doing physics. Why did I enjoy it? I used to play with it. I used to do whatever I felt like doing - it didn't have to do with whether it was important for the development of nuclear physics, but whether it was interesting and amusing for me to play with. I'd see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was rather easy to do. I didn't have to do it; it wasn't important for the future of science; somebody else had already done it. That didn't make any difference: I'd invent things and play with things for my own entertainment. When I felt I was burned out and will never accomplish anything, I decided I'm going to play with physics, whenever I want to, without worrying about any importance whatsoever. It was effortless. It was easy to play with these things. It was like uncorking a bottle: Everything flowed out effortlessly. I almost tried to resist it! There was no importance to what I was doing, but ultimately there was. A long time ago an older and well known number theorist made some disparaging remarks about Paul Erdos' work. You admire contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos' work could be reduced to a few tricks which Erdos repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert's collected papers contains Hilbert's papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert's beautiful results have been completely forgotten. But on reading the proofs of Hilbert's striking and deep theorems in invariant theory, it was surprising to verify that Hilbert's proofs relied on the same few tricks. Even Hilbert had only a few tricks! Math tricks: Many kids would learn to love mathematics if they had some investment in it. CONSIDER SET THEORY WITHIN LOGIC? [Math Overflow users resolve PhD thesis crisis | Hacker News](https://news.ycombinator.com/item?id=24049428) [co.combinatorics - Issue UPDATE: in graph theory, different definitions of edge crossing numbers - impact on applications? - MathOverflow](https://mathoverflow.net/questions/366765/issue-update-in-graph-theory-different-definitions-of-edge-crossing-numbers) [Knuth changes his mind on Bernoulli number B_1 | Hacker News](https://news.ycombinator.com/item?id=32281536) [Knuth: Recent News](https://www-cs-faculty.stanford.edu/~knuth/news22.html) [What does it mean for a monad to be strong? | Hacker News](https://news.ycombinator.com/item?id=37213739) [A Neighborhood of Infinity: What does it mean for a monad to be strong?](http://blog.sigfpe.com/2023/08/what-does-it-mean-for-monad-to-be-strong.html) [Ask HN: Whatever happened to Wolfram Alpha? | Hacker News](https://news.ycombinator.com/item?id=29131931) ## desired format I’m aiming for breadth, not depth. I don’t need to perform combinatorics in my head, but I do need to explain in plain English what the heck each math “thing” is, traced along a pseudo-path through the route of standard formalized education: Basic arithmetic Algebra Geometry/trigonometry Statistics Calculus Number theory, with a likely divergence into applied maths and game theory. Along the way, I’ll keep a [jargon-resistant dictionary](https://notageni.us/math-dictionary/) of the big math words. ## math branches [Mathematics and its branches | Britannica](https://www.britannica.com/summary/mathematics) [Lists of mathematics topics - Wikipedia](https://en.m.wikipedia.org/wiki/Lists_of_mathematics_topics) [ELI5: What is a slide rule, and why was it's invention such a big deal? : explainlikeimfive](https://old.reddit.com/r/explainlikeimfive/comments/wz1k8x/eli5_what_is_a_slide_rule_and_why_was_its/) ## philosophy of math ["When are we going to use this in our everyday life?" (2017) | Hacker News](https://news.ycombinator.com/item?id=33085532) [Flavoracle on Tumblr: It's sad how much of what is taught in school is useless to over 99% of the population. There are literally math concepts taught...](https://www.tumblr.com/flavoracle/167150535757/its-sad-how-much-of-what-is-taught-in-school-is) [468: Fetishes - explain xkcd](https://www.explainxkcd.com/wiki/index.php/468:_Fetishes) [Be good-argument-driven, not data-driven | Hacker News](https://news.ycombinator.com/item?id=32651763) [Be good-argument-driven, not data-driven - Richard Marmorstein](https://twitchard.github.io/posts/2022-08-26-metrics-schmetrics.html) [[0704.0646] The Mathematical Universe](https://arxiv.org/abs/0704.0646) ## history of math [Stone tablet shows Babylonians able to approximate √2 with 99.9999% accuracy | Hacker News](https://news.ycombinator.com/item?id=34178870) [Tivadar Danka on X: "This stone tablet from 1800-1600 BC shows that ancient Babylonians were able to approximate the square root of two with 99.9999% accuracy. How did they do it? https://t.co/ySDTvWjeYh" / X](https://twitter.com/tivadardanka/status/1608419325706391554) ### proofs and mathematicians [Gödel's first incompleteness theorem - an interactive tutorial | Hacker News](https://news.ycombinator.com/item?id=33236447) [Gödel's first incompleteness theorem | Busy Beavers!](https://busy-beavers.tigyog.app/incompleteness) [Goodbye Aberration: Physicist Solves 2,000-Year-Old Optical Problem | Hacker News](https://news.ycombinator.com/item?id=20369960) [Goodbye Aberration: Physicist Solves 2,000-Year-Old Optical Problem | PetaPixel](https://petapixel.com/2019/07/05/goodbye-aberration-physicist-solves-2000-year-old-optical-problem/) [Landmark computer science proof cascades through physics and math | Hacker News](https://news.ycombinator.com/item?id=22488927) [Quanta Magazine](https://www.quantamagazine.org/landmark-computer-science-proof-cascades-through-physics-and-math-20200304/) [New math book rescues landmark topology proof | Hacker News](https://news.ycombinator.com/item?id=28471159) [Quanta Magazine](https://www.quantamagazine.org/new-math-book-rescues-landmark-topology-proof-20210909/) [The existence of true one-way functions depends on Kolmogorov complexity | Hacker News](https://news.ycombinator.com/item?id=30934721) [Quanta Magazine](https://www.quantamagazine.org/researchers-identify-master-problem-underlying-all-cryptography-20220406/) [The Unparalleled Genius of John von Neumann (2019) | Hacker News](https://news.ycombinator.com/item?id=26695502) [The Unparalleled Genius of John von Neumann | Hacker News](https://news.ycombinator.com/item?id=21542753) [The Unparalleled Genius of John von Neumann | by Jørgen Veisdal | Cantor's Paradise](https://www.cantorsparadise.com/the-unparalleled-genius-of-john-von-neumann-791bb9f42a2d) [Turing Oversold? | Hacker News](https://news.ycombinator.com/item?id=28522761) [Turing Oversold](https://people.idsia.ch//~juergen/turing-oversold.html) ## math pedagogy [Teaching general problem-solving skills is not a substitute for teaching math [pdf] (2010) | Hacker News](https://news.ycombinator.com/item?id=40890847) [Relearning math as an adult | Hacker News](https://news.ycombinator.com/item?id=39047825) [How I'm (re)learning math as an adult](https://gmays.com/how-im-relearning-math-as-an-adult/) [Thoughts about what worked in math circles | Hacker News](https://news.ycombinator.com/item?id=37276502) [Thoughts about what worked in math circles • Buttondown](https://buttondown.email/j2kun/archive/thoughts-about-what-worked-in-math-circles/) [I rewired my brain to become fluent in math (2014) | Hacker News](https://news.ycombinator.com/item?id=40167163) [How I Rewired My Brain to Become Fluent in Math - Nautilus](https://nautil.us/how-i-rewired-my-brain-to-become-fluent-in-math-235085/) [A 'Rebel' Without a Ph.D. | Quanta Magazine](https://www.quantamagazine.org/a-math-puzzle-worthy-of-freeman-dyson-20140326) [Jim Simons has died | Hacker News](https://news.ycombinator.com/item?id=40320406) [Simons Foundation Co-Founder, Mathematician and Investor Jim Simons Dies at 86](https://www.simonsfoundation.org/2024/05/10/simons-foundation-co-founder-mathematician-and-investor-jim-simons-dies-at-86/) [Ramanujan's lost notebook | Hacker News](https://news.ycombinator.com/item?id=40044901) [Ramanujan's lost notebook - Wikipedia](https://en.wikipedia.org/wiki/Ramanujan%27s_lost_notebook) - math is the scrawlings of highly ordered crazy people [Programming Puzzles StackExchange](https://codegolf.stackexchange.com/questions/35569/tweetable-mathematical-art) Tweetable Mathematical Art [The Greatest Educational Life Hack: Learning Math Ahead of Time | Hacker News](https://news.ycombinator.com/item?id=40983734) [The Greatest Educational Life Hack: Learning Math Ahead of Time - Justin Skycak](https://www.justinmath.com/the-greatest-educational-life-hack-learning-math-ahead-of-time/) [How to choose a textbook that is optimal for oneself? | Hacker News](https://news.ycombinator.com/item?id=41016650) [mathematical pedagogy - How to choose a textbook that is pedagogically optimal for oneself? - Mathematics Educators Stack Exchange](https://matheducators.stackexchange.com/questions/27964/how-to-choose-a-textbook-that-is-pedagogically-optimal-for-oneself) [Göttingen was one of the most productive centers of mathematics (2019) | Hacker News](https://news.ycombinator.com/item?id=41087429) [How one German city developed – and then lost – generations of math geniuses](https://theconversation.com/how-one-german-city-developed-and-then-lost-generations-of-math-geniuses-106750) [If You Can't Learn Math, Maybe It's Not Your Fault](https://www.yearofmentalhealth.com/p/dyscalculia) - dyscalcia is real, but that doesn't mean math evades people - the symbols can be scary without context [Open Textbook Initiative | Hacker News](https://news.ycombinator.com/item?id=30270705) [Open Textbook Initiative | American Inst. of Mathematics](https://aimath.org/textbooks/) [Brilliant | Learn interactively](https://brilliant.org/) Online math brain puzzles. [Halfsies | Hacker News](https://news.ycombinator.com/item?id=38000551) [Brilliant | Halfsies](https://brilliant.org/challenges/halfsies/) ### guides [3Blue1Brown](https://www.3blue1brown.com/) ### books and courses [Euclid's Elements - Wikipedia](https://en.wikipedia.org/wiki/Euclid's_Elements) [Ask HN: The book that did it for you in math and/or CS? | Hacker News](https://news.ycombinator.com/item?id=30485544) [Ask HN: Math books that made you significantly better at math? | Hacker News](https://news.ycombinator.com/item?id=34439828) [Ask HN: Are there books for mathematics like Feynman's lectures on physics? | Hacker News](https://news.ycombinator.com/item?id=21346272) [120 Free Online Math Courses from the World's Top Universities](https://www.freecodecamp.org/news/math-online-courses-from-worlds-top-universities) [Free Math Resources Textbooks, Lectures Notes, Videos and Online Courses](https://realnotcomplex.com/) [GitHub - comp-physics/awesome-numerics: Resources for learning about numerical methods.](https://github.com/comp-physics/awesome-numerics)