# Math

Lean, Univalent foundations and Type Theory are interesting. And 3Blue1Brown has incredible explanations for various math topics.

So you want to study mathematics is a great intro. Riemann Hypothesis is a fun problem.

## Notes

- Math can be learned, it's just a bit silly to dive into a homotopy type theory book and expect to get a firm grasp of things. It's like picking up a Chinese news paper and expecting to get an idea of what its content is about.
- Without understanding the domain, you cant read a text well. Without understanding the grammar you cant read a text. Without understanding the symbols you cant even figure out the grammar.
- The prevailing view in modern mathematics is that structure is all that matters. i.e. isomorphic objects are really the same, they don't just look the same or act the same, they really are the same thing.
- There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices.
- Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically compute quantitative information about it (its value at 3 is blah, its integral over this interval is blah, its maximum value is blah).
- Linear algebra: vector spaces and linear operators, and their representation as vectors and matrices. Functions as forming infinite dimensional spaces, and Banach and inner product spaces.
- Differential equations and dynamical systems: extending what we did for calculus to differential equations. Phase space, orbits, Fourier and Laplace transforms, sets of linear differential equations, numerical integration, some partial differential equations. You do not need all the little tricks for special kinds of equations that you will find in, say, Boyce and dePrima. They're not helpful.
- Sets, groups, rings, and lattices. Mathematics today is written in terms of set theory. You need to understand the basics of manipulating sets and functions between them. Then you should know something about groups, rings, and lattices, which are the most ubiquitously useful algebraic structures besides vector spaces.
- After that, where you go is going to vary enormously. Based on what you're aiming to do.

- Some topics and books to learn from:
- Topology (book by Munkres)
- Real Analysis and Measure Theory (book series by Stein Shakarchi)
- Algebra (book by Aluffi)
- Linear Algebra (book by Friedberg Insel)
- Measure Theoretic Probability (book by Cinlar)
- Differential Geometry (book Smooth Manifolds by Lee)
- Numerical Analysis (book by Quarteroni)
- Set Theory and Propositional Logic (books by Goldrei)

- Or these books for more fundamental basic knowledge:
- Higher algebra - Hall and Knight
- Trigonometry I - Loney
- Coordinate geometry - Loney
- Calculus of one variable - Maron

- If you can, find someone else to work with, and someone else who already knows some math to ask occasional questions would probably help you a lot.
- Monad on Set is a way to describe some sort of gadget that's a set equipped with a bunch of operations obeying a bunch of equations
- There's no closed-form solution for the motion of 3 gravitational masses in space, because solving that would tell us whether some guy's pure math construct halts or not.
- To learn math/anything: 1. Grab a book/paper (try to understand what it's teaching). 2. Do exercises to understand (or build something). 3. Repeat.
- e^x is convex

## Links

- What does it feel like to invent math?
- How to Read Mathematics (HN)
- The Banach–Tarski Paradox
- John Conway: Genius Behind The Numbers
- Moiré pattern in JavaScript - Beautiful.
- Tools for thought
- Making things with maths
- The Map of Mathematics
- How to self learn math
- Math books
- Mathematics for Computer Science: Readings
- mathematicalmonk YouTube playlists - A lot of stuff covered.
- How Grothendieck Simplified Algebraic Geometry
- A new “Mathematician’s Apology”
- Interesting theorems
- Ask HN: How do I learn math/physics in my thirties?
- Napkin Project (Code) (PDF) (HN)
- Learn Math Fast - Curated list of awesome mathematics resources.
- Russian wiki on math & finance
- Math for CS MIT Course
- MSC2010 - Huge catalog of math topics.
- Cambridge math notes
- Mathematics as thought
- Penrose - Language-based platform for creating beautiful mathematical diagrams. (HN) (Web) (HN 2) (Lobsters) (Paper) (HN)
- Domain-Specific Languages of Mathematics
- A Programmer's Introduction to Mathematics book (Article) (HN)
- How to learn Math and Physics (2018)
- Ask HN: How to self-study mathematics from the undergrad through graduate level? (2019)
- How to Science [Part 2: Our Universe = Math?] - Amazing series.
- University of Oxford notes and exercises for math undergraduate courses
- MacTutor History of Mathematics archive
- Formalizing 100 Theorems
- Some Fundamental Theorems in Mathematics (2018) (HN)
- Some quotes from Robert Harper
- Hessix - HN-like math research discussion for Ph.D.s (HN)
- Curv - Language for making art using mathematics.
- Mathematics For Machine Learning book (2020) (PDF) (Code)
- Mathematics for Machine Learning (2018)
- All in One Mathematics Cheat Sheet (HN) (HN)
- too simple to be simple
- Self Studying the MIT Applied Math Curriculum (2019) (HN)
- Timeline of Mathematics
- Ask HN: Where can I do a self-study math degree online? (2019)
- Beautiful Math - New book shows off spectacular works of art inspired by mathematical principles.
- Cardano's formula (2019) (Lobsters)
- Collection of computational playthings (Code)
- Approach0 - Math-aware search engine. (Code)
- Think in Math. Write in Code. (2019) (HN)
- Analysing Mathematical Reasoning Abilities of Neural Models (2019)
- Cambridge Maths Outline (2019)
- Dexter Chua's math notes
- Math notes by Gareth Taylor
- How to Become a Pure Mathematician (or Statistician)
- Chicago undergraduate mathematics bibliography
- Ask HN: Best resources to gain math intuition? (2019)
- The Riemann Hypothesis (2019) (HN)
- Nonlinear Dynamics and Chaos lectures
- The Future of Mathematics? (2019)
- CoCalc - Collaborative Calculation in the Cloud.
- Prime Numbers and the Riemann Hypothesis (Code)
- nLab - Wiki-lab for collaborative work on Mathematics, Physics and Philosophy — especially from the n-point of view: in so far as these subjects are usefully treated with tools and notions of category theory or higher category theory.
- Understanding Mathematics
- Eigenvectors from eigenvalues (2019)
- Cantor's Paradise Math Blog
- Better intuition for information theory (2019)
- Mathematics Free and Open-Source Textbooks (HN)
- Stuff I learnt in 2019 (Reddit)
- T * sin (t) ≈ Christmas tree (2013) (HN)
- What was the hardest area of math for you? (2020)
- Expository papers by K. Conrad
- Math for the Layman
- Leonhard Euler | The Greatest Mathematician To Ever Live (2020)
- MIT Mathematics for Computer Science course (2015) - Serves as an introduction to discrete mathematics, probability, and mathematical thinking for computer scientists.
- Who are some lesser known mathematicians, and what are some of their accomplishments, or interesting facts about them
- Information theory
- Timothy Gowers: The Importance of Mathematics
- Oxford Undergraduate Courses in Math
- Oxford Postgraduate Courses in Math
- An inquiry into the Foundations of Mathematics (2020)
- Anthony Várilly-Alvarado, "The Geometric Disposition of Diophantine Equations." (2020)
- Mathpendium - Discover the world of mathematics.
- Where is the fashionable mathematics? (2020)
- The Map of Mathematics (2020)
- Introduction to Clifford Algebra (HN)
- Grant Sanderson's TED talk on effective teaching of math (2020)
- Fields Institute - International centre for scientific research in mathematical sciences at the University of Toronto.
- HN: Maryam Mirzakhani
- Mathematics for the adventurous self-learner (2020) (HN)
- Math is your insurance policy (2020) (HN)
- Discrete MathematicsAn Open Introduction
- Lecture notes from Cambridge maths (Code)
- Cambridge Maths Tripos notes
- Real Not Complex: Free Math Resources (HN)
- Short book recommendations on advanced topics? (2020)
- Are there any notable mathematicians/scientists that struggled with math but became prominent in their fields? (2020)
- Intuitive Explanations - Understand, don't memorize. (Code)
- Books to study math from first principles (2020)
- The formula for pi buried in a hydrogen atom (HN)
- Math Seminars
- Abstract Algebra Cheatsheet
- HN: Ladder of Algebraic Structures (2020)
- Math3ma - Math blog by Tai-Danae Bradley.
- Princeton Companion to Mathematics book (2007)
- Roger Penrose - Is Mathematics Invented or Discovered? (2020) (HN)
- What was the math that blew your mind the most, at any point in your physics career? (2020)
- Springer Math Books
- Advanced Math resources
- Math & Programming blog main content
- Project MOSAIC - Community of educators working to develop a new way to introduce mathematics, statistics, computation and modeling to students in colleges and universities.
- The Psychology of Advanced Mathematical Thinking
- Against Set Theory (2005) (HN)
- Discrete Mathematics: An Open Introduction (HN)
- Graduate Student Solves Decades-Old Conway Knot Problem (2020) (HN)
- What Made me a Mathematician: Off-the-Cuff Story Telling for a Story-Telling Class (2020)
- Mathematical Notes by Prof. Steven G. Johnson
- The Definitive Glossary of Higher Math Jargon
- On proof and progress in mathematics (1994)
- In Math, It Often Takes a Good Map to Find Answers (2020) (HN)
- karasu - Self-hosted remote communication system designed for math discussions. (Web)
- MathPages
- Bill Thurston's answer to “What's a mathematician to do?” (HN)
- Visual Math Tools
- Math History Books
- Ask HN: What kind of math do you study in your free time? (2020)
- Ask a Mathematician / Ask a Physicist
- Important concepts in numerical analysis and related areas
- Bi-whatever transformations
- The Mathematical Hacker (2012) (Lobsters)
- Ask HN: Good articles/courses for understanding mathematical notation? (2020)
- Small maths projects explored with Python
- BetterExplained - Clear, intuitive lessons about imaginary numbers, exponents, and more. (HN)
- Implementing the Exponential Function (2020) (HN)
- Feynman on Fermat's Last Theorem (2016)
- Ask HN: Recommend a maths book for a teenager (2020)
- Imaginary - Place for Open and Interactive Mathematics.
- Ridiculous Math Problems (2020) (HN)
- Discrete Math (Full Course: Sets, Logic, Proofs, Probability, Graph Theory, etc) (2020)
- Comprehensive List of Algebra Symbols
- Knuth - Two notes on notation (1992) (Lobsters)
- How Gödel’s Incompleteness Theorems Work (2020)
- The Marvels of Polynomials over a Field (2020)
- Mathematicians Discover the Perfect Way to Multiply (Lobsters) (HN)
- Bloom filters debunked: Dispelling 30 Years of bad math with Coq (2020) (Lobsters)
- Irrational Numbers (2020) (Lobsters)
- Terry Tao on some desirable properties of mathematical notation (HN)
- Linear Logic Flavoured Composition of Petri Nets (2020)
- How real are real numbers? (2004) (HN)
- Who Is Alexander Grothendieck? (2008) (HN)
- Taylor Series and Accelerometers (2020) (HN)
- What is the difference between a theorem, a lemma, and a corollary? (2008)
- Formalising Undergraduate Mathematics (2020) (Slides) (Tweet)
- Explanation of monad (2020)
- Compendium of Mathematical Symbols (2020)
- Math Vault - Learn Higher Mathematics The Online Way.
- Mathematicians Report New Discovery About the Dodecahedron (2020)
- MIT Math GitHub - Educational materials for MIT math courses.
- Journal of Logical and Algebraic Methods in Programming
- Metamath - Tiny language that can express theorems in abstract mathematics, accompanied by proofs that can be verified by a computer program.
- 3Blue1Brown Learn Math Recommendations
- Gurobi - Powerful mathematical optimization solver.
- Mathematics for programmers (2020) (Lobsters) (HN)
- Classic style and mathematical writing (2019)
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- Awesome Mathematical Notation Design
- Toward an API for the Real Numbers (2020) (HN)
- Making the Monty Hall problem weirder but obvious (2020) (HN)
- Euler's Formula: A Complete Guide (HN)
- Isomorphism - Mathematics of Programming book - Introduces the mathematics behind computer programming.
- Semigroups and Monoids (Lobsters)
- Favorite (potentially long) mathematician quotes (2020)
- Mathematics for Computer Science (2017) (HN)
- Ask HN: How do I improve my command of mathematical language? (2020)
- Ask HN: Which general programming language is good for learning/exploring math? (2020)
- Theta - Web app that graphs an equation and allows you to pan and zoom around the coordinate grid. (Code)
- Algebraic Stacks Project - Collaborative web-based project writing a text book on algebraic stacks and the algebraic geometry that is needed to define them. (Code) (Article) (HN)
- Why is there no equation for the perimeter of an ellipse‽ (2020)
- Open Textbook Initiative (HN)
- Open textbooks approved by the American Institute of Mathematics
- Intellectual Mathematics
- Notes for courses taken at Harvard (2015--2019) (HN)
- Paul's Online Math Tutorials and Notes (HN)
- Terence Tao's Career Advice for math
- Alexander Grothendieck: Memories of a Great Mathematician (2014)
- Learning Math for Machine Learning
- Fematika Video playlists on Math
- The Most Common Errors in Undergraduate Mathematics (HN)
- A History of Mathematical Notations by Florian Cajori
- Visualizing Algebraic Numbers
- The Simple Yet Powerful Math We Don't Talk About - Freya Holmer (2020)
- Math | Johns Hopkins University | Course Descriptions & Syllabus
- A Course of Pure Mathematics (1921)
- Learn Mathematics from start to finish (2020)
- MIT Mathematics (GitHub)
- NIST Digital Library of Mathematical Functions
- Mathematics and Computation (2018)
- Graph Toy - Interactive graph visualizer using mathematical functions. (HN)
- The Map of Mathematics (HN)
- Graphtoy (HN)
- Principle of Maximum Entropy (2020) (HN)
- The Riemann Hypothesis, Explained (2021) (Article)
- What is the Riemann Hypothesis? (2019)
- Math & Stats Uni Notes
- A Mathematician’s Lament (2009)
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- Kenneth Kuttler's Free Math Books
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- Formalising mathematics course (2021) (Code) (Intro Article)
- You Could Have Invented Homology, Part 1: Topology (2020)
- Art and Craft of Mathematical Problem Solving by Paul Zeitz (2010)
- CalcuLaTeX - Pretty printing calculator language. (Code) (HN)
- Defining Visual Narratives for Mathematics Declaratively (2019)
- Byrne’s Euclid - First Six Books of The Elements of Euclid With Coloured Diagrams and Symbols.
- Can One Hear the Shape of a Drum? (HN)
- Complete Proof of the Collatz Conjecture (2021)
- A Guide to Writing Mathematics (HN)
- How to Study Mathematics (2017) (HN)
- How to Solve It Book by George Pólya (1945)
- Vladimir Arnold on Teaching Mathematics (1997)
- How entropy can be seen as a volume - the geometric interpretation of information theory (2021) (Reddit)
- How (and why) to raise e to the power of a matrix (2021)
- ELI5 what is chaos theory, and what is it used for? (2021)
- What are the best/most interesting Springer Math Books? (2021)
- What Became of Moses Schönfinkel, Inventor of Combinators (2021)
- Mathematical Markup Language (MathML) (Code)
- Map of Mathematical Structures (MIT)
- Why do you like math? (2021)
- Oliver Heaviside (1850-1925) – Physical Mathematician (1983) (HN)
- Galois Theory (2021)
- Useful inequalities cheat sheet
- The Two Cultures of Mathematics (2000) (HN)
- Good books to relearn math (2021)
- Intuitive Guide to Hyperbolic Functions
- Cognicull - Learning site that pursues "ease of understanding" and "comfort".
- Which was the first book that gave you the taste of real mathematics? (2021)
- How to Code a Math Formula (2020) (Reddit)
- What's your favorite "strange" function? (2021)
- The Textbook That Unleashed Ramanujan's Genius (HN)
- Abakcus - Collection of the Best Mathematics and Science Stuff. (Twitter)
- The Essence of Mathematics Through Elementary Problems (HN)
- Zero: The Biography of a Dangerous Idea by Charles Seife (2000)
- Tau Day | No, really, pi is wrong (2010)
- What's your favorite proof? (2021)
- Math Memes Reddit
- Dr Peyam - YouTube
- Too simple to be simple
- Poisson's Equation is the Most Powerful Tool not yet in your Toolbox (2021) (HN)
- Three Months in Monte Carlo (2021) (Lobsters)
- The Utility of Mathematics (HN)
- Convergence of Infinite Products (2008)
- The Poincaré Conjecture, Explained (2021) (HN)
- Discrete Optimization Course
- A Bestiary of Functions for Systems Designers (2021)
- Steven Strogatz: In and out of love with math | 3b1b podcast (2021)
- Math texts, pi creatures, problem solving, etc. | 3blue1brown Q&A (2021)
- Ask HN: How can I learn to read mathematical notation? (2018)
- SageMath - Open-Source Mathematical Software System. (Code)
- Banach-Tarski and the Paradox of Infinite Cloning (HN)
- A Layman’s Guide to Recreational Mathematics Videos (2021) (HN)
- MIT Mathlets (HN)
- Elliptic Curve Explorer
- Visually stunning math concepts which are easy to explain (HN)
- Cheat-sheet for mathematical notation in code form
- Eleven proofs of the Gaussian integral (HN)
- A practical guide to radial basis functions
- Introduction to Higher Mathematics
- Helmhurts (2014)
- A mathematician's guided tour through higher dimensions (2021) (HN)
- The Dedekind tessellation
- Imagination and the Impossible (2021)
- Designer Maths (2021)
- Looking through Objects - How Tomography Works! (2021) (Tweet)
- Conterintuitive facts in mathematics, CS, and physics (2021) (HN)
- Math for Programmers: 3D graphics, machine learning, and simulations with Python Book (2020) (Code)
- 3Blue1Brown - Animated math videos. (YouTube) (Web Code) (Videos Code)
- Riding upon the fields on the horse of mathematics (2021)
- Math Foundations from Scratch (HN)
- AsciiMath - Easy-to-write markup language for mathematics. (Code)
- Self Studying the MIT Applied Math Curriculum (HN)
- How to learn mathematics: the asterisk method (HN)
- Measuring Mathematical Problem Solving With the MATH Dataset (2021) (Code)
- SoME1 results (2021) (HN)
- 3Blue1Brown - Puzzling through exact sequences (2021) (Tweet)
- Higher Math for Beginners (1987) (HN)
- Algebra: Chapter 0 by Paolo Aluffi (2009) (Tweet)
- A friendly introduction to Principal Component Analysis (2020)
- In simple English, what does it mean to be transcendental? (2021) (Lobsters)
- Introduction to stacks and moduli (2021)
- Mathematics and Computation Blog (Code)
- Ask HN: How did you learn math notation? (2021)
- Exploring the beauty of pure mathematics in novel ways (2021) (HN) (Paper) (Tweet) (Code)
- MathLingua - Language to create precise, automatically analyzed, beautifully rendered, collections of mathematical knowledge. (HN)
- On Induction (2021)
- Quaternions (2021) (HN)
- Algebra Notes
- Path to a free self-taught education in Mathematics
- Ask HN: I suck at math, where to start? (2022)
- Krater - Template to start building math-rich websites effortlessly. (Fork)
- A high-dimensional sphere spilling out of a high-dimensional cube (2022) (HN)
- Examples of common false beliefs in mathematics (HN)
- Математичный список полезных книг
- Solving some formal math olympiad problems (2022) (HN)
- Why isn't there a replication crisis in math? (2022) (HN)
- The mathematically optimal Wordle strategy (2022) (Tweet)
- insect - High precision scientific calculator with full support for physical units. (Code)
- Why won’t anyone teach me math? (2022) (HN)
- MathBook.Info - Интерактивные книги по математике.
- Math3d - User-friendly 3d graphing calculator for the web. Create, share, and animate 3d mathematical scenes. (Code)
- 1 thing in math you wish you had intuition for, but you don't
- Optics vs Lenses, Operationally (2022)
- So you want to study mathematics (2022) (HN)
- Savage Computer Algebra System
- The Open-Source Mathematics Degree
- The best books for mathematics enthusiasts
- My Mathematics PhD research workflow: LaTeX notes and instant PDF referencing (2022) (HN)
- LaTeX code for some of my lecture notes
- Interactive Videos to Explain Quaternions (HN)
- There’s more to mathematics than rigour and proofs (2007) (HN)
- Bad but interesting mathematical notation idea (2022) (HN)
- Things I would like to see in a computer algebra system (2022) (HN)
- Oldest mathematical text that you have seriously engaged with a substantial portion of?
- Ask HN: Books that made you fascinated to learn more mathematics? (2022)
- Are there any things in math you wish you could rename? (2022)
- Understanding Mathematics: A Study Guide
- ReX - Typesetting Mathematics.
- ProofWiki
- Jensen's inequality - Wikipedia (Tweet)
- Alexander Grothendieck: A Country Known Only by Name (2014)
- When We Cease to Understand the World (2021)
- Alexander Grothendieck: The Mysterious Disappearance of a Revolutionary Mathematician (2022)
- Math & CS Books/Resources collection
- Good math books to learn from ground up (2022)
- Less known YouTube channels with a lot of math content? (2022)
- Books on mathematical topics that deeply introduce you to the topic (2022)
- Mathematics for Machine Learning
- Ask HN: Serious mathematics books that can replace a good teacher? (2022)
- Mathigon - Mathematics education platform for students aged 12 to 18. (Code)
- Ask HN: How to learn math from zero for adults? (2022)
- Abstract math topics that made sense later (2022)
- Ask HN: Which mathematics books taught you the most in your career? (2022)
- What is your favorite thing about math? (2022)
- Daniel Quillen's Math Notebooks - Detailed notes of his day-to-day mathematical research.
- What's the most difficult mathematical concept you've encountered and had to understand so far? (2022)
- Ask HN: How to learn mathematical proofs from scratch? (2022)
- The Mathematical Universe - Max Tegmark (2007)
- What was the most remarkable mathematical achievement of the past 50 years? (2022)
- June Huh, High School Dropout, Wins the Fields Medal (2022)
- Fields Medals 2022 (HN)
- What is your favorite theorem in mathematics? (2022)
- Is the Kalman filter just a low-pass filter? (HN)
- Mathematical Tours of Data Sciences (Code)
- Combinatorics lecture notes
- Spaced Repetition for Mathematics (2021) (HN)
- How to Get a PhD in Mathematics in a Timely Fashion (HN)