Solving Mathematical Problems

What Hacker News says

1. adamsmith143Jan 2023
   A lot of comments about textbooks that helped in specific topics but I don't think that answers the spirit of OP's question. Sure working through ANY Linear Algebra textbook is going to improve your Linear Algebra skills.

   In the spirit of OP's question:

   How to Solve it by G. Polya

   Solving Mathematical Problems by Terrence Tao

   Introduction to Mathematical Thinking by Keith Devlin

   Are all amazing, How to Solve it in particular is an all time classic.

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2. photon_linesMay 2022
   Great introductory books along with videos:

     - Mathematics and Its History (John Stillwell)
     - Journey through Genius: The Great Theorems of Mathematics (William Dunham)
     - Proofs Without Words: Exercises in Visual Thinking (Roger B. Nelsen)
     - The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities (William Dunham)
     - Odds & Ends: Introducing Probability & Decision with a Visual Emphasis (Jonathan Weisberg)
     - Introduction to Probability (Joseph K. Blitzstein, Jessica Hwang)
     - The Secret Life of Equations: The 50 Greatest Equations and How They Work (Richard Cochrane)
     - Euler's Gem: The Polyhedron Formula and the Birth of Topology (David S. Richeson)
     - 3Blue1Brown Essence of Linear Algebra: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab 
     - 3Blue1Brown Differential Equations: https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6 
     - Dynamics: The Geometry of Behavior (Ralph Abraham, Christopher Shaw)
     - Geometry, Relativity and the Fourth Dimension (Rudolf Rucker)
     - 3Blue1Brown Essence of Calculus: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr 
     - Calculus for the Practical Man (J.E. Thompson)
     - Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Harry M. Schey)
     - Who Is Fourier? a Mathematical Adventure
     - Physics, Topology, Logic and Computation: A Rosetta Stone (John C. Baez, Mike Stay)
     - An Illustrated Guide to Relativity (Tatsu Takeuchi)
     - The Mathematics Of Quantum Mechanics (Martin Laforest)
     - A Student's Guide to Maxwell's Equations (Daniel Fleisch) or if you're also interested in a visual guide, I wrote one as well: https://github.com/photonlines/Intuitive-Guide-to-Maxwells-Equations 
     - Introduction to Elementary Particles (David J. Griffiths)
     - An Invitation to Applied Category Theory: Seven Sketches in Compositionality (Brendan Fong, David I Spivak)
     - Solving Mathematical Problems: A Personal Perspective (Terence Tao)

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3. Jun8Nov 2019
   Tao's Solving Mathematical Problems: A Personal Perspective (https://www.amazon.com/Solving-Mathematical-Problems-Persona...) mentioned here is an excellent book that provides keys to mathematical approach to problems and is now a classic ranking there with Polya's How to Solve It.

   One thing the article doesn't touch on is Tao's prominence in massively collaborative math research through his blog (latest such work from there was discussed on HN a day ago https://news.ycombinator.com/item?id=21542054). This new approach, first proposed by Tom Gowers (https://gowers.wordpress.com/2009/01/27/is-massively-collabo...) in 2009 has successfully solved a number of open problems in the past ten years (https://polymathprojects.org/2019/02/03/ten-years-of-polymat...).

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4. Jun8Oct 2019
   It's not wide-ranging but Terence Tao's Solving Mathematical Problems (https://books.google.com/books/about/Solving_Mathematical_Pr...) makes you understand mathematical thinking better than any other book I have seen at that level.
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5. Jun8Jul 2015
   The idea of a world class mathematician as as weird genius is common and may be said is not totally without base (Perelman comes to mind). Tao's interactions on his blog is always unassuming, trying to help, "super normal" as this article puts it.

   But to me the even more amazing thing is how good an educator he is. Most mathematicians I have encountered were of the "proof is left as an exercise" type, where if you didn't see how things done would't or couldn't help you in learning how to proceed. Tao is a big outlier in this regard. His How to Solve Mathematical Problems (http://www.amazon.com/Solving-Mathematical-Problems-Personal...) is more focused than Polya's famous book but is a much better help in learning how mathematicians think. You can get the first chapter for free (http://www.math.ucla.edu/~tao/preprints/problem.ps), an excellent way to finish off a Friday.

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6. Jun8Jun 2011
   For books i'd suggest Men of Mathematics that has inspired countless people to math careers or Terence Tao's Solving Mathematical Problems, which is excellent.

   Now, if money's no object, i.e. you're willing to spend around $200, the gomboc (http://en.wikipedia.org/wiki/Gömböc) is a wonderful object to buy, you can buy it at the gomboc-shop.com

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