Mathematics and plausible reasoning

What Hacker News says

1. eimrineAug 2024
   I was astonished that all the best and most important in methodical approach mathematical books are almost unknown to students here (and, seems to me, have not been translated into French). Among these are Numbers and figures by Rademacher and Töplitz, Geometry and the imagination by Hilbert and Cohn-Vossen, What is mathematics? by Courant and Robbins, How to solve it and Mathematics and plausible reasoning by Polya, Development of mathematics in the 19th century by F. Klein.
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2. sveronaAug 2023
   Mathematics and Plausible Reasoning, in two volumes also by Polya, is the closest thing I've ever found to an explanation of how a working mathematician goes about her business. It has exercises, too. Great fun when I was in undergrad.
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3. ggr2342May 2023
   They were not textbooks for any course. But I liked these:

   1. Mathematics and Plausible Reasoning by George Pólya

   2. Probability Theory: The Logic of Science by E.T. Jaynes

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4. steppiJan 2023
   Some favorites below. Books 0-3 are accessible. The remaining books are more difficult but I'd highly recommend them to math students.

   0. Jan Gullberg, Mathematics, From the Birth of Numbers. A highly accessible popular survey on different branches of higher mathematics. I read this over the Summer between high school and starting my undergraduate degree. It's what made me want to study math. Previously I'd wanted to be a guitar player, but had to find a new ambition after an injury left me unable to play.

   1. The high school mathematics series by Israel Gelfand. Algebra, Trigonometry, The Method of Coordinates, and Functions and Graphs. I didn't have much mathematics background in high school, but working through these really solidified my grasp on the basics.

   2. George Polya. How to Solve it. A short book giving excellent high level advice on mathematical problem solving.

   3. George E. Andrews, Number Theory. I worked through this freshman year contemporaneously with my first proof based class on simple logic and set theory. A very beautiful and accessible introduction to basic number theory. The combinatorial/geometric proofs of Fermat's Little Theorem and Wilson's Theorem are lovely. It also includes a very nice proof of Chebyshev's theorem on the asymptotic density of primes and even the Rogers-Ramanujan identities for integer partitions.

   4. Vladimir Arnold, Ordinary Differential Equations: Undergrad ODE classes are often taught in a cookbook fashion and if so, don't offer much enlightenment. This book explains what's going on at geometrical level. I didn't appreciate ODEs until I read this. See https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html for Arnold's views on teaching mathematics.

   5. E.C. Titchmarsh The Theory of Functions: Recommended by my undergraduate advisor because he noticed that I liked reading older books. It contains sections on complex analysis and real analysis with measure theory, but I've only read the complex analysis sections. It's not for everyone, if I recall correctly, there is not a single picture, but it is very lively and has a lot of material you won't find in a standard complex analysis book, including Dirichlet series. Excellent as a supplement to a standard complex analysis book.

   6. George Polya. Mathematics and Plausible Reasoning. An excellent expansion on Polya's ideas on How to Solve it. While the goal is to seek rigorous proofs, to get there it's powerful to be able to think based on intuition, heuristics, and plausible reasoning. A lot of math exposition is theorem/proof based and doesn't help develop these skills. In a similar vein, see also Terence Tao's classic post There's more to mathematics than rigour and proofs https://terrytao.wordpress.com/career-advice/theres-more-to-....

   7. H.S.M Coxeter, An Introduction to Geometry. A book of very beautiful classical geometry. Something typically not touched on at all in a typical mathematics curriculum.

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5. mmmmkayJan 2022
   came here to ask the same thing! :D

   "How to solve it" [1] was helpful for me in this regard, but it isn't really condensed like this blog post.

   After you finish "how to solve it", you can read the more advanced "Mathematics and Plausible Reasoning" [2]

   [1] https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie... [2] https://www.amazon.com/Mathematics-Plausible-Reasoning-Two-V...

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6. jacobolusJul 2020
   Instead of How To Solve It, which is organized dictionary–style with short sections on particular named problem solving topics, and is somewhat hard to interpret for novices without guidance, let me recommend Polya’s other two books (each 2 volumes), Mathematical Discovery and Mathematics and Plausible Reasoning.
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