Mathematical discovery

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1. 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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2. jacobolusApr 2018
   Concrete Mathematics is pitched at graduate students in computing. Spivak’s Calculus is an introductory real analysis book pitched at undergraduates who have gone through a computational calculus course already and want to study the subject more formally and rigorously; it has many difficult problems and would generally benefit greatly from the structure and expert feedback of a university course. Jaynes’s book is probably most relevant to science students who are at least at the advanced undergraduate level. How to Solve It is a dictionary of heuristic problem-solving techniques which is most useful to someone who is already (deeply) familiar with mathematical problem solving, and wants to codify their existing methods. Even advanced undergraduate math students who read it aren’t going to fully understand the book IMO; I would recommend Pólya’s other books (Mathematics and Plausible Reasoning, Mathematical Discovery), or maybe start with a gentler book like Mason, Burton, & Stacey, Mathematical Thinking.
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3. jacobolusFeb 2018
   Even more useful as a first step in my opinion is to think about problem solving per se. E.g. see:

   Mason, Burton, & Stacey, Thinking Mathematically

   Pólya, How to Solve It; Mathematics and Plausible Reasoning (2 vols); Mathematical Discovery (2 vols)

   Schoenfeld, Mathematical Problem Solving

   Larson, Problem-Solving Through Problems

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4. jonnybgoodJun 2017
   Read 'Proofs and Refutations, The Logic of Mathematical Discovery' by Imre Lakatos. That may provide a good answer to your question.
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5. JadeNBJul 2024
   > But then I read Imre Lakatos’s Proofs and Refutations. It is not, at first glance, a book about writing. It is a book of mathematical philosophy. By a Hungarian Stalinist, no less.

   I don't see what "Hungarian" has to do with it, and, though I do see what "Stalinist" might have to do with it, it probably shouldn't. (Someone's politics don't have to be good for them to make a valuable contribution to knowledge.) But, according to Wikipedia, this isn't true literally as written, unless one takes the view "once a Stalinist, always a Stalinist:"

   > After his release, Lakatos returned to academic life .... Still nominally a communist, his political views had shifted markedly, and he was involved with at least one dissident student group in the lead-up to the 1956 Hungarian Revolution.

   > ... He received a PhD in philosophy in 1961 from the University of Cambridge; his doctoral thesis was entitled Essays in the Logic of Mathematical Discovery, and his doctoral advisor was R. B. Braithwaite. The book Proofs and Refutations: The Logic of Mathematical Discovery, published after his death, is based on this work.

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6. sbDec 2012
   This year I either read, or started to read the following interesting books:

   Mathematics: Form and Function by Saunders Mac Lane. This is one of my favorite books concerning the "build-up" of mathematics (it also contains nice diagrams of "relatedness" of subjects). On HN somebody once recommended Mathematics: Its contents, methods, and meaning (from Russian mathematicians in the 50s) which is similar but without the cross references.

   Proofs and Refutations by Imre Lakatos. I have started reading this only recently and have to say that I find the approach and idea excellent. It would be great if we had something comparable for CS theory as well.

   Notes on Introductory Combinatorics by Polya, Tarjan, and Woods. Have not read this exhaustively, but the introduction with Pascal's triangle and some of Polya' legendary problem solving insights (paraphrased from my memory: "you are on to something once you find a pattern") are definitely highlights in this book.

   Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving by George Polya. Based on the previous book and my fond memories of reading "How to Solve it", I got this one from the library. Again I can't attest for all of the contents, but AFAICT now it's another gem from Polya.

   From HN advice in previous years I read The Tibetan Book on Living and Dying, which I can heartily recommend, too. It is an anti-thesis to Christian theology and I find it to contain many insightful comments and different views on leading a good, meaningful life. I disagree with some of the church-y comments on that it really is important to have a master and that only the master can do certain things, but that's probably just me being an atheist all along.

   I actually read some other books, but the list is already kind of long and might hold interesting pointers for other mathematically inclined readers, too. I for one am always fascinated on how much advice on problem solving in mathematics translates to CS.

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