Proofs and Refutations

As a child, I re-read The Number Devil constantly. It introduced me to some really cool mathematical ideas, couched in a cute story. The very last chapter includes a picture of the Principia Mathematica's proof of 1 + 1 = 2 -- mostly for shock value, I think, but also "even this is within your reach". I recently got to see a friend's copy of the Principia, and I realized just how much of it I actually did understand, which was a really nice closing of the loop. As an adult, Imre Lakatos' Proofs and Refutations gave me a much richer understanding of definitions in mathematics -- what job they're meant to do, and when it makes sense to change your definitions instead of adding premises to your theorems.

_Proofs and Refutations_ tops the list. Dialog and progressive discovery are crucial to solving problems, and these things happen even after (mostly after?) you arrive at a good solution. _P&R_ provides a working example of how this might go, and how a diligent and magnanimous mind might approach the process. I think it's a really fun and engaging book, too. Works on pragmatism by CS Peirce were really impactful, though I can't recommend a specific book. In a very focused way, Peirce advocates for evaluating ideas in terms of their real-world effects, often by employing a measured skepticism. On the psych/behavioral side, _Gut Feelings_ shows how to be an efficient thinker. Largely contrary to Kahneman's work, it vindicates intuition, but does so in a qualified way. It explains when and how to trust your instincts, and emphasizes the importance of cultivating them.

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.…

My list: - Pirkei Avot, it's a section of the Mishnah (a part of the Talmud), it summarizes the jewish moral thought (well, not for the Karaites Jews, they did not believe in the Talmud), I was born in a Orthodox Jewish family so it was important on my beliefs, even though I am an Atheist now. - Organon, it's a compilation of Aristotle's logic works, they are the basis for the logical thought of the Western civilization, I think it's easier to read the originals than compendiums. - Philosophical Investigations, the second book of Wittgenstein, it's dense, and obscure to some extent, but it's really eye-opener, some ideas that influenced the last century are introduced there. - Proofs and Refutations, a posthumous work by Imre Lakatos, it defied much of my established view on how Mathematics works. - Thus Spoke Zarathustra, a powerful book, it's dense and you can only drag many conclusions from it when you read other works from Nietzsche, which lead to my next book.…