HN Books @HNBooksMonth

israel gelfand, a top russian mathematician, has books aimed at high school students that served as education materials for distance learning in russia. the series covers algebra, functions, trigonometry, graphs, geometry, and more.

https://www.amazon.com/Functions-Graphs-Dover-Books-Mathemat...

https://www.amazon.com/Sequences-Combinations-Limits-Dover-M...

https://www.amazon.com/Geometry-Israel-M-Gelfand/dp/10716029...

https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/081763677...

https://www.amazon.com/Trigonometry-I-M-Gelfand/dp/081763914...

https://www.amazon.com/Method-Coordinates-Dover-Books-Mathem...

the books by sanjoy mahajan are also a treat and teach real-world applications of mathematical and scientific thinking.

https://www.amazon.com/Street-Fighting-Mathematics-Educated-...

https://www.amazon.com/Art-Insight-Science-Engineering-Compl...

I also like this guide: https://www.av8n.com/physics/thinking.htm "Learning, Remembering, and Thinking". I recommend checking out his other work for a model of working through problems coming from physicists.

One more thing. Oftentimes the key step to thinking is figuring out what you're questions are, and questions are always determined by what uncertainties you have in a domain, as specifically relevant as you can make them.

I'm gonna quote Venkat Rao (of Breaking Smart and Ribbonfarm fame) from an article he deleted years ago:

> Real questions, useful questions, questions with promising attacks, are always motivated by the specific situation at hand. They are often about situational anomalies and unusual patterns in data that you cannot explain based on your current mental model of the situation… Real questions frame things in a way that creates a restless tension, by highlighting the potentially important stuff that you don’t know. You cannot frame a painting without knowing its dimensions. You cannot frame a problem without knowing something about it. Frames must contain situational information. There are two types of questions. Formulaic questions and insight questions. …. Formulaic questions can be asked without knowing much. If they can be answered at all, they can be answered via a formulaic process. …. Insight questions can only be asked after you develop situation awareness. They are necessarily local and unique to the situation.

The world is /extremely/ information rich to the point of absurdity, and what fails is not the richness of our input data but rather our awareness of how we ought to use it. George Polya tried to teach his students how to problem solve in mathematics by means of getting people to ask questions. By verbalizing his thought process he hoped to convey these principles, as well as giving them a standard template to prompt their cycle of questions. But to adhere to a strict plan like that is to defeat the point. The real point is to maintain a conversation with yourself, giving yourself and refining your own questions until insight develops, and keeping yourself talking.

Ultimately I like to take an information-theoretic approach as the basis of my philosophy here. /Some/ information is /always/ going to be contained in /any/ comparison that I can make between two phenomena in the world. Most of this "information" would be considered noise relative to most reference frames. But it is always possible to extract /something/ from a situation by creating these tensions between yourself and your uncertainties in the world.

You can muddle around questioning things for awhile, but gradually things come up. The key is to let your uncertainty start off however it is and keep pruning away at it until your solution is sculpted from the clay. It can and will happen.

If you've ever tried doing Fermi Estimates (like those prescribed in https://www.amazon.com/Street-Fighting-Mathematics-Educated-... , https://www.amazon.com/Art-Insight-Science-Engineering-Compl... , https://web.archive.org/web/20160309161649/http://www.its.ca... , https://www.amazon.com/How-Measure-Anything-Intangibles-Busi...), then you'll be able to perceive the mindset that has significant transfer to many problems that have even just approximate answers.

⬐ mezod

thank you