Ask HN: Are there books for mathematics like Feynman's lectures on physics?

In addition to all the other good suggestions, the following are recommended (have not seen these mentioned so far);

- Concepts of Modern Mathematics - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

- Methods of Mathematics Applied to Calculus, Probability, and Statistics - https://www.amazon.com/Methods-Mathematics-Calculus-Probabil... (all books by Richard Hamming are recommended)

- Calculus: An Intuitive and Physical Approach - https://www.amazon.com/Calculus-Intuitive-Physical-Approach-...

For a Textbook reference, the following are quite good;

- Mathematical Techniques: An Introduction for the Engineering, Physical, and Mathematical Sciences - https://www.amazon.com/Mathematical-Techniques-Introduction-... (easy to read and succinct)

- Mathematics for Physicists: Introductory Concepts and Methods - https://www.amazon.com/Mathematics-Physicists-Introductory-C...

For General reading (all these authors other books are also worth checking out);

- Mathematics, Queen and Servant of Science - https://www.amazon.com/Mathematics-Queen-Servant-Science-Tem...

- Mathematics and the Physical World - https://www.amazon.com/Mathematics-Physical-World-Dover-Book...

- Mathematician's Delight - https://www.amazon.com/Mathematicians-Delight-Dover-Books-Ma...

It's extremely difficult to write a mathematics textbook in the intuitive style. There are some reasons for this.

Firstly, much of mathematics is symbolic and any description of equations in an intuitive style is unnecessarily verbose if it abandons the symbolic approach, essentially taking one back to descriptions like those used in ancient Greece before the invention of algebra, e.g. "and the third part of the first is to the second part of the first as the fourth part of the area is to the square on the gnomon".

The second reason is that an intuitive style supposes that one can answer natural questions that might arise, in an order that they are likely to arise in the mind of the student. Often the natural questions are much more difficult to answer mathematically, or the answers are not known.

The third reason is that concepts have arisen historically for non-obvious reasons, or reasons only known to experts with far more knowledge than the reader is expected to have, or the originator of the ideas did their best to obscure their motivation. This makes it extremely hard to motivate certain concepts naturally (intuitively) since such motivations are simply not known. For example, it is not hard to motivate solvable groups through a study of solubility of polynomial equations. But it is much harder to motivate the related concept of nilpotent groups, where the true motivations lie far deeper in the theory than the concepts themselves.

The fourth reason is that it is a massive effort to come up with good examples. Even the best textbook authors often struggle to come up with accessible examples for the concept they are trying to explain. Often, good examples require a really broad knowledge of mathematics that goes way beyond the narrow field being taught. Examples end up being very artificial, and neither intuitive nor typical, as a result.

Don't get me wrong. If someone told me something like the Feynman lectures existed for mathematics, I would salivate and spend a lot of money to acquire them. But having experimented with many styles of writing notes for myself on mathematics over the years, I well appreciate how hard, or perhaps impossible the task would really be. Of course there are some oases in mathematics where such an intuitive approach is possible.

Also libgen (since the authors are all dead):

http://gen.lib.rus.ec/search.php?req=Mathematics+methods+mea...

When I did analysis at university, I read Rudin instead of the recommended text. When the exam came around, I tried to generalise all the problems on the exam and gave answers which did not impress the person marking it, as they looked nothing like the standard solutions. As a result, I very nearly failed analysis. I wasn't particularly upset, as I used to consider my exam grade as a reflection of how well the course was taught, rather than a grading of my own comprehension. This is not a level of arrogance I would recommend to students looking to pass their exams.

ALL books by Yakov Perelman are a must read for every educated person. "Science popularization" at its finest! How i wish modern "authors" wrote books like these in their area of specializations.

Some of his Books;

* Physics for Entertainment Vols I & II

* Algebra for Fun

* Figures for Fun