I have a (bad) analogy here. Suppose that you write a program in a high-level language. This program will be compiled to a very verbose machine language. To understand why you have to slow down and go to the machine language level. Nevertheless, you are diluting the information if every time you perform this transformation and discard the high-level version. But yes, sometimes by not going to the compiled version, you will miss a critical detail, and it may be very hard to track down the cause. My analogy has an issue: compiling is deterministic, whereas your complaint is exactly that filling a math proof is not.

Going back to books, I guess that many mathematicians want in good faith to transmit what they believe to be "the important pieces". But while converting this mental representation to written proofs, we make serious mistakes. We could say this a "lossy compression". I have high hopes that (auto-)formalization can help us to keep a high-level human writing while being as lossless as possible.

Perhaps this is a great use case for AI? Generating the notes to fill in the elided details for a textbook.

Not only in mathematics, but also in physics texts and other sciences, mathematical demonstrations are often extremely cryptic, and a student may need hours to truly understand them.

Are there any math textbooks you've read that conform or are closest to your standard?

Hi Ángel, it is a bit hard for me to answer your question accurately because I don't remember the exact details or the difficulties posed by each book I read. My post here captures a general feeling I have had after reading several graduate-level textbooks in the last few years. I deliberately did not mention names because it is not just one or two specific books: nearly every graduate-level book presents proofs as high-level outlines. So I didn't want to single out any particular book in my post, especially since these books were written by very accomplished mathematicians and clearly embody a great deal of hard work, insight and mathematical wisdom.

That said, I did enjoy Introduction to Analytic Number Theory (Apostol, 1976) and A First Course in Coding Theory (Hill, 1986). Although these books are often used in graduate-level courses, the books themselves are marketed as undergraduate-level books.

If I had to give an example of a genuine graduate-level book with which I have had a mostly good experience, Algebraic Graph Theory (Godsil and Royle, 2001) would perhaps be a good one. There are some places where the proof style does resort to high-level outlines, but for the most part it is pretty good.

I don't mean to be overly critical of these books. They were written by very accomplished mathematicians. These are human works that must have required extraordinary effort. Humans are fallible, so no book is going to be perfect. My post was merely meant to lament that pedagogical problems are not limited to primary or secondary school but continue beyond that as well. Almost every book is bound to have omissions, for reasons explained in my post. Therefore someone who wants to convince themselves that the proofs really do work has to devote an extraordinary amount of time and effort to doing so.