"Why it works" is a very subjective notion. A very good example of the difference between proving something is true and showing why it's true comes in the under

"Why it works" is a very subjective notion. A very good example of the difference between proving something is true and showing why it's true comes in the undergraduate Real Analysis textbooks by Walter Rudin and Charles Pugh. If you look at reviews for Rudin's books, the word "magic" is used quite frequently. Rudin will cover the minimum amount of theory necessary and prove each theorem in a highly specific way. He tends to achieve the minimum number of words necessary for a proof. In contrast, Pugh focuses on building up the surrounding constructs and techniques to the point where you can apply the same tactics to a wide variety of problems. To me, this gives more of a sense of "why" each theorem is true.