When I started studying mathematics, I soon started to wonder what kind of superhumans make the mathematics up, for the mathematics was full of bizarre definitions, and unreadably long proofs. There was also some devilishly clever short proof, but they did not make it easier, for I knew that one had to posses divine intelligence to come up with them. For example, I did not understand how a mortal could read, much less invent, a statement of a theorem spanning three pages or its proof.

However, behind every definition, every theorem, every proof, there is an idea. Often the simplest implementation of the idea does not work, and the original approach has to be modified or supplemented with technical conditions. When the correct piece of mathematics is found, the original idea is buried deeply. Hence, explaining the idea involves far more than writing down logically correct mathematical statements. A good explanation requires description of the whole process of the discovery.

The process of discovery is always messy, and it is tempting to reveal as little of it as possible. An honest description of the convoluted twists of mind that culminate in a discovery is both long and embarrassing. The discoverer is thus tempted to impress the reader with the shortest argument possible. Since it is rarely possible, monstrosities, dense with equations, are born.

Natural mathematical explanations that are also well-written are thus extremely rare. Yet they are indispensable for appreciating, and learning mathematics. Below I have listed links to natural mathematical explanations that I know of. The topics they treat are wildly different, but the authors of all of them have managed to convey the joyful process of discovering mathematics. Please, suggest more...

The ultimate goal of mathematics is to eliminate any need for intelligent thought. [Alfred N. Whitehead]

- “Mathematician's Lament” by Paul Lockhart is a short essay, which talks about the most important in mathematics: the process of doing it.
- Timothy Gowers has a beautifully written discussions of undergraduate mathematical topics. There he explains how one could naturally be led to various mathematical concepts.
- “Concrete mathematics” by Ronald Graham, Donald Knuth and Oren Patashnik deserves to be on everyone's shelf. The book explains how to methodically approach a wide class of problems. Written in a witty and didactic style, this book does not bore with definition-lemma-theorem-proof style.
- George Pólya wrote several books on solving mathematical problems. The exposition of the derivation of Carleson's inequality in “Mathematics and Plausible Reasoning” is the best narrative of a solution of a mathematical problem there is.
- “Proofs from the book” by Martin Aigner and Günter Ziegler is a collection of beautiful mathematical pieces. Every page breathes with harmony and perfection. The beautfil ideas are distilled into the almost perfect proofs.
- “Making Transcendence Transparent: An intuitive approach to classical transcendental number theory” by Edward B. Burger and Robert Tubbs strips away the mystery surrounding the transcendence proofs, by explaining how one could naturally discover increasingly sophisticated techniques in the course of attempts to show that certain numbers are transcendental.
- “Real algebraic geometry” by Vladimir Arnold is a charming book about plane geometry. The exposition is both rigorous, and is unencumbered by formalism. Arnold smoothly goes from the problem of placement of jet engines to Riemann–Hurwitz formula. The Russian original is freely available online.
- Kalid Azad wrote nice explanations for several high-school math topics.

E-mail: