This is not so much "how to *do* mathematics" (to quote the OP),
but more how it feels to do mathematics.
A *New Yorker* review said this book "is less about math than about mathematicians—how they live, how they work, and how they talk to one another." I found it inspiring.

> Villani, Cédric. *Birth of a Theorem: A Mathematical Adventure*. Farrar, Straus and Giroux, 2015.
([*Guardian* review](https://www.theguardian.com/books/2015/feb/25/birth-of-a-theorem-mathematical-adventure-cedric-villani-review).
[*AMS* review](http://www.ams.org/notices/201402/rnoti-p172.pdf).)
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
[![Villani][1]][1]
<hr />

The *AMS* review ends: "The implicit task that Villani had set for himself, of explaining what it is all about, is a difficult one .... To my mind the book succeeds wonderfully."

[1]: https://i.sstatic.net/wQOlG.jpg