This is Part 7 on an ongoing series on mathematical culture.
Two years ago, I encountered some mathematician's work that caught my eye: Harold M. Edwards. My first impression of him was that he has a very unusual writing style setting him apart from the traditional theorem-proof style writing.
Other reviewers of his textbooks seem to agree with me:
As I have stressed time and time again, a textbook which does not fall neatly into homogenization for a course is unlikely to find widespread adoption; such a text suffers in popularity as a result. This book is one of them ! (Review of Advanced Calculus by Harold M. Edwards)
[...] but Edwards makes you understand how he approached it in a much more down-to-earth way than is now presented in textbooks. Instead of plodding thru theorem-proof without insight or motivation, this book actually works thru the relation between the algebra of polynomial solutions and the structure of their groups. This is the second book I have read from Harold Edwards. I found I learn an awful lot of the low-level details of a subject that I can't find anywhere else. (Review of Galois Theory by Harold M. Edwards)
Here is another mathematician who decided to deviate from traditional calculus proposing an unorthodox approach to analysis:
In fact, a lot of calculus books look very much the same, and there’s a lot of pressure on undergraduate lecturers to present a calculus course which is not dramatically different from other calculus courses because they feel that the engineers want this, and they require that. But we're kind of free from that, right? So, we are not having some agenda imposed on us. I can just develop the subject in a way that I think is best in terms of the actual logical structure of the subject. (Invitation to a more logical, solid and careful analysis - Wildberger)