While it is true that there are many examples of bad writing in mathematics, it is not true that unclear writing is glorified over clear exposition. To the cont

While it is true that there are many examples of bad writing in mathematics, it is not true that unclear writing is glorified over clear exposition. To the contrary, authors like Paul Halmos are rightly celebrated as great expositors, e.g. see his books like Naive Set Theory, Finite Dimensional Vector Spaces, Measure Theory etc. and his expository writing such as "How to Write Mathematics" and "How to Talk Mathematics": [PDF] http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.pdf http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.p... http://www.math.northwestern.edu/graduate/Forum/HALMOS.html http://www.math.northwestern.edu/graduate/Forum/HALMOS.html collected in his Selecta of expository writing. Some other books I like are Sheldon Axler's Linear Algebra Done Right, Real Mathematical Analysis by Charles Pugh, and the three books on manifolds (Topological Manifolds, Smooth Manifolds and Reimannian Manifodls) by John Lee, but there are many other well-known books that are clear in their exposition. Edit: Part of the issue is also the approach of the reader, i.e. is s/he there to learn mathematics, where the goal is to explore the mathematical world, or is s/he there to use the mathematics to explore the physical world? That should guide questions like "What is the point of studying vector spaces?" The motivation to study vectors spaces (and other mathematical topics) may need to come from other courses, e.g. on mechanics, and the course/book on vector spaces would serve to give in-depth knowledge of the particular topic.