Successful graduate students in mathematics are able to reach an advanced level in one or more areas. Textbooks are an important part of this process. A skilled lecturer is able to illuminate and clarify many ideas, but if the pace of a course is fast enough to allow decent coverage, gaps will inevitably result. Students will depend on the text to fill these gaps, but the experience of most students is that the usual text is difficult for the novice to read. At one extreme, the text is a thousand page, twenty pound encyclopedia which cannot be read linearly in a finite amount of time. At the other extreme, the presentation in the book is essentially a seminar lecture with huge gaps.
So it seems that improvements in readability of textbooks would be highly desirable, and the natural question is “What makes a text readable?” Is it possible to answer such a question concretely? I am going to try.
First, we need to be clear on exactly who is trying to read these books. Textbooks that are opaque for students may turn out to be quite useful to the research specialist. I will assume that the reader of the text is not already an expert in the area.The path to readability is certainly not unique, but here is some advice that may be useful.