Good Exposition

Timothy Chow in his article on forcing expresses his idea of an "open exposition problem". I think that his remarks about what makes a good exposition are seldom instantiated in the mathematical literature, I would like to record them here as a reminder to myself what I should aim for when giving talks and presentations and writing exposition.

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be “natural” in Donald Newman’s sense (Analytic Number Theory, Springer-Verlag, 1998): This term . . . is introduced to mean not having any ad hoc constructions or brilliancies. A “natural” proof, then, is one which proves itself, one available to the “common mathematician in the streets.”

Some other things to bear in mind are

• Every time you name something, explain why; if it is a later result reference it, if it is historical, annological, explain it.
• Every time you make a simplifying assumption explain how the general case can be reduced to it; if it cannot remark on this as well, it is ok to make assumptions just so we can get an idea, but be clear about WHY we make the assumption.