There is a fascinating and ever-growing body of literature comprised of works by mathematicians writing about what it means to be a mathematician. Sometimes these works touch on the topic while exploring the questions "what is mathematics?" or "how should math be taught?" but a significant number of mathematicians write about what they do without reference to these other questions.
Mathematicians who write in this vein are likely the best examples of reflective practitioners, and are probably better teachers, researchers, and scholars as a result of thinking carefully about what they do and taking the time to explain it to others. The "others" that this sort of literature is intended for are worth considering - who are these mathematicians trying to explain themselves to? Rarely are these works intended for complete outsiders, rather they are often intended as a set of Letters to a Young Mathematician, or are addressed to other academics (scientists, and engineers, mainly) that the authors feel they should have more in common with. Often these writings are addressed to fellow mathematicians, perhaps just to reassure them that they are all in the same boat. Sometimes these writings are a sort of meta-communication - mathematicians explaining to other mathematicians about how they should really be explaining things to the man in the street.
I started thinking about this kind of writing after seeing the film I want to be a Mathematician (mentioned here) about Paul Halmos, and reading some essays by Freeman Dyson and Timothy Gowers (mentioned here). Another excellent essay more sharply focused on the topic is Through a Glass Darkly by Steven Krantz.
Those who study the sociology of science, of academics, and of scholarly mathematicians in particular (yes, there are sociologists who study mathematicians), must have to treat these texts in a special way. Obviously, these are valuable documents, but are mathematicians the best observers of mathematicians?
