What does ‘self-explanatory’ mean? Is there something like a ‘self-explanatory’ exposition or (re-)presentation? Is ‘self-explanatory’ the same or similar as ‘self-evident’?

Explanation to oneself and/or to someone else is a part of the communication process viewed from the sender’s point of view which has the intention of achieving understanding on the side of the receiver. Explanation depends on both parties and hence cannot be formalized. It does not make sense to speak e.g. of an ‘objective’ ‘explanatory power’ of a proof. Explanation is more suited to be considered an art.

Like in every domain of the arts one can try to criticize specific cases of explanations or trends/styles in explanation. The issue of taste plays a necessary important role. Some aspects of an explanation might be more objective than others and one can speak of a mathematical item being explained well or badly. One and the same explanation might be good (explanatory) for one person and poor/bad for another.

Explanation is clearly a part of ‘living mathematics’. But isn’t it also something like a genre in writing, in literature? Is there something like ‘explanatory literature’?

Mathematicians are adapted, used, trained to understand specific modes of explanation. They receive this training from specific persons (teachers, professors, collaborators). This is part of the process of initiation into the mathematical community. Gradually one starts to explain in a similar manner as one’s teachers/colleagues. On the other hand explanation needs feedback from the receiver about its success which leads (or may lead) to a successive adaptation towards the specific audience (of receivers).

Is understanding more than just an exact repetition of the explanation? Isn’t understanding linked to the ability to modify the explanation, to produce an alternative explanation? A successful explanation would then mean the production of alternative explanations. The effort of the receiver has to be to produce a new explanation so that the sender says OK!