I don’t feel all that comfortable, talking about intention – I don’t think that people have fully-formed aims and objectives before they do something, and we seem mostly to bounce around in the world not quite knowing what we’ve done or what we’re doing next. It is however a useful thing to talk about to divide up the actors in a particular scenario and their different roles and outcomes.
The presenter wants to convince. They make an elaborate reconstruction of their own process of discovery, working on the assumption that their experience of understanding will be the best template for another person’s, often telling parts of the proof out of order because it helps understanding in a way that the formal version does not. The performance and the formalisation have different aims, it would seem. The presenter wants to convince an actual group of people, though their specific backgrounds are unknown, whereas the written paper is only for The Mathematical Community, a faceless entity that is unboreable, unconfusable, all-seeing and all-knowing. The performance also involves a social pressure to please. You do hear about mathematicians who just stand at the board and write, and don’t say a word. They’re the exceptions that prove the rule.
The audience wants to be taken along on a dance, though they have a critical eye – like a mega-fan reading the new book in a series, they might pipe up at one point and say, but this wouldn’t behave like that! In that instance, the author has created a character, but the mega-fan still feels justified in arguing back about what their creation would or wouldn’t do. A character’s a bunch of words, a series is a bunch of greek letters that someone’s writing down. Neither one will ‘do’ anything. But we give these things agency.
(So what could we say that a piece of maths wants? To be looked at? To lead somewhere? I ask this question in the same way that W. J. T. Mitchell asks What Do Pictures Want: as a thought experiment, a shift of focus to get away from questions of meaning)
The presenter might want to outline a technique, an approach, a way of cutting up a problem, like demonstrating a good tool. They might want to demonstrate the sturdiness of a structure that they’ve built, giving a tour and explaining their building techniques.