You can define abstractions very precisely in mathematics, that's what makes it useful. You have to get used to the fact that you're talking about the abstracti

You can define abstractions very precisely in mathematics, that's what makes it useful. You have to get used to the fact that you're talking about the abstraction and not any specific instance/use of it. In the case of monads, the things that are useful which we can say about them: unit is a left identity for bind and also the right identity for bind, etc. What makes an abstraction useful, and is kind of essential in my opinion, are the laws. These do include every case because we don't say anything about specific terms or types, just the laws of bind (in the case of monads). I agree that people keep making them out to be some kind of mountain to climb when, after building up an intuition and learning the fundamental, it turns out they are more of a hill. However I think it takes a good mix of both the theory and application to appreciate how elegant and simple it is in the end.