[Tainnor]

Metaphors are a very powerful tool in human cognition, and there has been a lot of research into their usage from a cognitive science perspective (see e.g. the works by George Lakoff).

Of course, an abstract mathematical proof is independent of any metaphors you may ascribe to them, but new maths is usually discovered by applying intuition to a problem and then verifying that the intuition is in fact correct. To do that, you need metaphors and analogies. The interesting part is when you can use several metaphors for the same concept and the key insight comes from a shift in perspective; e.g. some statements about complex numbers make more sense when viewed from a geometric (rotation-inspired) perspective. Similarly, thinking about e.g. monads as "containers that can be flattened" can lead to certain insights, but thinking about them in some other ways can maybe lead to different insights.

However, I agree with you in one point: we learn mathematics and programming by example; I think the key point is that people have to build their own intuitions and metaphors. It can be useful to guide people along that path, e.g. by pointing out some helpful analogies or dismissing some unhelpful ones, but ultimately, you have to start working with the concept and prove/disprove your own assumptions (this is as true of a new programming concept you're not familiar with yet as of mathematics, only that in the former case you're much less formal about it).