I think the problem I have with maths is that at the point where I start to realise that the simplifications are wrong, I lose trust in every area of my understanding and want to give up. I can't be comfortable feeling that there's a missing hole in my mental model.
OTOH, it doesn't work at all to give exhaustive mathematical proof because it doesn't construct a model in my head.
So I don't know what the solution is really.
To take complex numbers for example: the idea that -1 has a square root seemed like utter bullshit as it contradicts other models one has had drummed into one. If you look at it as a means to an end, however, it's a useful little bit of machinery that can help to make other problems easier to solve (e.g. converting a differential equation to a quadratic). That makes sense and explains why we're doing it whereas talking about SQRT(-1) starts one off with the feeling that everything is nonsense.
OTOH, it doesn't work at all to give exhaustive mathematical proof because it doesn't construct a model in my head.
So I don't know what the solution is really.
To take complex numbers for example: the idea that -1 has a square root seemed like utter bullshit as it contradicts other models one has had drummed into one. If you look at it as a means to an end, however, it's a useful little bit of machinery that can help to make other problems easier to solve (e.g. converting a differential equation to a quadratic). That makes sense and explains why we're doing it whereas talking about SQRT(-1) starts one off with the feeling that everything is nonsense.