@amiloradovsky@functional.cafe I would say it is the most interesting operation, but I wouldn't call it natural, and definitely not multiplication.

Types do make things clear, for the machine, but not always for humans that read the code. I want my compiler taking care of types, not doing it myself all the time. I can only see a point in defining matrix multiplication the way it usually is, if there is some generic code that multiplies and can accept both numbers and matrices, and does the "same" thing to both. I, thinking it's just as hard to come up with an example of that as of component wise multiplication of matrices... But the latter is such a trivial operation, that it has got to be useful for something, in practice. I don't know, maybe image processing, masking an image or modifying its intensity? And it's exactly the same for a number or a vector, both masking and scaling are applicable concepts, that makes sense for a multiplication to do.