* Counter-clockwise angles comes from making charts and graphs with the independent variable going from left to right, and the dependent variable from bottom to top. This is far too ingrained in the way we teach and learn about mathematics to change now, much more than the choice of pi or tau (though we do make y go from top to bottom in many computer graphics contexts). There’s no reasonable way that we could integrate direction of reading on a clock with direction of reading typical charts with direction of reading text into a uniform system, and I don’t consider changing all the clocks, texts, and charts in the world to be a reasonable possibility.
* What does the shape of the slash character have to do with anything? A slash is a fraction bar. Fractions and matrix diagonals are entirely unrelated, though in both cases numbers are read from top left to bottom right, in accordance with our typical reading direction in western texts.
* Matrix index ordering is a pain in the butt and will be confusing whichever way they’re labeled. The logic behind the current system is to use the first index for the component that results when multiplying by a vector, and using the second index within that component. Picking the opposite convention would also end up confusing. Figuring out the proper ordering when dealing with non-commutative “number” systems in general is a pain, and I don’t think there’s any easy answer. We have matrices multiply column vectors on the left, because that’s typically how we notate operators acting on some input. But it means that composition is multiplication from left to right, which is a bit confusing. There’s no way to make the order be always left-to-right or always right-to-left. But much more importantly, matrices are a kind of painful abstraction to use in general. Mathematics education would be much improved in many ways if we used Geometric Algebra instead of matrix representations a lot more of the time. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
* It’s much easier to just call these “concave up” and “concave down”. Problem solved.
* What does the shape of the slash character have to do with anything? A slash is a fraction bar. Fractions and matrix diagonals are entirely unrelated, though in both cases numbers are read from top left to bottom right, in accordance with our typical reading direction in western texts.
* Matrix index ordering is a pain in the butt and will be confusing whichever way they’re labeled. The logic behind the current system is to use the first index for the component that results when multiplying by a vector, and using the second index within that component. Picking the opposite convention would also end up confusing. Figuring out the proper ordering when dealing with non-commutative “number” systems in general is a pain, and I don’t think there’s any easy answer. We have matrices multiply column vectors on the left, because that’s typically how we notate operators acting on some input. But it means that composition is multiplication from left to right, which is a bit confusing. There’s no way to make the order be always left-to-right or always right-to-left. But much more importantly, matrices are a kind of painful abstraction to use in general. Mathematics education would be much improved in many ways if we used Geometric Algebra instead of matrix representations a lot more of the time. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
* It’s much easier to just call these “concave up” and “concave down”. Problem solved.