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When someone comes up with a way to explain the same concepts using a simpler and easier to understand way, over time it becomes the new notation. But it rarely is world changing, mostly it’s small, incremental changes.

One big example of a major notation change is Grothendieck’s effect on algebraic geometry. It used to be about solutions of sets of polynomial equations. Now it’s all about sheaves and schemes and moduli spaces and representable functors and stacks and… This new notation changed the way mathematicians think. Did it simplify things and made it more accessible to lay people? I’d argue, not at all, in fact you probably need a year or two of university math education to fully understand what a scheme even is.

I guess category theory is example in the other direction: it greatly simplified and unified thinking in algebraic topology and geometry. However, it’s not that big of a help to a complete newbie: if you are at a complete loss when you encounter a cohomology theory for the first time, framing it in terms of graded functors won’t be of huge incremental help.

Point here is that notation is there to help mathematicians, not regular people, but regular people neither are able to understand all of it, nor do they care, so it’s not much of a loss.