It isn't, it takes the derivative of the derivative operator, not just applying the derivative to a function several times. Fractional calculus is the study of viewing derivatives as a continuous quantity rather than discrete, so there taking derivative of the derivative operator makes sense.
You could very well mean something different than what that article is talking about, but it isn't like I am just saying that a second order derivative is the same thing as what you are talking about. But if you mean something different than "the derivative of the derivative operator" then you aren't very good at explaining what you mean, you have to be more precise as math is a wide field and if you describe an object imprecisely then people will misunderstand.
> I also have a PhD in maths, thanks.
You yourself argued that people with PhD in math gets this wrong, so I don't see why you would bring this up. But the most likely scenario is that you just failed to communicate your thoughts properly since you used imprecise language. Maybe mathematicians should get taught more how to be precise with their statements, but usually they can rely on the crutch of old notation. I did invent new notation and solved some old unsolved problems that way in grad school, and the other mathematicians had no problems understanding what I wrote, so mathematicians have no problems understanding new things from my experience.
It IS unusual to worry about type signatures as a mathematician. Sounds to me like your PhD topic was closer to computer science than math? Or at least constructive mathematics ;-)
I got tired of having units mismatch in physics equations so I picked up a ton of type theory and used it for everyday work. Using it, rather than talking about it, has given me a very different perspective on it than anyone else I've talked to.
If you keep walking down that path you'll soon be labelled a crank by the old guard.
Developing a practical rather than a theoretical understanding of type theory rapidly makes you express intuitions which other people can't lex/parse.
Because you begin to think in functors/compositions e.g constructively; and as you've already pointed out many of those functors don't have corresponding English nomenclature in a classical setting.
> Find some Category Theorists to talk to instead.
Most mathematicians are also category theorists today. They were the ones who invented category theory in the first place and today it is a basic topic most takes in grad school and then used just about everywhere.
I also have a PhD in maths, thanks.