When you enter truly new grounds mathematicians don't even agree if the distinctions being made have a meaning, let alone if they are true.
What examples are you thinking of? I can think of only one case where this has been the case (Mochizuki and the ABC Conjecture), but that turned out to have so much fanfare precisely because mathematicians did not agree on what distinctions were being made and this was the only time in living memory that this had occurred (the general consensus is that Mochizuki's proof is simply too obfuscated to make heads or tails of). As such the ABC Conjecture is not considered solved.
However, that is almost always not the case, even on the cutting edge of mathematics and even when making other breakthrough discoveries (e.g. Fermat's Last Theorem).
> E.g. it would be like a psychologist asking how an results of a well proven result would be different if all participants wore red shoes.
And to be clear the major reason why fields like psychology are termed to have a "replication crisis" is because well-known results are being overturned, not just cutting-edge ones.
EDIT: I see now that OP also refers to the ABC Conjecture.
Back in grad school I was very much into types which let me do things sideways to how most mathematicians did them.
The worst one was the derivative of a derivative - NOT the second order derivative. Using R for the real numbers, the type signature of first (and any) order derivative is (R -> R) -> (R -> R). The type signature for what I was talking about was ((R -> R) -> (R -> R)) -> ((R -> R) -> (R -> R)). It didn't have very many interesting properties I could find but trying to explain it to anyone else in the department was like pulling teeth. They'd start thinking about second order derivatives every time and there is no common notation for talking about third order functions like there is for first order and (some) second order functions (derivative, integral, transforms, etc).
Mathematics only works as well as it does because mathematicians are all working on pretty much the same thing, not because there is some innate quality in mathematics that pushes mathematicians towards truth.
Usually notation is defined for higher order functionals when the need arises - when a new result is found to have interesting properties. Topological proofs on function spaces can use third order functions.
> Mathematics only works as well as it does because mathematicians are all working on pretty much the same thing
I don't think any person alive can understand all the major branches of mathematics well. By that measure mathematics is very broad.
What you are talking about already exists, in many different versions at that. I think you really underestimate how varied objects mathematicians works with, they work with so many things that it is hard to come up with original ideas, this idea was already explored close to 200 years ago.
https://en.wikipedia.org/wiki/Fractional_calculus#Fractional...
I also have a PhD in maths, thanks.
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.
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.
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.
Except like, instead of the 2nd order function returning a number, where the functional derivative would be ((R -> R) -> R) -> ((R -> R) -> R), uh, instead, the thing you got.
Ok, but, differentiation is linear, so, (\frac{d}{dx}(u(x) + \varepsilon \eta(x)) - \frac{d}{dx}(u(x)) )/\varepsilon is just \frac{d}{dx}(\eta(x)) (and so there's not even much need to take the limit as \varepsilon goes to 0, as it doesn't depend on \varepsilon anyway) ..
Uh, that seems a little odd, that the derivative of differentiation in the direction of a function would be the derivative of that function? Like, the derivative of something linear should be a constant, ah, but, the directional derivative of 5 x_1 + 3 x_2 also depends on the direction in which it is taken, though it doesn't depend at all on where it is taken.
Ok, yeah, same situation here then.
Is that the sort of thing you are talking about?
Though, you said "of a derivative" not "of the derivative", so I guess maybe you mean like, more generally things that satisfy Liebniz's law?
If everyone commits to using approximately the same definitions then (at the social scale) you get normative semantics as an unintentional by-product. Everybody in your tribe uses the same terminology/notation and means the same thing by "==".
It's a desirable property because it minimises miscommunication and it enable effective asynchronous communication. Good luck trying to read a paper by somebody using different semantics.
And then computer scientists come along and point out that all definitions (and by proxy - all Mathematics) are arbitrary.
Because the chosen axioms are arbitrary.