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> 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

> 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.

thrown_22 5y ago
>What examples are you thinking of? 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.
- grungegun 5y ago
  >It didn't have very many interesting properties I could find 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.
- Jensson 5y ago
  > The worst one was the derivative of a derivative 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_derivatives https://en.wikipedia.org/wiki/Fractional_calculus#Fractional...
  - thrown_22 5y ago
    The type of that derivative is still (R->R)->(R->R). I also have a PhD in maths, thanks.
    - auggierose 5y ago
      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 ;-)
      - thrown_22 5y ago
        Mathematical physics oddly enough. 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.
        ukj 5y ago
        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.
- drdeca 5y ago
  So like, a functional derivative of differentiation? 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?
- ukj 5y ago
  This pretty much nails it. 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.

thrown_22 5y ago

>What examples are you thinking of? 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.

grungegun 5y ago
>It didn't have very many interesting properties I could find 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.
Jensson 5y ago
> The worst one was the derivative of a derivative 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_derivatives https://en.wikipedia.org/wiki/Fractional_calculus#Fractional...
- thrown_22 5y ago
  The type of that derivative is still (R->R)->(R->R). I also have a PhD in maths, thanks.
  - auggierose 5y ago
    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 ;-)
    - thrown_22 5y ago
      Mathematical physics oddly enough. 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.
      - ukj 5y ago
        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.
drdeca 5y ago
So like, a functional derivative of differentiation? 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?
ukj 5y ago
This pretty much nails it. 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.