1

I have been going down the YouTube mathematics rabbit hole. I learned about something called Willans' (horribly inefficient) formula for calculating the nth prime, which made use of the Sigma (summation) operator, as well as some other axioms, and the floor function.

Then I watched a video about extending the definition of factorial! to the real numbers, which makes use of the Pi (product) operator.

Then it hit me. Is there a more abstract operator than Pi and Sigma, one which takes a binary function as an additional parameter? Say for example, Omega. Omega((+)) would then equal Sigma, and Omega((*)) would equal Pi. I was curious if there exists a higher order operator like this in formal mathematics.

To somewhat answer my own question, I have dabbled quite a bit in Haskell programming. In Haskell, there are two folding functions, foldl and foldr, which do something similar, except folds in Haskell operate on foldable data structures rather than a function and a range.

Any insight would be greatly appreciated!

user3079361
  • 21
  • 1
    How would it be any different than assigning two random new symbols to $\displaystyle\sum$ and $\displaystyle\prod$? – insipidintegrator Sep 26 '22 at 10:23
  • @insipidintegrator it would take one additional parameter: a binary function. I guess it might be a stupid question but I was just curious so I thought I'd ask. – user3079361 Sep 26 '22 at 10:28
  • 2
    In programming this would be reduce(). Of course it is not a problem to define something like this formally in math but the question is why? $\Sigma$ and $\Pi$ are notational shorthands, just like $\bigoplus$, $\bigcup$, $\bigcap$ and many others used in the same way. – Michal Adamaszek Sep 26 '22 at 10:33
  • 1
    https://en.m.wikipedia.org/wiki/Hyperoperation – CrabMan Sep 26 '22 at 10:48
  • @Michael thank you for your comment. It's full of insight. I can't imagine in what case such an operator would be useful. Reading the formal definition on Wikipedia, I see that a "default" value would have to be supplied, e.g. 1 for multiplication or 0 for addition, in cases where b < a. I thought maybe such an operator would be useful so less language would be needed to express ideas in math, but I see that this might just be too abstract, and cause more confusion than concision, to have a legitimate place in formal mathematics. I was just wondering if it exists. – user3079361 Sep 26 '22 at 10:51
  • 1
    if there exists a higher order operator like this in formal mathematics --- I'm not sure whether you're asking a question related to programming/CS issues (of which I would know almost nothing about) or a question about a mathematical issue. If the latter, then certainly functions can have other functions as inputs. Even in a school-level first calculus course one can make the observation that taking the derivative of a (differentiable) function is itself a function. (continued) – Dave L. Renfro Sep 26 '22 at 11:17
  • 1
    Moreover, even in applied fields (such as quantum mechanics) the notion of a function whose input can be the operation of taking the derivative (among other possibilities) is used. For more about this, see my answer to Is there a such thing as an operator of operators in mathematics?. – Dave L. Renfro Sep 26 '22 at 11:17

0 Answers0