Before I even begin my question, I'm going to start off with a couple disclaimers:
Disclaimer 1: I'm not sure this really belongs on Mathematics, but I'll go for it anyways since it's the closest SE site that I can think of on which this would belong.
Disclaimer 2: This is a hypothetical question, as such please ignore questions of physical practicality or plausibility.
Disclaimer 3: I'm not good with words, and I lack mathematical expertise, if you need clarification, tell me and I'll do my best to fix the problem. But more importantly, I'm only 100% comfortable with Calculus 3 level stuff and below, I can understand linear algebra, basic topology and basic number theory concepts, but if you pull out the equations I'm likely get confused quickly.
Okay, so with those out of the way I'll start my question.
My personal views on calculus and math in general.
Bare with me for a while, I think that my thought process might be important to better understanding my question even if it seems tangential and/or wrong. In my mind, I have two categories for abstract mathematics. Arithmetic and Calculus.
Arithmetic takes in numbers for inputs and outputs numbers. It does this through the usage of little black boxes called functions. I consider algebra to (mostly) fall into arithmetic. The most basic functions of arithmetic are addition and subtraction, and combonations thereof.
Calculus, on the other hand, inputs functions and outputs functions, and instead uses little black boxes refered to as transformations. The most basic transformations of calculus include those of arithmetic in addition to concatination, derevation, and antiderevation.
My views on Turing Machines
Having laid out my ideas of calculus and arithmetic, I want to turn now to the other half of my question: Turing machines. Turing machines are the very literal little black boxes known as functions, they do everything a function does, with the additional ability to store data. Basically, Turing machines are physical functions. And as we can see in the modern world, Turing machines are capable of a lot ranging from estimating solutions to differential equations to creating neural networks, to playing Fortnite.
Universial Calculus Machine?
My question is more or less the following: A Turing Machine is to Arithmetic as what is to calculus? What would a Turing Machine, that took continous functions as input and returned continous functions as output using transformations, be capable of, more specifically what would they be capable of that a Turing Machine would not? I would imagine that these machines would be capable of solving differential equations much quicker but normal Turing Machines can come to close enough estimates given enough time. So what are some practicle things my machine could do, that Turing's just can't or is just impractical for?
Best Answer
What you are describing sounds very close to some form of lambda calculus. Lambda calculus is a way of modelling computation making use of mappings that transform inputs into outputs. This inputs and outputs can include other lambda transforms. You can implement a Turing machine in lambda calculus and you can use a Turning machine to implement lambda calculus so anything one can do the other can also do.
Lambda calculus is the basis of functional programming languages like lisp, scheme, Haskell and javascript.