Are there alternative definitions/conceptions for the function concept?

Question

The function is a mapping between elements of sets.

However,

I was thinking that would it be philosophically possible to come up with a new concept that's not a mapping between elements of sets, but is "function-like".

Do any of such "function-like" concepts exist already?

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By "function-like" I'm thinking that it must be a "transformation". It transforms something into something else.

My philosophical motivation is to see whether it's philosophically feasible to think "about the world" in other ways than "mappings between elements (of sets)" or "input-output".

Group action (https://en.wikipedia.org/wiki/Group_action) is one example into the direction that I'm looking for. It's a function, but it deals with different sort of abstractions (such as rotations).

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It does seem a bit difficult though. The function is so general concept and it seems to be able to describe a lot of things mathematically imaginable, such as different shapes.

Answer 1

The closest I can think of goes something along the lines of inputs and outputs.$$inputs\to outputs$$I mean, that's pretty much the same thing as mapping, but this is indeed what a function does. Takes some inputs, gives out outputs.

Answer 2

This is quite a vague question, so I'm going to give one possible suggestion that you may find interesting.

First, instead of thinking of a function as a map from one set to another, there's another (mathematically equivalent) way of thinking about functions; rather than thinking of it as sending one element to another, you can think of it as an 'assignment' of values, or a pairing of numbers. For example, consider $f(x)=x^2$. We would typically say $f$ 'sends' $3$ to $9$, $10$ to $100$ etc. But instead you could use the language of assignment, that is more closely aligned to, say, plotting a function on a graph. Then you would say you 'assign' the number $100$ to the number $10$. It's just a change in language, but it does have different implications. For a discussion on different ways to formalise and conceptualise notions of function, I highly recommend a book by one of my professors: 'Topoi', by Robert Goldblatt. See Chapter 2.

Another example that comes to mind is something that again, can be defined as a function. There is a concept of a group action in mathematics. A group is a very abstract collection of symbols, where you can combine the symbols according to some basic rules. The standard example is the whole numbers, where you combine them by adding. This satisfies all the rules of groups; there is a 'zero' that leaves other numbers alone, and every number has a negative, that cancels it out.

Then there's more abstract groups, like the group of square rotations; this is the collection of rotations by $90^\circ$, $180^\circ$, $270^\circ$ and $360^\circ$. Now turning by $360^\circ$ doesn't actually do anything, it just put the shape back where it started! So this element is a bit like $0$ in the integers.

Here, you can combine elements just by thinking of doing one rotation after another, so $180^\circ +270^\circ=450^\circ$, but overall, that's the same as doing a rotation by $90^\circ$, because the full-turn is the same as doing nothing, overall. Now so far I've just talked about this group as a bunch of symbols you can combine, but obviously they have a more meaningful interpretation - this group can 'act' on a shape! This means there is an obvious way for each element to 'transform' a shape; $90^\circ$ 'acts' on a square by turning it $90^\circ$. This group could 'act' on an octagon, or a circle, or even the whole infinite plane. This is another example of a way of thinking about mathematics without thinking of input-output.

I think the central problem you will face, and the reason this question is so hard to answer, is that pretty much any notion of 'transformations' can be thought of as functions, or something very close to functions, namely an arrow in category theory. But that doesn't mean you have to think of it that way; these are just languages for interpreting the world. Lots of mathematics and plenty of mathematicians don't think in terms of input-output, even though they could. There's lot's of other perfectly useful ways to discuss these concepts.