If definitions themselves are informally just maps from words to collections of other words. Then in order for one to define anything, they must inherently already have a notion of a function. I mean of course one could ask what the exact "things" our functions are mapping to and from, and then revert back to set theory or something else to represent those "things" but why not just define everything to be a function so we are mapping functions to functions? (perhaps with some axiom guaranteeing the existence of at least one function which we can use to recursively build up a bunch of other functions allowing us to express most of contemporary mathematics in the same way I've seen this done iteratively with the emptyset in ZFC)

I mean how can one read any language, much less a mathematics textbook unless they are capable of mapping words to other words i.e. using functions (albeit even if one doesn't mentally recognize it, their brain is still implementing something akin to a function).

Even in formulations of ZFC it seems functions are implicitly being used, for example the Axiom of intersection says given any two sets $A$ and $B$ there exists a set containing all the elements they have in common that one denotes by $A\cap B$.

However we essentially have just defined a function $\cap$ of two variables, taking any two sets to another set - their intersection. Similarly in propositional logic when one defines the logical conjunction $\land$ we are again defining a two variable function from truth values to other truth values.

Pre-facing a statement about mathematical objects that are being mapped to something with the words "we represent this by" or "we denote it by" merely conceals the fact one has just defined a function by hiding it as a statement in the English language. So re-iterating I don't see how anyone can learn *anything* without having some mental grasp of what a map/function/arrow etc. is.

For example a dictionary can be crudely expressed as a functional relation $f$ where we might have:

$$f(\texttt{apple})≡\texttt{a round fruit that grows on trees}$$

So how is it one can accept any mathematical axioms without first taking functions as a primitive, when reading and interpreting words/symbols in of itself requires their usage. If to formally define a function you must use functions then isn't that circular reasoning?

How can a person understand a simple map or visual diagram when interpreting objects, color-intensity etc. requires mentally creating a bijective map between the object being labeled and what it represents. I mean the use of any variables for that matter requires creating what amounts to a bijection, at least mentally between the letters/symbols on paper and the objects/ideas they represent. If understanding everything from written language to cave drawings requires some mental notion of a functional relation, then shouldn't it be used as a starting point?

Also from an aesthetic point of view, it seems a lot simpler to just accept some mathematical variant of a map/arrow/morphism/function etc. then to define a large number of other objects or "syntactic abbreviations" (I'm not familiar with the proper term but this is what Peter Heinig calls it) which appear to me as objects that hold almost all the same characteristics of functions. Lastly if one gets slightly looser with this idea, couldn't you argue understanding any cause and effect relationship essentially requires some mental abstraction that when modeled symbolically is functional? This could be argued as much simpler as then even non-humans would be capable of grasping similar notions, e.g. gorillas capable of rudimentary sign language can understand that by "inputting" a configuration of their hands they can "output" a banana from their owner. In any case though before I get off topic and venture into philosophy, I want to add I'm not too well versed in mathematical logic, so I'm guessing this exact thing has been covered elsewhere. In which case I would appreciate any references.

Axiom of Union. The existence of the intersection of an arbitrary family of sets has to be, and is,provedlater on when one develops the theory. $\endgroup$ – Peter Heinig Aug 30 '17 at 10:43mere syntactic abbreviationfor the symbols $\{x\colon x\in A\ \wedge\ x\in B\}$. Yes, sometimes (notably in Jech's opening chapter), $\cap$ is called an 'operation on classes', but please note that Jech himself calls it"an informal notion". $\cap$ isn't a function. $\endgroup$ – Peter Heinig Aug 30 '17 at 10:48