[and] + [for] -- functions
Mapping inputs to outputs is how humans make sense of the world. For example —
Input = fire
Output = heat
Input = food
Output = satiety
This intuitive definition satisfied mathematicians for some time. However, concerns about rigor led thinkers in the late 19th century to redefine a function as a set of ordered pairs that obeyed certain criteria. The examples above become —
(input, heat)
(food, satiety)
Viewed in this way, functions are really just concatenation devices, taking two separate entities and making them one. “Assigning a value” to something means creating an ordered pair.
This definition is formally unimpeachable, but something seems to be lost. Cognitively, concatenation differs significantly from the idea of input and output, which implies some kind of process or transformation. Substitution may be a better way to model this phenomenon. For example, since heat is so closely associated with fire, we gradually learn to *substitute* heat for fire in our minds. Food is so closely tied to satiety that it eventually *becomes* satiety. Substitution also accurately models the most common mathematical operation with functions, evaluation.