Revisiting the Notion of Function

Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege's Begriffschrift [17], Russell's Ramified Type Theory [42] and the lambda-calculus (originally introduced by Church [12, 13]) showing that the lambda-calculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambda-calculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads..