ABSTRACT
Functions are everywhere, and combinatory logic is one of the theories that make them their objects of study. To start with, let us consider a simple function such as + , the addition of integers. + can be applied to numbers; for instance, 46+ 68 is the result of this application, which is also denoted by 114. Then we can view + as a certain map from ordered pairs of integers into integers, or, further, as a set of ordered triples. From another perspective, + can be thought of as a rule to compute the sum of two integers — indeed, this is how each of us first became acquainted with + . In both cases, knowing that + is commutative and associative (and has other properties) is useful in identifying seemingly different versions of + . Just as light has a dual character — being a wave and being a beam of particles — functions have two sides to them: they are collections of input-output pairs, and they are rules for computation from inputs to the output.
