Categorical Monads and Computer Programming

The categorical notion of monad was first introduced into computer science as a way of structuring mathematical models of programming languages. The idea has subsequently been transferred back into computing practice, influencing the design of widely-used languages and frameworks. Expressions in conventional imperative programming languages bear a superficial similarity to conventional mathematical notation, but they behave quite differently. For example, if f and g are both ‘functions’ or ‘methods’ that take an integer argument and return an integer result (we say they have type int -> int and write f : int -> int), the two expressions f(3) + g(4) and g(4) + f(3) are not generally equivalent, as they would be if f and g were ordinary mathematical functions. The issue is not that integer addition in programming languages is not commutative (it almost always is), but that calling a ‘function’ can do many things as well as, or instead of, computing a result depending solely on the arguments. And the order and multiplicity of these ‘other things’ – which are called side-effects – affect the behaviour of one’s program. The ability to perform input/output operations is one example of a side-effect. If, say, both f and g just return their argument, but also print it to the screen, then the expressions will both have value 7, but we would see “34” printed in the first case, and “43” in the second, assuming that the operands of addition are evaluated left-to-right. Another ubiquitous side-effect is mutable state. If the definitions (in C-like syntax) were