# On the relationship between mathematical functions and program functions

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By: John Clements

NOTE: this text is a brief overview intended for instructors of a CS course that we’re developing; it is not technical. It tries to make the case that our early courses should steer students toward understanding problems in terms of pure functions. If you have suggestions or feedback, I’d love to hear them/it.

This section of the course introduces functions, a crucial topic in the field of computer science AND in the field of math.

Programmers and mathematicians sometimes think about the term “function” somewhat differently. Furthermore, some people who are familiar with both fields assign different meanings to the word “function” in the two fields.

The definition of a function in the mathematical domain is fairly well specified, though of course things get a little fuzzy around the edges. We’re going to define functions as the arrows in the category Set, more or less (if that’s not helpful, ignore it). That is, a function has a specified domain and a specified codomain, and it maps every element of the domain to a particular element in the codomain. There’s no requirement that it map every element of the domain to a different element of the codomain (one-to-one) OR that there be some element of the domain that maps to any chosen element of the codomain (onto). This (I claim) is the standard notion of “function” in math.[*]

Programmers also use and love functions. Nearly every programming language has a notion of functions. Of course, they’re sometimes called “procedures” or even “paragraphs” (I believe that’s COBOL. Yikes.). In programming, functions are often thought of as being elements of abstraction that are designed to allow repetition. And so they are. But it turns out that they also, in most modern programming languages, can be thought of as mathematical functions. Well, some of them can.

For many functions, this is totally obvious. If I consider the function from numbers to numbers that we might write in math class as f(x) = 14x + 2, then I can write that as a function in most programming languages. (If you disagree with me, hold that thought.)

But… things aren’t always so clear. What about a function that doesn’t return at all? What about a function that takes input, or produces output? What about a function that mutates an external variable, or reads a value from a mutable value? What about a function that signals an error? All of these present problems, some more substantial than others. None of these have a totally obvious mapping to mathematical functions.

There certainly are ways to fit these functions into mathematical models, but in general, the clearest lesson is that when there is a natural way to express a problem using functions that map directly to mathematical functions, we should. These are generally called “pure” or “purely functional” functions.

So, why should it matter whether our functions are pure? What benefits do we gain when we express functions in purely functional ways?

The clearest one is predictability, also known as debuggability and testability. When I write a pure function that maps the input “savage speeders” to 17, then I know that it will always map that string to 17; I don’t need to worry that it will work differently when the global foo counter is less than zero, or when it’s run in parallel, or on Wednesday, or when the value in memory location 0x3342a7 is less than the value in memory location 0x3342a8.

Put differently, pure functions allow me to reliably decompose problems into sub-pieces. When I’m debugging and testing, I don’t need to worry about setup and teardown to establish external conditions.

Another way to understand this is to dip into functions that use mutation. If we want to model these as mathematical functions, we need to understand that in addition to their stated inputs, they take additional hidden inputs. In the simplest case, this may be the value of a global counter. Things get much more complex when we allow mutation of data structures; now we need to worry about whether two values are the “same” value; that is, whether mutating one of them will change the other. Worse still, mutating certain values may affect the evaluation of other concurrent procedures.

For these reasons and others like them, pure functions are vastly easier to reason about, debug, and maintain. Over time, many of our programming domains and paradigms are migrating toward primarily-pure settings. Examples include the spread of the popular map-reduce frameworks, and the wild explosion of popularity in deep learning networks. In both cases, the purity spreads downward from a mathematical framework.

Note that it is not always the case that pure approaches are the most natural “first choice” for programmers, especially introductory programmers, for whom programs are often imagined as a “sequence of changes”; do this, then do this, then do this, then you’re done. In this model, the program is performing a series of mutations on a larger world. Helping introductory programmers move to a purer model is a challenge, but one with substantial payoff.

For this reason, this section focuses directly on pure functions, and invites students to conceive of programs using the models that they’ve been taught in elementary and secondary school, most particularly tables mapping inputs to outputs.

[*] The only reason I mention the category Set is to draw attention to the distinction between “codomain” and “range”; every function has a named codomain, regardless of whether its range covers it. For instance, the “times-two” function from Reals to Reals is a different function from the “times-two” function from integers to integers, and the “times-two” function from integers to reals is yet a third function.