Some years ago - never mind how long precisely - having little or no money I found myself as a teaching assistant for an introductory course in algebra. In an event that I still recall, the instructor gave a lecture about the fundamentals definitions of a topic known as **group theory**. For those that don’t know, a group is an algebraic structure that consists of a set of elements together with an operation that combines any two elements to produce a third element; and where the operation satisfies four equational laws. The lecture was straightforward, but afterwards a student walked up to me and asked me question:

So what’s a group … actually?

That’s easy I thought, I’ll use the usual metaphor one uses to describe groups: modular arithmetic or clock arithmetic. If the hour hand of a clock is at 9 and 4 hours is added to it it loops back around to 1. One can never add time to the position outside of the 12-hour cycle. The addition of time always proceeds equally forward if equal intervals are added. If one advances the clock by no time, it remains at the same position. And regardless of what time you’re at there’s always some interval of time you can add to get to any other point in time.

This is a vague metaphor-heavily description of what a group is. More precisely it’s defined to be a combination of a set \(\text{G}\) and an operation \(\star\) written as \((\text{G}, \star)\) with four laws:

Law | Description |
---|---|

Closure |
For all \(a\), \(b\) in \(G\), the result of the operation, \(a \star b\), is also in \(G\). |

Associativity |
For all \(a\), \(b\) and \(c\) in \(G\), \((a \star b) \star c = a \star (b \star c)\). |

Identity |
For any element \(a\) in \(G\) there is an element \(e\) where \(e \star a = a \star e = a\). |

Invertibility |
For each element \(a\) in \(G\) there is an element \(b\) where \(a \star b = b \star a = e\), where \(e\) is the identity. |

Groups are an extremely important concept that are integral to next generation of elliptic curve cryptography that protect our internet transactions and banking, and on top of that show up constantly as part of our underlying description of the physical laws of the universe itself.

After a half hour explanation of clock arithmetic, examples over the integers and hand waving wildly at blackboard, the student then asked me:

I understand the equations, but what’s a group actually?

At this point, I didn’t have much else to say. There really is nothing I can say, other than that a group *is the set of equations*, I can’t point at something in the classroom and say that this object fully embodies “groupiness”, nor can I pull out a dictionary and find a synonym for group that would in any way convey the essence of what a group is. Common English simply lacks a word for “set with operation satisfying closure, associativity, identity and invertibility”.

The concept is not particularly hard in retrospect, and indeed after a bit of quiet contemplation the topic eventually clicked for the student and she eventually went on to take other classes in higher mathematics, in particular Galois theory which builds on these foundations. Since then I’ve left teaching math behind me and gone on to programming in industry.

So why do I bring this up? Groups, like many concepts in functional programming, are often some of the first concepts that we encounter that defy a reduction down to our everyday experience; and are a constant point of confusion because of it. Yet, for as long as I’ve been involved with industrial programmers it has been a constant point of contention that the names used in describing concepts like *Monad, Functor*, and *Category* are problematic because they don’t convey immediate (partial) understanding by reducing them down to concepts from our everyday experience. This thinking is not all that dissimilar from the mental gap that the student studying groups for the first time had to overcome.

I understand the functions and laws, but what’s a monad actually?

A monoid (or pick any of your favorite Haskell abstractions) is typically a small interface defined over a set of types that satisfies certain laws. This style of designing abstractions is often quite foreign in programming in the large, and other schools of thought (see Gang of Four) actively encourage weaving cryptic metaphors and anthropomorphising code as a means to convey structure.

Law | Description |
---|---|

Left Identity |
`mempty <> x = x` |

Right Identity |
`x <> mempty = x` |

Associativity |
`(x <> y) <> z = x <> (y <> z)` |

The argument that **Monoid** should be called something else (maybe **Appendable**) is about as convincing as the proposition that a **Group** should be **Clock**. A clock in a contrived sense can be considered a group and perhaps helps with some initial intuition, but the term is ultimately misleading. Similarly, if one expects that *all* constructs in programming be modeled on everyday concepts one will eventually hit up against the limitations of everyday experience to model higher abstractions. We would never arrive at complex numbers by counting scratches on a clay tablet, nor will would come up with Galois theory or elliptic curve cryptography by considering groups purely in terms of clocks.

The algebraic terminology was invented often hundred of years ago and is effectively arbitrary. Therein lies the strength though, it’s intentionally precise because it doesn’t come muddled in the baggage of everyday experience, which can confuse and mislead (i.e only special monoids have an append-like operation in their definition). Using the terminology of mathematics opens up hundreds of years of progress done by thousands of people discovering results we would never think of on our own. And on a larger scale often opens up surprising results mapping between different disciplines and computer science.

Dijkstra’s quote is quite apt: “The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise.” Programming with precise algebraic names and equational reasoning is here to stay, and the edifice of abstraction is only going to grow as programming becomes more precise and refined.