1 Introduction

What is addition? On Mathstodon and Bluesky I asked people to decide which of the following options best, for them, encapsulated the meaning of $2+3$:

1. $5$

2. $3$ more than $2$

3. the sum of $2$ and $3$

I also put an "other" option and got a few variations on these in the comments.

One thing I tried to make clear in my posts was that this wasn't a trick question: all three (and "other") interpretations were valid. This was more a question of whether people felt one was more "them" than the others.

The final ordering in both Bluesky and Mathstodon was: sum, $5$, more than.

Is that right? Is the concept of addition as finding the sum of some numbers the best viewpoint? Or is it a bit more nuanced than that?

And what does category theory have to do with something that many of us probably learnt so long ago that we don't remember learning it?

This post was sparked by reading Atul Rana's commentary on BIDMAS, and how thinking about the nature of operations is a deeper concept. I completely agree.

2 So What Is It?

One of my favourite pedagogical videos is a scene in the Red Dwarf episode Stasis Leak. In the scene, the main characters have found a way to travel in time. One of the characters – usually a little slower on the uptake – asks the others to explain what's going on.

Cat:

What is it?

Rimmer:

It's a rend in the space-time continuum.

Cat:

What is it?

Lister:

The stasis room freezes time, you know, makes time stand still. So whenever you have a leak, it must preserve whatever it's leaked into, and it's leaked into this room.

Cat:

What is it?

Rimmer:

It's singularity, a point in the universe where the normal laws of space and time don't apply.

Cat:

What is it?

Lister:

It's a hole back into the past.

Cat:

Oh, a magic door! Well, why didn't you say?

Let's apply this to addition. What, actually, is addition?

2.1 Counting On

When we first encounter addition we probably do so as the notion of "counting on". To switch from SciFi to Fantasy, in the Truckers Trilogy then Terry Pratchett has an ongoing metaphor about a type of frog that lives its whole life inside a flower, except that one small group of frogs leaves its flower and ventures forth to find another one. During their journey they have to struggle with the concept of "otherness"; namely, that the flower they leave and the flower they arrive at are different flowers. There's a moment in the story where the frogs see that the one over here and the one over there are different ones and so they need a word for one more than one.

And thus they take the first step on the journey to arithmetic.

Addition begins with that concept of more than. It is taught as counting on. How do we figure out what $2+3$ actually is? Well, we start at $2$ and count on $3$ more to get to $5$.

So the interpretation "$3$ more than $2$" is nothing more than our initial encounter with addition.

Which – to me – makes it interesting that it is the least favoured option. Perhaps we think that because it was first and we have since learnt other interpretations then it is somehow the lesser of them. Whereas in actual fact, it is the most fundamental.

2.2 An Alternative Description

Once we've learnt to count on, we are naturally drawn to the view that the "answer" to "what is $2+3$?" is $5$, since that is where we arrive by counting on $3$ from $2$. This viewpoint is reinforced by all the questions we get asked in primary school and beyond. Or, rather, it is reinforced by all the answers to those questions. We learn that if the question reads "What is $2+3$?" then the answer that gets the tick is $5$. Moreover, we learn lots of seeming synonyms for "What is" such as "What is the value of" and "Evaluate" and even just "$2+3=\dots$".

With this view we think of $2+3$ as synonymous with $5$. After all, if we have two hippos and then three more hippos arrive then clearly there are five hippos altogether. The story of how they get there may make for a fun children's book (it does!) but, as that book concludes, what is important is how many there are in total.

This viewpoint gets reinforced in more subtle ways. There are times when we want to rewrite a calculation and knowing that $2+3$ is the same as $5$ can be very useful. For example, if we had been asked to add $8$ and $5$ then knowing that $5$ is $2+3$ means that we can do $8+2+3=10+3=13$ instead of having the "count on" from $8$. Being able to split numbers makes it easier to exploit patterns such as "number bonds to $10$" as here.

We see this idea exploited time and again with more complex operations. The "sneaky one", whereby a fraction gets multiplied by, say, $\frac{3}{3}$ to turn it into something more palatable; or adding a zero pair to make a subtraction more straightforward. Even things like rewriting $16$ as $2^4$, these are all examples of this interpretation.

2.3 In Summary

At the end, though, we learn that $2+3$ is a description of a process but that in said process there is no precedence between the $2$ and $3$. After all, we've learnt that $3$ more than $2$ leads to the same place as $2$ more than $3$, so as $2+3$ and $3+2$ are the same we should have a viewpoint in which they are of equal import.

Thinking of $2+3$ as the sum of $2$ and $3$ takes us back to the idea that we're doing something (finding the sum) but in a way that allows us the flexibility of commutativity. We can then go on to talk of its other properties, such as associativity, and how it interacts with other things like multiplication and powers.

This also lends itself more readily to the generalisation technique in algebra. Initially we might regard a letter, say $x$, as an unknown but – crucially – fixed quantity. All those questions about solving an equation to find the value of $x$ reinforce this idea. But eventually we want to get to the idea that $x$ is a generic quantity and that we want to consider expressions such as $x^2+5x+6$ where $x$ could be any real number.

In this case we want to regard $x^2+5x+6$ as a set of instructions: if I were to give you a number then you would square it, add five lots of it, and then add a further $6$. If we want to lean in to this viewpoint of expressions then we need an interpretation of addition as an action or process. And one in which there is no favouritism between the two things being added seems to fit well with the algebraic use where we don't know which side the known and unknown quantities might end up.

3 Counting the Differences

So there are our three interpretations, all with different strengths and uses. Do we need to decide? I guess that's the "trick" in my question – though I hope it wasn't actually a trick since I did lead with the comment that all three were valid interpretations.

All three are more than valid in that all three bring a perspective to addition that they don't share with the other two. All three allow us to think of addition in a particular way that the others slightly suppress. So choosing one over the others is perhaps not as good idea as my question might imply.

But do we need to? Is there some context in which we can move between these interpretations? Some way in which mathematically we can switch back and forth?

I say "mathematically" because of course we can do so externally to mathematics. Indeed, I've just done so above in going through the descriptions of these interpretations. But can that be matched within mathematics? If so, that would be a powerful idea where we can encode not just addition but also these interpretations of addition.

The bigger question here is: can mathematics talk about itself?

This feels like such a "teacher question": I wouldn't be asking it if the answer were "no", would I? But to figure out what's going on we need a little detour.

4 Arity, Arity, All is Arity

It's time to learn some vocabulary.

A set is a mathematical collection. A set has elements, and for this discussion you can just think of sets of numbers such as $\{1,2,3,\dots ,10\}$. In fact, the only set we'll need for this is the set of natural numbers, which we'll write as $\mathbb{N}$. For the purposes of this article I don't care if your version of the natural numbers includes $0$ or not1.

We also need the concept of tuples (of natural numbers). These are ordered lists (of natural numbers). The set of $k$–tuples (so, ordered lists of natural numbers of length $k$) is written ${\mathbb{N}}^k$. There is a special (and unique) tuple of length $0$ which I will write as $\star$ (I find that writing it as an empty list – $()$ – is too open to confusion). So ${\mathbb{N}}^0=\{\star \}$.

An operation on $\mathbb{N}$ is simply a function ${\mathbb{N}}^k\to \mathbb{N}$, for some $k$. The value of $k$ is called the arity of the operation. For some small $k$ then there are special names: zeroary, unary, binary, ternary. It's also possible to think of the arity as the number of inputs (each being a natural number) of the operation.

Zeroary (also known as nullary) operations are a bit special. In the sense of operations, they appear have no input in that you don't put a natural number into a zeroary operation. However, you do still put $\star$ into a zeroary operation so you can still invoke the function. But because you only have one input then you can only get one output. Therefore, a zeroary operation is completely determined by its single output, which in this case is just a natural number. So a zeroary operation is a fancy way of viewing a number as an operation.

With this language we can examine our notions of addition.

Let's start with our initial definition of $2+3$ as "$3$ more than $2$". This means we have an operation $\mathbb{N}\to \mathbb{N}$ which is "count on $3$ from where you started". This operation is unary as it takes in a single number. We could write this as ${\alpha }_3$.

Clearly, there's nothing special about $3$ here. Indeed, for any natural number we have a "count on that number" function. That is to say, for any $k$ we have a unary operation ${\alpha }_k:\mathbb{N}\to \mathbb{N}$.

Here's the bit of magic. Using this family of unary operations we can define a single operation ${\mathbb{N}}^2\to \mathbb{N}$. This works by:

Namely, given two numbers we "count on" the first number starting from the second.

So our family of operations, all the ${\alpha }_k:\mathbb{N}\to \mathbb{N}$, can be subsumed into a single operation, let's call it $\alpha :{\mathbb{N}}^2\to \mathbb{N}$. The difference being that the ${\alpha }_k$ each take in one argument but $\alpha$ takes in two, hence is a binary operation.

Here, we swap a family of unary operations for a single binary one.

There's more.

We can go the other way: to zeroary operations.

So, remember that we have the family of operations ${\alpha }_k$, one for each $k\in \mathbb{N}$. We can say that instead of thinking of ${\alpha }_k$ as an operation, we can think of it itself as a family of numbers, let's write them as ${\alpha }_{k,n}$, where ${\alpha }_{k,n}$ is the number ${\alpha }_k(n)$. As mentioned above, this is the same as a zeroary operation.

Summing up (sorry, this topic seems made for puns) then our three viewpoints on $2+3$ are:

1. $5$: here, $2+3$ is the zeroary operation ${\alpha }_{3,2}$,

2. $3$ more than $2$: here, $2+3$ is the unary operation ${\alpha }_3$ applied to $2$,

3. the sum of $2$ and $3$: here, $2+3$ is the binary operation $\alpha$ applied to the pair2 $(3,2)$.

   2Don't worry about the shift in order here

So all of our different concepts of addition can be mathematically realised as different operations, and these operations have different arities.

5 And Where's That Soggy Plain?

But it's all the same thing!

It's all just addition, so while it might be useful to be able to talk mathematically about addition from all three perspectives we should also be able to say that they are also all the same thing.

Can we do this3?

"Yes, we can!"

What we want is a way to say that these distinct things are equivalent without forcing them to be the same. This is a central concept in category theory: that when things are equivalent we don't force them to be identical but rather explore how they are equivalent.

This isn't meant as a primer on category theory. I will put the technical details in as an appendix for those who know a little about category theory but haven't met this specific part (sometimes called universal algebra or general algebra).

For those not familiar, I will confine myself to saying that category theory says that the three things: the family of zeroary operations ${\alpha }_{n,k}$, the family of unary operations ${\alpha }_k$, and the binary operation $\alpha$ are not just related, they are naturally related. It gives us a context in which we can easily switch between the three while also keeping track of which one we mean at the time.

It is, in fact, an example of a natural isomorphism between adjoint functors.

So, when teaching small children how to add, whether by "counting on", by learning to write an answer to a sum, or by abstracting out the concept of "addition", what we are really doing is teaching them about natural isomorphisms.

6 To Sum Up

So which is it? What is the true meaning of addition?

Why, all of them, of course! And category theory allows us to have our cake and eat it – up to natural isomorphism, that is.

A The Gory Details

Let's see how this looks in category theory. I'll assume familiarity with the basics of category theory, namely objects, morphisms, functors, and natural transformations but not universal algebra.

We're interested in the category $Set$, whose objects are sets and whose morphisms are functions between those sets. I'll write $Set(A,B)$ for the morphisms from $A$ to $B$.

Now, this is an ordinary category so the morphisms from $A$ to $B$ themselves form a set. That is, we can view $Set(A,B)$ as again an object in $Set$. It's useful to have an alternative notation for it when viewed as an object and the usual notation is the exponential one: $B^A$.

There's another important construction that we'll need: that of the cartesian product of two sets, written $A\times B$. This is the set of ordered pairs whose first term is from $A$ and second is from $B$.

These two are closely related. We can regard $B\mapsto B^A$ and $B\mapsto A\times B$ as functors on $Set$ (the details are not difficult to check), and it turns out that they are a pair of adjoint functors.

The unit is ${\eta }_B:B\to (A\times B{)}^A$, where ${\eta }_B(b)(a)=(a,b)$. The counit is ${\epsilon }_B:A\times B^A\to B$, where ${\epsilon }_B(a,f)=f(a)$.

When worked through, that means that there are natural isomorphisms:

Let's see what this means for the "counting on" viewpoint of addition. For each $k\in \mathbb{N}$ we have the operation of "counting on by $k$" which we wrote as ${\alpha }_k$. This is a function $\mathbb{N}\to \mathbb{N}$, so is in $Set(\mathbb{N},\mathbb{N})$. But we have one of these for each $k\in \mathbb{N}$, so we have an assignment $k\mapsto {\alpha }_k$, which looks awfully like a function in its own right with domain $\mathbb{N}$. But for that to be a function, its codomain needs to be a set so here's where we need to use ${\mathbb{N}}^{\mathbb{N}}$ for the set of functions $\mathbb{N}\to \mathbb{N}$.

That is to say, the family of functions ${\alpha }_k$ fit together to give a morphism in $Set(\mathbb{N},{\mathbb{N}}^{\mathbb{N}})$. But as we've just seen, $Set(\mathbb{N},{\mathbb{N}}^{\mathbb{N}})$ is naturally isomorphic to $Set(\mathbb{N}\times \mathbb{N},\mathbb{N})$ so our family of ${\alpha }_k$ define a single morphism $\mathbb{N}\times \mathbb{N}\to \mathbb{N}$. This is our old friend $\alpha$.

To get the zeroary morphisms we need to know that there is a natural isomorphism between the identity functor on $Set$ and the functor $B\mapsto B^{\{\star \}}$. Applied to $\mathbb{N}$, this leads to an isomorphism:

and ${\mathbb{N}}^{\{\star \}}$ is the set of zeroary operations on $\mathbb{N}$ so our one binary operation $\alpha$ defines a family of zeroary operations ${\alpha }_{k,n}$ for each pair $(k,n)\in {\mathbb{N}}^2$.