The mathematics and philosophy of hyperoperations

In "On numbers, structure and induction" I give an account on how the concept of numbers is inextricably tied to mathematics. This might seem obvious to the untrained eye, and I agree holds true for arithmetics; however, as far as mathematics are concerned, taking "things that matter™", such as numbers, for granted, is undeniably childish, for the simple reason that mathematics and taking things for granted don't go well together. Thus, we explore not only how numbers are purely mathematical objects, but also how they can be expressed in more than one formalism, as there are usually many ways in which one can view "things that matter™", such as numbers.

Also in my previous article, I deliberately avoid discussing anything other than "succession", i.e. the natural unary operation from which the infinite set of numbers arises, and addition. I do this only for the sake of simplicity, when in fact the avoided is unavoidable: the "predecessor" is only a partial morphism, and this partiality is reflected in more than mere addition, which is why we have integers, fractionals, reals, imaginaries, and, moreover, transcedentals, primes, perfects, and so on and so forth. The theory of numbers is anything but simple to teach and learn.

It would therefore seem that, both for theoretical and practical purposes, we require higher-order operations: we want to be able add numbers multiple times, and so we get multiplication, and we often also want to be able to multiply numbers multiple times, and so we get exponentiation. This is not a child's plaything: addition, multiplication and exponentiation, together with the relations arising between them, rule Life, The Universe and Everything in more ways than the brain of the average human can comprehend. But I am preaching to the choir, I do not need to convince the educated reader of the truth.

Let us now shift our focus on the following problem: mathematics, or at least certain branches of mathematics deal with generalizations, that is, the unification of sometimes seemingly unrelated concepts into a single theory. This is especially desired by formally-inclined mathematicians, also (but not only) because formalizing a single mathematical object is orders of magnitude easier than formalizing a million of them; and this is where we get back to our story about "numbers and stuff we are able to do with them".

It is natural to pose the following questions: given the conceptual sequence of succession, addition, multiplication and exponentiation, what comes after? We must be, and indeed we are, able to exponentiate numbers multiple times, and so we get tetration. But then what comes after that? Well, we can tetrate numbers multiple times, and so we get pentation. But then what comes after that? And this is where I guess you got the general idea, an idea which mathematicians way more competent than myself have already explored: what is the generalization of these operations?

Said mathematicians¹²³ apparently identify this abstraction through what we know as hyperoperations. Quite isomorphically to natural numbers, the set of these binary operations goes from "zeration" (what I previously named "succession", in fact a unary operation), to addition, multiplication, exponentiation, and then to tetration, pentation, sextation and so on, ad infinitum. I propose that we explore them ourselves from a computational perspective, by defining them mathematically and in Haskell, then attempting to (intuitively) find some basic general properties, and then, finally, by ranting on this subject, hoping that his would help us to make sense of the whole that are hyperoperations.

Generalizing exponentiation

Going back to our previous account, we defined natural numbers recursively, or using the principle of induction, i.e. by stating that zero is a natural number, and the "successor" of any given natural number is also a natural number. This enumeration generates the set of natural numbers, which I will refer to as \(\mathbb{N}\) in the remainder of this post.

In Haskell, \(\mathbb{N}\) is represented by the following algebraic data type:

data Nat = Zero | Succ Nat

over which we define the function (.+) (addition) as an infix operator:

(.+) :: Nat -> Nat -> Nat
n .+ Zero       = n
n .+ (Succ n')  = Succ $ n .+ n'

We know that multiplication involves repeated additions of the same number, i.e. \(3 \times 4 \equiv 3 + 3 + 3 + 3\). We therefore define (.*) recursively in terms of (.+):

(.*) :: Nat -> Nat -> Nat
_ .* Zero       = Zero
n .* (Succ n')  = (n .+) $ n .* n'

We can use the same recipe to define exponentiation:

(.^) :: Nat -> Nat -> Nat
_ .^ Zero       = Succ Zero
n .^ (Succ n')  = (n .*) $ n .^ n'

Note how the recursion steps for the three operations look very similar: they all use partial application of the immediate lower-order operator, with the exception of (.+), where Succ is unary. Also note that all three definitions are right-associative, that is, the recursion is done on the second number, the number to the right. Intuitively, this must go the same for all hyperoperations.

The base case, i.e. when the second number is zero, however differs for each of the three definitions. Addition naturally defines the base case as the number to the left, while multiplication and exponentiation yield constant numbers, namely zero and one respectively. This fails to give us a clear intuition for how to define higher-order operations: multiplication is repeated addition, so a number added to itself zero times amounts to zero; exponentiation is repeated multiplication, so a number multiplied by itself zero times amounts to one⁴; what does a number exponentiated by itself zero times amount to then? As per Goodstein⁵, we assume that \(a\;\text{op}\;0 = 1\), where \(\text{op}\) is a hyperoperation of higher order than multiplication, or \(\text{op} \equiv \text{op}_n\), where \(n \ge 3\).

The observation that \(\text{op} \equiv \text{op}_n\) isn't at all arbitrary. In fact, it's telling us that we can assign a number to each hyperoperation, making the order of a given hyperoperation the property of countability; in other words, the enumeration leads us to a family of (countable) hyperoperations! However, for the sake of consistency with our Haskell implementation, we choose to use functions to define the semantics of hyperoperations.

We will therefore implement a Haskell function called hyper n a b, which has the following properties:

• Zeration (hyper where \(n = 0\)) is defined as the right-associative Succ constructor (i.e. applied on the second parameter of the operation).
• Addition (hyper where \(n = 1\)) of a number \(a\) to \(0\) yields \(a\).
• Multiplication (hyper where \(n = 2\)) of a number \(a\) to \(0\) yields \(0\).
• hypers where \(n \ge 3\) of a number \(a\) to \(0\) yield \(1\).
• hypers where \(n \ge 1\) of a number \(a\) to a number \(b\), \(b \ge 1\), are computed recursively, i.e. they are formally defined as a recurrence, solved through induction⁶.

The properties are equivalent to the following Haskell implementation:

hyper :: Nat -> Nat -> Nat -> Nat
hyper Zero _ b                  = Succ b
hyper (Succ Zero) a Zero        = a
hyper (Succ (Succ Zero)) _ Zero = Zero
hyper (Succ _) _ Zero           = Succ Zero
hyper (Succ n) a (Succ b)       = hyper n a $ hyper (Succ n) a b

Assuming that we have a fromNum function that converts Haskell numbers to Nats, we can now check our implementation:

> hyper (fromNum 1) (fromNum 7) (fromNum 0)
7
> hyper (fromNum 1) (fromNum 7) (fromNum 17)
24
> hyper (fromNum 3) (fromNum 3) (fromNum 2)
9
> hyper (fromNum 4) (fromNum 3) (fromNum 2)
27

The full source code is available in Hyper.hs, feel free to mess around with it.

Basic properties of hyperoperations

To illustrate and prove the properties of hyperoperators, we will make use of equational reasoning, assuming only the Haskell implementation provided in the previous section. First we will show that the first hyperoperation (addition) is associative. In the process, we will also show its commutativity⁷⁸.

Theorem (T1). Succession is associative. This is quite trivial, following from the fact that Succ is in fact unary, yielding unique natural numbers. If we apply induction, first Succ Zero is unique, then if we assume that n' is unique, then n = Succ n' is unique. \(\square\)

Lemma (L1). \(\forall n \in \mathbb{N}\), Zero .+ n = n. We show this by induction on n. For n = Zero, we get:

Zero .+ Zero = Zero

which simplifies using the base case of (.+). For n = Succ n', we assume that Zero .+ n' = n' and iterate on the recursion step, obtaining:

   Zero .+ Succ n' = Succ n'
=> Succ (Zero .+ n') = Succ n'
=> Zero .+ n' = n'

which coincides with the induction assumption. \(\square\).

Lemma (L2). \(\forall a, b \in \mathbb{N}\), (Succ a) .+ b = Succ (a .+ b). Similarly, we perform induction on b.

For b = Zero:

   Succ a .+ Zero = Succ (a .+ Zero) -- (.+) base case
=>         Succ a = Succ a

For b = Succ b', the induction hypothesis is Succ a .+ b' = Succ (a .+ b'):

   Succ a .+ Succ b' = Succ (a .+ Succ b') -- (.+) recursion
=> Succ (Succ a .+ b') = Succ (Succ (a .+ b'))
=>        Succ a .+ b' = Succ (a .+ b') -- ind.

\(\square\)

Theorem (T2). Addition is commutative, \(\forall a, b \in \mathbb{N}\),

a .+ b = b .+ a

We perform induction using b. For b = Zero, we get

   a .+ Zero = Zero .+ a -- (.+) base case
=>         a = Zero .+ a -- L1
=>         a = a

For b = Succ b', we assume that a + b' = b' + a and we prove that:

   a .+ Succ b' = Succ b' .+ a -- (.+) recursion
=> Succ (a .+ b') = Succ b' .+ a -- L2
=> Succ (a .+ b') = Succ (b' .+ a)
=>        a .+ b' = b' .+ a -- ind

\(\square\)

Theorem (T2). Addition is associative, \(\forall a, b, c \in \mathbb{N}\),

a .+ (b .+ c) = (a .+ b) .+ c

We'll take the same proof strategy as before. We choose b for induction to attempt reducing the parenthesis to a reflexive proposition, or to the induction hypothesis. First, for b = 0 we get:

   a .+ (Zero .+ c) = (a .+ Zero) .+ c -- L1, (.+) base case
=>          a .+ c = a .+ c

Then, for b = Succ b', with a .+ (b' + c) = (a .+ b') .+ c:

   a .+ (Succ b' .+ c) = (a .+ Succ b') .+ c -- L2, (.+) recursion
=> a .+ Succ (b' .+ c) = Succ (a .+ b') .+ c -- (.+) recursion, L2
=> Succ (a .+ (b' .+ c)) = Succ ((a .+ b') .+ c)
=>        a .+ (b' .+ c) = (a .+ b') .+ c -- ind.

\(\square\)

I'll let the reader explore other properties of hyperoperations through the following exercises:

Exercise (E1). Prove commutativity and associativity for multiplication. You might have to use other lemmas in addition to the ones presented here. Hint: First prove the following particular case of distribution of multiplication over addition: a + a * b = a * (1 + b).

Exercise (E2). Try to use the same steps to prove associativity for higher-order hyperoperations, with exponentiation as a prime trial. Find some counter-examples. Indeed, it would seem that hyper ns for \(n \ge 3\) aren't associative! This once again goes beyond intuition, illustrating that hyper is a weird function, or rather that it behaves differently than we'd expect it to.

Exercise (E3). Prove that \(\forall n, \in \mathbb{N}\), hyper n 2 2 = 4.

Conclusion and basement philosophy

This post explored the mathematics of hyperoperations from a computational, i.e. layman's, perspective, providing a toy implementation in Haskell, such that it is roughly equivalent to the formal description. We also showed a few basic properties of the more commonly-used operations and gave a perspective to explore further properties of the less common ones, or of hyperoperations in general.

One question that lingers in the back of my mind is: how many of these properties stem directly from the properties of numbers, and how many follow from the properties of the definition of hyperoperations? I'm inclined to say that numbers by themselves are useless, and it's the things that we do to them that define their properties; for example we define primality in terms of divisibility, divisibility in terms of division, division in terms of subtraction and so on.

What's more baffling is that numbers, whether seen in this unary form given by the Peano axioms or in some other form⁹, are fundamental to our logical-mathematical view of the world in ways that are hard to understand. We made use of a bit of induction to prove stuff about numbers in this post -- and it was in fact structural induction, even though it looks strikingly similar to the mathematical one --, even though properties about numbers are used to define induction!

Getting back to our hyperoperations, we have some other interesting things to say about them too. Firstly, operations from pentation onwards "explode" very quickly even for small numbers; that means they can be used to express numbers that are sensibly larger than the number of atoms in the observable universe¹⁰. Secondly, the hyper function viewed as a relation on four numbers (the three parameters and the result) is isomorphic to the set of natural numbers. Finally, as far as hyper goes, infinity is way larger than that, which goes to prove how little our feeble brains can actually encompass.