Peano's Forte

A hundred-ish years ago, long before Pokémon and the Slap Chop, there lived a clever one named Giuseppe Peano, who came up with a neat way to describe the natural numbers (0, 1, 2, 3, ...):

  • The first one is 0, which we’ll write as Z.

  • The number after any x is its successor, S x.

Using these two rules, we can write every number in the series:

Friendly Peano
0 Z
1 S Z
2 S (S Z)
3 S (S (S Z))
. . . . . .

Maybe it’s not the prettiest, but I think it’s pretty cool. A number is either Z or the S of another number, which is either Z or the S of another number, which is… well, you get the picture! We call this a recursive definition.

Because of this, we can define functions on the Peano numbers using recursion, too! Here’s a function for testing equivalence:

A B A == B
Z Z true
S x Z false
Z S y false
S x S y x == y

Rule 3 might seem like a duplicate of 2, but we have to define both in case the order of arguments matters (e.g. with subtraction, 2 - 3 is not the same as 3 - 2).

Rule 4 is the magical recursive bit: if x and y are still successors, we run rule 4 again, and we keep doing that until one of the other three conditions is met. We can write out the process for determining 2 == 3 with our Peano numbers:

(S (S Z)) == (S (S (S Z)))
  => (S Z) == (S (S Z)) -- By rule 4
  => Z == (S Z)         -- By rule 4
  => false              -- By rule 2

Loads of brackets, but it’s hopefully clear enough to see what’s going on: for as long as both of the arguments are successors, we remove one S at each step until one Z. If the other reaches Z at the same time, then the two numbers are equal! Ooer.

Defining addition is also fairly straightforward:

A B A + B
S x Z S x
Z S y S y
S x S y S (S (x + y))

Rules 1, 2, and 3 define our base cases: adding Z to any value produces that same value. Rule 4, of course, is where we define our recursion. Let’s look at an example with 3 + 2. We’ll use square brackets to make things clearer, but just think of them as regular brackets:

(S (S (S Z))) + (S (S Z))
  => S (S [(S (S Z)) + (S Z)]) -- By rule 3
  => S (S (S (S [(S Z) + Z]))) -- By rule 3
  => S (S (S (S (S Z))))       -- By rule 1

And there you have it! One more example that’s worth mentioning before we go is how to convert our Peano numbers back into the integers that we know and love:

A toInt A
Z 0
S x 1 + toInt x

We only need one argument for this, so the function is super straightforward. Yay! Here’s the function working with 3:

toInt (S (S (S Z)))
  => 1 + toInt (S (S Z)) -- By rule 2
  => 1 + 1 + toInt (S Z) -- By rule 2
  => 1 + 1 + 1 + toInt Z -- By rule 2
  => 1 + 1 + 1 + 0       -- By rule 1
  => 3                   -- By addition

Ok, so this maybe isn’t the sexiest concept, but we’ve covered some important stuff:

  • We can use recursion to write simple definitions of functions that can just call themselves to deal with each “step” of the problem.

  • We can use induction to show that, if a function works for Z (or some other base case) and S x, it can work for any Peano number!

  • Running a function is just substituting one thing for another! Functions shouldn’t do anything - they should just swap an input for an output.

That’s all from me today! If you want to play around with the concepts, you can use this gist to play with the code. Otherwise, I hope this was at least a little interesting, and I’ll talk to you soon!

Take care ♥