Words About A Wordless Induction Proof

This pair of tweets came across my feed. And who doesn’t like a good visual proof of a mathematical fact? I hope you enjoy.

So here’s the proposition.

This is the sort of identity we normally try proving by induction. Induction is a great scheme for proving identities like this. It works by finding some index on the formula. Then show that if the formula is true for one value of the index, then it’s true for the next-higher value of the index. Finally, find some value of the index for which it’s easy to check that the formula’s true. And that proves it’s true for all the values of that index above that base.

In this case the index is ‘n’. It’s really easy to prove the base case, since 13 is equal to 12 what with ‘1’ being the number everybody likes to raise to powers. Going from proving that if it’s true in one case — 1^3 + 2^3 + 3^3 + \cdots + n^3 — then it’s true for the next — 1^3 + 2^3 + 3^3 + \cdots + n^3 + (n + 1)^3 — is work. But you can get it done.

And then there’s this, done visually:

It took me a bit to read fully until I was confident in what it was showing. But it is all there.

As often happens with these wordless proofs you can ask whether it is properly speaking a proof. A proof is an argument and to be complete it has to contain every step needed to deduce the conclusion from the premises, following one of the rules of inference each step. Thing is basically no proof is complete that way, because it takes forever. We elide stuff that seems obvious, confident that if we had to we could fill in the intermediate steps. A wordless proof like trusts that if we try to describe what is in the picture then we are constructing the argument.

That’s surely enough of my words.