The big argument: What's the best way to end a proof?

20 May 2024

QED, argues Sam Kay

Quod erat demonstrandum: that which was ‘to be demonstrated’. Doesn’t it just sound cool? If you’re not already convinced, let me guide you through this page viz the correct way to sign off your proofs.

This glorious Latin abbreviation was integrated into society rather early on, with its roots in ties with Greek mathematicians Euclid, Archimedes et al (ca 300 ACN). Although Latin is less-widely used today, it still serves purpose in lots of secondary school mottoes.

Okay, sure, some school kids might not like the use of Latin phrases as it makes them sound ‘pretentious’, ‘pompous’, etc. Well, to those claiming this is the MO of pretentious people, would you be one to strike your PhD from your CV? In this day and age, AD 2024?

Once I was typing a proof by contradiction into Overleaf. But the font I used didn’t have a symbol, and was instead printed as the universal ‘missing character symbol’, ie $\square$!

There is so much more freedom when using Latin abbreviations. In PDE theory we often talk about the existence of solutions. After one of these proofs, one may want to use QEC, which was to be constructed. Other equivalent forms are QEF, which was to be done, or QEI, which was to be found out.

And that concludes my argument. QED.

PS: this argument should be titled QED vs $\square$.

$\square$, argues Clare Wallace

Obviously the square is the better choice. Look at it: $\square$. It’s elegant. It’s understated. It says, “oh, this old proof? I just had it lying around.”

Paul Halmos introduced the $\square$ notation for the end of proofs, inspired by end marks in (non-maths) magazines, in the 1950s—and lots of people still call it the ‘halmos’. Honestly, I’d be on board with anything he has to say about writing and typesetting maths.

And as I keep telling my fellow Chalkdust editors, I think the more we can do to make maths look more appealing, the better. I’d even argue that we should take more inspiration from magazines and newspapers: more colour! more pictures! more clickbaity article titles!

OK, maybe not that last one. But the point is: QED feels needy. It’s trying to show off with abbreviations. “Look at me,” it says. “I’ve stuck some unnecessary Latin at the end of this proof, because I’m clever.”

On the other hand, $\square$ is just cool. It’s not impressed by dead languages. It’s here to do a job, and it’s done it. It’s letting the maths speak for itself. It’s got nothing to prove.

$\boldsymbol{\square}$

Sam Kay

Sam is a maths student at Durham University where he hosts
Chalkboard Ultra podcast and spends too much time thinking about spinors. Outside of maths, Sam runs one of the university’s jazz bands.

Clare Wallace

Clare (like gare) is a Chalkdust editor. In her spare time, she’s an assistant professor at Durham University. She likes Skittles, probability, and making the magazine look more like Cosmo.