How to End a Proof

Here’s a proof for you.

  • Premise #1: Mathematical proofs are perfect.
  • Premise #2: Perfect things are good.
  • Premise #3: All good things must come to an end.
  • Conclusion: All mathematical proofs must come to an end.

Does that bring a tear to your logical eye? Fear not, my blog-reading friend. The end of a proof is not an occasion for mourning, but for celebration. The proof is done, which means the theorem may live forever.

How do we mark this festive occasion? I offer you eight ways.

First, the classic:

1 classic

It’s a Latin abbreviation; the Greek equivalent goes back to antiquity. It stands not for “Quite Easily Done” (as my high school math teacher Mr. Sherry insisted) or “quantum electrodynamics” (as the heathens in the Physics department would insist).

No, if you want to know what it stands for, we must move to Proof Ending #2: the pretentious flourish.

2 pretentious

This translates to “what was to be shown.” In other words: “We did it! We proved the thing!” For constructions, Euclid and others occasionally used the alternative “Q.E.F.” which stands for “quod erat faciendum,” i.e., “what was to be done.”

But please don’t think that, just because proofs are timeless, we must necessarily end them with tired old ritualistic Latin abbreviations. In fact, the 20th century gave us a delightful alternative, a 3rd way to end a proof: the conclusion of maximal concision.

3 concise

Originally a way to end magazine articles, this little box was first used for mathematics by Paul Halmos, in 1950. (Some have called it the “halmos” in his honor.)

And it goes to show: one way to end a proof is with your quirky personal stamp. When I first taught mathematical proof, my students in Oakland chose to invent their halmos equivalents. One of their favorites was Option #4:

4 with pizzazz

One student preferred the variant “Alabama!” which, stripped of its geographic associations, does indeed sound like an enthusiastic interjection (or possibly a spell from Harry Potter). My current students in Saint Paul prefer “Boom!” (or, for reasons I cannot fully decipher, “Big Brain!”).

Less appropriate for students is Option #5, the conclusion one might give to a particularly sexy proof:

5 sexy proof

(Note: This blog does not endorse cancer. Perhaps you can just let an unlit cigarette dangle between your lips, like the heartthrob fellow from The Fault in Our Stars.)

Which proofs merit a post-coital cigarette? I might nominate one of the many proofs of the divergence of the harmonic series, or perhaps some proofs without words.

(And I’d love to hear your nominations in the comments!)

What if you’re proving not a theorem, but (Option #6) a mere lemma?

6 lemma

A lemma is a stepping-stone theorem, a base camp from which you ascend to a larger, more formidable theorem.

At least, that’s the idea. Some lemmas are pretty impressive in their own right. Proving the “Fundamental Lemma” from the Langlands Program, for example, was enough to win Ngô Bảu Châu a Fields medal.

(I’m reminded of watching Battlestar Galactica, Season 2. Most episodes ended with gut-wrenching cliffhangers. But every so often, an episode would leave the characters in a relatively safe and comfortable place. These episodes, and only these, would end with “To Be Continued…”, as if to say, “Don’t worry! We would never let the characters you love actually stay happy! Anyway, this oxymoronic usage of “To Be Continued” echoes the oxymoron of a “Fundamental Lemma.”)

Now, Option #7: for “a proof from the book”:

7 from 'the book'

Itinerant math-loving alien Paul Erdos loved to refer to “The Book,” where God kept all of the most beautiful and perfect proofs. “You don’t have to believe in God,” he said, “but you should believe in the book.”

Which proofs belong? G.H. Hardy heaped praise on Euclid’s proof of the infinitude of the primes, as well as the classic proof of the irrationality of the square root of two. Many mathematicians might add the proof of Euler’s identity.

Any other suggestions?

(The mathematicians Martin Aigner and Gunter M. Ziegler attempted to write their own version of the book. It no doubt suffers from the limitations of every human attempt at divine scripture, but as with organized religion, it makes a decent starting point.)

And finally, Option #8, for sneaky proofs:

8 sneaky trick

For some examples, check out my post “The Joy of Slightly Fish Proofs”, including the ridiculous, adorable idea of using Fermat’s Last Theorem to prove the irrationality of the cube root of 2.

Anyway, that’s all, folks. For now. I once pledged to write a book titled “101 Ways to End Your Proof, from QED to Boomshakalaka.” Only 93 more to go.

24 thoughts on “How to End a Proof

  1. You left out one of my favorites, the Andrew Wiles finale: “I think I’ll stop here.”
    (…although I gotta admit I’d like it just as well if he had concluded with, “Boomshakalaka!”

    1. I’m certain the “boomshakalaka” is popularized (if not coined) by Rev. J.D. Manning, who uses is frequently in his sermons and speeches.

  2. My calc professor would end particularly long or hard proofs with “Result Is Proven”, abbreviated RIP.

  3. I thought at first that this was going to be your farewell posting: I very much hope it is not.

    I think the Book probably contains the proof that horses have an infinite number of legs:

    1. A horse has two legs in the back and forelegs in front.

    2. That makes six legs

    3. Six is an odd number of legs for a horse to have.

    4. But six is even.

    5. The only number that is both odd and even is (affine) infinity.

    6. Horses have an infinite number of legs. ∎

    Disclaimer: This proof may be inoperable in Scotland, Ireland, certain Caribbean islands, and India, and among older Southern Americans, New Englanders, and African Americans.

    1. No, he wrote in Greek. QED and QEF are how his proofs were ended in Latin translation, and because of the continuing popularity of the Elements and the reach of the Roman Empire, they’re the version that proliferated.

  4. My classmates and I convinced the class next door that QED was Latin for “In your FACE!”. Not sure how that came about as an expression of triumph, but loved it.
    Alternative translations included “Yay, we did it!”, ” Boom!” and “Phew!”.

  5. Wondering if the “Big Brain” idea is a reference to PULP FICTION, when Samuel L Jackson says, “Check out the big brain on Brett,” after Frank Whaley’s character gleans that the metric system accounts for the fact that French McDonald’s serve “Royales With Cheese” rather than Quarter Pounders.

  6. My preferred ending to a proof is the open square (in LaTeX, this is the `\square` command provided by the amssymb package) rather than the filled square (`\blacksquare` in LaTeX). This is particularly useful when reading a paper or proofreading my own work, as I can put a little checkmark in the box to indicate that it is correct, or that I have understood it, or whatever. I also sometimes use baseball scorekeeping symbols (a single for “I need to spend a lot more time on this”, a home run for “ah! I got this”, and a backwards K for “WTF? I’m never gonna get this!”).

    On the other hand, for particularly long and tedious proofs, where getting to the end feels like a real accomplishment, I like to end with QEMFD, which, of course, stands for “*quod erat demonstrandum*”.

  7. 1. When I first introduce QED to my students, I tell them that it’s Latin for “dilly dilly.”

    2. Instead of lighting a cigarette after a major proof, I do the Cristiano Ronaldo goal celebration: a small jog in front my students, and then a 180-degree jump with a twirl of the right arm before landing with arms extended in exultant celebration. Sadly, there’s no symbol for this in LaTeX.

  8. You forgot these:

    “Obviously.” The universally accepted truths for lazy mathematicians.
    “Yeah!” My student’s favorite ending.
    “The End.” A classic. Some prefer the shorter “Fin”.
    “This proof was written by .” I get this a lot during tests.
    “Fsiufusqqqqyyyyy.” I was teaching my students how to write in LaTeX, and I vaguely leaned my elbow on the keyboard, and this was produced.
    “YEAH!!!!!!!!!!!!!!!!!” Another version of the aforementioned “Yeah!”
    “End of the proof. Your brain goes Kabloof!” An aspiring poet (apparently)’s favorite proof.
    And of course, “In your FACE!” Also known as QED.

  9. I can’t believe that I missed the opportunity to say this in October, but my favorite way to end a proof is with a smiley face. This was how Gerald Edgar finished all of the proofs in the first edition of his book on fractal geometry and measure theory. Unfortunately, I only own the second edition, and my masters advisor was unwilling to trade. :\

    1. That’s great! FYI, I added this to my document of notes for a future book on mathematical notation, and found that I’d already copied and pasted your earlier comment on this post, too.

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