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:
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.
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.
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:
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:
(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?
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”:
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:
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.