What are the advantages of ending a proof with “QED”?

Question

I know “QED” indicates the end of a proof. We also use $\square$ or similar.

How did we start doing it? It’s a very old practice passed down from Greek mathematicians like Euclid over twenty centuries ago.

But what are the advantages of doing so? I’d like an answer that addresses several contexts, ranging from books to papers to even posts here.

I can make guesses and persuade myself one way or another, but I’d like to hear from those who are more experienced.

Answer 1

I'll offer a somewhat different answer that contends we sometimes have a net benefit from such a strategy, but sometimes don't. So when is it beneficial? Well, that's complicated.

Let's contrast two very different ways of explaining how we know something. Example 1:

Positive integers have prime factorisations (since a minimal counterexample couldn't be prime and hence would be a product of two smaller positive integers whose prime factorisations force a contradiction), and up to permuting prime factors such factorisations are unique (since if $n=\prod_i p_i=\prod_j q_j$ are distinct factorisations for a minimal counterexample $n$ each $p_i$ divides, and hence is equal to, some $q_j$, whence $n/p_1$ or $n/q_1$ is a smaller counterexample unless both products are empty and hence identical).

This one-sentence proof of the fundamental theorem of arithmetic would be easy enough for a reader who knows (1) how to present a proof by induction in terms of hypothetical minimal counterexamples and (2) that primes divide at least one factor of products they divide, which follows from Bézout's lemma. If you're confident your readers can manage this, you might think it'd be a waste of everyone's time to write

Theorem, with name: blah blah blah

Proof: several sentences $\square$

I almost never write solutions on this website in such a format, partly because I fear verbosity can stop a reader seeing the forest for the trees. Even in my PhD thesis, when I could prove something succinctly I simply "reasoned out loud" in sentences that make it seem more like a string of casual observations, where $\square$ doesn't belong (and, if it were used, you'd feel like it was a strange choice when a full stop would do). Sometimes, that's a good way to do it, if only because (I think) it mirrors the way people understand things. People are accustomed to thinking in sentences, not in a particular formatting style exclusive to text. And while this strategy very rarely uses words such as proof or theorem, in many cases it doesn't hurt the rigour and formality of the proof.

(Mind you, I'll admit if I'm trying to defend such a writing style, the lengthy sentence in my example is "pushing it".)

But you asked why we would use such formatting, right? Well, let's look at a few things that can derail the above style:

• The proof is long enough a new paragraph ought to start when it's finished; and, just in case the next paragraph might be expected to also be a part of the proof, you need to make clear it isn't. Proofs can be long because no shorter option exists, because you as author don't know of one, or because the sort of compression I used above asks too much of the reader. (I suspect my example above would be a bit much for most people if it was the first time they ever see the FTA proven.) Heck, even needing to make one or more equations display-line can break the eye's definition of a paragraph too much to get away with the "conversational" approach I described.
• The proof needs to be stretched out to familiarise the reader with techniques it's meant to illustrate. Example 2 below shows how this looks when we're helping people learn induction (but after a while, you can literally just write the theorem and say it follows by induction, and the entire proof will instantly form in the reader's head):

Theorem: for all integers $n\ge 0$, $\sum_{j=1}^n(2j-1)=n^2$.

Base step of proof by weak induction: $\sum_{j=1}^0(2j-1)$ is an empty sum, equal to $0=0^2$ as desired.

Inductive step: if $\sum_{j=1}^k(2j-1)=k^2$ then $\sum_{j=1}^{k+1}(2j-1)=k^2+2(k+1)-1=(k+1)^2$. $\square$

• The result is so important it has a name, and needs front-and-centre attention. It might give its name to the chapter you're reading, even if most of the chapter discusses its significance while the proof itself is half a page. If I had appended "This is called the fundamental theorem of arithmetic" to example 1, I would have buried the lede. (And the reader might wonder whether both parts or just the second one were the theorem; and if I'd said "these two results", some readers wouldn't have parsed it so as to know what two results I mean. So either way, I can't win.) You're welcome to invent your own examples of when the name needs to be up front, but the reasons why would vary. Formatting is of course an art, where any number of concerns can come up from time to time. But once you're committed to theorem-with-name followed by the proof... well, you've seen where that leads in terms of spacing.

As I said, I sometimes see a benefit in making a proof seem more like an obvious observation than something that needs formatting carved out of the rest of the document's flow. I said that not doing this can stop one seeing the forest for the trees; one wants to know the "main point" of the proof, the "reason why" a theorem is true. (Mathematicians rarely think of proofs as explanations, but they can be succinct enough to be comparable to the explanations we find elsewhere in life.) However, sometimes you really need to take the trees one at a time, or there are too many to take in the whole forest at once. Sometimes, a proof is even presented as:

We will prove the A theorem, which is B. We begin by proving C. We then prove D. We then show this implies A.

No wonder you need an ending signal after all that, before we move onto the next theorem.

Having said that, you could argue some long mathematical publications, be they PhD theses or Andrew Wiles's proof of Fermat's last theorem, are essentially one long proof with a lot of lemmata (which may or may not be explicitly highlighted as such). The end of that proof doesn't have the same rationale for such signposting, precisely because nothing is next.

Answer 2

It is part of the grammar of mathematical writing/discourse.

A proof is a key unit of mathematical discourse. It is important therefore to have efficient markers of the beginning and end of the unit. It doesn't matter what these are - the specifics are arbitrary.

You mention the end of the proof. There are conventional beginnings too: eg "Theorem 2" or "Proposition 5.6" or "Lemma 3.2" followed by a statement of what is to be proved. Why not just leave these out?

Why is it not redundant: well sometimes people write or say things like "in the proof of proposition 5" - and if we have markers of the beginning and end of the proof we know what the point of reference is.

The key thing is that it costs little and adds to efficiency and accuracy of communication.

Answer 3

QED is short for the Latin phrase quod erat demonstrandum and means, loosely, "that which was to be shown". As noted in the comments, it's a summary way of saying that the proof is concluded. The $\square$ symbol (often filled in) seems to have originated with magazines in the early nineteen hundreds, but (as was also noted by a commentor) the History of Maths Stackexchange would be a better place to pursue that.

Why indicate the end of a proof? Several reasons:

1. In a book where there is both discussion and proof indicating the end of the proof prevents a reader from thinking that the following discussion is still part of the proof. If a reader is having trouble understanding the proof this stops them from getting more confused by thinking that an introduction to the next idea is part of the proof they're struggling with.

2. Not all books are text-books. Some are reference works, and for a researcher who is skimming a text looking for a specific theorem, proof or idea the end-of-proof marker is really helpful for jumping down the page in blocks.

3. It's sort-of traditional. Look at books written 100 or 150 years ago and you'll find practically no "Theorem", "Lemma" or even "Proof" in there. Ideas are brought up, and maybe a theorem is italicised (but not often) and then proven. The end-of-proof marker is much more useful in texts like this because it's now an end-of-idea marker and is very helpful in navigating the text. But humans tend to be tidy, so when the Lemma-Theorem-Proof style formed, the end-of-idea marker was kept as it helped with clarity.