To end a proof, I often write "as was to be shown" or "q.e.d". Both of these terms make sense to me as a reader. On the other hand, I feel a little strange to put down $\square$ although I saw it many times here and there. In fact, I learned $\square$ notation here. I wonder if anyone could give me a brief explanation of $\square$ notation in mathematics. Where does it come from? More importantly, how does it logically mean "end" of a proof? Thank you.
Why does drawing $\square$ mean the end of a proof?
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11I think Halmos started using that box for "QED". – 2011-08-09
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1I think it first appears in his Measure Theory book. Halmos, by the way, was a great lecturer. Went to many meetings, always taking pictures. Unfortunately, he went through a Minox (tiny spy camera) phase, so many of the pictures were very low res. – 2011-08-09
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3Strictly speaking, "q.e.d." (as stated) means something like "as was to be shown", so (strictly) it is only appropriate if the last thing in your proof, indeed, was the thing to be shown. In Euclid, for example, the last thing is every proof is a re-statement of the theorem. Heath's translation often just has "Therefore etc." and doesn't include the actual re-statment. On the other hand, if the halmos means just "end of proof" then there is no such quibble. – 2011-08-09
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1A late addendum: John L. Kelley's *General Topology* (1955) says: "The end of each proof is signalized by ∎. This notation is also due to Halmos." – 2012-07-05
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0Throughout my academic career (fairly recently in history, mid/late 2010's), I have always only used QED or the solid black square tombstone $\blacksquare$. Most instructors prefer these as well. @GEdgar provides useful insight there on a possible distinction. Though I always prefer to restate my conclusion anyway, coming full circle - in my view it truly finishes off a proof. I only ever see the empty square $\square$ on stackexchange where I have presumed people were lazy with their LaTeX. A friend uses solid black diamond $\Diamond$. – 2018-04-21
4 Answers
It just means the same thing as q.e.d. Its introduction is usually attributed to Paul Halmos:
"The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ▯, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the 'halmos'.", Paul R. Halmos, I Want to Be a Mathematician: An Automathography, 1985, p. 403.
(This is quoted in Wikipedia)
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1Thank you. But is there a logic relation between `q.e.d` and $\square$? – 2011-08-09
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28@Chan: of course not. There is no logical relation between the number two and the character $2$, either... – 2011-08-09
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0Nice example, now it makes sense to me ;). – 2011-08-09
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0I'm glad that the "rudin" $////$ didn't get adopted more widely :) – 2011-08-09
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7@Theo Buehler I love the //// . I should try ending my proofs with :D or xD for a change. – 2011-08-09
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1@Olivier: Well, *de gustibus*... You could do as a lecturer of mine did: the number of slashes indicates what kind of proof ends so you can even interlace them: two slashes for a claim, three slashes for a lemma, four and five slashes for propositions and theorems. Immensely practical if you organize your lectures as he did: Theorem ... Proof: ... Lemma ... Proof ... /// ...(continuation of proof of theorem)... Claim ... proof ... // ...(continuation of proof of theorem) ... ///// – 2011-08-09
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11@Olivier: or, try what Paul Sally does! http://books.google.com/books?id=0mOzVTs2vmwC&lpg=PP1&dq=paul%20sally&pg=PA6#v=onepage&q&f=false instead of the box, he uses a picture of himself smoking a cigar. – 2011-08-09
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0@Theo: If you could satisfy my (morbid) curiosity, why are you glad about that fence of slashes not being more widely adopted? (Personally, I would likely feel dizzy reading one fence of slashes after another, but maybe your reasons are better/more logical...) – 2011-08-10
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3@J. M. I'm afraid that this will disappoint you, but I don't have a better reason than: I simply don't like the look of them (maybe this is because I didn't particularly like the lectures I mentioned in my previous comment). Be that as it may, I'm not a big fan of these end of proof signs in general, as good layout should be able to do without them entirely, but this may not be a popular position. I'm using the halmoses out of conformism more than by conviction. – 2011-08-10
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1@theo: halmoi? :) – 2011-08-10
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0:) ${}{}{}{}{}$ – 2011-08-10
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0My favorite alternative "end of the proof" notation is the tiny badger picture in Using the Borsuk-Ulam Theorem by Jiri Matousek (Borsuk means badger in Polish). I have stared using little panthers. – 2017-03-16
See
When typesetting was done by a compositor with letterpress printing, complex typography such as mathematics and foreign languages were called "penalty copy" (the author paid a "penalty" to have them typeset, as it was harder than plain text).[8] With the advent of systems such as LaTeX, mathematicians found their options more open, so there are several symbolic alternatives in use, either in the input, the output, or both. When creating TeX, Knuth provided the symbol ■ (solid black square), also called by mathematicians tombstone or Halmos symbol (after Paul Halmos, who pioneered its use). The tombstone is sometimes open: □ (hollow black square).
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0@Chan: You are welcome! – 2011-08-09
I have been told that it had a practical application. When a referee has read through the proof and checked its accuracy they could check the box.
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0Nice example ;). Thank you. – 2011-08-09
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12Well, I don't think that this is very plausible. When I referee a paper it is very rare that a proof gets away with a single checkmark. More to the point: how do you check a big fat black rectangle as the sign originally was in Halmos's *Measure Theory* or Kelley's *General Topology*? – 2011-08-09
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1@t.b: Plausibility need not be the issue. Take it as a MNEMONIC, for crying out loud. – 2011-10-26
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0How could this be a mnemonic? What are you trying to memorize with this story as the mnemonic? Do you have trouble remembering that the "end of proof" symbol is a box? – 2017-04-23
Perhaps it comes as a stretch, but consider the natural deduction proofs of Jaskowski. You find a sequence of statements within boxes with the last statement outside of any of the boxes... see here. So, you could interpret the box symbol as indicating the last statement as falling outside of the proof boxes, were the proof to get written in that style. Or in other words, it indicates the last statement as a theorem. This isn't to say that's a historically correct interpretation of this symbol though.
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8Your last sentence understates the reality... this is of course *not* the historically correct interpretation of the symbol. – 2011-08-09
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0@Mariano No question about that. But perhaps this answer gives us an interpretation as to how the box symbol "logically" means end of proof. Your historically correct answer, unfortunately, doesn't seem to do this as the OP wanted. – 2011-08-12
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3The symbol does not "logically" *mean* "end of proof", it simply does. How does the glyph `2` "logically" mean the number two? It doesn't. Some things just *are*, and inventing an "explanation" is at best silly. – 2011-08-12
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0@Arturo Yes, the symbol, as interpreted historically does not mean "end of proof". However, I do not see anything silly in inventing an explanation which makes such a symbol logical per se. People have engaged in storytelling for centuries. The explanation I gave here highlights proof boxes, which *do* help keep track of the scope of assumptions in proofs... and it does come as a good idea to keep track of the scope of assumptions in a proof. – 2011-08-12
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3@Doug: It is patently clear that what you consider silly and what I consider silly are disjoint sets, just like what you consider clear and useful is disjoint from my own. If we don't intersect again, I don't think it will be my loss. Or yours, since you care for little beyond your idiosyncratic interpretation of what the universe "should be", as opposed to what it is. – 2011-08-12
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1@Arturo I would very, very much doubt them disjoint. Remember, if there exists a single element where out sets of what we consider silly match, the sets are not disjoint. – 2011-08-12
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2Inventing explanations may seem harmless. However it is the source of disinformation. Read 1984, and find out how it was actually Big Brother who invented the Halmos box. – 2011-08-16
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0@Asaf Yes inventing explanations comes as the source of disinformation, however, this only happens when invented explanations don't get claimed as invented. I simply haven't claimed this explanation as anything but invented. – 2011-08-16
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2@Mana: Would you P L E A S E come here and say like you did before, “Guys, calm down, calm down.” Doug was simply creating a MNEMONIC, for crying out loud. As he pointed out, there is no harm if the story is openly acknowledged as invented. The mnemonic for SOS follows the same pattern as Doug’s story. That is, unlike most acronyms, the meaning (“Save our souls!”) came AFTER the acronym was created (which was picked simply because it was easy to key), and therefore was an “invented” meaning. (An example of a harmful story is the supposed benefit to eyesight from eating carrots.) – 2011-10-22
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0@MarianoSuárez-Alvarez:@xxx: Understating reality is a large part of what mathematics is all about. Failing to realize that makes you sound like the average high school physics teacher. To refresh, or kindle, this realization, read the answer given by user
to the question “Why do mathematicians use single-letter variables?” As – 2011-10-23points out, this understatement of reality (“ideas are not bound by physical reality”) is what allows for freedom and creativity in mathematics. -
0@Mike: this question is not a mathematical one. Even *if* it were true that understating reality is a large part of what mathematics is all about, it would be completely irrelevant here. – 2011-10-23
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0@MarianoSuárez-Alvarez: You're picking up the wrong end of the stick. I'm not saying that if you are doing mathematics you are understating reality, I am saying that if you are understating reality, you are doing mathematics:) – 2011-10-23