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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.

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I think Halmos started using that box for "QED". –  Jonas Teuwen Aug 9 '11 at 21:36
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I 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. –  André Nicolas Aug 9 '11 at 22:05
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Strictly 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. –  GEdgar Aug 9 '11 at 22:11
    
A 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." –  MJD Jul 5 '12 at 2:54
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4 Answers 4

up vote 23 down vote accepted

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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Thank you. But is there a logic relation between q.e.d and $\square$? –  Chan Aug 9 '11 at 21:39
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@Chan: of course not. There is no logical relation between the number two and the character $2$, either... –  Mariano Suárez-Alvarez Aug 9 '11 at 21:40
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@Theo Buehler I love the //// . I should try ending my proofs with :D or xD for a change. –  Olivier Bégassat Aug 9 '11 at 22:59
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@Olivier: or, try what Paul Sally does! books.google.com/… instead of the box, he uses a picture of himself smoking a cigar. –  Eugene Bulkin Aug 9 '11 at 23:27
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@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. –  t.b. Aug 10 '11 at 3:54
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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).

http://en.wikipedia.org/wiki/Q.E.D.#Electronic_forms

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Thanks a lot for the reference. –  Chan Aug 9 '11 at 21:42
    
@Chan: You are welcome! –  Américo Tavares Aug 9 '11 at 21:43
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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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Nice example ;). Thank you. –  Chan Aug 9 '11 at 22:13
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Well, 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? –  t.b. Aug 9 '11 at 22:34
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@t.b: Plausibility need not be the issue. Take it as a MNEMONIC, for crying out loud. –  Mike Jones Oct 26 '11 at 3:48
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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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Your last sentence understates the reality... this is of course not the historically correct interpretation of the symbol. –  Mariano Suárez-Alvarez Aug 9 '11 at 23:25
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The 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. –  Arturo Magidin Aug 12 '11 at 17:30
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@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. –  Arturo Magidin Aug 12 '11 at 21:20
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Inventing 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. –  Asaf Karagila Aug 16 '11 at 14:29
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@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.) –  Mike Jones Oct 22 '11 at 3:47
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