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Who replaced the Halmos solid QED symbol $\blacksquare$ with the open square $\square$? In his 1955 "general topology" book, page vi, John Kelley attributed the original solid square to Paul Halmos.

The end of each proof is signalized by $\blacksquare$. This notation is also due to Halmos.

To be precise, Kelley used a solid rectangle, about 7 points height above the line, 2 points depth below the line, and a width of 3.2 points. But I can't find such a solid rectangle in this TeX symbol manual for use in Mathjax: http://www.math.harvard.edu/texman/node21.html.

In 1958, Howard Eves in "Foundations and fundamental concepts of mathematics", page 149, wrote:

The modern symbol $\square$, suggested by Paul R. Halmos, or some variant of it, is frequently used to signal the end of a proof.

Even though Eves used the open square, he didn't attribute the change from solid to open to any particular author. The other books where I have seen any historical comments at all on this subject all simply refer to Halmos. No one gives any credit for who used the open square first.

Q1. Does anyone have any references which use the open square earlier than 1958?

Q2. Does anyone know of a definitive attribution for the open square?

Q3. Is it possible that the open square came first, before the solid square?

As you would expect, there are several mentions in wikipedia, but none of them definitive. The closest to an explanation is this apparently self-contradictory page: https://en.wikipedia.org/wiki/Tombstone_(typography). I don't have full confidence in the accuracy of the Halmos quote on that page. There's also Q.E.D., Mathematical proof and List of mathematical symbols.

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  • $\begingroup$ Is this really about history of science and math ? $\endgroup$
    – Alexandre Eremenko
    Oct 8, 2015 at 19:43
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    $\begingroup$ Handwriting a filled square is no fun, I write open squares on the blackboard (and filled ones by LaTeX' AMS theorem environments). $\endgroup$
    – vonbrand
    Oct 8, 2015 at 21:53
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    $\begingroup$ @Alexandre Eremenko: Yes indeed. It is a question about the history of mathematical notation. In mathematics more than in any other subject, it is generally recognized that the choice of notation has an enormous effect on the ability to express ideas and to think clearly about ideas. Many times in the history of mathematics, the invention of a new notation had substantial benefits. Conversely, the English insistence on using the Newtonian dot-notation for derivatives is generally believed to have held back English mathematics for a hundred years. I can give references for that if you like. $\endgroup$
    – Alan U. Kennington
    Oct 9, 2015 at 5:49
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    $\begingroup$ @vonbrand: That makes me wonder if it's something like the development of the blackboard bold fonts as a way of indicating bold on a blackboard. Now BB is part of regular typography. My personal preference for the open square in printed mathematics is because printers sometime have difficulties printing solid black. It shows up the imperfections on the cylinder of a laser printer. But possibly it was the blackboard which incubated the open square. The quote from Halmos was 1985, and no doubt by then, the open square had become more usual. $\endgroup$
    – Alan U. Kennington
    Oct 9, 2015 at 5:56

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Interesting, I thought that the origin was a Latin pun: "quod" in "quod erat demonstrandum" is similar to "quad", the root in "quadratus", square. It is possible that Halmos did not specify whether the square needed to be filled, and Kelley, or his typographers, simply chose the filled version. End marks in articles that Halmos refers to as inspiration were not necessarily filled, or squares. Reporters often use -30- to sign off, and "in the late 1930s, the popular weekly Woman used a right-facing double chevron to show when its fiction continued over a page (», like a guillemet or French punctuation mark) and simply said ‘THE END’ at, well, the end. Algamated’s news magazine Pictorial Weekly also said ‘THE END.’ on its long pieces (1933)".

Halmos' quote is accurate, and appears on p. 403 of his I Want to Be a Mathematician. The symbol in the text is a hollow square. Here is the full paragraph: "My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented "iff", for "if and only if" - but I could never believe that I was really its first inventor. I am quite prepared to believe that it existed before me, but I don't know that it did, and my invention (re-invention ?) of it is what spread it through the mathematical world. 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"". The use of "iff" was also pioneered in Kelley's General Topology (1955).

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  • $\begingroup$ I guess there could be another pun there too on the word "square", as in "Now we're square.", meaning that now we have cleared our debts. In a sense, the statement of a theorem creates a kind of debt which must be cleared. One makes a promise, and one must keep it. On the other hand, early examples were more rectangular in shape. I've seen many other QED symbols, like a very bold right square bracket for example. $\endgroup$
    – Alan U. Kennington
    Oct 9, 2015 at 6:03
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    $\begingroup$ @Alan U. Kennington Sometimes I wonder if such confabulation while not historically accurate nonetheless contributes to wide adoption of symbols that lend themselves to it, and that is why square "won". Another example is the radical symbol which Euler speculated was stylized "r". Cajori finds it highly unlikely since the original sign had no vinculum and looked more like "v", but perhaps it contributed to the adoption before Euler and by Euler himself. Many modern sources cite Euler's speculation as fact. math.stackexchange.com/questions/15787/… $\endgroup$
    – Conifold
    Oct 11, 2015 at 23:07
  • $\begingroup$ Absolutely! The invention of anecdotes has great mnemonic value. I'm sure this is why so many fables have survived since ancient times. They assist the teaching of any subject. When students ask "Where did this notation [or concept] come from?", it must always be very tempting to offer a story which will keep the students quiet, but also assist their memorization of the notation/definition. And then the real historians come along and spoil it all!!! In my opinion, one must ask not "who first used something?", but "Why did it persist through history?" Fables help concepts to persist. $\endgroup$
    – Alan U. Kennington
    Oct 12, 2015 at 4:22

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