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The Americans and the French use a different notation for open intervals: The Americans use (x, y) while the French use ]x, y[. How did this notational divergence appear?

Ali Caglayan
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Franck Dernoncourt
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    The square brackets notation is due to Bourbaki. – Andrés E. Caicedo Oct 30 '14 at 16:56
  • @AndresCaicedo Thanks, I had no idea it was so recent. Do you know why they chose this notation? – Franck Dernoncourt Oct 30 '14 at 17:42
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    Some discussion on this topic here and here. One commenter suggest that the backward brackets might have been introduced by Bourbaki to prevent confusion with ordered pairs. I am still at a loss for a documented history, but it is at least a part of an old ISO standard. I am not seeing it in the latest standard ISO 80000-2. – J. W. Perry Oct 30 '14 at 18:06
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    I guess it is intuitive (inclusion/exclusion of the endpoints depends on the direction of the bracket), but I have not found anything written by them stating this or some other motivation explicitly. – Andrés E. Caicedo Oct 30 '14 at 18:18
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    @J.W.Perry Thanks! I had indeed went through the standard for that reason, see notation 2-6.10 last column. – Franck Dernoncourt Oct 30 '14 at 23:54
  • @FranckDernoncourt Good eye, hence still in the standard. Not sure how I missed it, I must not have been looking hard enough. I would still like to see the documented history, or at least see the primary document text trace of the ]a, b[ notation. Where's a Florian Cajori when you need him?! – J. W. Perry Oct 31 '14 at 00:16
  • I've heard in a math class some years ago, that the motivation behind the $]\cdot,\cdot[$ notation is that the $(\cdot,\cdot)$ notation is reserved for ordered pairs. – user153012 May 26 '15 at 11:38
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    The ] [ notation is used in Belgium too, also in the Dutch-speaking part. – Dominique Feb 27 '23 at 13:27

1 Answers1

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Notation $()$ is traditional, and $].[$ was introduced by Bourbaki.

Much of the Bourbaki notations and terminology became standard, but English speaking people are the most conservative ones in this respect:-) (Recall the history of the metric system:-)

Another example of the same is "injection", "surjection", "bijection". Many English authors still write "one-to-one", "onto" and "one-to-one and onto".

Another example: Bourbaki taught us that "positive" is $\geq 0$, and "strictly positive" is $>0$.

But many people still prefer "positive" to mean $>0$ and "non-negative" for $\geq0$.

Remark. I am educated in Ukraine in 1970-s, and I experienced a strong influence of Bourbaki on education. But I still like $(,)$, perhaps just for aesthetic reasons.

Alexandre Eremenko
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    Thanks, very interesting, I had no idea that the definition "positive" is $\geq 0$ was from Bourbaki as well, I am always having trouble with that in the US. – Franck Dernoncourt Nov 01 '14 at 20:14
  • When you say Bourbaki taught the meaning of positive as $\geq 0$, do you really mean that pre-Bourbaki in France the word positif in math meant $> 0$ rather than $\geq 0$? In my experience, положительный means $> 0$, but did you ever have teachers in the USSR suggest it should mean $\geq 0$? – KCd Apr 30 '15 at 02:01
  • @KCd: I suppose "positif" meant in pre-Bourbaki France the same as "positive" in English. Concerning Soviet teachers of 1960-70th, some of them were Bourbakists, others not. Yes, I had teachers who promoted Bourbaki terminology, but I understand that this was not very common. I studied in Western Ukraine, not in Moscow. – Alexandre Eremenko Apr 30 '15 at 11:44
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    A friend of mine asked a member of Bourbaki, let's call him X-X. X, about this and indeed the usage is due to Bourbaki. X-X. X said Bourbaki wanted to allow the notation $\subset$ to include the possibility of equality, and not just mean a strict subset. Compatibly with that, they wanted $<$ to mean less than or equal to and $>$ to mean greater than or equal to. This is why Bourbaki started using the word positif to mean greater than or equal to 0. – KCd May 01 '15 at 04:16
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    Yes, and Bourbaki had partial success: everyone uses $\subset$ nowadays in their sense. – Alexandre Eremenko May 01 '15 at 11:29
  • "Positive" positively means greater than 0. A positive zero has its place in economy and commerce and unexact speaking. Bourbaki's opinion is completely irrelevant in this and other respects.. – Franz Kurz May 30 '17 at 13:12
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    @AlexandreEremenko: From my, admittedly rather limited, experience I'd say the opposite is (still) true: since $<$ is usually interpreted as a strict inequality, I do prefer writing $\subseteq$ for (not necessarily strict) inclusion. – Jules Lamers May 30 '17 at 14:17
  • @AlexandreEremenko: You experienced a strong influence of Bourbaki on education? This is in contrast with a statement of Murray Gell-Mann: "Nature Conformable to Herself", Bulletin of the Santa Fe Institute, 7 (1992) 7-10: "Pure mathematics and science are finally being reunited and, mercifully, the Bourbaki plague is dying out. (In the late Soviet Union they never succumbed to it in the first place)" – Franz Kurz May 30 '17 at 20:35
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    "Yes, and Bourbaki had partial success: everyone uses ⊂ nowadays in their sense." Who is everyone? If you see him, tell him that he is wrong. – Franz Kurz May 30 '17 at 20:37
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    @Otto: I use it in that way, and I am not wrong. – timur Sep 28 '17 at 02:39
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    @Otto: And I. In fact, I don't think I EVER used $\subseteq$ in my long career, and had to look to my LaTeX manual, when typing this comment. Same applies to most authors of books and papers that I read. – Alexandre Eremenko Nov 11 '21 at 03:19