A reader of one of my drafts found fault with my use of parentheses; I put the word "bounded" in parentheses in a statement of a certain theorem, and he replied "But the statement isn't true if the assumption of boundedness is dropped!"

That reader seemed to be thinking that parentheses mark things that are in some way inessential (as is sometimes the case in non-mathematical prose). But, as I wrote to him:

Here I am using parentheses to mean "Of course the interval must be bounded! In case some of you are nodding off, I'll include the stipulation of boundedness, but I might not include it next time." I wonder if that use of parentheses has a name?

Does this use of parentheses have a name, or any sort of pedigree that might dignify it, within or beyond mathematical writing?

I have no idea how to tag this post; it's a question about the (possibly nonexistent) subfield of modern Rhetoric that is concerned with the ways mathematicians use language to communicate ideas to other mathematicians. I'll be grateful if someone will suggest appropriate tags and add them (and I'll make a note of what the tag is, in case I need it again).

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    $\begingroup$ It seems to me that these two uses of parentheses are not necessarily different. Ideally before you use parentheses to indicate "I won't tell you this again" you will say something like "all widgets are henceforth assumed to be bounded" and then when you write "(bounded) widget" it is an inessential reminder of this global assumption. $\endgroup$ – Trevor Wilson Jul 20 '12 at 19:16
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    $\begingroup$ I agree with Trevor. The parentheses around "bounded" should indicate that the theorem is true without that word, probably because some earlier convention said that boundedness is always tacitly understood. The reason for including the redundant word in parentheses would usually be that the convention was stated so long ago that the reader might have forgotten it. $\endgroup$ – Andreas Blass Jul 20 '12 at 19:24
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    $\begingroup$ In short, be explicit, as explicit as you can without becoming painful: the seconds you save by not writing things out will be charged to your readers in terms of time and unease. $\endgroup$ – Mariano Suárez-Álvarez Jul 20 '12 at 19:25
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    $\begingroup$ A shriek, "(bounded!)", says "reminder" and removes the possibility of the "inessential" interpretation at the cost merely of a single extra character. $\endgroup$ – David Feldman Jul 20 '12 at 22:13
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    $\begingroup$ Is it really worth making a convention to be able to write "interval" instead of "bounded interval"? You should remember that most readers won't read your paper linearly, i.e. they may jump directly to section 3.14 since that's all they care about. Not being able to read section 3.14 without having read sections 1.1-3.13 is a BIG disservice to most readers. $\endgroup$ – Arend Bayer Jul 21 '12 at 16:42

Re: "Does this use of parentheses have a name?",

preterition |ˌpretəˈri sh ən|

noun (...) the rhetorical technique of making summary mention of something by professing to omit it.

ORIGIN late 16th cent.: from late Latin praeteritio(n-), from praeterire ‘pass, go by.’

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    $\begingroup$ That's similar but not quite it, is it? The Eco's artiluge mentioned at the end of that section, though, would make for a fun paper :) $\endgroup$ – Mariano Suárez-Álvarez Jul 20 '12 at 20:06
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    $\begingroup$ @Mariano: to me that's exactly it, i.e. unless I'm missing something his "(bounded)" is synonymous with "(I won't mention bounded)". $\endgroup$ – Francois Ziegler Jul 20 '12 at 20:18
  • $\begingroup$ I think a more typical use is found in this gem from Cicero's Against Catiline (quoting from the 1856 trans. on perseus.tufts.edu): "What? when lately by the death of your former wife you had made your house empty and ready for a new bridal, did you not even add another incredible wickedness to this wickedness? But I pass that over, and willingly allow it to be buried in silence, that so horrible a crime may not be seen to have existed in this city..." Cicero draws attention to the crime making himself appear generous for doing so. $\endgroup$ – Adam Saltz Jul 20 '12 at 20:43
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    $\begingroup$ (This is on the verge of becoming offtopic but) I think that preterition is the artiluge of saying that one is not saying something in order to say it, but the «(bounded)» does not carry that intention. It would be different if the theorem were something like «Don't get me started on the fact that we are assuming that our intervals are bounded, and let us just say that continuous functions on an interval are integrable.» $\endgroup$ – Mariano Suárez-Álvarez Jul 20 '12 at 20:50
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    $\begingroup$ Isn't one of the standard examples of preterition "I come to bury Caesar, not to praise him...", followed by much praising? Or, for a more recent exmplar, Peter Cook's sketch cvillewords.com/2007/11/09/entirely-a-matter-for-you $\endgroup$ – Yemon Choi Jul 21 '12 at 8:49

I think it might be beneficial to see the actual context in which the comments were made (by me; not as a referee, but just someone that Jim wrote to and asked for comments on his nice paper, which by the way, has a fair bit of its provenance in various MO threads).

The work in question is on the arxiv here. Various properties of an ordered field $R$ are being considered and compared. The last two are:

(17) The Shrinking Interval Property: suppose $I_1 \supset I_2 \supset \ldots$ are (bounded) closed intervals in $R$ with lengths decreasing to zero. Then the intersection of the $I_n$'s is nonempty.


(18) The Nested Interval Property: Suppose $I_1 \supset I_2 \supset \ldots$ are (bounded) closed intervals in $R$. Then the intersection of the $I_n$'s is nonempty.

I was not thrilled with the use of (bounded) in (17), but I let it go. I objected to the use of (bounded) in (18).

Note that "(bounded)" is playing different roles in the two statements. In (17), it is a superfluous hypothesis: if the lengths of the intervals are decreasing to zero then necessarily all but finitely many of them are bounded. In (18) it certainly isn't. I found this lack of parallelism especially confusing: so confusing that the first time I read it I honestly did arrive at the (ridiculous) conclusion that Jim Propp was unaware that e.g. $\bigcap_{n=1}^{\infty} [n,\infty) = \varnothing$.

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    $\begingroup$ I agree with Pete's comment completely. (In my earlier email correspondence with him, I confused (17) with (18), and missed the salient difference between them: in the first case the boundedness follows from the other hypotheses, and in the second it doesn't.) The "praeteritional" ("praeterite"?) use of parentheses is allowable for (17), but not for (18). Anyway, the responses I've received to this question have convinced me that in mathematical writing it's best to avoid confusion by being more explicit (e.g. "for the rest of this proof, all intervals are assumed to be bounded"). Thanks! $\endgroup$ – James Propp Jul 25 '12 at 2:57

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