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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).

James Propp
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    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. – Trevor Wilson Jul 20 '12 at 19:16
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    Indeed, if you want to assume from some point onwards that your intervals will be bounded, say so explicitly. In your parethetical reminder, be explicit about the fact that you are reminding the reader, as in «(bounded, as per our conventions established in the introduction)», for otherwise you end up causing more problems than by not saying nothing: if the «(bounded)» shows up at a place where the convention is active, then the reader will wonder where exactly did it stop holding... and will have to start going back checking, &c. – Mariano Suárez-Álvarez Jul 20 '12 at 19:23
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    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. – Andreas Blass Jul 20 '12 at 19:24
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    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. – Mariano Suárez-Álvarez Jul 20 '12 at 19:25
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    I myself consider your usage legitimate and useful: reminder/emphasis of standing assumptions. Yet I am also aware that some readers are confused by this use, or are hostile to it. A different sort of negative feature, to my mind, is the visual grittiness of the parentheses, and the possible perceived insinutation that there are other implicit assumptions that are not being recalled, but which the paranoid reader will fret over. :) – paul garrett Jul 20 '12 at 19:26
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    You'll likely get knowledgeable answers at http://english.stackexchange.com/search?q=parentheses&submit=search – Joel David Hamkins Jul 20 '12 at 19:28
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    (FWIW, IMHO there are very, very few real reasons for parenthesis to be used, and whenever someone gives me a manuscript to proof read I systematically suggest all of them be removed...) – Mariano Suárez-Álvarez Jul 20 '12 at 19:29
  • The syntax of parentheses can get somewhat intricate in theorem statements. Consider situations involving duality, like "the pullback (pushforward) of f is injective (surjective) when f is open (closed)" which saves space but becomes extremely annoying to read. In general, instead of using (bounded) in the theorem, maybe it is preferable to declare at the top of the section "all spaces are assumed to be bounded henceforth unless explicitly stated otherwise". – Vidit Nanda Jul 20 '12 at 19:46
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    In terms of whether the use has a name, there was a similar question (from a mathematical source, no less!) at http://english.stackexchange.com/questions/49809/what-is-the-proper-grammatical-terminology-to-describe-this-parenthetical-remark -- maybe that would be a good place to take the present question. – Barry Cipra Jul 20 '12 at 20:28
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    I personally have never seen parentheses used to mean "this assumption will be implicit from now on," and it strikes me as imprecise usage. But I do understand your frustration at a fussy referee. On my first ever submitted paper the referee criticized several mathematical grammatical constructions because he personally didn't like them, despite their widespread usage. – Jim Conant Jul 20 '12 at 21:02
  • By the way, I've got my own pet peeve: http://mathoverflow.net/questions/70241/terminology-question-transverse-v-transversal – Jim Conant Jul 20 '12 at 21:14
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    A shriek, "(bounded!)", says "reminder" and removes the possibility of the "inessential" interpretation at the cost merely of a single extra character. – David Feldman Jul 20 '12 at 22:13
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    @Barry Cipra Research-level mathematics makes distinctive demands on language that engender idiomatic usages. Garden-variety grammar mavens (say, at english.stackexchange.com) usually just don't get mathematics. I once had a professional editor try to change "number theoretical statements" to "theoretical statements about numbers." While questions like Jim's should never form the core of MO, there may exist no better forum then MO for getting them answered. – David Feldman Jul 20 '12 at 22:29
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    @Jim Conant Lack of editing propagates degenerate constructions and makes them widespread and ineradicable, e.g., my pet peeve, "impact" as a transitive verb. – David Feldman Jul 20 '12 at 22:29
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    David, I disagree. There are currently 141 detailed questions at http://english.stackexchange.com/search?q=parentheses&submit=search on the proper use of parentheses. These are people who think carefully about correct grammar. Mathematicians, in constrast, are often sloppy and inconsistent, or just plain wrong. – Joel David Hamkins Jul 20 '12 at 23:56
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    ...You, for example, should write of "number-theoretic statements" rather than "number theoretic statements", and this distinction is exactly what underlies your editor's comment. To understand it, look up the proper use of the hyphen in iterated adjectival phrases in the Chicago Manual of Style, which explains the difference between a red-bike factory and a red bike factory. It is perfectly logical. – Joel David Hamkins Jul 21 '12 at 00:15
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    @Mariano: I am in complete agreement with you. When I read papers that have too many parentheses, I find that my eyebrows go "down" for every open parenthesis and back "up" for every closed parenthesis, and pretty soon my eyebrows are very tired. – Lee Mosher Jul 21 '12 at 00:39
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    @Joel, I noticed you (unconsciously?) changed David Feldman's "theoretical" to "theoretic." There is an interesting (to grammarians, if not mathematicians) discussion of ic's and ical's at http://english.stackexchange.com/questions/6581/why-is-it-geometric-but-theoretical – Barry Cipra Jul 21 '12 at 01:15
  • Barry, thanks for the link! I had made it parallel because I had wanted to highlight only the hyphen difference (or actually en-dash). But I also admit that I don't really understand proper "ic" versus "ical" usage. – Joel David Hamkins Jul 21 '12 at 02:29
  • Slightly off-topic, but what do people think about "jointly convexity", as in the abstract and the body of http://www.pnas.org/content/108/18/7313.full ? – Yemon Choi Jul 21 '12 at 08:45
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    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. – Arend Bayer Jul 21 '12 at 16:42

2 Answers2

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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.’

Francois Ziegler
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    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 :) – Mariano Suárez-Álvarez Jul 20 '12 at 20:06
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    @Mariano: to me that's exactly it, i.e. unless I'm missing something his "(bounded)" is synonymous with "(I won't mention bounded)". – Francois Ziegler Jul 20 '12 at 20:18
  • 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. – Adam Saltz Jul 20 '12 at 20:43
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    (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.» – Mariano Suárez-Álvarez Jul 20 '12 at 20:50
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    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 http://cvillewords.com/2007/11/09/entirely-a-matter-for-you/ – Yemon Choi Jul 21 '12 at 08:49
  • The fit isn't perfect, but it's close enough for me. – James Propp Jul 24 '12 at 15:46
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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.

and

(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. $\bigcap_{n=1}^{\infty} [n,\infty) = \varnothing$.

Pete L. Clark
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    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! – James Propp Jul 25 '12 at 02:57