Punctuation question

The Princeton Companion to Mathematics is even more nearly nearly finished than it was last time I said it was nearly finished. In fact, this time I can give a date — July 13th — past which it will be too late for me to do any work on it. The book will be printed in September and available in November.

As an example of the important issues we now face, here is a question about hyphens: I’m fairly sure there will be a small but passionate minority of mathematicians who care about these, and a question has come up. I am curious to know what other people think, so I’m not going to say what I think: I’ll just try to present the question as neutrally as possible. And here it is.

Consider the following four phrases: “intermediate value theorem”, “travelling salesman problem”, “twin prime conjecture”, “minimal surface equation”. Do they need hyphens? And how about “three body problem”?

Brief argument in favour: the general rule is that if you have a phrase made up out of an adjective, a noun and a noun then you put in a hyphen. (An example is “brute-force search”.)

Brief argument against: the cases above are exceptions because they are a bit like proper names: the first two words are not really functioning as an adjective that describes the main, third, word; rather, the three words form a single phrase. In addition, the lack of hyphen does not lead to any conceivable ambiguity.

If you buy something like the second argument, can you make it more precise?

Also, if you wouldn’t write the hyphen in “intermediate-value theorem”, do you find it positively wrong and strange looking, or merely not necessary?

In other words, should there definitely be hyphens, should there definitely not be hyphens, or would either decision be acceptable? (To judge from Google, standard mathematical practice seems to be not to put hyphens, but that alone is not a sufficient argument.)

I’ve had a more mathematical blog post planned for months, but it’s still going to have to wait. But once again I’d like to make clear that this blog’s still just about alive and will I hope become more active again before too long.

40 Responses to “Punctuation question”

  1. D. Eppstein Says:

    The only one that I would hyphenate from your list is three-body problem — in general I hyphenate phrases of the form number-noun, but not adjective noun, and I don’t think the “twin” in “twin prime” is sufficiently number-like to be an exception. I wouldn’t strongly object if I saw any of the others hyphenated, but it would come across as a little antiquated and pedantic to me. And I would probably spell it “traveling salesman problem” with only one L in the first word, though the spelling you give is also acceptable.

  2. t8m8r Says:

    I’d like to see as few hyphens as possible.

  3. r Says:

    Exactly what David says, plus I prefer “Traveling Salesperson Problem”.

    I might go a bit farther in thinking that hyphenating the first four would look a little weird.

  4. Nathan Dunfield Says:

    The phrase “intermediate-value theorem” looks very odd to me, as do the others except for “three-body problem” which, like David, I would hyphenate if left to my own devises. I’m not sure now I would justify my choice formally. Perhaps the issue is that when I say aloud “intermediate value theorem” the words come out evenly spaced, whereas a hyphen indicates that I should be condensing the first would into some kind of compound word. Compare this with saying aloud the phrase “Cauchy-Schwartz inequality” (though this is not the same kind of example as we’re discussing here.)

  5. D. Eppstein Says:

    A related typographic peeve: “Cauchy–Schwartz inequality” should be spelled with an en-dash, not a hyphen. Compare, e.g., “Birch–Swinnerton-Dyer” conjecture: the compound name Swinnerton-Dyer is hyphenated but the two conjecturers’ names are separated by a dash.

  6. .mau. Says:

    For what a non-English speaker’s opinion is worth, I’d prefer not to have any hyphen in the names you wrote. It looks better. Yes, I know that I wrote “non-English” hyphenated, but I never said I am consistent 🙂

  7. Nathan Dunfield Says:

    Hmm, I had always thought that “Birch–Swinnerton-Dyer” was a special case, though; when you put a hyphen between two words where one already has a hyphen, then the new hyphen becomes an en-dash, at least some of the time.

  8. John Armstrong Says:

    D. Eppstein: don’t forget that those fractions should each have a solidus instead of a slash.

  9. New reader Says:

    Better explain the problems instead of talking about such irrelevant junk.

  10. Jason Dyer Says:

    I agree that the three-body problem is the only one that needs a hyphen. I have no overall argument, just that it matches what I’ve seen in books and now I’m used to it.

  11. Nitpicker Says:

    Since we’re removing the extra “l” in “traveling”, could we please also remove the extra “t” in “Cauchy–Schwarz”?

  12. gowers Says:

    I should have made it clear earlier that “travelling” is the correct English spelling, but that the Princeton Companion will use US spelling and will therefore have a single L.

  13. Ben Fairbairn Says:

    “Also, if you wouldn’t write the hyphen in “intermediate-value theorem”, do you find it positively wrong and strange looking, or merely not necessary?”

    The IVT with the hyphen looks strange to me. Their use in general does too. It is probably best to try and stick with the conventions used in the rest of the book, assuming the chief editor(s) have some standardisation rules in place. This approach (delegation) has the advantage of saving you the time of worrying about such trivialities leaving you free to worry about the real meat of the article.

  14. Charles Greathouse Says:

    I agree with the first comment: only the three-body problem needs a hyphen.

  15. Chris Says:

    Usually I am a strong hyphen advocate, mostly because people forget them so often. Nevertheless, regarding the argument in favor, a hyphen in an adjective-noun-noun trio (where the first two words are functioning as a compound adjective) is only strictly necessary when the a-n-n combination is not in any sense a variation on or specialization of the n-n combination, for instance in particular when the n-n combination alone is nonsense or otherwise topically misdirecting. By this standard, only the “three-body problem” requires a hyphen, which seems to fit with the other impressions posted here. Of the remaining four, “intermediate value theorem” seems most borderline. I am sympathetic to the argument against, provided it isn’t applied too broadly: “intermediate value theorem” or “traveling salesman problem” is a set phrase, as evidenced by its ability to function in turn as an adjective: “intermediate-value-theorem techniques” or “traveling-salesman-problem research”. Indeed, the necessity of the hyphenation in these last two phrases suggests that the a-n-n combination _oughtn’t_ take hyphens. If “traveling-salesman problem” were appropriate, then “traveling-salesman problem research” wouldn’t look and sound quite so wrong.

  16. Charles Wells Says:

    I seem to be marching to another drummer than your other commenters. The example that needs a hyphen in my opinion is “minimal-surface equation”. The others can have hyphens or not, I don’t care. Hyphens in these situations certainly don’t irritate me.

  17. Anon Says:

    Brief argument against: the cases above are exceptions because they are a bit like proper names: the first two words are not really functioning as an adjective that describes the main, third, word; rather, the three words form a single phrase. In addition, the lack of hyphen does not lead to any conceivable ambiguity.

    I go with this. I think convention has a important role to play too — most readers are used to reading the above mentioned titles as proper names. Using the hyphen might throw in some confusion.

  18. Anon Says:

    Disclaimer: Off topic.

    You need/ought to get a “bot checker”, before the comments section gets flooded with garbage. I’ve had the problem myself.

  19. Mike Borns Says:

    I’m a non-mathematician who edits mathematics articles for a living. My rule of thumb for the hyphenation of three-word compounds is: hyphenate only when the first word can be construed as modifying more than one of the other two words.

    My non-mathematical example: California history teacher.

    In this example, “California” can conceivably modify “history,” “teacher,” or both. Therefore, a hyphen is called for. “California-history teacher” is a teacher of California history, “California history-teacher” is “a history teacher from California, and “California-history-teacher” is a teacher of California history from California.

    I don’t take hyphens lightly. If there’s no ambiguity, I don’t use them!

    As for “intermediate value theorem,” “traveling salesman problem,” “twin prime conjecture,” and “minimal surface equation,” I leave these to you mathematicians to determine whether or not they contain ambiguities. (“Three-body problem” is hyphenated by the always-hyphenate-if-the- first-word-is-a-number-and-it-modifies-the-second-word-in-a-three- word-compound rule.)

  20. Gil Says:

    Trying to apply Mike Born’s extremely rational proposal, I think that for a name of a theorem, conjecture, problem, or equation, most usually hyphens are not needed. In a phrase “X Y Theorem,” usually ‘X’ modifies ‘Y’. Only rarely ‘X’ modifies ‘theorem’ (like “Hard Lefschetz theorem”), and I am not aware of cases of ambiguity. In other math cases, applying Mikes’ rule, you do need hyphens. Like “quantum error-correction” or “spontaneous error-synchronization”.

  21. Bill Watson Says:

    Less is more.

  22. harrison Says:

    I agree that all except possibly “three-body problem” should be left unhyphenated. I wouldn’t find something like “intermediate-value theorem” to be wrong and strange-looking, but I have been put off by typographical or nomenclative deviations before. (See e.g. David Mermin’s continued use of “Qbit” for what everyone else calls a qubit.)

    OT: Is there anything new to report on the “tricks wiki?” I’ve been wondering about that for months, but there’s been no sign of whether the project is even alive, which is disappointing.

  23. Mike Borns Says:


    It’s a rational proposal, but I don’t always stand by it. I wouldn’t insist on hyphenating a phrase that is ambiguous to the layman but not to the specialist if no one else does. Convention trumps rationality.

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  25. JSE Says:

    My opinions about punctuation, usually idiosyncratic, agree here with everyone else’s, so I’ll use this space just to say — congratulations on the near-end of this immense project! I’m eagerly awaiting the final product.

  26. Matt Says:

    Three-body problem is fine. All the others would look weird to me if they were hyphenated.

  27. gowers Says:

    Harrison, just in answer to your query about the Tricks Wiki, I must apologize that that is not yet up and running — the Princeton Companion has taken longer to be completely finished than I thought possible — but I can assure you that the project is not dead. A lot of work has gone into the technical side, so even if I weren’t interested in the idea any more (which I definitely am) I would be morally obliged to press on with it. I think it’s likely to be another couple of months before it gets going, but before then I may well find myself asking the blogosphere for further advice about how precisely it should work.

  28. Hypatia Says:

    I like this quote and I hope Eric Schechter doesn’t mind my using it:

    The English language was not designed for mathematical clarity. Indeed, most of the English language was not really designed at all – it simply grew. It is not always perfectly clear. Mathematicians must build their communication on top of English [or replace English with whatever is your native or local language], and so they must work to overcome the weaknesses of English. Communicating clearly is an art that takes great practice, and that can never be entirely perfected.
    Eric Schechter, Associate Professor
    Department of Mathematics
    Vanderbilt University

    Certainly leave out the hyphens, if you prefer. If questioned, you can always state that it is standard mathematical form. Really, who is going to question a Fields medalist on standard mathematical form! But please, do not leave out the ‘extra’ l in travelling. That is simply a uniquely American spelling.

  29. Anonymous Says:

    It seems natural to me. Apostol does this in his book ‘Mathematical Analysis’.

  30. Peter Shor Says:

    I think these expressions are slightly easier for the reader to parse with hyphens, so I’d go with hyphens (forget about strict rules of grammar; ease of reading should be the actual criterion in ambiguous cases like these).

  31. Mark Bennet Says:

    If akin to proper names, why not use capital letters for all three words when they appear together? Although strictly unnecessary to decode the meaning, this might make it evident at first reading (though this may be inconsistent with other naming conventions in the book).

  32. Henry Wilton Says:

    I’m with Peter (and against the majority, apparently). They seem easier to parse with hyphens, convention notwithstanding.

  33. Ivan Says:

    why don’t you just ask your grandfather?

  34. Catleigh Says:

    In Modern English Usage Fowler makes an elaborate study of the hyphen. He begins engagingly by pointing out that “superfluous hair-remover” can only mean a hair-remover that nobody wants, and he proceeds to work out a code of rules for the proper use of the hyphen. He admits that the result of following his rules “will often differ from current usage”. But, he adds, “that usage is so variable as to be better named caprice”. The author of the style-book of the Oxford University Press of New York (quoted in Perrin’s Writer’s Guide) strikes the same note when he says “If you take hyphens seriously you will surely go mad”.

    I have no intention of taking hyphens seriously. Those who wish to do so I leave to Fowler’s eleven columns. If I attempted to lay down any rules I should certainly go astray, and give advice not seemly to be followed. For instance, the general practice of hyphening co when it is attached as a prefix to a word beginning with a vowel has always seemed to me absurd, especially as it leads to such possibilities of misunderstanding as unco-ordinated must present to a Scotsman. If it is objected that ambiguity may result, and readers may be puzzled whether coop is something to put a hen in or a profit-sharing association, this should be removed by a diaeresis (coöp) not a hyphen (co-op). That is what a diaeresis is for.

    I will attempt no more than to give a few elementary warnings.

    (i) Do not use hyphens unnecessarily. If, for instance, you must use overall as an adjective (though this is not recommended) write it like that, and not over-all.

    But if you do split a word with a hyphen, make sure you split it at the main break. Though you may write self-conscious, if you wish to have a hyphen in the word, you must not write unself-conscious but un-selfconscious.

    (ii) To prevent ambiguity a hyphen should be used in a compound adjective (e.g. well-written, first-class, six-inch, copper-coloured). The omission of a hyphen between government and financed in the following sentence throws the reader on to a false scent:

    When Government financed projects in the development areas have been grouped.
    But remember that words which form parts of compound adjectives when they precede a noun may stand on their own feet when they follow it, and then they must not be hyphened. “A badly-written letter” needs a hyphen, but “the letter was badly written” does not. There must be hyphens in “the balance-of-payment difficulties” but not in “the difficulties are over the balance of payments”.

    (iii) Avoid as far as possible the practice of separating a pair of hyphenated words, leaving a hyphen in mid-air. To do this is to misuse the hyphen (whose proper function is to link a word with its immediate neighbour) and it has a slovenly look. The saving of one word cannot justify writing

    Where chaplains (whole- or part-time) have been appointed
    instead of “where chaplains have been appointed, whole-time or part-time.

  35. beans Says:

    I read through (about) half of the comments, and realised that I am another one of the few who would put a hyphen for all three! I don’t really know much about whether they need hyphens (which I aim to find out) and I don’t mind without, but the hyphen is asking to be put there.

    Then again I do have an issue with placing dashes and semi colons in silly places…

  36. Gil Says:

    Dear Tim, What was decided at the end regarding hyphenation? –Gil

  37. gowers Says:

    Gil, I myself was slightly torn, because there were arguments in both directions. But in the end I was encouraged by something I found on a reputable page on the internet (connected with the Guardian newspaper) that gave examples of phrases such as “triple jump champion” (that wasn’t one of their examples but it will do), where the first two words go so naturally together that a hyphen seems excessive. Once it became clear that this situation arises in non-mathematical language as well, I felt happier about the idea of doing the same for things like “fixed point theorem”. Since my instincts about when hyphens are needed were the same as those of most people who have commented above (it was virtually everybody at first, but then a few serious hyphen fans made themselves known), and appeared to conform to standard mathematical practice (to judge from Google searches), we have gone for not always hyphenating. In particular, just “three-body problem” has been hyphenated, out of the examples I gave. But we argued about it quite a lot before coming to that decision, and didn’t really reach complete agreement. Incidentally, for the last couple of weeks it has been too late to make any changes to the Princeton Companion, which is scheduled to appear in November.

  38. A Beautiful Garden of Hypertrees « Combinatorics and more Says:

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  39. weierstrass Says:

    One of my lecturers started a course by saying “The problem with language is that it’s not associative.” and went on to explain that while algebraic number theory could be the theory of algebraic numbers, it was in fact the algebraic theory of numbers.

    The best non-mathematical example I could think of was “Third World War”.

  40. Matthew Emerton Says:

    Dear Weierstrass,

    Actually, while it is true that algebraic number theory is (in part) the
    algebraic theory of numbers, it is just as much the theory of algebraic
    numbers. (And its study involves methods that are not algebraic, as
    well as methods that are.) So perhaps the ambiguity is not actually so
    bad in this particular instance.

    Best wishes,


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