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If a formula appears in the main body of the text, there seems no doubt that punctuation should be given to that formula when necessary. But when a formula appears displayed (i.e. solely taking up a line), there seems no consensus on whether punctuation is needed for it. I wonder if there will be any potential ambiguities if punctuation is (or not) included in this case.

This is essentially a question about functions of punctuation for displayed formulas. I don't want an answer about rules from certain style guide...

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In mathematics there is a near universal consensus that displayed mathematics should not be treated differently from inline mathematics with regards to punctuation. By this I mean that the vast majority of papers on the arXiv follow this convention, most journals will add punctuation according to this principle if it's not already there, and so on.

Thus whenever you're unsure whether or not to include punctuation at the end of a displayed equation, try to replace \[ ... \] with $ ... $ - whatever punctuation mark looks natural when it's inlined should be included also at the end of the displayed equation.

I don't think that texts without punctuation after displayed equations are necessarily ambiguous, but then again, a text without punctuation after inlined equations would probably not be ambiguous either. The main purpose of punctuation is after all not to reduce ambiguity but to increase readability and "flow" of the text.

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    ... maybe not-quite-universal... but, yes, the very fact that anyone at all would believe that there was such a consensus is the data point. I don't mean to disparage anyone by expressing doubt that there's "near consensus", and, indeed, there may be, but I do wish to register at least a small dissent, as in my "answer"... – paul garrett Nov 26 '13 at 2:42
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    @paulgarrett "But I have come to the conclusion that that eliminating visual junk from the printed page is more important than punctuatory pedantry, so that when the same formulas is displayed, for example $$t^2+1$$ then it looks silly if the comma is included, like this, $$t^2+1,$$ and everything is much cleaner and less ambiguous without punctuation."--Preface of Galois Theory (third edition) by Ian Stewart. – tqw Nov 26 '13 at 16:01
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    @paulgarrett In effect I also wonder when this "convention" formed. 150 years ago, there seemed no such convention. For instance, see Riemann's 1859 manuscript: claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/… Some of the displayed formulas have punctuation attached, but some don't. Maybe only most English papers today follow this "convention"? – tqw Nov 26 '13 at 16:11
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    @tohecz Yes! See claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/… Notably, the translator does not change the way how Riemann used punctuation for displayed formulas. (The last formula on the first page certainly can have a comma attached.) – tqw Nov 27 '13 at 20:07
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    @tohecz Also see the German version: claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/… – tqw Nov 27 '13 at 20:10
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As a probably-minority opinion, I try to avoid having English-language punctuation juxtaposed to mathematical notation, especially anything complicated, whenever possible. Also, I try to avoid beginning a sentence with mathematical notation, trying, instead, to begin with an obvious English word that is capitalized.

In that vein, I do not put periods or commas at the right edge of displayed purely-formulaic expressions, but have the next line start with a capitalized English word, signifying new-sentence.

My objection to juxtaposition of English-punctuation with formulas is the visual noise, small though it may be. At least my own perception of my own scanning of English+mathematics is that I think of English in a somewhat different manner than I think of the mathematics (apart from small naming-phrases), and everything's easier if the two functions of "comma" and "period" are clearly distinguished.

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    Maybe the customs are field-dependent, and not so universal outside maths: I have checked my own papers (in both physics and chemistry journals), and it's a mixed bag of punctation and no-punctuation… So not all publishers favor punctuation in displayed formulas! – F'x Nov 25 '13 at 15:20
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    I try to punctuate displayed formulas, but I'll take out the punctuation is it's really distracting. But I'm religious about not starting sentences with mathematical notation, so I have lots of sentences that start "The set S..." or "The graph G...". More generally I try to separate any two "phrases" of mathematical notation with at least one English word. (Commas do not mean "and" or "where" or (shudder) "for all"!) – JeffE Nov 25 '13 at 16:56
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    I am amused that there was a down-vote, mostly because I am acquainted with people who have VERY strong feelings about this... apart from house-styles at journals, which have brow-beaten me into adding marks. Naturally, "clarity" is context-dependent, ... but I am entirely unconvinced that "rules" truly enforce "clarity", etc. But, perhaps, the lesson is that the people who believe that displays must be punctuated as though they were English have powerful beliefs, while those who doubt ... are not as aggressive... not burning the crops and villages. :) – paul garrett Nov 26 '13 at 2:10
  • @paulgarrett I'm strongly tempted to downvote this too, because I see this opinion as quite wrong. Yes, there are publishers who consider punctuation in math wrong, but as well, there are publishers who publish journals double-spaces, so I don't think that this is a good measure. – yo' Nov 27 '13 at 17:22
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    @JeffE Two of the basic rules for mixing math and text say: 1) never math at a beginning of a sentence, 2) each two in-line formulas should be seperated by a word. Bad example: Theorem: $H(v)=H(u)$. If $G(v)=G(u)$, $K=0$. Better example: Theorem: We have $H(v)=H(u)$. If moreover $G(v)=G(u)$, then $K=0$. – yo' Nov 27 '13 at 17:29
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Short answer: Yes.

Longer answer: Yes, of course. No matter what is the formatting on the page, the text is linear, with the only exceptions being floating objects (figures and tables), which obviously float. The fact whether the formula is on display or not should have no implication on the punctuation used.

This issue exactly is not much addressed by Knuth in his Mathematical writing. Still, him being a profesionnal typographer, a mathematician and an author of many books, his opinion on this (which is clear if you open any of his works) is IMHO quite valuable, being a strong reference for proper punctuation. The linked article is definitely worth reading.

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The reason for including the punctuation is that text with math in it is still text. The reason for leaving it out is that it looks ugly because we're juxtaposing elements of two writing systems in which symbols have completely different meanings. Either possibility can be jarring to the reader.

A good way to deal with these problems is to leave some white space between the equation and the punctuation.

The Pythagorean theorem,
  A^2+B^2=C^2    ,
has been known since ancient times.

In LaTeX, I use a \qquad for this.

In my personal style, I also sometimes end a sentence with a displayed equation set off by a colon, without a period after the equation.

Thus from Euclid's five postulates we arrive at our final result,
known as the Pythagorean theorem:
  A^2+B^2=C^2

Here I feel that the colon acts like a signal on the tracks that tells the train conductor we're nearing the end of the sentence. The construction of the sentence also reinforces the reader's subconscious expectation that the sentence will not continue after the equation. Grammatically, the equation does not function as any part of speech; the style is similar to what one would use in introducing a diagram that was in-line in the body of the text and had no caption or figure number.

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    Ick. I find the \qquad considerably more distracting than just putting the comma against the formula. – JeffE Nov 25 '13 at 16:57
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    I can see the good intent, but the non-standardness itself (as innocent as it is) may be more distracting than the original-bad-thing. Dunno. – paul garrett Nov 26 '13 at 2:08
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    Downvote. Never . Ever . Put . A . Space . Before . Punctuation . (Unless you are French.) – yo' Nov 27 '13 at 17:10

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