Periods and commas in mathematical writing

I just realized that I am a barbarian when it comes to writing. But I am not entirely sure, so this might be the right place to ask. When typing display-mode formulae do you guys add a period after the formula ends a sentence?

Like:

This is the formula for a circle $$x^2 + y^2 = r^2.$$
Therefore blabla...

or

This is the formula for a circle $$x^2 + y^2 = r^2$$
Therefore blabla...

My supervisor has been complaining a lot that I don't use period and commas in my display-mode formulae. But I get uneasy doing that because it doesn't feel natural to me, I took a look at two books at random and both of them so far do the punctuation in their display formulae.. I know this is stupid of me and its amazing I have never noticed that.

Edit: This would be a fantastic opportunity to see what people actually like as opposed to what they think they like. Everyone who has an opinion on what the punctuation should be should provide an illustrative example of such so that by the voting it can be seen what is actually preferred. If you do this, make your answer just the example (so provide any general homilies in another answer) so that the voting truly reflects the community view of the example.

-
I would object to both variants :) Notice that if you read either one aloud (imagine yourself reading the text to someone on the phone, so that you cannot point to the equation), you do not find a sentence: IMO both should start with «The formula for the circle is». – Mariano Suárez-Alvarez Nov 24 2009 at 13:29
@Mariano: or, since the focus is on punctuation, "This is the formula for a circle:". – Andrew Stacey Nov 24 2009 at 13:34
Would anybody mind if this gets closed due to having aquired enough responses? – quid Sep 5 at 22:23
Similar questions abound: nonnegative or non-negative? – Dan Fox Sep 6 at 7:50

Plain text needs some punctuation. Otherwise it can easily appear confusing. But a displayed formula should be burnt into the brain of the reader as it is. Every additional punctuation is disturbing this aim. I think that the new paragraph starting below the formula is sufficient to mark an interruption if there is any.

-

N. David Mermin wrote an article, What's wrong with these equations, Physics Today, October 1989, p.9, reprinted in his book, Boojums all the Way Through, in which he gave three rules concerning displayed equations in papers.

$\it Rule\ 3$ (Math is Prose Rule). The Math is Prose Rule simply says: $\it End\ A\ Displayed\ Equation\ with\ a\ Punctuation\ Mark$.

Mermin goes on to discuss this at some length.

If anyone is curious, Rule 1 is $\it Number\ All\ Displayed\ Equations$, and Rule 2 is $\it When\ Referring\ to\ an\ Equation\ Identify\ It\ by\ a\ Phrase\ as\ Well\ as\ a\ Number$.

-
A reprinted version of Mermin's essay can be read at cgiss.boisestate.edu/~billc/Writing/equations.pdf . I especially like the anti-simile in his reasoning: "...unlike dog turds on a lawn, the equations you display are embedded in your prose, and constitute an inseparable part of it." – Barry Cipra Sep 5 at 23:37

I disagree with the convention to punctuate formulas. What does it bring to the reader, besides confusion? (one time, I confused a comma and a prime, and I wasted a lot of time).

The reader does not need punctuation after a formula anyway, because usually, he stops to understand it.

Math formulas already harbor a lot of indexes and signs, so adding punctuation does not help.

-

On a technical side, if you want to avoid cluttering your display formulas with punctuation in the LaTeX code, but still be able to show it inside the formula, take a look at this brilliant answer by Alexey on stackoverflow.

You may even later disable the display of that punctuation if you think that it gets in the way (which is my point of view), look at the comments to the answer linked above.

For me this is the perfect solution: I can send a paper with punctuation for publishing, or compile it without punctuation for me, and I can always copy-paste the display formulas without having to remove that pesky punctuation.

-
 This is indeed brilliant. Go Alexey, and thanks Olivier! – Scott Morrison♦ Feb 10 2010 at 17:55

Mathematics is part of a text in the same way that poetry might be part of a literary essay. When citing poetry, a set off (i.e., displayed as a quote) part of a verse almost universally keeps exactly the punctuation from the original and nothing more, with the exception of putting in ellipses to mark elided text within (i.e., not at the beginning or at the end) the excerpt.

Poetry is not exactly analogous to mathematics in this respect, since if punctuation was added, it would not be clear whether the additional punctuation were part of the original, and this is pretty crucial to the metre of the poem. But I think the analogy does show that it does not follow from "A mathematical text is, before everything else, a text." that we should add punctuation to mathematics that is set off.

There's a reasonable issue of taste here that is not settled by dogma. Publisher style nearly always trumps other considerations; if you have the luxury to choose, balance issues of consistency with your closer colleagues with practical issues of layout and your own sense of style.

-

My view on this seems to be contrary to most of the other opinions expressed here. I think periods and commas in display mode are so ugly that they should never be used. Display mode is something removed from text mode, in another dimension as it were, so vestiges of text mode like punctuation should never appear in display mode.

Granted this aesthetic judgment, what should one do instead? For a start, one can choose not to display things that don't really have to be displayed. As I see it, there are only two reasons for displaying something: Either it is too large and unwieldy to put in text mode, or it is a short formula that is so important that one wants to make it stand out on the page by displaying it. The latter situation should only occur rarely, otherwise the author seems to be constantly shouting, like writing half of one's message in ALL CAPS. Historically there may have been typographical reasons for displaying all math longer than a couple characters, but I don't think that's the case any more. As an illustration of this principle of avoiding unnecessary displays, I think the equation for a circle in the original post is something that could easily be put in text mode within the paragraph rather than in display mode. (Unless the equation for a circle was the main new result in your paper that you wanted to highlight, of course!)

So if one only displays things that really have to be displayed, the problem is somewhat ameliorated. For the large displays that remain, one can argue that their sheer size alone provides enough of a separation for the reader that putting punctuation after them is unnecessary. Certainly this is the case for commas in most situations. For periods a solution that might placate the purists is the following: If the sentence does not continue after the display, then warn the reader that this is happening by putting a colon right before the display. This might involve a slight rephrasing to make a colon fit in gracefully. In my own writing I have often ended a sentence with an unpunctuated display, with a clear conscience, but I may try using this colon rule more systematically in the future.

-
Aha, I couldn't have put it better myself. – Jose Capco Nov 25 2009 at 19:11
Agreed. I also don't like to use punctuation at the end of displayed formulas. Indeed LaTeX to me suggests that such a period or comma does not belong. The punctuation is certainly not part of the formula to be displayed and so doesn't belong inside the double dollar signs, and if you place it outside, then it goes to the next line! – Allan Edmonds Nov 26 2009 at 2:13
Upvoted and concur: I do not use commas, semi-colons, or periods in displays. They are noise. I do make sure to begin the next line with a capitalized English word, so that it is clear that a new sentence has begun. – paul garrett Sep 6 at 0:21
@Allan: But Knuth himself punctuates displayed formulas in his books. – Algernon Sep 6 at 6:39
This answer describes my feelings exactly. – Olivier Sep 6 at 12:10
show 1 more comment

Displayed formulas can serve two roles in a math paper: as abbreviations for text that would otherwise be unreadable, and as figures (or illustrations) that are referred to by the text but are not part of it grammatically. My opinion is that in the former case they should be punctuated, but in the latter they should not.

Here are some examples.

1) If $x$ and $y$ are points on a circle of radius $r$ then

$x^2 + y^2 = r^2$.

2) The points of a circle of radius $r$ satisfy the following equation.

$x^2 + y^2 = r^2$

3) The following diagram commutes.

(diagram without any punctuation)

I think these examples demonstrate the necessity of distinguishing the two roles a displayed equation can play. As Simon already pointed out above, there is no reasonable place to put a punctuation mark in a commutative diagram, presumably because a commutative diagram can't be read aloud. On the other hand, it's difficult to view the sentence in the first example as complete without a period at the end of the equation.

I suggest the following rule of thumb: if the formula can be removed from the text without breaking the flow of a sentence, then it does not need to punctuated. Otherwise, it should be punctuated as it would be if the symbols were expanded into words.

Many authors use a colon where I used a period in the second example and follow the equation with a period.

2') The points of a circle of radius $r$ satisfy the following equation:

$x^2 + y^2 = r^2$.

I don't consider this incorrect, but I do consider it a completely different sentence from 2).

-

Mathematics embeds in natural language inheriting the grammatical structure of its 'host'. So mathematical constructions (eg. sets) are nouns, relations (like $\le$, $\in$ and $=$) are verbs, properties are adjectives and quantifications like $\forall x$ are adverbial phrases. The usual rules of grammar of the host natural language, including punctuation rules, now extend naturally.

So, for example, the sentence after this one is a perfectly good self-contained sentence so it should end with a period. $(x+1)^2>x^2+2x$.

-
I can be sure that there would be very few disagreements that it is bad form to start a sentence with a mathematical statement. – alekzander Nov 24 2009 at 23:52
Serre (and many other mathematicians I know) would say that one should not use relations or equations as verbs or something containing a verb. – Jose Capco Nov 25 2009 at 1:18

A math paper should follow all the usual rules of grammar, so in particular there should be subjects and verbs and the sort of punctuation you'd expect to find in a piece of nontechnical writing. I would prefer to write the following:

The formula for a circle is $$x^2+y^2=r^2.$$

If I had to use the wording in the original question, I would write

This is the formula for a circle: $$x^2+y^2=r^2.$$

Occasionally, the aesthetics of the page make punctuation look awkward. For example, one might write:

Therefore, the following diagram commutes:

M×N -> M⊗RN
\    |
\   |
\  |
v v
A

with no punctuation after the diagram. There isn't any sensible location for a period at the end of a sentence, so I'd leave it out.

-
In the latter situation, I'd follow Kingsley Amis's maxim: "Recast your sentence!" If there's no sensible place for the correct punctuation, then you have written a bad sentence. – HW Nov 24 2009 at 16:57
@Henry: Usually that's good advice. But trying to work a commutative diagram, of all things, into the "flow" of a sentence is sort of ridiculous. Simply the amount of time it takes to parse even a simple commutative triangle or square, let alone "read it aloud" if such a thing is possible, indicates that diagrams don't work like ordinary sentence components. – Sam Lichtenstein Nov 24 2009 at 17:04
Commutative diagrams have punctuations in many books I saw, an example is the famous "Categories for the working mathematician" by Mac Lane – Jose Capco Nov 24 2009 at 17:49
@Sam, Sure. Actually, I think even trying to include commutative diagrams in sentences is doomed to failure. So I might have rephrased your example as: "Therefore, the following diagram commutes. <diagram> The diagram has no punctuation." Or something like that. – HW Nov 24 2009 at 20:41
It is very common to make commutative diagrams part of a sentence. Look at page 60 of Categories for the Working Mathematician (Mac Lane) or page 202 of Sheaves in Geometry and Logic (Mac Lane and Moerdijk). They have many other examples of the form "therefore the diagram [diagram (set off)] commutes" and similar situations. You don't have to pronounce math to read it, although I am aware that some people think the opposite. At the board, I would pronounce a complicated formula or diagram as "this" (pointing at it). That is very common. – SixWingedSeraph Nov 24 2009 at 21:13

This is something I've never paid attention to until graduate school, but virtually every book uses the convention that formulae in display mode are part of the text. Every Springer text for instance uses these conventions.

If we define the function $f:\mathbb{R}\rightarrow\mathbb{R}$ by

$$f(x) = e^x,$$

then we can place a comma after the definition to indicate a pause one might take if speaking such a sentence. We could also have defined the function by

$$f(x) = \sin(x).$$

As this last definition was the end of a sentence, it ought to have a period. Finally we could also have

$$|f(x) - f(x_0)| < \varepsilon$$

whenever $|x - x_0| < \delta$. Here, no punctuation was needed.

There are exceptions: Spanier's Algebraic Topology doesn't follow these conventions, but Hardy does, and all modern books that I've read do. Unless I pay attention, I don't even notice the punctuation.

-
Commas don't indicate pauses. – Kevin Lin Nov 24 2009 at 16:07
Sometimes they do. There are times when there can be no argument - a comma is either right or wrong - and there are times when it's a stylistic choice. Here is a story (taken from Lynne Truss's "Eats, shoots and leaves") about a long battle between a New Yorker journalist, Thurber, and his editor, Ross, over the editor's insistence on commas. Thurber was asked by a correspondent: "Why did you have a comma in the sentence, `After dinner, the men went into the living room'?" His response: "This particular comma was Ross's way of giving the men time to push back their chairs and stand up." – Joel Fine Nov 24 2009 at 23:01
I was never taught punctuation in english, but, in french, the comma following "After dinner" is there because you have used an inverted form of the sentence. The simple form of the sentence is "The men went into the living room after dinner." (no comma). Placing the complement at the beginning of the sentence is an inversion requiring a comma. English is rather unpunctuated compared to french (or at least in its usage). All I mean to point out is that in french (at least) most of the punctuation is not a matter of style, but rather a matter of rules. – Rob Harron Nov 24 2009 at 23:24
There are, of course, many differences between punctuation in French and English, some of which I have been slowly learning since moving to Belgium! In English, however, there are certain situations when the decision to place a comma is up to the author or, more likely, the journal's house rules. Incidentally, if you want to learn more about punctuation in English I can recommend Lynne Truss's book. It's short and you'll pick up a lot (although her style becomes a bit annoying after a while). – Joel Fine Nov 24 2009 at 23:42
In any case, (in English at least) a commas is always used to separate the clauses in an "if-then" sentence. – mdeland Jan 13 2010 at 22:08

Whichever rule you follow, the journal you send it to will want the opposite.

-
So the real answer is to define a LaTeX macro which will allow you to add or remove the punctuation according to the journal's house rules. – Andrew Stacey Nov 24 2009 at 13:33
The journal's copy-editors will put it into house style. There's generally no need to try to approximate it yourself. It's just annoying when they take things like punctuation out after you fight with your coauthors to put it in... – Andy Putman Nov 24 2009 at 13:46
I have never seen that, actually. I did once publish in a UK journal, and they changed my decimal points to commas. – Gerald Edgar Nov 25 2009 at 18:01
I had a journal insert commas at every displayed equation that was not at the end of a sentence. Surely that is a bad rule! – Richard Kent Nov 25 2009 at 18:31
@Gerald maybe they compiled the TeX with a language package that did that. – Jose Capco Nov 26 2009 at 18:22

My meta-guide with respect to that is

Tautology 2.3.1 — A mathematical text is, before everything else, a text.

from Michèle Audin's Conseils aux auteurs de textes mathématiques, which you can get from her webpage.

A corollary is that when one writes a mathematical text one is writing sentences, to which all rules which apply to sentences of course apply. And, say, sentences end in a period.

-
-1: Good quote, bad argument. There are plenty of cases where it is justified to end full sentences without punctuation. – Charles Stewart Jan 14 2010 at 13:28
Charles, I certainly agree that there are sentences that end without periods. (Can you find one?) However, I am hard pressed to think of a case where we don't need any punctuation. Could you give an example? – L Spice Feb 25 2010 at 16:57
@L Spice, I can think of at least one good reason for not ending this full sentence with any punctuation – Barry Cipra Apr 29 at 19:15

The problem is compounded by the following facts:

• There are different style sheets for different disciplines and even subdisciplines that use mathematics. For example, do you put a punctuation mark that happens to follow a quoted item inside or outside the quotation marks? Once upon a time, it was a universal rule, so far as I know, to put the punctuation inside, but computer folk began to depart from this convention due to the importance of exact quotation in formal languages and full-text searching.

• Different implementations of TeX and its kin, by design or oversight, force different choices with respect to: $\operatorname{Blah}, \operatorname{Blah},$ on the one hand, and $\operatorname{Blah}, \operatorname{Blah}$, on the other hand, at least, if you want to avoid the risk of having a punctuation badly split to the next line every now and then.

-
For your first point, there is a difference between British and American style (en.wikipedia.org/wiki/…). – Harald Hanche-Olsen Nov 24 2009 at 16:42
Sure, even within the U.S. there are variations between journal and journalistic style with respect to every jot and tilde down to serial commas. Here's a fun blog to read if you like that sort of thing — johnemcintyre.blogspot.com – Jon Awbrey Nov 24 2009 at 21:25
"Every jot and tilde" ... I like that variant, especially for mathematical text. – Gerald Edgar Jan 12 2010 at 14:51
A pleasant aspect of English is that there are not so many clear rules. Although from the US, I long ago started punctuating quotations in what is apparently the British style, because to me it seemed more sensible. When I learned Spanish, I discovered that one does not have this flexibility in Spanish. There is a correct way to write and spell, ordained by some official academy. Something that muddies mathematical writing is that the writers are mostly not native English speakers, and bring to their writing notions typical of their own native languages, but quality copy-editing is not common. – Dan Fox Sep 6 at 7:47
"Tilde" does come from "Medieval Latin titulus tittle" according to m-w.com; but this "jot" is not the usual "to write briefly or hurriedly" but a variant of "iota". So, "every $\iota$ and tilde"? – Noam D. Elkies Apr 29 at 20:27

The rule is: there is no rule.

Punctuation is intended to add clarity to a body of text. If punctuation after a formula does so, put it in. If it does not, leave it out. My default is to put it in on the principle that the mathematics expressions are part of the text and therefore subject to its rules. However, this can conflict with comprehension particularly where the punctuation can be mistaken for a part of the formula.

As a guideline, I would say that for short formulae, put it in (for example, in your example in your question) since the reader can read those quickly enough that they don't lose the thread of the text. However, the average reader cannot parse larger formulae so quickly and so will effectively stop reading in order to understand the mathematics, then start reading again afterwards. Thus the formula itself acts as a sort of punctuation mark and so does not need any further adornment.

Of course, there are always grey areas (even gray ones) and that's where you'll find the most vociferous eraser fights. But the zeroth law (or, if you prefer, Rule 42) is: the one that makes it clearest is the right choice.

-
I'm wondering in what situations the punctuation could be mistaken for part of the formula. The only one I can think of is mistaking the factorial for an exclamation point at the end of a sentence. I'll admit I don't write factorials at the end of sentences for this reason. – Michael Lugo Nov 24 2009 at 13:55
Maybe I phrased it badly. It's not that the punctuation could reasonably be mistaken for part of the formula in that the formula would make sense if the punctuation were part of it, but rather that one has to do a double-take to realise that the punctuation is not part of the formula. If you've switched to "math mode" when parsing, then the punctuation doesn't parse and so jars, and so should not be there. If you haven't switched to "math mode" then the punctuation parses properly and isn't conciously noticed. – Andrew Stacey Nov 24 2009 at 14:43
A sentence without a full stop always jars. – HW Nov 24 2009 at 16:53
My point is that you've forgotten that it was a sentence because you've had to take so long to figure out the mathematics. Therefore it jars to be taken back into the flow before you are ready for it. – Andrew Stacey Nov 24 2009 at 18:56
And my point is that if I'm reading a sentence and I don't eventually encounter a full stop then I get confused, regardless of the mathematics. If the mathematics is so complicated that it interrupts the flow of your sentence, then you shouldn't include it in the sentence. – HW Nov 24 2009 at 20:44