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The ellipsis "$...$" sometimes seems to be seen as a bit informal. Its use is often justified for cases "where the intention or meaning is clear". And of course, one can go arbitrarily far with nitpicking, intentionally misunderstanding a notation, or suggesting ambiguities. In many cases, the intention really is clear. But for me, it still looks like the writer was handwavingly saying: "Yeah, and so on, y'know what I mean". Usually, the ellipsis can easily be replaced with a more rigorous notation - often involving some sort of indexing over $\mathbb{N}$. And I wonder why this is often not done.

So my question is:

How acceptable is an ellipsis "$...$" in formal mathematics?

Of course, this does not refer to textbooks where the natural numbers are introduced as "$\{1,2,3,4,...\}$". It rather refers to mathematical research, or as a specific example: A paper about a proof where the correctness of the proof crucially depends on the right interpretation of an ellipsis, even if it is only used in a basic definition of something "trivial and obvious" like "$a_1 + ... + a_n$".

How far should one go with trying to avoid the use of the ellipsis, in order to not be confronted with the possible ambiguities or lack of rigour?

I found two questions that are related to this one:

They refer to a particular use of ellipsis "$...$", and how to replace it with a more rigorous notation. Further search reveals attempts to formalize the ellipsis - for example, Proofs About Lists Using Ellipsis (A. Bundy, J. Richardson) states

A notation often used in informal mathematical proofs is ellipsis (the dots in $a_1 + ... + a_n$)

...

The first problem in formalising ellipsis is its inherent ambiguity. The reader of a formula containing ellipsis has to induce a pattern from the expressions on either side of the dots. [...] One can try todisambiguate ellipsis by putting in more context [...] but some ambiguity will always remain. More importantly, it is hard to see how we can ensure that a “proof” is in fact a proof unless it can be expressed in an unambiguous internal representation

But this refers to a very specific context, and not to how acceptable the ellipsis is in proofs and definitions in general.

Xander Henderson
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Marco13
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    I think that this is a good question for an editor or reviewer, but that it is not really a good fit for this site. – Xander Henderson Sep 30 '19 at 13:07
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    I always find this notation when the intention of the writing is mostly educational. Then the writer may sacrifice some rigor to make things easier to understand. – nicomezi Sep 30 '19 at 13:08
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    To answer your question, I think that ellipsis are often just fine. When you type them in LaTeX, however, you should use \ldots, rather than .... In the context of a sum, you should use \cdots to produce $a_1+\cdots+a_n$. – Mark McClure Sep 30 '19 at 13:10
  • @XanderHenderson There are several math-related sites (mathoverflow, math, matheducators...) but wonder where it could fit better...? – Marco13 Sep 30 '19 at 13:22
  • @nicomezi Of course, it wouldn't make sense, for example, to replace $a_1 + \cdots +a_n$ with a $\Sigma$ in an introductory textbook where the meaning of the $\Sigma$ is supposed to be explained with the ellipsis as an example. The question specifically aims at things like proofs in scientific papers. – Marco13 Sep 30 '19 at 13:27
  • One can think the ellipsis to be an IQ test for those who will continue reading the text opposed by those who quit. – A.Γ. Sep 30 '19 at 13:29
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    in my experience, the 3 dots are fine if the the object is reached in finite number of steps, e.g by indexing $a_i, i=1,\ldots, n$. I personally think it's bad style to use the 3 dots in infinite expressions a la $$e^x = 1+\frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots $$ – AlvinL Sep 30 '19 at 13:36
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    @AlvinLepik I don't see why it should make a difference whether, for example, a sum (that is supposed to be represented with the ellipsis) runs up to $n \in \mathbb{N}$ or up to $\infty$, but maybe that's subjective to some extent. – Marco13 Sep 30 '19 at 13:44
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    @Marco13 Because the meaning of the symbol $\sum_{n=0}^\infty$ is not the same of $+$. – AlvinL Sep 30 '19 at 13:48
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    On the off-chance that @postmortes sees this: I thought your answer was fine and there was no need to delete it. If you had edited in a sentence to say that the important bit was thinking about the red flag (rather than always removing ellipses) then it would have been fine [that is, clarify the point which lead to my comment there]. – user1729 Sep 30 '19 at 13:48
  • @AlvinLepik Sorry, I'm not a "true" mathematician, but couldn't the difference be ironed out with a $\lim_{n\to\infty}$ at the right place? The point is: I think for both cases there are variants of writing it without the ellipsis. Which one is deemed acceptable in which context is part of the question. (BTW: user1729 and postmortes: I also didn't understand the deletion, but that's up to the answerer...) – Marco13 Sep 30 '19 at 13:58
  • @user1729 thank-you, I think that's a very fair statement. But it's still true that it doesn't help the OP, so I hope someone else can write them an answer that they'll be happy with :) – postmortes Sep 30 '19 at 14:00
  • @postmortes I thought it would help the OP, as their comment to your answer reads "But the main point is how critical the "red flag" is that you mentioned." I am basically suggesting that you address this. [The comments to the answer would be an appropriate place to discuss the specific example brought up in the OP's comment, and I don't think it necessarily needs to be addressed in the answer itself.] – user1729 Sep 30 '19 at 14:09
  • @user1729 well, I'd be happier if there were at least one alternate answer to this question, but refusing now would be ill-mannered of me. I've edited the post to focus on exactly that, and I sincerely hope that someone else will provide an answer better than mine. – postmortes Sep 30 '19 at 14:27
  • @A.Γ. I can’t tell if you’re being serious or kidding – gen-ℤ ready to perish Sep 30 '19 at 14:35
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    @MarkMcClure The amsmath package (or one of the related packages) actually has \dotsc (for dots between commas), \dotb for dots between binary operators, and so on. See, for example, this post on TeX.se. ;) – Xander Henderson Sep 30 '19 at 15:13
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    @Marco13 As I said in my first comment, I think that this is a good question for an editor or reviewer. If you are writing something which will be neither edited nor reviewed, then do what feels right to you. ;) – Xander Henderson Sep 30 '19 at 15:14
  • @gen-zreadytoperish This kind of questions normally means that I have already been way beyond the acceptable level of MSE, so I better stop. – A.Γ. Sep 30 '19 at 16:25
  • @AlvinLepik This usage is very standard and is not ambiguous. – Jair Taylor Oct 01 '19 at 20:29
  • This is mathematical notation. It's not nitpicking, it's appropriate rigour. And it's not nice to accuse readers of intentionally misunderstanding; their misunderstanding is due to ambiguity in the text which should have been expressed clearly. I once had an exam question with an expression like $$\frac1m+\cdots+\frac1{mn}.$$ – Rosie F Oct 02 '19 at 06:17
  • @RosieF The degree to which there is potential for misunderstanding is certainly subjective and debatable. An example like 1/n + ... + m/1 also came to my mind (i.e. the case when there are two variables, and they may even run into opposite directions). But in any case, as it has been sorted out in the answers+comments, in critical contexts (important proofs or exams), such ambiguities can be sorted out by showing the ellipsis-free form alongside the one where the ellipsis helps to convey something like a pattern that can be observed in the terms. – Marco13 Oct 02 '19 at 12:03

3 Answers3

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Discussing whether ellipsis are inherently good or bad is not that productive - that's a decision made in reference to particular writing and particular purpose. It is better to recognize that written mathematics is meant to communicate both rigor and intent and to understand the way in which elements like ellipses serve that purpose.

Note that ellipses introduce elements into the text that sums do not:

  • They explicitly substitute for the initial (and, if finite, terminal) segments of a sequence, which is useful if you want to make a point about those terms or if those values help clarify that bounds of the sum are sensible.

  • They show the ordering of the terms. This is useful if you want to make an argument involving adjacent terms cancelling - and if you're in a non-commutative setting, this is often less ambiguous than a symbol like $\prod$.

  • They create some space on the page for each term. This is fantastic when you're dealing with something like generating functions where you might need pointwise operations on the coefficients of multiple series, because you can lay out the coefficients of multiple functions in a grid and can also integrate worked examples of small cases with general calculation by using notation such as $$1+2x+3x^2+\cdots+(n+1)x^n+\cdots.$$

There are also tangential benefits that depend on the audience and purpose - for instance, if you're trying to express a formal argument to an audience without so much mathematical maturity, ellipses can be a nice way to do that. Of course, ellipses also don't do some things that you might want them to:

  • Ellipses don't always pin down what the summands are. If the pattern is just "counting, with a function applied unevaluated" - that is an expression like $f(1)+f(2)+f(3)+\cdots+f(n)$ - it's probably safe, but one has to be careful not to frustrate readers. Of course, you can always include a general term to clarify or explicitly state your intention in the text preceding the equation (and, hopefully, the equation should not come from nowhere! If it does, you haven't written enough words to introduce it!)

  • Ellipses do not indicate the indices of the sum. This can be relevant if you need to split up the sum in some way, as often happens in analysis - there's no good way to say "here's the set of big terms, and here's the set of small terms, let's look at them separately."

  • Ellipses cannot represent sums without order. If you're summing over the set of partitions of some set, you'd better use summation notation.

There are surely more subtle things, but these are the most striking aspects of the notation that come to mind that would most often persuade me to use ellipses or to avoid them - and there are definitely situations where a creative use of notation can contradict what I wrote here and situations where it doesn't really matter what notation you choose.

Milo Brandt
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  • These are well thought-through points, thank you! Particularly the "terms cancelling" and "what are the indexes (sic)" points are clear cons and pros for one notion or the other. But when using an ellipsis for the reasons that you mentioned, could or should it not also be written with a formal notation? Roughly as in "We have these terms [some summation] which take this form [terms with ellipsis] where terms cancel". Basically using the ellipsis "only" for the illustrative purposes, but still unambiguously defining what the ellipsis means there, exactly, using the "more formal" form? – Marco13 Sep 30 '19 at 18:40
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    @Marco13 I've sometimes seen people write things like $$\sum_{i=1}^n(2i-1)=1+3+\ldots+(2n-1).$$ The function of this is usually at the start of a series of simplifications on the right-hand-side alone (probably aligned as a multi-line equation) - it's a sort of transition step which does nothing formally, but guides the reader. So yeah, you can have both notations around - but, as with anything in your writing, you need a productive reason for doing so. I wouldn't write a sum just because I felt an ellipsis was ambiguous - I'd work on making the ellipsis less ambiguous with text or remove it. – Milo Brandt Sep 30 '19 at 22:26
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    Very good answer! But can you please change $\ldots$ (\ldots) to $\cdots$ (\cdots) in your examples? – Matthew Leingang Sep 30 '19 at 23:54
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    @MatthewLeingang Done. – Milo Brandt Oct 01 '19 at 00:23
  • Great answer. I would add a paragraph about ellipsis used in infinite series, which was addressed in the question in one of the examples. – Rad80 Oct 01 '19 at 07:23
  • After you laid out the pros and cons of both forms so nicely, I think this is the most acceptable answer. Subjectivity or the (lack of) "good will for the right interpretation" aside: Both forms have their justification, and the ellipsis can have undeniable benefits for guiding the reader. And, as I realized later, and as you confirmed in response to my comment: In doubt, it is still possible to show the transition step to the more "rigurous" form, if someone expects it, or when it could help to resolve ambiguities. – Marco13 Oct 01 '19 at 13:02
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    Fantastic! You vaguely allude to 2-dimensional sums that can be both pleasing to the eye and easy on the head when written using ellipses, whereas the nice pattern vanishes when we use summations, which is unfortunate but necessary for true rigour. For an example right here on Math SE, consider this post, and see how illustrative it is to use actual terms and dots for the examples, even if I didn't use them in any of the formal statements. =) – user21820 Oct 02 '19 at 14:42
  • There is a less ambiguous way: use recursion.

    $S=-1$. $-S=1$. $(1-2)S=1$. $S-2S=1$. $S=1+2S$. $-1=1+2(-1)=1+2+4(-1)=1+2+4+8(-1)$. $S=1+2+4+8+\cdots+2^{n}+2^{n+1}S$. $S=Sum(S)+Tail(S)$. $S-Tail(S)=Sum(S)=(-1)(1-2^{n+1})=2^{n+1}-1$.

    Notice that all I did was expand a recursive pattern starting from a chosen number. You can use it to expand $-\frac{1}{12}$ as sum of powers of 13. There's no notion of infinity here at all; other than letting you expand $n$ times. It even solves for the sum of $n$ terms. Recursion makes patterns. Keep the recursive tail, or it is ambiguous.

     – Rob Sep 16 '23 at 00:31
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I have two points. Firstly, this question is inherently subjective. Different people have different opinions. You should be aware of this, and form your own opinions about what you do and do not care about. You should also be aware that your opinions may annoy other people no matter what you do. For example, if you use an ellipsis you may annoy someone but another person may look favourable on your choice, while not using one may produce the polar opposite effect.

Secondly, I do not believe that research papers are the correct place for "formal mathematics". A paper is a means of communication, and therefore should be readable. I believe that it is more helpful that the reader should be able to reconstruct the formal mathematics from the arguments presented. The skill in writing good mathematics is to write something readable, and where this "reconstruction" is easy. Therefore, ellipses have a place in research papers.

(Note that the above paragraph has lots of caveats missing. Hopefully you can reconstruct my intended point from it though...)

user1729
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  • The point about subjectivity is true (subjectively), but I thought that it finds it limits at the core of formal, mathematical rigour, where subtle differences in the interpretation of a notation make the difference between "correct" and "wrong". Of course, one can argue: "Interpreted this way, the proof is correct, and that way, the proof is wrong - so let's assume the author meant this". I thought that (unless it's in some educational context or so), going the extra mile of a "..."-free notation could be appreciated or even expected, to minimize subjectivity and ambiguities. – Marco13 Sep 30 '19 at 18:34
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    @Marco13 You do want to minimize ambiguity, but you must maximize the readers' ability to understand. A correct machine checkable proof would convey little meaning. Mathematicians always assume that an informed reader can reconstruct any necessary formality given a well written human readable exposition. – Ethan Bolker Oct 01 '19 at 00:30
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    See how Gauss used ellipses here http://www.archive.org/stream/abhandlungenmet00gausrich#page/n23/mode/2up or the use of [etc.] here http://www.archive.org/stream/abhandlungenmet00gausrich#page/n39/mode/2up. – Michael Hoppe Oct 01 '19 at 19:23
  • @MichaelHoppe I am not convinced that it is healthy to look at old papers for advice on writing style. Mathematics, and in particular how we communicate it, has changed greatly since the time of Gauss. Moreover, being a great mathematician does not make one a great writer of mathematics! – user1729 Oct 01 '19 at 20:02
  • @MichaelHoppe It's clear that ellipses are used frequently, and even though the work by Gauß is not an "introductory textbook", it certainly focuses on conveying an idea. As it has been sorted out in the comments and answers here: That's certainly a case where they are appropriate. The question was supposed to cover that, but with a tilt towards things like e.g. proofs in high-ranked papers, or maybe task descriptions in exams, where the stakes are high, and I thought that people might expect or demand an unambiguous, rigorous notation. – Marco13 Oct 01 '19 at 21:57
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The following ten aspects are relevant concerning the question:

1. If there is (almost) no ambiguity for an informed reader and if it increases the readability, there is no problem with using an ellipsis.

...

10. If you can replace an ellipsis by a more rigorous notation without increasing the complexity / decreasing the readability, then you should do so.

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