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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.

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    $\begingroup$ 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. $\endgroup$ – Xander Henderson♦ Sep 30 '19 at 13:07
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    $\begingroup$ 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. $\endgroup$ – nicomezi Sep 30 '19 at 13:08
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    $\begingroup$ 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 $a_1+\cdots+a_n$. $\endgroup$ – Mark McClure Sep 30 '19 at 13:10
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    $\begingroup$ 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 $$e^x = 1+\frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots $$ $\endgroup$ – Alvin Lepik Sep 30 '19 at 13:36
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    $\begingroup$ @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. $\endgroup$ – Marco13 Sep 30 '19 at 13:44
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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.

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  • $\begingroup$ 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? $\endgroup$ – Marco13 Sep 30 '19 at 18:40
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    $\begingroup$ @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. $\endgroup$ – Milo Brandt Sep 30 '19 at 22:26
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    $\begingroup$ Very good answer! But can you please change $\ldots$ (\ldots) to $\cdots$ (\cdots) in your examples? $\endgroup$ – Matthew Leingang Sep 30 '19 at 23:54
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    $\begingroup$ @MatthewLeingang Done. $\endgroup$ – Milo Brandt Oct 1 '19 at 0:23
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    $\begingroup$ 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. =) $\endgroup$ – user21820 Oct 2 '19 at 14:42
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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...)

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  • $\begingroup$ 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. $\endgroup$ – Marco13 Sep 30 '19 at 18:34
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    $\begingroup$ @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. $\endgroup$ – Ethan Bolker Oct 1 '19 at 0:30
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    $\begingroup$ See how Gauss used ellipses here archive.org/stream/abhandlungenmet00gausrich#page/n23/mode/2up or the use of [etc.] here archive.org/stream/abhandlungenmet00gausrich#page/n39/mode/2up. $\endgroup$ – Michael Hoppe Oct 1 '19 at 19:23
  • $\begingroup$ @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! $\endgroup$ – user1729 Oct 1 '19 at 20:02
  • $\begingroup$ @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. $\endgroup$ – Marco13 Oct 1 '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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    $\begingroup$ Although I like the humor, I hesitate to upvote this one (but no downvote either :-)) $\endgroup$ – Marco13 Oct 1 '19 at 21:46

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