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:
- I've been told that the use of ellipsis in "$S = x_1 + x_2 + x_3 + x_4 + \dots$" is ambiguous and meaningless. Is it?
- More rigorous notation than "ellipsis" for "and so forth?"
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.
Best Answer
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.