Should one use "above" and "below" in mathematical writing?

Question

I started thinking about this question because of this discussion:

http://sbseminar.wordpress.com/2010/08/10/negative-value-added-by-journals/

about how journals often change a paper (for the worse) after acceptance.

Here's my question: Since it doesn't make sense to number every single equation (especially if you will only refer to it once in the text), I often use the phrase "the equation above" or "the equation below" to refer to the previous or next displayed equation. I am well aware that after compiling, said equation may be on the previous or next page, respectively, and not strictly above or below. I thought this was common practice. One journal publisher changed every single one of my "the above equation" phrases to "the earlier equation". I didn't bother protesting, but in my mind it definitely made the text worse.

Do other people avoid using "above" and "below"?

Answer 1

Since it doesn't make sense to number every single equation...

I used to number only equations I referred to, but then someone pointed out the following. When you write a paper and make it public, you are de facto allowing other people to talk about your paper and simply because you in your paper see no reason to refer to some equation does not mean that other people reading your paper will not do so. Hence as a friendly gesture to your readers, you should number all your equations, and in this way allow them to refer to, say, equation (n) in your paper, instead of having to come up with some complicated reference. Since that time I have tried to number all my equations.

Answer 2

No. See above.

Answer 3

First of all, the question is a small matter that only a journal editor should take seriously. There are much deeper reasons that a paper might be well written or poorly written. Even among the concerns at this level, it is more important to use concise, regular, correct grammar than to follow particular journal conventions.

That said, in my opinion it's okay to say "above" and "below" to refer either to an equation, or more often to a paragraph of the paper. E.g., you might say "By the above discussion...". But it may be clearer to do something else for equations. If you just say "above", it may not be clear if it means immediately above (or previous) or something else; but if you say "immediately above", that's a bit strained. My rule is to say "this" equation if it is immediately previous, and otherwise to number the equation even if I only explicitly cite it once. If the equation is not explicitly cited, but is just part of the narrative, then like you I wouldn't number it. But there are authors who number every equation.

For paragraphs, I sometimes say "above" and "below", if the cited paragraph is in the same section and subsection.

Answer 4

I think, for the reasons Jose points out, that you should err on the side of numbering more. In particular, an equation which you refer to more than a line or two after it appears should probably be numbered. I've also found this makes editing easier: If you insert a new paragraph, with several equations in it, you don't need to look for all of the references you've broken.

I agree with you, though, that it never would have occurred to me that "above" couldn't refer to a line on a previous page. Perhaps this is a difference between those of us who read on screens, with our scroll bars, and those who page through print outs?

Answer 5

Of course, different areas tend to be more or less equations-heavy, but an author's individual style matters too. Two people could write up the exact same result with dramatically different number of equations. So if you observe that your papers tend to be dense with equations, it makes sense to me to try to differentiate by numbering only the ones that will be referred to regularly (my previous comment notwithstanding, because I shy away from using too many equations).

That being said, I would avoid the "above" and "below" as not artful. "Previous" is a good word, or making use of \pageref would allow you to be more precise. It's difficult to be more precise without a concrete example, but in most cases you should be able to refer to the equation by describing it. E.g. "in the long exact sequence we just established", "the previous norm inequality implies that", ...

Answer 6

Like Greg Kuperberg, I think of this as a small stylistic point compared to many others. For me it's more important to choose numbering of sections, subsections, equations, definitions, and such to maximize the reader's ease of understanding what is going on. Some built-in LaTeX styles used in books and journal articles work against readability. Typical is a reference to Proposition 7 or 3.7, which may be hard to go back and find quickly (especially if Proposition 3.7 occurs in Section 3.4). Similarly, referring back to equation (38) may send the reader on a lengthy hunt through the previous 40 pages of a long article. Also of dubious help to the reader is consecutive default numbering of the sort: Definition 1, Theorem 2, Remark 3, Lemma 4, Definition 5, etc.

Answer 7

Practically, numbering equations works, sequenced in the article from beginning to end. While it is a courtesy to readers and arbiters, later, who discuss lines that the author did not, it most certainly adds specificity, without tortuous sentence structure or word choice. "Equation 24" will always be just that, regardless of its placement on the page, and regardless of reprint formulation or quality.

As an editor, my choice is to number all equations, and bold those that are referred to by the text of the article.

Equation identification, when there is textual sectioning, by chapter or other means, can include the section as predicate: e.g.: 5-24 refers to equation 24 in chapter 5. Texts that use this system (or any system) wisely place an explanation at the forward.

Obviously, longer papers/articles, or complex sectioning, both complicate the system for the reader, the ultimate consumer for the publication (e.g.: 5.3-24). Personally I would avoid it; I can't imagine the necessity of referring to large counts of equations in other chapters or sections, and in those rare cases, for a reference such as "see equation 5-24", I would consider adding a postscript: "in Chapter 5.3" or "at page 137".

Answer 8

I think it is a matter of style whether to keep numbered formulas to a minimum or to a maximum, or to opt to something in between. I usually try to have as few, as possible. "The above equation" seems to me like a slight abuse of both grammar and common sense: one usually has lots of equations above the point of reference! I may occasionally write this myself, but normally I prefer something like "the last equation".

Answer 9

Every equation ought to be numbered in print publications or fixed-format electronic publications; if an equation is not important enough to be included as a numbered equation in the article, it ought not be included at all. As for "above" and "below", I've learned them contextually as meaning "prior" and "later" in the current article. I've never understood it to mean exactly one equation above or one equation down. In fact, I've even seen absolute and relative references together, as in "see equation 12 above." If an equation reference goes too far forwards or backwards, it makes sense to repeat the equation renumbered with a new number in this location. It's much easier to look at it on the same page rather than have to flip back and forth.

Relative references make some sense for fixed publication media such as printed copies of journals. Absolute references, such as pointers and index numbers and URLs, make more sense for variable view-model media such as electronic publications (HTML particularly).

I agree with the other answers (above, and below) for mathematical writings for print publications such as journals. However, there is an extra consideration for electronically published items in electronic journals or particularly in forums like this web-site, Mathoverflow.

Users have the option of controlling their viewing model on electronic publication systems and changing what appears at the "top" of their electronic page. They may choose chronological order in order to view comments in the same order they were submitted, allowing ease in understanding the flow of commentary. They may choose reverse chronological order, for example when they are revisiting a question just to see what the latest entries have been. They may also choose to order the results by popularity or relevance (with popularity of votes being an electronic self-selected polling of relevance by other readers).

On forums like Mathoverflow, references to other writer's contributions as "the answer above" or the "answer below" are rendered meaningless and confusing by the fact that the physical ordering of the answers is different for different readers and at different times. Reader preference can re-order the answers according to time submitted (oldest first, newest first) or by popularity (votes thus far); the popularity is evanescent as the number of votes will also change over time.

Certain options lead to difficulty in following threads. For example, comments tend to be initally shown in descending order of votes, destroying the temporal ordering of conversations or the ordering of comments spread out amongst multiple entry boxes. I would have expected that long comments spread out amongst multiple boxes would be discouraged; but they seem to be rather prevalent among mathoverflow. I find that I always have to click on the "show additional comments" button in order to be able to follow the unfolding of the comments and understand the conversation in the commentary.

Answer 10

I would use the following equation in a sentence immediately preceding the equation. I would use the equation below if there were more text in between.

I would tend to avoid these formulation to be frank, I would re-organise the text to avoid forward references to an equation in general, unless I speak about a celebrated equation. For example The law or large numbers below has first appeared in... , Einstein's equations below describe.... Compare with The following equation is the strong law of large numbers: [Displayed Eq.].

If I refer to an equation above, I number it and refer to the number: In Eq. (xx), $a$ is the acceleration, ...