# A suggestion for mathematical English

I constantly find myself writing phrases like “the number of pairs $(i,j)$ with $i>j$, $i \in I$ and $j \in [n] \setminus I$“. And then recasting them, because of the rule of mathematical typesetting that you should not have two formulas separated by a mere comma.

I used to think that the typesetting rule was a useless nuisance, but I have been persuaded by experience that it is too hard on human eyes to see that the comma is not internal to the formula. Tonight, I realized there is another alternative. What if we all collectively decided that it was grammatical to write “the number of pairs $(i,j)$ with $i>j$ and $i \in I$ and $j \in [n] \setminus I$“? Not only would this solve the typesetting problem, but it would be helpful to be reminded of which logical connector we were using throughout the list, rather than just at the end.

Is there a downside to this construction which I am missing?

## 17 thoughts on “A suggestion for mathematical English”

1. An alternative (more conventional, but maybe worse in other ways) is to write
“with $i > j$, such that $i \in I$ and $j \in [n]\setminus I$”.

2. Marcel says:

what about writing something like “$(i,j)\in I \times [n]\setminus I$ with $i>j$”? I guess it is maybe worse to read?! I think one could also talk about “strictly ordered pairs of numbers”?!

3. DC says:

For what it’s worth, I’ve never thought that it wasn’t grammatical to write that, and never realized that the consensus might be that it isn’t.

I’m not sure I’ve used this kind of formulation in my own writing (if I have, I haven’t noticed, and nobody has pointed it out to me), but when I see it in papers, it rarely strikes me as ungrammatical. I certainly prefer it to symbolism that I have to interpret, or English-language rephrasing that seems to have been chosen for the sole purpose of reducing the number of ands.

In my mind there is a kind of distinction between “mathematical and” and “English and”, with mathematical grammar taking slight precedence over English grammar when the two are in conflict. Within reason, anyway. (To me, it is a little like how in non-mathematical English, generally it is bad style to use the same words over and over— but good mathematical prose very frequently makes heavy use of a rather small vocabulary. The rules are just different.)

4. try inserting small descriptive terms: “the number of pairs of positive integers $(i,j)$ for which the index $j<i,$ the number $i\in I$ — the index set — and the index j is in the complement: $j\in [n]\setminus I$." My solution is suboptimal at the moment, but you should remember that your reader *does not* remember all of your notational conventions and English words help identify them.

5. Anonymous says:

I do this all the time and don’t think there is anything wrong with it in either mathematical or ordinary english grammar.

6. I tend to just separate things by commas. I agree that the small separation between formulas is undesirable, but usually the alternate (incorrect) ways of parsing the symbols are nonsensical, so there’s no real ambiguity. Ordinary English is like this too. If you look at just a fragment of a sentence there are often multiple ways to interpret it, but usually only one of those ways makes sense in the larger context.

I think the only downside of using multiple “and”s is that it reads a little funny. To me, this disadvantage is comparable to the disadvantage of commas.

Another alternative is to replace “such that A, B and C” with “such that A, and also B and C”.

7. I think it’s almost always better to put these things on separate lines:

The number of pairs (i,j) with:

(a) i>j
(b) i \element I
and (c) j \element [n]\I

8. The trouble, both with the comma splice of formulae and with the “and… and” construction is that they don’t read well as English. Each creates a snag for the reader.

In the case of the comma splice, the snag comes when the reader realizes that a change of formula has just happened without enough of a signpost. In the case of the “and… and,” the snag occurs later, as the mind processes “too many” conjunctions (indeed, the associativity of “and” makes the reading more difficult, so that in standard English one almost always specifies the order of evaluation).

The grammatical name for either is a run-on sentence. The formula $i>j$ is an independent clause. So is $i \in I$, and so forth. In standard English, we do not splice independent clauses together with a mere comma, and the same rule applies in mathematical English. One should, as a rule, depart from the standard grammar of the ambient language only for really good reason.

9. I’ve also struggled with this a lot, and I eventually gave in and started splicing together things with “and”‘s. However, if you’re going to have more than 3 conditions or so then I think it is good to use LaTeX “itemize” to separate them into separate lines.

10. I’d add my vote against the awkward “X and Y and Z” construction, though there is no ideal solution. Note that LaTeX does have commands which add various amounts of space, such as \: or \quad, to avoid crowding of symbols.
Old-fashioned typesetting was worse in many ways, breaking mathematical expressions in arbitrary ways at the end of a line, etc. Even with LaTeX you have to intervene. For example, I found once that I had ended a sentence with the mathematical abbreviation St (in roman font) for the Steinberg module, but started the next sentence with “Lusztig….” The effect was instant sainthood, as pointed out by a reader.

11. Hi. I often encounter similar problems, especially since I’m not a native speaker, so I couldn’t resist commenting even though I don’t know the context.

On the one hand, as Wesley Calvert and Andy P. mentioned: lists are always an option. When in doubt, itemize.

On the other, with this special problem you could just use plain English to help you out. For pairs (i,j) with i n is clearly nonsense (from context), $i \in I, j \notin I$ could be much easier for the reader’s flow.

12. Sorry, I’m reposting the parts that were cut off (in a strange fashion).

On the other, with this special problem you could just use plain English to help you out. For pairs (i,j) with i n is clearly nonsense (from context), “i \in I, j \notin I” could be much easier for the reader’s flow.

13. Ok, before you think I’m trolling around, let me try one last time to post the first part correctly… Let me know if you care for the rest ;)

On the other, with this special problem you could just use plain English to help you out. For pairs (i,j) with i strictly less than j why not simply write ordered pairs? This is most likely intuitive and can be mentioned the first time it occurs.

14. Because that’s not what “ordered pair” means. Saying that (i,j) is an ordered pair means that we remember the order of (i,j). So $(i,j)$ and $(j,i)$ are two different ordered pairs, but $\{ i,j \}$ and $\{ j,i \}$ are the same set.

As for enumeration, that is an excellent solution a lot of the time, but it makes the paper extremely long if you break out every example of this form in an eumerated environment. Possibly, the difficulty here is that I don’t like symbols enough. I find it difficult to read a paper which is filled with expressions like $\# \{(i,j): \ i \in I, \ j \in [n] \setminus I, \ i, but it does evade the typesetting rule by putting everything inside an equation.

By the way, Peter, I tried to fix your LaTeX but I couldn't figure out what you meant to write. You might find https://sbseminar.wordpress.com/including-equations-in-comments/ useful.

15. David, that was extremely silly of me. Yes, I do know the meaning of ordered pair. But obviously I blanked… I really did mean what I wrote though. Maybe ‘in order’ is a better substitute to refer to the order on the base set.

With unordered pairs (say, of ordinals) the convention often is to assume that a two-element set is always written ‘in order’ (especially when indices come into play). This somewhat applies here, since if you presume $i you don't really need ordered pairs — you already have an order (which, of course, led to my amusing error; gee, and I thought I'd just write non-native speaker to be on the safe side…).

One other thing I had written was that if it is clear from context that $j< n$, why not just write $i\in I, j\notin I$ for smoother reading?

Some other (possibly bad) spontaneous thoughts. What about $(i? I use that for pairs of finite sets of integers a lot (to indicate something else, of course).

The last thing I had written in incorrect wordpress-latex was: what about $I \times_< [n] \setminus I$ (possibly dropping the subscript)?

If nothing else works I (as a reader) always prefer conventions. Just call them suitable pairs. After all, LaTeX makes it easy to have \suitable make an automatic reference (with an internal link in the pdf) to the place where you introduced you convention.

On a side note, does latex.stackoverflow accommodate such a type of conversation? I'm guessing not, but I wish there was a central place where people would converse about good (mathematical) writing.

16. My guiding principle in these situations is that I write as I would speak. When I’m reading maths I tend to vocalize in my head, so it has to sound right when read out. I don’t think I would use the double ‘and’ when speaking. I might possibly go with “the number of pairs $(i,j)$ with $i>j$ where $i \in I$ and $j \in [n] \setminus I$“.

17. Richard Séguin says:

Like Simon, I also tend to write mathematics so that it reads the same as I would speak it, even if it makes the text slightly longer. I didn’t do this when I was younger, but I’ve come to realize that it generally makes the text easier for someone else to read. If your goal is to communicate your ideas to others, why make it more difficult for them?