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This is more of a mathematician question than a mathematics question, but I hope it is still appropriate.

Several times now, when friends have been editing my mathematical writing, they have pointed out instances where I began a sentence with a mathematical symbol, such as $X$, or $a\in A$, etc. This is the sort of thing that never struck me as inherently bad writing, since it is often exactly how I would speak the sentence aloud. However, I can also see their point, in that a sentence beginning with a mathematical symbol can look a little weird (especially when it starts with a lowercase letter), and doesn't quite read the same as a sentence beginning with an English word.

However, now that I have been trying to avoid doing this, more and more I find myself writing awkward sentences to avoid the most natural way of speaking, and I can't quite decide if I am increasing or decreasing the quality of my writing.

Do people have strong opinions on this? Do people have tricks for dealing with this, besides a succession of empty and synonymous clauses ("Thus", "Then", "Therefore", "We find", "Looking at"...)?

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I do hope you've seen Serre's lecture on mathematical writing. Remember, write "The map f", or "The function f" instead of just "f". "The manifold M", "The space X", "The scheme S", "The element e", etc. –  Harry Gindi May 15 '10 at 21:27
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"Now" is useful. –  Gerald Edgar May 15 '10 at 21:27
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There is a link to modular.fas.harvard.edu/edu/basic/serre from Serre's wikipedia page. The linked page has a link to a file serre.avi that just might be the lecture in question. (It's 459 megabytes – I am downloading it now … 23 minutes download time remaining.) –  Harald Hanche-Olsen May 15 '10 at 22:55
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I'd also recommend Knuth's "Mathematical Writing" (ISBN 088385063X). Page 1, point 2: "Don't start a sentence with a symbol." –  Blue May 15 '10 at 23:04
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I've hit this question with the wiki hammer. Any question that comes down to "who has strong opinions about X?" should be wiki, please. –  Scott Morrison May 16 '10 at 19:34
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8 Answers 8

I also try to avoid starting sentences with a mathematical symbol. I feel strongly about it, but I can't quite articulate why.

One trick I use that you haven't mentioned yet is something like, "The space $X$ ..." or "The function $f$...".

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Greg, there is a difference between speaking and writing, so you shouldn't worry that writing doesn't always match speaking or that writing may take more effort than speaking. One concrete reason to avoid symbols at the start of a sentence is that it can be hard to see where a sentence ends and begins if you use symbols to start them. (Imagine a symbol ends a sentence, then a period, then another symbol.) Also, Serre has a list of writing tips which includes this advice. Amazingly (to me), some mathematicians don't mind starting sentences with symbols. But please stick to avoiding that! –  KConrad May 15 '10 at 21:32
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There was an earlier MO question on punctuation in math formulas at mathoverflow.net/questions/6675/… with a strong opinion expressed by Allen Hatcher. Greg, I think you can get any answer you wish on your writing questions by asking enough faculty in your department. –  KConrad May 15 '10 at 21:46
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A friend of mine often wrote sentences like "Since for all $x>0$, $f(x)>0$, $\int_0^1 f>0$. Despite they didn't begin with a symbol and all the punctuation was in place, I usually had to spend a few seconds figuring out the meaning (though, if that had been read aloud to me with proper pauses, I would have no trouble with it). So, in my humble opinion, the more words between the formulae, the better... –  fedja May 16 '10 at 1:55
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@fedja: "Let X be an E.C. with. C.M.", where "with." of course stands for "without". –  Harry Gindi May 16 '10 at 6:44
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You shouldn't do it. This is covered in Halmos's justifiably famous article "How to write mathematics". (You'll find lots of copies of this on the web.) It's very closely related to the rule that you should avoid having notation separated by punctuation.

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If one stoops to starting sentences with a symbol, then one soon descends to finishing a sentence with a symbol and starting the next with a symbol. Then one is liable to finish a sentence with a symbol and start a sentence with the same symbol. To the reader (if you still have one) madness beckons.

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I share the dislike that many people have expressed for starting sentences with symbols. In trying to explain why, I wanted to come up with an example that would illustrate the temptation to do so, but I find it surprisingly hard. Suppose, for instance, that f is so clear from the context that to use Tara's trick and start a sentence with "The function f" would be ridiculous. (In general, by the way, using Tara's trick can add clarity by reminding the reader what is what, so I am talking about a special circumstance here.) Now suppose that we want to say something about f, such as that it is a homomorphism. It is very hard to think of a natural context in which one would actually want to start a sentence with "f is a homomorphism". Nearly always there would be a reason for saying that, or a justification for it, such as, "It is not hard to check that", or "We have already seen that". And if it came out of the blue somewhere, then one would want to signal a slight change of subject with a word such as "Now" (a trick that has already been mentioned). So I'm tempted to say that if your prose is flowing properly the problem shouldn't arise, or should arise very infrequently.

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I think I agree. I can't actually construct a situation where it doesn't seem somehow nicer to add "Now" or "Then" or "Consider when" or something similar. –  Robby McKilliam May 15 '10 at 23:01
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Examples can be found if you think about statements of theorems. Someone not used to avoiding symbols at the start of a sentence may be inclined to write an equation as the theorem (with quantifiers following rather than preceding it). For example, Mike Rosen told me that he doesn't have a concern about starting a sentence with a symbol and if you look at Ireland & Rosen's "A Classical Introduction to Modern Number Theory" (2nd ed.) you'll see Theorem 3 on p. 212 (or, for that matter, most of the lemmas which precede that theorem) illustrates the possibilities. –  KConrad May 15 '10 at 23:17
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The best examples I've found occur when you have a list of conditions. For example, "Theorem 1. The function f has property X if and only if one of the following is true: (1) f is surjective. (2) f is open. (3) f is bounded." –  Vectornaut May 16 '10 at 1:52
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@Vectornaut: If you're giving the conditions in an ordered list, I think that it's fine to put the symbol at the beginning of the sentence because it's clearly set off from the prose. –  Harry Gindi May 16 '10 at 6:48
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Robby: "I can't actually construct a situation where it doesn't seem somehow nicer to add "Now" or "Then" or "Consider when" or something similar" I think one reason for this is, that many mathematicians often insert a "Now" to avoid beginning a sentence with a symbol, so you are used to this kind of constructions. If you asked a non-mathematician, I think he might prefer a construction that began with a symbol. I agree that you shouldn't begin a sentence with a symbol, but I don't think that these constructions are "nicer" English. –  Sune Jakobsen May 16 '10 at 10:36
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This seems widely considered as bad form within mathematics papers. In some cases, it might not be clear that the previous sentence has ended, particularly if it also ends in mathematical typesetting, or the punctuation itself might be considered as part of the mathematics. This applies more generally, in that one should avoid placing too much mathematics around all punctuation marks.

There are also capitilisation considerations, e.g. starting sentences with f(x).

As already noted, it's usually very easy (and often desirable) to avoid this situation.

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Since you almost never have a sentence like "x is in A" without an additional clause, it shouldn't be hard to reverse the order of the sentence. For example, "u and v are harmonic since f is analytic" could be changed to "As f is analytic, u and v are necessarily harmonic" or a number of variants thereof (since, as a consequence of, by virtue of).

If there is some case where you would actually want a standalone like "x is in A" --- for example after a long lemma establishing that x is in A --- one could use flashier phrases, e.g. to wit, that is to say, whence etc.

Good writing is a lot like good mathematics: it's more art than science.

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I personally find that it is occasionally convenient and natural to start sentences with a mathematical symbol, but I have had coauthors who do not like it, and in this case I typically modify the sentence (into what I sometimes feel is more awkward).

Tara has mentioned a nice trick for amending sentences starting with a symbol and this is typically the approach I have taken. Another approach that I find sometimes works is to consider merging the sentences (perhaps with a little restructuring) so that the symbol follows a comma or a conjunction such as `and'.

I probably have an overly relaxed attitude towards the written language, but in my opinion, if it works and it is clear, do it. I can envisage situations where starting a sentence with a symbol makes the text confusing to read. Obviously it should be avoided in this case.

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Under no circumstances can you have symbols then punctuation then other symbols. As a special case you can never have symbols at the end of one sentence and the beginning of the next. Outside of the above case, starting sentences with symbols is bad, but occasionally all other options are worse.

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When x > 0, x^3 > 0 too. Hmm.. –  KConrad May 16 '10 at 5:17
    
Writing m = [G:H], mG is a subgroup of H. (Here groups are abelian.) –  KConrad May 16 '10 at 5:36
    
Personally I must admit that both examples would take longer (though far from impossible of course) to parse than versions following Noah's rule. That I think should be the reason for doing it one way or other. (It is very tricky to apply such a principle however as it is very individual. I realised that after having done some programming I tended to put in more parentheses in formulas which probably makes them more difficult to read for some, perhaps most, mathematicians.) –  Torsten Ekedahl May 16 '10 at 8:22
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I would avoid writing either of Keith's examples. For example "Writing m=[G:H], we see that mG is a subgroup of H" or "When x>0, certainly x^3>0." –  Noah Snyder May 17 '10 at 2:43
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How about "For every x∈A, f(x) is prime"? I agree with "avoid", not with "[u]nder no circumstances": the issues are that (i) there's usually poor phrasing going on, and (ii) the typography is tricky. –  Charles Stewart May 19 '10 at 12:29
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