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Many English grammar books state that in an "if ... then" conditional sentence, the word then should be preceded by a comma. However, in mathematical writing I often see that this rule is ignored.

Here is a sentence taken from Higham's Handbook of Writing for Mathematical Sciences:

If $\alpha>-1$ then the integral does not exist.

Since the book is intended for writing mathematics correctly and this structure has appeared in many places inside it, lack of a comma in the above sentence cannot be a simple typo. Does anyone know the reason for this?

Added in Edit: As a separate, but related question, would it be possible that the omission of comma is due to the conditional sentence ending with a mathematical formula?

EPS
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    One thing you will learn as you progress... mathematicians are not English grammar experts. Not usually. – Clarinetist Jan 26 '15 at 03:19
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    Further, mathematics is written in a technical sort of grammar where words and sentence structures can mean radically different things than in plain English.

    In this case, the comma placement "If $\alpha > -1$, then..." does not seem to serve to clarify any part of the statement. There might be cases where meaning is lost if the comma is omitted, but I can't think of any examples off the cuff.

     – walkar Jan 26 '15 at 03:21
  • @Clarinetist I mentioned in my post that Higham's book is about mathematical writing so your comment is not applicable. – EPS Jan 26 '15 at 03:26
  • @Clarinetist: mathematicians are English grammar experts. More precisely, mathematicians are the most knowledgeable about foreign languages among any group of people having university degrees, including professional linguists. I'm not sure about clarinetists, however... – Georges Elencwajg Jan 26 '15 at 10:08
  • Personally, I would always use a comma there. It's not necessary to clarify the meaning of the sentence, but I notice its absence anyway. (Maybe I'm not the best person to give advice though. While I get most punctuation stuff, I could never get commas straight.) – Randy E Jan 28 '15 at 03:50
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    Certainly the sentence as it stands can't be misunderstood; in that sense, it's not a serious error. But it is a grammatical error: the dependent clause preceding the main clause requires a comma. (If the dependent clause comes after the main clause, then no comma is needed: "The integral does not exist if $\alpha > -1$.") – mjqxxxx Jan 30 '15 at 18:29
  • @mjqxxxx: Absolutely right. Also, in mathematical writing, always use then to follow a prior if, after the comma. – John Bentin Jan 31 '15 at 17:40
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    -1. Why would English grammar make an exception for sentences involving math? Either way, this question belongs to some of the English SEs, rather than Math SE, since English grammar rules is what purely determines whether to put a comma or not. – user26486 Feb 03 '15 at 21:00
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    @user314 Of course, you are free to upvote or downvote, but the article-writing tag is the reason this question is posted here. – EPS Feb 03 '15 at 23:09

2 Answers2

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If the premise of a conditional statement is so tediously, extraordinarily, and inordinately long that it is easy to lose track of its train of thought, especially when the premise contains many, many, many, many commas and subordinate clauses, then using a comma before the concluding clause may well be beneficial.

If the premise is short then the lack of comma doesn't really hinder understanding.

If it were up to me, I would always use a comma.

By the way, the 16th edition of The Chicago Manual of Style suggests that “[a] dependent clause that precedes a main clause should be followed by a comma,” and uses the following example (see 6.30, p. 317):

If you accept our conditions, we shall agree to the proposal.

I believe this rule applies also when a “then” is added to the beginning of the concluding clause.

triple_sec
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    Fantastic response! – amcalde Jan 28 '15 at 04:06
  • I would note that professor I have had that went to Oxford tend to refuse to put commas after dependent clauses. I don't think it is regional since I tried looking it up. The British grammar style guide agrees with putting commas after dependent clauses. – dustin Jan 30 '15 at 22:12
  • @dustin Maybe it's just unintentional. If he does this practice on purpose, it might be a matter of choice of personal style—which I think is perfectly fine! – triple_sec Jan 31 '15 at 04:46
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    My supervisor at Oxford, Alex Wilkie, follows the one-line statement of his most famous theorem with an explanation beginning "If...". The sentence is eight lines long (with three nested parentheses (like this (!), for example) at one point) before it reaches the "then". There's a comma before "then", but putting it in italics also really helps. I'd have gone up the punctuation scale with a semicolon or even a colon, myself. – HTFB Jan 31 '15 at 14:44
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    @HTFB: This one-sentence style for complicated theorems is not to be recommended. Much clearer is something along the lines of "Let $X$ be a ... satisfying the following seven conditions: (1) ... ; ... ; (7) ... [note the full stop here]. Then [statement]. Beginning a statement with then, even after a full stop, necessarily links it to the previous text. – John Bentin Jan 31 '15 at 17:54
  • @JohnBentin It's even better to write: “Then,” with a comma. When you enumerate a long list of premises in spoken language and finally arrive to the conclusion, it's best to say: “Then” with an emphasis and a short pause after it to allow the listeners to adjust their minds to hearing about the conclusion. Putting a comma after a sentence-initial “then” would nicely reflect this pause of adjustment in writing. – triple_sec Jan 31 '15 at 18:50
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    @triple_sec: OK. The lack of a closing quotation mark after the closing bracket prior to "Beginning" in my comment is an instance of Bierce's law. – John Bentin Jan 31 '15 at 21:00
  • John Bentin: Thank you for your kindly-meant advice, but since the paper has already been published and won prizes, and since it wasn't mine in the first place, I fear that there is little we can now do. – HTFB Feb 01 '15 at 00:44
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My answers to the question' Why do mathematical if-then statements often miss out the comma after the if clause?' would be:

1) Mathematicians try to keep things simple & clear, and in scruffy handwriting a comma might be mistaken for something else by the reader;

2) The comma rule is not so strictly adhered to among other English users anyway;

3) Typical if-then statements in ordinary life have more contingency and uncertainty to both sides of the if-then statement than they do in maths. For example, in a legal if-then sentence both sides can be debatable ...only true with certain clarifications. In "If the client's property is damaged by an act of terrorism, then...", for example, there can easily be debate in court as to whether a given event qualified as matching the content of the if clause. In contrast, maths readers are typically willing to accept the if clause as "transparent", not needing debate (if it is unclear, they often reject the whole sentence or document as unclear), so that "If x > 2" requires no special consideration, no discussion as to whether x is or is not greater than 2. The maths reader simply accepts this clause and moves straight on, hence "If x > 2 then y ...whatever" does not look strange without a comma;

4) A commonly stated rule of thumb in English is that commas give the reader a chance to take a breath. For reasons similar to those in (3), maths readers are less likely to wish to take a breath or a pause. The if-then statement has the feel of a complete unit for mathematicians, and they want to "take it whole".

Strangely, most children in most English-speaking countries attend school without ever being given grammar lessons in their own language. To clarify reason (2) above it would be interesting to hear from mathematicians who write their first drafts (or all drafts!) in a language where grammar of the native language is a compulsory subject at school: German or French or Russian speakers on here? Do you automatically carry over all the punctuation rules of your first language into the mathematical sentences you write?

Travelling Salesman
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  • English went through a regrettable phase when a generation or two of academics decided they could write a set of grammatical rules for English without actually worrying about how English was really used. It took a long time to throw off these so-called rules (there are skirmishes still) and teaching grammar is unfortunately still often viewed as an exercise in lying about how English actually works. – Joffan Feb 03 '15 at 22:10
  • Agreed, Joffan. There's also a big range of ways in which English is used - a problem the Germans solve by distinguishing High German (a common medium for educated people from Saxony or Switzerland or Bavaria to talk to each other) and Low German dialects which you might say represent the version of German people in a certain area speak and "really use" in your sense.

    I'm not suggesting we launch a High English, although someone talking about the widely varying Gypsy dialects across Europe did tell me once they thought the Roma need a kind of 'High Gypsy' version of their language(s)!

     – Travelling Salesman Feb 04 '15 at 11:54