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I am not a native english speaker. I learnt about defining and non-defining relative clauses from english grammar books. Grammar books tell me not to use commas in defining relative clauses, so I don't understand why there is a comma preceding "where" in these examples:

From Theorem 4.59 (Sylow Theorems) in Anthony W. Knapp's Basic Algebra, Digital Second Edition:

Let $G$ be a finite group of order $p^mr$, where $p$ is prime and $p$ does not divide $r.$

From Hungerford's Algebra:

Theorem 6.7 (Fundamental Theorem of Arithmetic) Any positive integer $n \gt 1$ may be written uniquely in the form $n = p_1^{t_1}p_2^{t_2} \cdots p_k^{t_k}$, where $p_1 \lt p_2 \lt \cdots \lt p_k$ are primes and $t_i \gt 0$ for all $i.$

I think the clause "where $p_1 \lt p_2 \lt \cdots \lt p_k$ are primes and $t_i \gt 0$ for all $i$" is a defining relative clause, since it gives essential information about the form $n = p_1^{t_1}p_2^{t_2} \cdots p_k^{t_k}.$

I looked up the word "where" in three mathematical textbooks, and in similar situations, they all use commas between formulae and the words "where". Is this a convention just in mathematical writing?

ryang
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lightweight
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    This isn't exclusive to math. See http://ell.stackexchange.com/questions/32382/comma-and-where . As a native English speaker I find lots of commas in mathematical writing helpful to break up the long sentences. – mathematician Jan 10 '17 at 09:12
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    For me, this is a clear, simple grammar convention. – DonAntonio Jan 10 '17 at 09:12
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    I find it quite praiseworthy that a non-native speaker shows interest in such fine grammatical points, while some native speakers here dare write about a polynomal that "it's roots are distinct" ! – Georges Elencwajg Jan 10 '17 at 10:43
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    In another direction, Halmos (How to Write Mathematics, though I don't own the book and can't give a page reference) advised against the "where" construction, arguing that notation should be explained before its first use. For instance, Halmos might have suggested, "For every integer $n > 1$, there exist unique primes $p_1 < \cdots < p_k$ and positive integers $t_i$ such that $n = p_1^{t_1} p_2^{t_2} \cdots p_k^{t_k}$" for the fundamental theorem, or "Let $r$ be a positive integer, $p$ a prime not dividing $r$, and let $G$ be a finite group of order $p^{m}r$" for the theorem in Knapp. – Andrew D. Hwang Jan 10 '17 at 13:38
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    I'm no English expert, but it's not clear to me that this is a defining relative clause. It seems that the thing being defined is "form" and the thing that defines it is "$n = p_1^{t_1}\cdots p_k^{t_k}$", not the following "where" clause. Maybe someone with better expertise can run with that and find some references or something. – David Z Jan 10 '17 at 19:38
  • Halmos is right, of course, if you can give the information at the start. If not, considerations of grammar and logic aside, custom demands the comma. – BruceET Jan 21 '17 at 06:51

2 Answers2

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Consider an example:

(1)$\qquad$"We can find $x\in A$ such that $f(x,y)=0$, where $g(y)\in B$ "

(2)$\qquad$"We can find $x\in A$ such that $f(x,y)=0$ where $g(y)\in B$ ".

The first statement could be rewritten, with a change of emphasis, as

$\qquad$"We can find $(x,y)\in A\times \{y:g(y)\in B\}$ such that $f(x,y)=0$ ".

The second statement has the same structure as "There is fire where there is smoke" and might be interpreted as

$\qquad$"We can find $x\in A$ such that $f(x,y)=0$ whenever $g(y)\in B$ ".

Usually in mathematics, this latter type of interpretation is unintended, and the comma is needed.

Ideally, all notation should be defined before it is used. However, it often happens that the defining condition—for example, "where $c$ is some positive constant"—is not the focus of interest of the statement; so we may not wish to preface our statement as "There is some positive constant $c$ such that ... ". This is especially the case when the notation and condition are routine and conventional. In such cases, the where clause (preceded, of course, by a comma!) is unobjectionable.

John Bentin
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Grammar books tell me not to use commas in defining relative clauses, so I don't understand why there is a comma preceding "where" in these examples:

Let $G$ be a finite group of order $p^mr$, where $p$ is prime and $p$ does not divide $r.$

Any positive integer $n \gt 1$ may be written uniquely in the form $n = p_1^{t_1}p_2^{t_2} \cdots p_k^{t_k}$, where $p_1 \lt p_2 \lt \cdots \lt p_k$ are primes and $t_i \gt 0$ for all $i.$

A relative clause (for example, in "The dog that she adores is brown") applies to nouns or noun phrases or pronouns, and contains no standalone sentence. In each case above, since the "where..." clause applies to the entire sentence preceding it and does contain a full sentence, it is actually not a relative clause, and the comma is grammatically correct.

To more clearly see that the "where..." clauses above are actually referring to the entire sentences preceding them, observe that the "where..." clause in the first example is implictly two universal quantifications, while in the second example is implictly two existential quantifications. (And in this example, it is implicitly mixed quantification, which has the additional ambiguity of hanging quantifiers! As Andrew Hwang's above comment suggests, relying on the "where..." clause to implicitly declare quantification is bad writing!)

Finally, as pointed out in John's answer and corroborated in Are "where" and "such that" interchangeable?, without a preceding comma, the word "where" typically technically means "whenever/if".

ryang
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