Much of mathematics is written in a peculiar pidgin. If you wish to
write in standard English, here are a few things to avoid.
Don't use "so that" when you mean "such that"
Here are the definitions.
so that
- In order that, as in I stopped so that you could catch up.
- With the result or consequence that, as in Mail the package now so that it will arrive on time.
- so ... that. In such a way or to such an extent that, as in The line was so long that I could scarcely find the end of it.
From dictionary.com
such that
- adj : of a degree or quality specified (by the "that" clause); their anxiety was such that they could not sleep. (dictionary.com)
- A condition used in the definition of a mathematical object. For example, the rationals can be defined as the set of all m/n such that n is nonzero and m and n are integers . (mathworld.wolfram.com)
Examples
- We require $x$ to be a rational number so that $mx$ is an integer for some $m$. [Correct]
- We require $x$ to be a rational number so that $3x$ is an integer. [Incorrect; should be such that 3x is an integer.]
- Let $H$ be a discrete subgroup of the Lie group $G$ so that $G/H$ is compact.
[Incorrect --- not all discrete subgroups of Lie groups have compact quotient; this is from
the Annals of Math., 107, p313.]
- Let $N$ and $N'$ be submodules of a module $M$ such that
$N$ contains $N'$, so that $N/N'$ is a submodule of $M/N'$. [Correct!
From Steps in Commutative Algebra.]
Or simply note that "so" is an adverb and "such" an adjective.
You won't find "so that" among lists of commonly misused expressions because only mathematicians commonly misuse it.
Probably the error arose from the influence of German on American (mathematical) English, since both are
expressed by "so dass" in German, but distinguished with a comma:
Sei $x$ in $\mathbb{Q}$, so daß es ein $m$ in $\mathbb{Z}\setminus\{0\}$ mit $xm$ in $\mathbb{Z}$ gibt.
Sei $y$ in $\mathbb{Q}$ so, daß $3y$ in $\mathbb{Z}$ liegt.
Write "Gowers's Weblog", not "Gowers' Weblog" or "Gower's Weblog" (Gowers)
[Incidentally, The Princeton Companion to Mathematics is written in unusually good English.]
See:
Use an apostrophe plus -s to show the possessive form of a
singular noun, even if that singular noun already ends in -s.
(grammar.about.com)
Form the possessive singular of nouns by adding 's.
(Strunk and White, The Elements of Style, Elementary Rules of Usage, Rule 1.)
The possessive of most singular nouns is formed by adding an apostrophe and an s.
(The Chicago Manual of Style, 7.17)
It is always correct to add an apostrophe and s to a
possessive of a singular noun in formal English (you don't
have to pronounce the s). Some sources allow you to omit the
s in some circumstances, but this only adds confusion
and ambiguity (without the s, "Langlands' car" could be the car
of two people called "Langland" or of one person called "Langlands").
For a truly gruesome error, see BAMS 46.4 (2009), p601, line 8, where
one finds: "the Langland's program". (Should be "the Langlands program"
or "Langlands's program".)
Don't write "verify" when you mean "satisfy"
"Verify" is often misused by mathematicians for "satisfy",
especially by those whose native language is French. For
example, Roos (2006) writes: "The reason for this is that
AB4* is rarely verified for them." He means satisfied. It is
possible for a condition to be always satisfied but rarely
verified (for example, the commutativity of a certain class
of diagrams).
verify means
to prove the truth of;
satisfy means
to fulfill the requirements of.
For example, I believe the function satisfies the Leibniz
condition, but I haven't verified [i.e., checked] this.
Don't use "associate to"
Instead use "associate with" or "attach to", whichever is more appropriate. In English, you may associate
with gangsters, or attach yourself
to the Crips,
but you may not associate
to either: "associate to" is
not English (native French and Italian speakers please take note).
Alas,this particular illiteracy has
become almost standard in scientific journals -- where once we had two
expressions "attach to" and "associate with" with distinct uses, we now have
only one "associate to = attach-to-associate-with". [Even Google Translate
gets this right: it translates
"associé à" correctly as "associated with".]
Normally pure mathematicians are relatively respectful of grammar,
but many of them have adopted the habit of using the dreadful phrase "associated to"
when they seem to feel that "associated with" has not a sufficiently specific flavour.
I am at a loss to understand why they do not use the perfectly grammatical "assigned to" instead. (Penrose,
The Road to Reality, p.354.)
The verbs "permit", "allow", "reduce" normally require objects
For example, the following are incorrect:
The preceding result permits (or allows) to assume that
x is positive. [Should be: permits (or allows) us to assume.]
By taking a shortcut, we reduce to 10 miles. [Should be: By taking a shortcut, we reduce the distance to 10 miles.]
We use the local theory to construct
the formal group associated to the Neron-model, which then allows
to simplify the global steps (Faltings 2008, p93). [Should be: ... formal group attached to... allows us...]
Let now us praise famous men.
Only mathematicians write "Let now" --- real people write "Now let" or "Let us now".
Be careful with your use of "any"
The word "any" can mean "one, some, every, all" (see any dictionary).
In mathematics, it can usually be replaced by one of these more precise words.
Compare (in increasing order of ambiguity):
It is true for every x if it is true for one.
It is true for any x if it is true for one.
It is true for every x if it is true for any.
It is true for any x if it is true for any.
The next sentence, from an actual article, is truly ambiguous:
... he introduced the "associate form" of a cycle, apparently the first treatment valid in any
characteristic.
Does it mean "valid in all characteristics" or "valid in some characteristic"?
Halmos went so far as to decree that "any" should never
be used in mathematical writing.
[Of course, if your intention is to obfuscate, you can exploit the ambiguity. State a theorem:
"If any zero of the zeta function lies on the critical line, then the Riemann hypothesis is true". Prove the theorem by interpreting "any"
as "every". Then interpret "any" as "some", exhibit a zero on the critical line, and claim to have proved the Riemann hypothesis.]
Be careful with quantifiers and negatives
The following statements are equivalent:
- $a(n) \neq 0$ for all $n$.
- $a(n)$ is not equal to zero for all $n$.
- $a(n)$ is not zero for all $n$.
- There exists an $n$ for which $a(n)$ is not zero.
So don't use the first to mean "no $a(n)$ is zero"; instead write "no $a(n)$ is zero" (or at
least put a comma before the "for all").
Write sentences that can be parsed by a nonexpert
Serre gives the example of an article that announces its main result in the form:
formula A = formula B = formula C
(no words). Only someone very expert in the field will recognize that the main result is
"formula A = formula B" --- the second equality is only an aside.
Don't write "issue" or "challenge" when you mean "problem"
The misuse of "issue" for "problem"
began in the computer industry (there are no problems, only issues),
and spread like a virus. For example, GM requires employees to replace
"problem" with the euphemism "issue" when referring to its products.
Even Google Translate has been corrupted: it
translates "sur ces questions" as "on these issues"
[update 2020: it got it right, "on these questions"].
The word "issue" means:
An important topic or problem for debate or discussion:
'the issue of racism' 'raising awareness of environmental issues' (OED).
The misuse of "challenge" for "problem" is feel-good English: not all problems are challenges; some are just problems.
See
You have problems, not issues.
Since I wrote this, the euphemisms "issue" and "challenge" have
almost entirely replaced "problem" in American English.
Don't write "address" when you mean "solve", "examine", "study", or ...
Instead, write "solve", "examine", "study", or whatever it is you actually mean. For example,
don't write "the conjecture is successfully addressed"; write "the conjecture is proved" (if
that is what you mean).
address: People in the business of not really meaning
what they say love this word for its soothing vagueness. When they
undertake to "address the issue of ... " you can be sure that nothing
much will happen, and that said issue will speedily be kicked into the
long grass. (The Guardian, 10 April 2015.)