Mathlish - J.S. Milne, Top
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

1. In order that, as in I stopped so that you could catch up.
2. With the result or consequence that, as in Mail the package now so that it will arrive on time.
3. 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

1. adj : of a degree or quality specified (by the "that" clause); their anxiety was such that they could not sleep. (dictionary.com)
2. 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

1. We require x to be a rational number so that mx is an integer for some m. [Correct]
2. We require x to be a rational number so that 3x is an integer. [Incorrect; should be such that 3x is an integer.]
3. 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.]
4. 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.]

Briefly, 'such that' expresses consequence, and 'so that' purpose. 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:
Sei x in Q, so daß es ein m in Z\{0} mit xm in Z gibt.
Sei y in Q so, daß 3y in 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). Sometimes it is clear from the context which of these you mean, but usually it is better to choose a more precise word. Compare (in increasing order of ambiguity):
The statement is true for every element in the set if it is true for one.
The statement is true for any element in the set if it is true for one.
The statement is true for every element in the set if it is true for any.
The statement is true for any element in the set 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.
Because of its ambiguity, Halmos decreed that "any" should never be used in mathematical writing.
[Of course, if your intention is to confuse, 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:
1. a(n) \neq 0 for all n.
2. a(n) is not equal to zero for all n.
3. a(n) is not zero for all n.
4. 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 has spread. 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". There is some overlap between the two words, but "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.

## 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.)

## Be wary of using :=

Personally, I consider the use of "A := B" for "B is defined to be A" or "B equals A by definition" to be ugly programming jargon (especially ugly if the symbols aren't vertically centred--use the package mathtools to fix this), but if you do choose to use it, use it only when the colon genuinely adds something. Note that "We define A:=B" is shorthand for "We define A is defined to be B".