Common Errors to Avoid in Writing Mathematical English (by David Kramer)

I. Mathematical English Is English Prose
The most important error to avoid in writing mathematics in any language is to forget that mathematical writing is a highly specialized form of communication that requires above all accuracy and clarity. Although mathematical writing has its own particular set of requirements, keep in mind that when you are writing mathematics in English, you are writing English prose, and all the rules of English—in particular grammar and punctuation—apply.

While we cannot go into all the intricacies of English in this brief article, here are a few guidelines for making your mathematical writing conform to the norms of English prose.

1. Sentence Structure
Use complete sentences: Since the primary goals of mathematical writing are accuracy and clarity, always write in complete English sentences, following all the rules of English grammar and punctuation. Mathematical symbols are abbreviations for words and should be treated as such.
Treat mathematical expressions as a single unit: To avoid confusion, treat an entire mathematical expression as a single grammatical unit. Thus if the subject of a sentence is a word, then the main verb should be a word as well. It should not be a symbol and should above all not be part of a mathematical expression.

The following sentence is confusing.
incorrect: The total output Xtotal = 4.5, as can be seen from the following calculation.
The author of the above sentence has attempted to combine the subject of the sentence, “total output,” with a verb extracted from the mathematical expression, namely the equal sign =. Not knowing this, the reader reaches the end of the sentence still expecting the appearance of a verb, something like this:
The total output Xtotal = 4.5, as can be seen from the following calculation, is a local maximum.
Taken by surprise when the sentence ends abruptly, the reader has then to go back and figure out that the equal sign is functioning as the main verb in the sentence. Therefore
Treat mathematical expressions as a single unit. If a word or words form the subject of a sentence, the main verb must be a word or words as well, not a symbol.
correct: The total output Xtotal is equal to 4.5, as can be seen from the following calculation.

No extra punctuation around symbols:

The following, with an added colon, is incorrect: The equation: x2 = –1 has no real solutions.

Correct: The equation x2 = –1 has no real solutions.

Do not treat independent typographical units such as theorems and definitions as part of a sentence:
incorrect: We have thus proved the following Theorem 4.5. Let G be a …
correct: We have thus proved the following theorem. Theorem 4.5. Let G be a …

2. Joining sentences:
The simplest sentences consist of a single independent clause:
The group G is nonabelian. The subgroup H is cyclic and therefore abelian.
Two such independent clauses may be combined into a single sentence by linking them with a conjunction or a semicolon:
The group G is nonabelian, but the subgroup H is cyclic and therefore abelian.
The group G is nonabelian; the subgroup H is cyclic and therefore abelian.
They cannot be joined with a comma:
incorrect: The group G is nonabelian, the subgroup H is cyclic and therefore abelian.

They cannot be joined with an adverb:
incorrect: The group G is nonabelian, however the subgroup H is cyclic and therefore abelian.
correct: The group G is nonabelian; however, the subgroup H is cyclic and therefore abelian.
correct: The group G is nonabelian. However, the subgroup H is cyclic and therefore abelian.
incorrect: Let X be a set of positive integers, then X has a least element.
correct: Let X be a set of positive integers; then X has a least element.
correct: Let X be a set of positive integers. Then X has a least element.

The following are not conjunctions and therefore cannot be used to link two clauses: therefore, however, thus, then, hence. You must use a conjunction, a semicolon, or form two sentences:
We see that x must be zero, and therefore the theorem is proved.
We see that x must be zero; therefore the theorem is proved.
We see that x must be zero. Therefore the theorem is proved.
We see that x must be zero. However, y might be nonzero.
We see that x equals zero; hence y must be zero as well.

II. Grammar

1. Defining or Nondefining? Restrictive or Nonrestrictive?
This section discusses what is perhaps the most important type of error to avoid, since unlike many errors, this type may be undetectable and therefore leave your reader confused. By a defining or restrictive expression is meant a word or phrase or clause that limits the meaning or scope of the word or words to which it refers.

Example: The mathematician Archimedes was one of the greatest mathematicians of all time.
Here “Archimedes” restricts the scope of “mathematician” to the individual Archimedes.
A nonrestrictive element, which does not limit scope but merely provides additional information, is indicated in English by being set off by commas:
Example: The greatest mathematician of the ancient world, Archimedes, was also a physicist, engineer, inventor, and astronomer.
Here, the scope of “greatest mathematician of the ancient world” is already limited to a single individual (there can be only one greatest), and “Archimedes” provides additional information, namely that greatest mathematician’s name.

Relative clauses: that/which and commas/no commas
The most significant source of error in distinguishing defining from nondefining elements is in relative clauses (a clause is a sentence element with its own subject and verb). Put simply, a relative clause follows the noun (antecedent) that it modifies, and in mathematical writing:

A restrictive relative clause is introduced with “that” and is not set off by commas:
Example: The real numbers that cannot be expressed as the quotient of two integers are called “irrational.”

Here the relative clause (in italics) restricts the scope of all real numbers to those that cannot be expressed as the quotient of two integers.

A nonrestrictive relative is introduced with “which” and is always set off by commas.
Example: The set of rational numbers, which is an abelian group under addition, is in fact a subfield of the real numbers.

Here the relative clause (in italics) provides additional information about the set of rational numbers without restricting the scope.

In the following two examples, it is only the use of commas and the that/which distinction that determines two very different meanings, particularly if one is unfamiliar with the terminology (which here we have invented):

A grpp that is also a mrpp is a verpp.
A grpp, which is also a mrpp, is a verpp.

In the first example, only those grpps that that additionally are mrpps are verpps, while in the second example, every grpp is a verpp; the fact that a grpp is also a mrpp is simply additional information.

“Such that” as a restrictive/defining phrase
The phrase “such that” is generally used in definitions as a restrictive/defining element. Therefore it is not set off by commas:

Let S denote set of all x in X such that x is irrational.
Choose x in G such that x also belongs to the subgroup H.
(See “such that vs. so that” below for a usage problem.)

2. The Subjunctive Mood in English
The subjunctive mood, used to express, among other things, a wish, requirement, or possibility, is used less in English than in such Eurpoean languages as French, German, Italian, and Russian. Nonetheless, there are several common uses of the subjunctive in mathematical writing of which one should be aware:

Most sentences expressing a requirement employ the subjunctive.
We require that the exponent be a nonnegative integer.
It is necessary that the exponent be a nonnegative integer.
We insist that the exponent be a nonnegative integer.

Note, however, that the subjunctive is not generally used with such words as assume and suppose:

We assume that the exponent is a nonnegative integer.
We hypothesize that the exponent is a nonnegative integer.

Since it is used to express something counterfactual, the subjunctive often appears in proofs by contradiction.
Suppose x were not an element of Z. Then x would not commute with some element z, and we would have xz \(\neq\) zx, a contradiction.

It is often desirable to avoid the complexities of the subjunctive. In English, the preceding sentences may be correctly cast in the indicative mood:
Suppose x is not an element of Z. Then x does not commute with some element z, and we have  xz \(\neq\)  zx, a contradiction.

3. Dangling Modifiers
A common error is the so-called dangling modifier, a word or phrase associated by its placement with a word other than one it is intended to modify, or with no such word at all.

Incorrect: On stepping into the street, the bus ran over the pedestrian.
In this example, it seems to be the bus that stepped into the street. The problem is usually solved by placing the word to be modified, in this case “pedestrian,” near the phrase that modifies it:
On stepping into the street, the pedestrian was run over by a bus.

Here is a mathematical example:
Incorrect: Summing over all elements of H, the result is zero.
Here “summing” refers to the person doing the summing, not to the result;
correct: After summing over all elements of H, we find that the result is zero.

Here are a few more examples:
After proving the following theorem, it will be clear that …
After proving the following theorem, we shall see that…
or
After we have proved the following theorem, it will be clear that…

Even if fn is initially a signed distance function, fn+1 will generally not be a signed distance function after solving the linear system. (Here “solving” is the dangling modifier.)
Even if fn is initially a signed distance function, fn+1 will generally not be a signed distance function after the linear system has been solved.
or
Even if fn is initially a signed distance function, fn+1 will generally not be a signed distance function after we have solved the linear system.

4. Parallel Structure
For clarity, similar items in a sentence are given the same form.

Examples with either/or.
Use the same structure after “or” as was used after “either”:
We need to prove either that p is prime or that p is divisible by 4.
(incorrect: We need to prove either that p is prime or p is divisible by 4.)
We have seen that either the group is abelian or it is simple.
We have seen that the group is either abelian or simple.
(incorrect: We have seen that either the group is abelian or is simple.)

The same principle applies with both/and
We must prove both that the group is abelian and that it is not cyclic.
(incorrect: We must prove both that the group is abelian and is not cyclic.)
Parallel structures are also used with constructions such as “greater than” and “as well as”:
(incorrect: Its total run time may be greater than the Dostoevsky algorithm.)
Its total run time may be greater than the run time of the Dostoevsky algorithm.
Its total run time may be greater than that of the Dostoevsky algorithm.
incorrect: We must be able to handle this case as well as maintaining the sparsity of the original matrix.
correct: We must be able to handle this case as well as maintain the sparsity of the original matrix.
correct: We must be able to handle this case as well as to maintain the sparsity of the original matrix.

III. Usage

The following usages often cause trouble for nonnative speakers writing mathematics in English.

1. Present Perfect or Simple Past?
In contrast to some languages in which the verb form “she has lived” or “we have proved” can represent an action completed in the past (gestern haben wir bewiesen…, hier nous avons démontré, ieri abbiamo dimostrato…), in English this form expresses completed action in the present. If a reference to a definite past time appears in a sentence or is even implied, then generally the simple past must be used:

simple past:
In last week’s seminar we proved the prime number theorem.
Although a jurist by profession, Fermat was a brilliant mathematician.
Research on this topic was conducted for many years. [But it is no longer a topic of interest.]
In earlier chapters we considered only positive numbers.

present perfect:
Since x is nonzero, we have proved the following theorem.
Thus far in this chapter we have considered only positive numbers.
Research on this topic has been conducted for many years. [And continues to be conducted.]

2. Negating Adjectives
The prefix “non” is generally used to form a negative of an adjective (nonabelian, nonzero, nonintegrable). Certain words do not follow this pattern: aperiodic, inhomogeneous, indecomposable.

3. Plurals
The following are the standard forms:
lemmas, not lemmata;
formulas, not formulae.
The plural of phenomenon is phenomena.

4. Choosing between Alternatives

if or whether?: The English word if is more or less equivalent to German "wenn",  while whether corresponds to "ob". French uses "si"  for both these concepts.

In English, we use if to indicate a condition:
An integer n greater than 1 is prime if its only factors are n and 1.

Use whether as an indicator of alternatives:
We are unsure whether the set of twin primes is infinite.

whether vs. whether or not: Since whether implies that alternatives are involved, adding or not is usually superfluous and should be omitted, as in the example above. Occasionally, the two alternatives must be expressly stated:
The proof of the theorem goes through whether the group is finite or not.

obtain vs. find: We find something that we are looking for, while we obtain something through manipulation of some sort.

Thus when we look for a mathematical object successfully, we find it.
Example: To implement the RSA encryption algorithm, we must find two large prime numbers.
However, when we perform a mathematical operation, we obtain a result.
Example: Reversing summation and integration, we obtain the solution x = –2.
A rough equivalent to “obtain” is the phrase “find that”:
Reversing summation and integration, we find that x = –2.

arbitrary vs. arbitrarily: The former is an adjective, while the latter is an adverb:
Let x be an arbitrary element of the set X.
Let x be selected arbitrarily from the elements of X.

assure vs. ensure vs. insure: The word assure means to give confidence to, and always applies to a person or persons:
The fact that x is positive assures us that there is at least one solution.

On the other hand, ensure means to guarantee:
The fact that x is positive ensures the existence of a solution.
Finally, insure means to provide insurance and is not generally used in mathematical writing:
The author insured his manuscript for 5000 dollars.

permit vs. allow vs. make possible: These have similar meanings in mathematical writing. However, the first two of these cannot be used with an infinitive without reference to a person:
Thus Lemma 4.3 allows us to conclude that…
Thus Lemma 4.3 permits us to conclude that…
Thus Lemma 4.3 makes it possible to conclude that…
incorrect: Thus Lemma 4.3 allows to conclude that…

as vs. since: Both of these words have, among other senses, the meaning “because.” However, because as is not used in this sense in American English, is considered bad form even by many English stylists, and because it has a number of other uses in mathematical writing, it should be avoided in mathematical writing.

incorrect: As the group G is finite, every element of G has finite order.
correct: Since the group G is finite, every element of G has finite order.

like vs. as: The former is a preposition, while the latter is frequently a conjunction:
The function has a graph like that of the exponential function.
It grows like the exponential function.
It is always positive, as is the exponential function.

unlike: This preposition has no counterpart to the conjunction as. Thus the following is incorrect:
incorrect: Unlike in the previous situation…
correct: In contrast to the previous situation…
However, unlike can be used as a preposition:
Unlike the previous situation, this situation is different:

percent vs. percentage: Use the former with a number:
Ten percent (or 10%) of the population.
Otherwise, use percentage:
A large percentage of the population…

alternate vs. alternative: The synonym for a substitute is alternative:
We provide an alternative proof.
We choose one of the following alternatives. 

comprise vs. consist of vs. constitute: The whole comprises its parts. A synonym is consist of:
The subject of abstract algebra comprises (consists of) group theory, ring theory, and field theory.
The parts constitute the whole:
Group theory, ring theory, and field theory constitute the subject of abstract algebra.

analog vs. analogue: Use the former as an adjective meaning as the opposite of digital and the latter as a noun for something analogous.

denote vs. define: The word denote is used to indicate that a symbol will stand for a concept:
Let Z denote the center of the group G.
We denote by Z the center of the group G.
The center of G will be denoted by Z

In contrast, define means to make a definition:
Let us define f to be the function that maps…
Define X = x3.
incorrect: Denote X = x3.

less vs. fewer: Use fewer when a discrete number of items is involved, while less refers to an undenumerable quantity:
The set A contains three fewer elements than the set B.
The size of A is less than that of B.

only: For clarity, place the word only near the word or phrase that it modifies.
The function f depends only on the parameters a and b.
(incorrect: The function f only depends on the parameters a and b.)
The local height functions are determined only up to bounded functions.
(incorrect:The local height functions are only determined up to bounded functions.)
This condition may be satisfied only for divisors in this group.
(incorrect:This condition may only be satisfied for divisors in this group.)
We show that changing the model modifies the function f by only a bounded amount. (the amount is not unbounded)
(incorrect:We show that changing the model modifies only the function f by a bounded amount. (in contrast, the function g is not modified))

such that vs. so that:
The phrase such that always modifies a noun:
Choose c such that the equation f(x) = c has a solution. (choose such a c)
Equivalently: Let c be chosen such that the equation f(x) = c has a solution.
On the other hand, “so that” has two different meanings. One of these is “with the result that,” as in the following example:
We see that n is prime, so that its only factors are n and 1. (In this sense, “so that” is preceded by a comma.)
The other is “in order that,” as in the following example:
We have insisted that n be an even integer so that when we divide by 2, we will again have an integer. (In this sense, “so that” is not preceded by a comma.)

the case vs. the case that vs. the case in which: The following examples indicate how one can refer to various cases that may occur. They are similar and often interchangeable. The main point is the following are incorrect:
the case where…
the case when…
These are correct examples:
In the case x = 0, the result is trivial.
We consider the case that x is odd.
The case in which x is a member of Z and relatively prime to y.

in vs. when: Use “in” instead of “when” in impersonal constructions like the following:
The basic strategy in (not when) proving this theorem is to …
However, when “we” are involved, then use when:
When using this function, we employ the following strategy.

prove vs. show: To show means to exhibit:
We show the plot of the function in Figure 3.4.
To prove means to give a formal verification:
We shall prove the following result.
incorrect: We shall show the following result.
We shall prove that the set S is finite.
Or slightly less formally: We shall show that the set S is finite.